Subgroups — C₃₁×#34494 #21491253

Order 7936.

2660 conjugacy classes of subgroups.

The subgroup lattice has 758 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#550	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2377	C₂	1	4	yes	maximal, complemented
3	C₃₁×#2366	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2366	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2361	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2355	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2349	C₂	1	2	yes	maximal, complemented
8	D₄×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#2339	C₂	1	2	yes	maximal, complemented
10	C₃₁×#2328	C₂	1	2	yes	maximal, complemented
11	C₃₁×#2325	C₂	1	4	yes	maximal, complemented
12	C₃₁×#2324	C₂	1	2	yes	maximal, complemented
13	C₃₁×#2319	C₂	1	2	yes	maximal, complemented
14	C₃₁×#2311	C₂	1	2	yes	maximal, complemented
15	C₄×C₃₁×#127	C₂	1	1	yes	maximal, characteristic, complemented
16	C₃₁×#2295	C₂	1	1	yes	maximal, characteristic, complemented
17	C₃₁×#2293	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂²×#41×C₃₁	C₂²	1	2	yes	—
19	C₃₁×#550	C₄	1	16	yes	complemented
20	C₂×C₃₁×#127	C₂²	1	2	yes	—
21	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#121	C₂²	1	2	yes	—
23	C₂×C₃₁×#121	C₂²	1	4	yes	—
24	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
25	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
26	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
27	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	—
28	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
29	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
30	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
31	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
32	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
33	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
34	C₂×C₃₁×#117	C₂²	1	2	yes	—
35	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
36	C₃₁×#391	C₂²	1	1	yes	characteristic
37	C₃₁×#391	C₂²	1	2	yes	—
38	C₃₁×#391	C₂²	1	1	yes	characteristic
39	C₃₁×#390	C₂²	1	2	yes	—
40	C₃₁×#390	C₂²	1	2	yes	—
41	C₃₁×#390	C₂²	1	2	yes	—
42	C₃₁×#390	C₂²	1	2	yes	—
43	C₃₁×#390	C₂²	1	4	yes	—
44	C₃₁×#390	C₂²	1	1	yes	characteristic
45	C₃₁×#390	C₂²	1	1	yes	characteristic
46	C₃₁×#390	C₂²	1	2	yes	—
47	C₃₁×#389	C₂²	1	1	yes	characteristic
48	C₃₁×#389	C₂²	1	2	yes	—
49	C₃₁×#389	C₂²	1	1	yes	characteristic
50	C₃₁×#389	C₂²	1	1	yes	characteristic
51	C₃₁×#389	C₂²	1	1	yes	characteristic
52	C₃₁×#388	C₂²	1	4	yes	—
53	C₃₁×#388	C₂²	1	4	yes	—
54	C₃₁×#388	C₂²	1	2	yes	—
55	C₃₁×#388	C₂²	1	2	yes	—
56	C₃₁×#387	C₂²	1	2	yes	—
57	C₃₁×#387	C₂²	1	4	yes	—
58	C₃₁×#387	C₂²	1	2	yes	—
59	C₃₁×#386	C₂²	1	4	yes	—
60	C₃₁×#386	C₂²	1	4	yes	—
61	C₃₁×#386	C₂²	1	4	yes	—
62	C₃₁×#386	C₂²	1	4	yes	—
63	C₃₁×#386	C₂²	1	4	yes	—
64	C₃₁×#386	C₂²	1	4	yes	—
65	C₃₁×#386	C₂²	1	4	yes	—
66	C₃₁×#385	C₂²	1	4	yes	—
67	C₃₁×#385	C₂²	1	4	yes	—
68	C₃₁×#385	C₂²	1	4	yes	—
69	C₃₁×#384	C₂²	1	1	yes	characteristic
70	C₃₁×#382	C₂²	1	2	yes	—
71	C₃₁×#381	C₂²	1	4	yes	—
72	C₃₁×#380	C₂²	1	4	yes	—
73	C₃₁×#380	C₂²	1	4	yes	—
74	C₃₁×#380	C₂²	1	2	yes	—
75	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
76	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
77	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
78	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
79	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
80	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
81	C₃₁×#376	C₂²	1	2	yes	—
82	C₂×C₃₁×#95	C₂²	1	4	yes	—
83	C₄³×C₃₁	C₂²	1	1	yes	characteristic
84	C₂×#41×C₃₁	—	4	4	no	—
85	C₂×#41×C₃₁	—	4	4	no	—
86	C₂×#41×C₃₁	C₂×C₄	1	16	yes	complemented
87	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
88	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
89	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
90	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
91	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
92	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
93	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
94	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
95	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
96	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
97	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
98	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
99	C₃₁×#127	C₂×C₄	1	8	yes	complemented
100	C₃₁×#127	C₂×C₄	1	16	yes	complemented
101	C₃₁×#121	C₂×C₄	1	16	yes	complemented
102	C₃₁×#121	C₂×C₄	1	32	yes	complemented
103	C₄×Q₈×C₃₁	C₂×C₄	1	8	yes	complemented
104	C₄×D₄×C₃₁	—	4	4	no	—
105	C₄×D₄×C₃₁	—	4	4	no	—
106	C₄×D₄×C₃₁	—	4	4	no	—
107	C₄×D₄×C₃₁	—	4	4	no	—
108	C₄×D₄×C₃₁	—	4	4	no	—
109	C₄×D₄×C₃₁	—	4	4	no	—
110	C₄×D₄×C₃₁	—	4	8	no	—
111	C₄×D₄×C₃₁	C₂×C₄	1	8	yes	complemented
112	C₄×D₄×C₃₁	C₂×C₄	1	16	yes	complemented
113	C₃₁×#117	—	4	4	no	—
114	C₂×C₄⋊C₄×C₃₁	—	4	4	no	—
115	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
116	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
117	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
118	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
119	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
120	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
121	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
122	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
123	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
124	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
125	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
126	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
127	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
128	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
129	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
130	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
131	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
132	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
133	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
134	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
135	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
136	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
137	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
138	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
139	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
140	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
141	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
142	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
143	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
144	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
145	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
146	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
147	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
148	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
149	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
150	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
151	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
152	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
153	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
154	C₃₁×#95	C₂³	1	2	yes	—
155	C₃₁×#95	C₂³	1	2	yes	—
156	C₃₁×#95	C₂³	1	4	yes	—
157	C₃₁×#95	C₂³	1	4	yes	—
158	C₃₁×#95	C₂³	1	2	yes	—
159	C₃₁×#95	C₂³	1	2	yes	—
160	C₃₁×#95	C₂³	1	2	yes	—
161	C₃₁×#95	C₂³	1	2	yes	—
162	C₃₁×#95	C₂³	1	2	yes	—
163	C₃₁×#95	C₂³	1	2	yes	—
164	C₃₁×#95	C₂³	1	4	yes	—
165	C₃₁×#95	C₂³	1	4	yes	—
166	C₃₁×#95	C₂³	1	2	yes	—
167	C₃₁×#95	C₂³	1	2	yes	—
168	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
169	C₂⁴×C₃₁	C₂⁴	1	4	yes	—
170	#41×C₃₁	—	4	32	no	—
171	C₂×Q₈×C₃₁	—	2	2	no	—
172	C₂×Q₈×C₃₁	—	2	2	no	—
173	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
174	C₂×D₄×C₃₁	—	2	2	no	—
175	C₂×D₄×C₃₁	—	2	4	no	—
176	C₂×D₄×C₃₁	—	2	4	no	—
177	C₂×D₄×C₃₁	—	2	4	no	—
178	C₂×D₄×C₃₁	—	2	2	no	—
179	C₂×D₄×C₃₁	—	2	4	no	—
180	C₂×D₄×C₃₁	—	4	2	no	—
181	C₂×D₄×C₃₁	—	4	4	no	—
182	C₂×D₄×C₃₁	—	4	2	no	—
183	C₂×D₄×C₃₁	—	4	4	no	—
184	C₂×D₄×C₃₁	—	4	4	no	—
185	C₂×D₄×C₃₁	—	4	4	no	—
186	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
187	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
188	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
189	C₂×D₄×C₃₁	C₂²×C₄	1	16	yes	—
190	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
191	C₂×D₄×C₃₁	C₂²×C₄	1	16	yes	—
192	C₂²×C₄×C₃₁	—	2	4	no	—
193	C₂²×C₄×C₃₁	—	2	4	no	—
194	C₂²×C₄×C₃₁	—	2	8	no	—
195	C₂²×C₄×C₃₁	—	4	4	no	—
196	C₂²×C₄×C₃₁	—	4	4	no	—
197	C₂²×C₄×C₃₁	—	4	4	no	—
198	C₂²×C₄×C₃₁	—	4	4	no	—
199	C₂²×C₄×C₃₁	—	4	4	no	—
200	C₂²×C₄×C₃₁	—	4	4	no	—
201	C₂²×C₄×C₃₁	—	4	4	no	—
202	C₂²×C₄×C₃₁	—	4	4	no	—
203	C₂²×C₄×C₃₁	—	4	4	no	—
204	C₂²×C₄×C₃₁	—	4	4	no	—
205	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
206	C₂²×C₄×C₃₁	C₂²×C₄	1	16	yes	—
207	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
208	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
209	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
210	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
211	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
212	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
213	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
214	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
215	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
216	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
217	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
218	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
219	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
220	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
221	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
222	C₄⋊C₄×C₃₁	—	2	2	no	—
223	C₄⋊C₄×C₃₁	—	2	4	no	—
224	C₄⋊C₄×C₃₁	—	2	4	no	—
225	C₄⋊C₄×C₃₁	—	2	2	no	—
226	C₄⋊C₄×C₃₁	—	2	2	no	—
227	C₄⋊C₄×C₃₁	—	2	4	no	—
228	C₄⋊C₄×C₃₁	—	2	2	no	—
229	C₄⋊C₄×C₃₁	—	4	4	no	—
230	C₄⋊C₄×C₃₁	—	4	2	no	—
231	C₄⋊C₄×C₃₁	—	4	2	no	—
232	C₄⋊C₄×C₃₁	—	4	2	no	—
233	C₄⋊C₄×C₃₁	—	4	2	no	—
234	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
235	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
236	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
237	C₂²⋊C₄×C₃₁	—	2	4	no	—
238	C₂²⋊C₄×C₃₁	—	4	2	no	—
239	C₂²⋊C₄×C₃₁	—	4	2	no	—
240	C₂²⋊C₄×C₃₁	—	4	2	no	—
241	C₂²⋊C₄×C₃₁	—	4	4	no	—
242	C₂²⋊C₄×C₃₁	—	4	4	no	—
243	C₂²⋊C₄×C₃₁	—	4	4	no	—
244	C₂²⋊C₄×C₃₁	—	4	4	no	—
245	C₂²⋊C₄×C₃₁	—	4	4	no	—
246	C₂²⋊C₄×C₃₁	—	4	4	no	—
247	C₂²⋊C₄×C₃₁	—	4	4	no	—
248	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
249	C₂²⋊C₄×C₃₁	C₂²×C₄	1	16	yes	—
250	C₄²×C₃₁	—	2	2	no	—
251	C₄²×C₃₁	—	2	4	no	—
252	C₄²×C₃₁	—	2	2	no	—
253	C₄²×C₃₁	—	2	2	no	—
254	C₄²×C₃₁	—	2	2	no	—
255	C₄²×C₃₁	—	2	4	no	—
256	C₄²×C₃₁	—	2	2	no	—
257	C₄²×C₃₁	—	4	1	no	—
258	C₄²×C₃₁	—	4	4	no	—
259	C₄²×C₃₁	C₂²×C₄	1	8	yes	—
260	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
261	#34494	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
262	C₂³×C₃₁	—	2	2	no	—
263	C₂³×C₃₁	—	2	2	no	—
264	C₂³×C₃₁	—	2	4	no	—
265	C₂³×C₃₁	—	2	2	no	—
266	C₂³×C₃₁	—	2	4	no	—
267	C₂³×C₃₁	—	2	4	no	—
268	C₂³×C₃₁	—	2	4	no	—
269	C₂³×C₃₁	—	2	2	no	—
270	C₂³×C₃₁	—	2	2	no	—
271	C₂³×C₃₁	—	2	4	no	—
272	C₂³×C₃₁	—	2	2	no	—
273	C₂³×C₃₁	—	2	4	no	—
274	C₂³×C₃₁	C₂³×C₄	1	4	yes	—
275	C₂³×C₃₁	C₂³×C₄	1	8	yes	—
276	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
277	Q₈×C₃₁	—	2	8	no	—
278	D₄×C₃₁	—	2	16	no	—
279	D₄×C₃₁	—	2	8	no	—
280	D₄×C₃₁	—	2	16	no	—
281	D₄×C₃₁	—	4	8	no	—
282	D₄×C₃₁	—	4	16	no	—
283	D₄×C₃₁	—	4	16	no	—
284	C₂×C₄×C₃₁	—	2	1	no	—
285	C₂×C₄×C₃₁	—	2	1	no	—
286	C₂×C₄×C₃₁	—	2	1	no	—
287	C₂×C₄×C₃₁	—	2	2	no	—
288	C₂×C₄×C₃₁	—	2	2	no	—
289	C₂×C₄×C₃₁	—	2	2	no	—
290	C₂×C₄×C₃₁	—	2	2	no	—
291	C₂×C₄×C₃₁	—	2	2	no	—
292	C₂×C₄×C₃₁	—	2	1	no	—
293	C₂×C₄×C₃₁	—	2	2	no	—
294	C₂×C₄×C₃₁	—	2	16	no	—
295	C₂×C₄×C₃₁	—	2	1	no	—
296	C₂×C₄×C₃₁	—	2	1	no	—
297	C₂×C₄×C₃₁	—	2	1	no	—
298	C₂×C₄×C₃₁	—	2	2	no	—
299	C₂×C₄×C₃₁	—	2	1	no	—
300	C₂×C₄×C₃₁	—	2	1	no	—
301	C₂×C₄×C₃₁	—	2	2	no	—
302	C₂×C₄×C₃₁	—	2	2	no	—
303	C₂×C₄×C₃₁	—	2	2	no	—
304	C₂×C₄×C₃₁	—	2	2	no	—
305	C₂×C₄×C₃₁	—	2	4	no	—
306	C₂×C₄×C₃₁	—	2	4	no	—
307	C₂×C₄×C₃₁	—	2	2	no	—
308	C₂×C₄×C₃₁	—	2	2	no	—
309	C₂×C₄×C₃₁	—	2	2	no	—
310	C₂×C₄×C₃₁	—	2	4	no	—
311	C₂×C₄×C₃₁	—	2	2	no	—
312	C₂×C₄×C₃₁	—	2	1	no	—
313	C₂×C₄×C₃₁	—	2	1	no	—
314	C₂×C₄×C₃₁	—	2	2	no	—
315	C₂×C₄×C₃₁	—	2	1	no	—
316	C₂×C₄×C₃₁	—	2	1	no	—
317	C₂×C₄×C₃₁	—	2	2	no	—
318	C₂×C₄×C₃₁	—	2	2	no	—
319	C₂×C₄×C₃₁	—	4	16	no	—
320	C₂×C₄×C₃₁	—	8	8	no	—
321	C₂×C₄×C₃₁	—	8	8	no	—
322	C₂×C₄×C₃₁	—	8	8	no	—
323	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
324	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
325	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
326	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
327	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
328	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
329	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
330	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
331	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
332	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
333	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
334	C₂×#550	C₂×C₃₁	1	1	yes	characteristic
335	#2377	C₂×C₃₁	1	4	yes	complemented
336	#2366	C₂×C₃₁	1	2	yes	complemented
337	#2366	C₂×C₃₁	1	1	yes	characteristic, complemented
338	#2361	C₂×C₃₁	1	2	yes	complemented
339	#2355	C₂×C₃₁	1	1	yes	characteristic, complemented
340	#2349	C₂×C₃₁	1	2	yes	complemented
341	D₄×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
342	#2339	C₂×C₃₁	1	2	yes	complemented
343	#2328	C₂×C₃₁	1	2	yes	complemented
344	#2325	C₂×C₃₁	1	4	yes	complemented
345	#2324	C₂×C₃₁	1	2	yes	complemented
346	#2319	C₂×C₃₁	1	2	yes	complemented
347	#2311	C₂×C₃₁	1	2	yes	complemented
348	C₄×#127	C₂×C₃₁	1	1	yes	characteristic, complemented
349	#2295	C₂×C₃₁	1	1	yes	characteristic, complemented
350	#2293	C₂×C₃₁	1	1	yes	characteristic, complemented
351	C₂²×C₃₁	—	2	4	no	—
352	C₂²×C₃₁	—	2	8	no	—
353	C₂²×C₃₁	—	2	4	no	—
354	C₂²×C₃₁	—	2	8	no	—
355	C₂²×C₃₁	—	2	4	no	—
356	C₂²×C₃₁	—	2	8	no	—
357	C₂²×C₃₁	—	4	2	no	—
358	C₂²×C₃₁	—	4	4	no	—
359	C₂²×C₃₁	—	4	4	no	—
360	C₂²×C₃₁	—	4	2	no	—
361	C₂²×C₃₁	—	4	2	no	—
362	C₂²×C₃₁	—	4	4	no	—
363	C₂²×C₃₁	—	4	4	no	—
364	C₂²×C₃₁	—	4	2	no	—
365	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
366	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
367	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
368	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
369	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
370	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
371	C₂²×C₃₁	#585	1	1	yes	characteristic
372	C₄×C₃₁	—	2	4	no	—
373	C₄×C₃₁	—	2	4	no	—
374	C₄×C₃₁	—	2	2	no	—
375	C₄×C₃₁	—	2	4	no	—
376	C₄×C₃₁	—	2	2	no	—
377	C₄×C₃₁	—	4	2	no	—
378	C₄×C₃₁	—	4	4	no	—
379	C₄×C₃₁	—	4	1	no	—
380	C₄×C₃₁	—	4	2	no	—
381	C₄×C₃₁	—	4	2	no	—
382	C₄×C₃₁	—	4	1	no	—
383	C₄×C₃₁	—	4	2	no	—
384	C₄×C₃₁	—	4	1	no	—
385	C₄×C₃₁	#518	1	4	yes	—
386	C₄×C₃₁	#520	1	4	yes	—
387	C₂²×#41	C₂²×C₃₁	1	2	yes	—
388	#550	C₄×C₃₁	1	16	yes	complemented
389	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
390	C₂×#127	C₂²×C₃₁	1	2	yes	—
391	C₂×#121	C₂²×C₃₁	1	2	yes	—
392	C₂×#121	C₂²×C₃₁	1	4	yes	—
393	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
394	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
395	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
396	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
397	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
398	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	—
399	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
400	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
401	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
402	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
403	C₂×#117	C₂²×C₃₁	1	2	yes	—
404	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
405	#391	C₂²×C₃₁	1	1	yes	characteristic
406	#391	C₂²×C₃₁	1	2	yes	—
407	#391	C₂²×C₃₁	1	1	yes	characteristic
408	#390	C₂²×C₃₁	1	2	yes	—
409	#390	C₂²×C₃₁	1	2	yes	—
410	#390	C₂²×C₃₁	1	2	yes	—
411	#390	C₂²×C₃₁	1	2	yes	—
412	#390	C₂²×C₃₁	1	2	yes	—
413	#390	C₂²×C₃₁	1	4	yes	—
414	#390	C₂²×C₃₁	1	1	yes	characteristic
415	#390	C₂²×C₃₁	1	1	yes	characteristic
416	#389	C₂²×C₃₁	1	1	yes	characteristic
417	#389	C₂²×C₃₁	1	2	yes	—
418	#389	C₂²×C₃₁	1	1	yes	characteristic
419	#389	C₂²×C₃₁	1	1	yes	characteristic
420	#389	C₂²×C₃₁	1	1	yes	characteristic
421	#388	C₂²×C₃₁	1	4	yes	—
422	#388	C₂²×C₃₁	1	4	yes	—
423	#388	C₂²×C₃₁	1	2	yes	—
424	#388	C₂²×C₃₁	1	2	yes	—
425	#387	C₂²×C₃₁	1	2	yes	—
426	#387	C₂²×C₃₁	1	4	yes	—
427	#387	C₂²×C₃₁	1	2	yes	—
428	#386	C₂²×C₃₁	1	4	yes	—
429	#386	C₂²×C₃₁	1	4	yes	—
430	#386	C₂²×C₃₁	1	4	yes	—
431	#386	C₂²×C₃₁	1	4	yes	—
432	#386	C₂²×C₃₁	1	4	yes	—
433	#386	C₂²×C₃₁	1	4	yes	—
434	#386	C₂²×C₃₁	1	4	yes	—
435	#385	C₂²×C₃₁	1	4	yes	—
436	#385	C₂²×C₃₁	1	4	yes	—
437	#385	C₂²×C₃₁	1	4	yes	—
438	#384	C₂²×C₃₁	1	1	yes	characteristic
439	#382	C₂²×C₃₁	1	2	yes	—
440	#381	C₂²×C₃₁	1	4	yes	—
441	#380	C₂²×C₃₁	1	4	yes	—
442	#380	C₂²×C₃₁	1	4	yes	—
443	#380	C₂²×C₃₁	1	2	yes	—
444	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
445	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
446	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
447	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
448	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
449	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
450	#376	C₂²×C₃₁	1	2	yes	—
451	C₂×#95	C₂²×C₃₁	1	4	yes	—
452	C₄³	C₂²×C₃₁	1	1	yes	characteristic
453	C₂×C₃₁	—	4	4	no	—
454	C₂×C₃₁	—	4	8	no	—
455	C₂×C₃₁	C₂×#518	1	1	yes	characteristic
456	C₂×C₃₁	C₂×#520	1	1	yes	characteristic
457	C₂×C₃₁	C₄×H(5,𝔽₂)	1	1	yes	characteristic
458	C₂×C₃₁	#3543	1	1	yes	characteristic
459	C₂×C₃₁	#3546	1	1	yes	characteristic
460	C₂×C₃₁	#3548	1	1	yes	characteristic
461	C₂×C₃₁	#3554	1	1	yes	characteristic
462	C₂×#41	—	4	4	no	—
463	C₂×#41	—	4	4	no	—
464	C₂×#41	C₂×C₄×C₃₁	1	16	yes	complemented
465	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
466	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
467	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
468	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
469	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
470	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
471	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
472	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
473	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
474	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
475	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
476	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
477	#127	C₂×C₄×C₃₁	1	8	yes	complemented
478	#127	C₂×C₄×C₃₁	1	16	yes	complemented
479	#121	C₂×C₄×C₃₁	1	16	yes	complemented
480	#121	C₂×C₄×C₃₁	1	32	yes	complemented
481	C₄×Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
482	C₄×D₄	—	4	4	no	—
483	C₄×D₄	—	4	4	no	—
484	C₄×D₄	—	4	4	no	—
485	C₄×D₄	—	4	4	no	—
486	C₄×D₄	—	4	4	no	—
487	C₄×D₄	—	4	4	no	—
488	C₄×D₄	—	4	8	no	—
489	C₄×D₄	C₂×C₄×C₃₁	1	8	yes	complemented
490	C₄×D₄	C₂×C₄×C₃₁	1	16	yes	complemented
491	#117	—	4	4	no	—
492	C₂×C₄⋊C₄	—	4	4	no	—
493	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
494	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
495	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
496	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
497	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
498	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
499	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
500	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
501	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
502	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
503	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
504	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
505	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
506	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
507	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
508	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
509	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
510	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
511	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
512	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
513	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
514	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
515	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
516	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
517	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
518	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
519	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
520	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
521	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
522	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
523	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
524	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
525	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
526	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
527	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
528	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
529	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
530	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
531	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
532	#95	C₂³×C₃₁	1	2	yes	—
533	#95	C₂³×C₃₁	1	2	yes	—
534	#95	C₂³×C₃₁	1	4	yes	—
535	#95	C₂³×C₃₁	1	4	yes	—
536	#95	C₂³×C₃₁	1	2	yes	—
537	#95	C₂³×C₃₁	1	2	yes	—
538	#95	C₂³×C₃₁	1	2	yes	—
539	#95	C₂³×C₃₁	1	2	yes	—
540	#95	C₂³×C₃₁	1	2	yes	—
541	#95	C₂³×C₃₁	1	2	yes	—
542	#95	C₂³×C₃₁	1	4	yes	—
543	#95	C₂³×C₃₁	1	4	yes	—
544	#95	C₂³×C₃₁	1	2	yes	—
545	#95	C₂³×C₃₁	1	2	yes	—
546	C₃₁	#34494	1	1	yes	characteristic, direct product factor, Sylow
547	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
548	C₂⁴	C₂⁴×C₃₁	1	4	yes	—
549	#41	—	4	32	no	—
550	C₂×Q₈	—	2	2	no	—
551	C₂×Q₈	—	2	2	no	—
552	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
553	C₂×D₄	—	2	2	no	—
554	C₂×D₄	—	2	4	no	—
555	C₂×D₄	—	2	4	no	—
556	C₂×D₄	—	2	4	no	—
557	C₂×D₄	—	2	4	no	—
558	C₂×D₄	—	2	2	no	—
559	C₂×D₄	—	4	2	no	—
560	C₂×D₄	—	4	4	no	—
561	C₂×D₄	—	4	2	no	—
562	C₂×D₄	—	4	4	no	—
563	C₂×D₄	—	4	4	no	—
564	C₂×D₄	—	4	4	no	—
565	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
566	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
567	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
568	C₂×D₄	C₂²×C₄×C₃₁	1	16	yes	—
569	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
570	C₂×D₄	C₂²×C₄×C₃₁	1	16	yes	—
571	C₂²×C₄	—	2	4	no	—
572	C₂²×C₄	—	2	4	no	—
573	C₂²×C₄	—	2	8	no	—
574	C₂²×C₄	—	4	4	no	—
575	C₂²×C₄	—	4	4	no	—
576	C₂²×C₄	—	4	4	no	—
577	C₂²×C₄	—	4	4	no	—
578	C₂²×C₄	—	4	4	no	—
579	C₂²×C₄	—	4	4	no	—
580	C₂²×C₄	—	4	4	no	—
581	C₂²×C₄	—	4	4	no	—
582	C₂²×C₄	—	4	4	no	—
583	C₂²×C₄	—	4	4	no	—
584	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
585	C₂²×C₄	C₂²×C₄×C₃₁	1	16	yes	—
586	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
587	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
588	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
589	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
590	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
591	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
592	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
593	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
594	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
595	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
596	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
597	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
598	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
599	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
600	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
601	C₄⋊C₄	—	2	2	no	—
602	C₄⋊C₄	—	2	4	no	—
603	C₄⋊C₄	—	2	4	no	—
604	C₄⋊C₄	—	2	2	no	—
605	C₄⋊C₄	—	2	2	no	—
606	C₄⋊C₄	—	2	4	no	—
607	C₄⋊C₄	—	2	2	no	—
608	C₄⋊C₄	—	4	4	no	—
609	C₄⋊C₄	—	4	2	no	—
610	C₄⋊C₄	—	4	2	no	—
611	C₄⋊C₄	—	4	2	no	—
612	C₄⋊C₄	—	4	2	no	—
613	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
614	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
615	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
616	C₂²⋊C₄	—	2	4	no	—
617	C₂²⋊C₄	—	4	2	no	—
618	C₂²⋊C₄	—	4	2	no	—
619	C₂²⋊C₄	—	4	2	no	—
620	C₂²⋊C₄	—	4	4	no	—
621	C₂²⋊C₄	—	4	4	no	—
622	C₂²⋊C₄	—	4	4	no	—
623	C₂²⋊C₄	—	4	4	no	—
624	C₂²⋊C₄	—	4	4	no	—
625	C₂²⋊C₄	—	4	4	no	—
626	C₂²⋊C₄	—	4	4	no	—
627	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
628	C₂²⋊C₄	C₂²×C₄×C₃₁	1	16	yes	—
629	C₄²	—	2	2	no	—
630	C₄²	—	2	4	no	—
631	C₄²	—	2	2	no	—
632	C₄²	—	2	2	no	—
633	C₄²	—	2	2	no	—
634	C₄²	—	2	4	no	—
635	C₄²	—	2	2	no	—
636	C₄²	—	4	1	no	—
637	C₄²	—	4	4	no	—
638	C₄²	C₂²×C₄×C₃₁	1	8	yes	—
639	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
640	C₂³	—	2	2	no	—
641	C₂³	—	2	2	no	—
642	C₂³	—	2	4	no	—
643	C₂³	—	2	2	no	—
644	C₂³	—	2	4	no	—
645	C₂³	—	2	4	no	—
646	C₂³	—	2	4	no	—
647	C₂³	—	2	2	no	—
648	C₂³	—	2	2	no	—
649	C₂³	—	2	4	no	—
650	C₂³	—	2	2	no	—
651	C₂³	—	2	4	no	—
652	C₂³	C₂³×C₄×C₃₁	1	4	yes	—
653	C₂³	C₂³×C₄×C₃₁	1	8	yes	—
654	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
655	Q₈	—	2	8	no	—
656	D₄	—	2	16	no	—
657	D₄	—	2	8	no	—
658	D₄	—	2	16	no	—
659	D₄	—	4	8	no	—
660	D₄	—	4	16	no	—
661	D₄	—	4	16	no	—
662	C₂×C₄	—	2	1	no	—
663	C₂×C₄	—	2	1	no	—
664	C₂×C₄	—	2	1	no	—
665	C₂×C₄	—	2	2	no	—
666	C₂×C₄	—	2	2	no	—
667	C₂×C₄	—	2	2	no	—
668	C₂×C₄	—	2	2	no	—
669	C₂×C₄	—	2	2	no	—
670	C₂×C₄	—	2	1	no	—
671	C₂×C₄	—	2	2	no	—
672	C₂×C₄	—	2	16	no	—
673	C₂×C₄	—	2	1	no	—
674	C₂×C₄	—	2	1	no	—
675	C₂×C₄	—	2	1	no	—
676	C₂×C₄	—	2	2	no	—
677	C₂×C₄	—	2	1	no	—
678	C₂×C₄	—	2	1	no	—
679	C₂×C₄	—	2	2	no	—
680	C₂×C₄	—	2	2	no	—
681	C₂×C₄	—	2	2	no	—
682	C₂×C₄	—	2	2	no	—
683	C₂×C₄	—	2	4	no	—
684	C₂×C₄	—	2	4	no	—
685	C₂×C₄	—	2	2	no	—
686	C₂×C₄	—	2	2	no	—
687	C₂×C₄	—	2	2	no	—
688	C₂×C₄	—	2	4	no	—
689	C₂×C₄	—	2	2	no	—
690	C₂×C₄	—	2	1	no	—
691	C₂×C₄	—	2	1	no	—
692	C₂×C₄	—	2	2	no	—
693	C₂×C₄	—	2	1	no	—
694	C₂×C₄	—	2	1	no	—
695	C₂×C₄	—	2	2	no	—
696	C₂×C₄	—	2	2	no	—
697	C₂×C₄	—	4	16	no	—
698	C₂×C₄	—	8	8	no	—
699	C₂×C₄	—	8	8	no	—
700	C₂×C₄	—	8	8	no	—
701	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
702	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
703	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
704	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
705	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
706	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
707	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
708	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
709	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
710	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
711	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
712	C₂²	—	2	4	no	—
713	C₂²	—	2	8	no	—
714	C₂²	—	2	4	no	—
715	C₂²	—	2	8	no	—
716	C₂²	—	2	4	no	—
717	C₂²	—	2	8	no	—
718	C₂²	—	4	2	no	—
719	C₂²	—	4	4	no	—
720	C₂²	—	4	4	no	—
721	C₂²	—	4	2	no	—
722	C₂²	—	4	2	no	—
723	C₂²	—	4	4	no	—
724	C₂²	—	4	4	no	—
725	C₂²	—	4	2	no	—
726	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
727	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
728	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
729	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
730	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
731	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
732	C₂²	C₃₁×#585	1	1	yes	characteristic
733	C₄	—	2	4	no	—
734	C₄	—	2	4	no	—
735	C₄	—	2	2	no	—
736	C₄	—	2	4	no	—
737	C₄	—	2	2	no	—
738	C₄	—	4	2	no	—
739	C₄	—	4	4	no	—
740	C₄	—	4	2	no	—
741	C₄	—	4	2	no	—
742	C₄	—	4	1	no	—
743	C₄	—	4	1	no	—
744	C₄	—	4	2	no	—
745	C₄	—	4	1	no	—
746	C₄	C₃₁×#518	1	4	yes	—
747	C₄	C₃₁×#520	1	4	yes	—
748	C₂	—	4	4	no	—
749	C₂	—	4	8	no	—
750	C₂	C₂×C₃₁×#518	1	1	yes	characteristic
751	C₂	C₂×C₃₁×#520	1	1	yes	characteristic
752	C₂	C₄×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
753	C₂	C₃₁×#3543	1	1	yes	characteristic
754	C₂	C₃₁×#3546	1	1	yes	characteristic
755	C₂	C₃₁×#3548	1	1	yes	characteristic
756	C₂	C₃₁×#3554	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₃₁
Frattini, cosocle	C₂³	C₂⁵×C₃₁
prime radical	C₂×C₃₁×#550	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₃₁×#2377	C₂	semidirect
3. C₃₁×#2366	C₂	semidirect
4. C₃₁×#2366	C₂	semidirect
5. C₃₁×#2361	C₂	semidirect
6. C₃₁×#2355	C₂	semidirect
7. C₃₁×#2349	C₂	semidirect
8. D₄×C₄⋊C₄×C₃₁	C₂	semidirect
9. C₃₁×#2339	C₂	semidirect
10. C₃₁×#2328	C₂	semidirect
11. C₃₁×#2325	C₂	semidirect
12. C₃₁×#2324	C₂	semidirect
13. C₃₁×#2319	C₂	semidirect
14. C₃₁×#2311	C₂	semidirect
15. C₄×C₃₁×#127	C₂	semidirect
16. C₃₁×#2295	C₂	semidirect
17. C₃₁×#2293	C₂	semidirect
19. C₃₁×#550	C₄	semidirect
86. C₂×#41×C₃₁	C₂×C₄	semidirect
99. C₃₁×#127	C₂×C₄	semidirect
100. C₃₁×#127	C₂×C₄	semidirect
101. C₃₁×#121	C₂×C₄	semidirect
102. C₃₁×#121	C₂×C₄	semidirect
103. C₄×Q₈×C₃₁	C₂×C₄	semidirect
111. C₄×D₄×C₃₁	C₂×C₄	semidirect
112. C₄×D₄×C₃₁	C₂×C₄	semidirect
261. #34494	C₃₁	direct (N × H)
335. #2377	C₂×C₃₁	semidirect
336. #2366	C₂×C₃₁	semidirect
337. #2366	C₂×C₃₁	semidirect
338. #2361	C₂×C₃₁	semidirect
339. #2355	C₂×C₃₁	semidirect
340. #2349	C₂×C₃₁	semidirect
341. D₄×C₄⋊C₄	C₂×C₃₁	semidirect
342. #2339	C₂×C₃₁	semidirect
343. #2328	C₂×C₃₁	semidirect
344. #2325	C₂×C₃₁	semidirect
345. #2324	C₂×C₃₁	semidirect
346. #2319	C₂×C₃₁	semidirect
347. #2311	C₂×C₃₁	semidirect
348. C₄×#127	C₂×C₃₁	semidirect
349. #2295	C₂×C₃₁	semidirect
350. #2293	C₂×C₃₁	semidirect
388. #550	C₄×C₃₁	semidirect
464. C₂×#41	C₂×C₄×C₃₁	semidirect
477. #127	C₂×C₄×C₃₁	semidirect
478. #127	C₂×C₄×C₃₁	semidirect
479. #121	C₂×C₄×C₃₁	semidirect
480. #121	C₂×C₄×C₃₁	semidirect
481. C₄×Q₈	C₂×C₄×C₃₁	semidirect
489. C₄×D₄	C₂×C₄×C₃₁	semidirect
490. C₄×D₄	C₂×C₄×C₃₁	semidirect
546. C₃₁	#34494	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃₁×#550	2
C₃₁×#2377	2
C₃₁×#2366	2
C₃₁×#2361	2
C₃₁×#2355	2
C₃₁×#2349	2
D₄×C₄⋊C₄×C₃₁	2
C₃₁×#2339	2
C₃₁×#2328	2
C₃₁×#2325	2
C₃₁×#2324	2
C₃₁×#2319	2
C₃₁×#2311	2
C₄×C₃₁×#127	2
C₃₁×#2295	2
C₃₁×#2293	2
#34494	31

Subgroups — C31×#34494 #21491253

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#34494 #21491253