Subgroups — C₃₁×#34384 #21491143

Order 7936.

2318 conjugacy classes of subgroups.

The subgroup lattice has 738 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#529	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2383	C₂	1	4	yes	maximal, complemented
3	C₃₁×#2361	C₂	1	2	yes	maximal
4	C₃₁×#2360	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2355	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2354	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2352	C₂	1	2	yes	maximal, complemented
8	Q₈×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	D₄×C₄⋊C₄×C₃₁	C₂	1	2	yes	maximal, complemented
11	C₃₁×#2330	C₂	1	2	yes	maximal, complemented
12	C₃₁×#2326	C₂	1	2	yes	maximal, complemented
13	C₃₁×#2322	C₂	1	4	yes	maximal, complemented
14	C₃₁×#2317	C₂	1	2	yes	maximal, complemented
15	C₃₁×#2314	C₂	1	2	yes	maximal, complemented
16	C₂×C₃₁×#384	C₂	1	1	yes	maximal, characteristic, complemented
17	C₃₁×#2283	C₂	1	1	yes	maximal, characteristic, complemented
18	C₃₁×#529	C₄	1	16	yes	complemented
19	C₂×C₃₁×#126	C₂²	1	2	yes	—
20	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#123	C₂²	1	2	yes	—
23	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×#121	C₂²	1	1	yes	characteristic
25	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
26	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
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	1	yes	characteristic
31	C₂×C₃₁×#117	C₂²	1	2	yes	—
32	C₂×C₃₁×#117	C₂²	1	2	yes	—
33	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
34	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
35	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
36	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
37	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
38	C₃₁×#391	C₂²	1	2	yes	—
39	C₃₁×#390	C₂²	1	2	yes	—
40	C₃₁×#389	C₂²	1	4	yes	—
41	C₃₁×#389	C₂²	1	4	yes	—
42	C₃₁×#389	C₂²	1	4	yes	—
43	C₃₁×#389	C₂²	1	2	yes	—
44	C₃₁×#389	C₂²	1	4	yes	—
45	C₃₁×#389	C₂²	1	2	yes	—
46	C₃₁×#388	C₂²	1	4	yes	—
47	C₃₁×#388	C₂²	1	4	yes	—
48	C₃₁×#388	C₂²	1	4	yes	—
49	C₃₁×#388	C₂²	1	4	yes	—
50	C₃₁×#387	C₂²	1	4	yes	—
51	C₃₁×#387	C₂²	1	4	yes	—
52	C₃₁×#387	C₂²	1	4	yes	—
53	C₃₁×#387	C₂²	1	4	yes	—
54	C₃₁×#386	C₂²	1	4	yes	—
55	C₃₁×#385	C₂²	1	4	yes	—
56	C₃₁×#385	C₂²	1	4	yes	—
57	C₃₁×#385	C₂²	1	4	yes	—
58	C₃₁×#385	C₂²	1	4	yes	—
59	C₃₁×#385	C₂²	1	4	yes	—
60	C₃₁×#384	C₂²	1	2	yes	—
61	C₃₁×#384	C₂²	1	2	yes	—
62	C₃₁×#384	C₂²	1	2	yes	—
63	C₃₁×#384	C₂²	1	1	yes	characteristic
64	C₃₁×#384	C₂²	1	1	yes	characteristic
65	C₃₁×#384	C₂²	1	2	yes	—
66	C₃₁×#383	C₂²	1	2	yes	—
67	C₃₁×#382	C₂²	1	1	yes	characteristic
68	C₃₁×#382	C₂²	1	1	yes	characteristic
69	C₃₁×#382	C₂²	1	2	yes	—
70	C₃₁×#381	C₂²	1	2	yes	—
71	C₃₁×#380	C₂²	1	2	yes	—
72	C₃₁×#380	C₂²	1	2	yes	—
73	C₃₁×#380	C₂²	1	1	yes	characteristic
74	C₃₁×#380	C₂²	1	1	yes	characteristic
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	2	yes	—
78	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
79	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
80	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
81	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
82	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
83	C₃₁×#376	C₂²	1	2	yes	—
84	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
85	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
86	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
87	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
88	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
89	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
90	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
91	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
92	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
93	C₃₁×#126	C₂×C₄	1	16	yes	complemented
94	C₃₁×#125	C₂×C₄	1	8	yes	complemented
95	C₃₁×#124	C₂×C₄	1	8	yes	complemented
96	C₃₁×#123	C₂×C₄	1	16	yes	complemented
97	C₃₁×C₂²⋊Q₈	C₂×C₄	1	8	yes	complemented
98	C₃₁×#121	C₂×C₄	1	8	yes	—
99	C₄×Q₈×C₃₁	—	4	4	no	—
100	C₄×Q₈×C₃₁	C₂×C₄	1	8	yes	complemented
101	C₄×D₄×C₃₁	—	4	4	no	—
102	C₄×D₄×C₃₁	—	4	8	no	—
103	C₄×D₄×C₃₁	C₂×C₄	1	8	yes	complemented
104	C₄×D₄×C₃₁	C₂×C₄	1	16	yes	complemented
105	C₃₁×#117	—	4	4	no	—
106	C₃₁×#117	C₂×C₄	1	16	yes	complemented
107	C₂×C₄⋊C₄×C₃₁	—	4	4	no	—
108	C₂×C₄⋊C₄×C₃₁	—	4	4	no	—
109	C₂×C₄⋊C₄×C₃₁	—	4	4	no	—
110	C₂×C₄⋊C₄×C₃₁	—	4	4	no	—
111	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
112	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
113	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
114	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
115	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
116	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
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	4	yes	—
121	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
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	4	yes	—
125	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
126	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
127	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
128	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
129	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
130	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
131	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
132	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
133	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
134	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	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	2	yes	—
139	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
140	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
141	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
142	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
143	C₂×C₄²×C₃₁	C₂×C₄	1	8	yes	complemented
144	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
145	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
146	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
147	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
148	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
149	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
150	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
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	2	yes	—
157	C₃₁×#95	C₂³	1	2	yes	—
158	C₃₁×#95	C₂³	1	4	yes	—
159	C₃₁×#95	C₂³	1	2	yes	—
160	C₃₁×#95	C₂³	1	2	yes	—
161	C₃₁×#95	C₂³	1	4	yes	—
162	C₃₁×#95	C₂³	1	4	yes	—
163	C₃₁×#95	C₂³	1	2	yes	—
164	C₃₁×#95	C₂³	1	2	yes	—
165	C₃₁×#95	C₂³	1	4	yes	—
166	C₃₁×#95	C₂³	1	4	yes	—
167	C₃₁×#95	C₂³	1	4	yes	—
168	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
169	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
170	C₂×Q₈×C₃₁	—	2	2	no	—
171	C₂×Q₈×C₃₁	—	2	2	no	—
172	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
173	C₂×D₄×C₃₁	—	2	4	no	—
174	C₂×D₄×C₃₁	—	2	4	no	—
175	C₂×D₄×C₃₁	—	2	2	no	—
176	C₂×D₄×C₃₁	—	2	2	no	—
177	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
178	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
179	C₂²×C₄×C₃₁	—	2	2	no	—
180	C₂²×C₄×C₃₁	—	2	2	no	—
181	C₂²×C₄×C₃₁	—	2	2	no	—
182	C₂²×C₄×C₃₁	—	2	2	no	—
183	C₂²×C₄×C₃₁	—	2	4	no	—
184	C₂²×C₄×C₃₁	—	2	2	no	—
185	C₂²×C₄×C₃₁	—	2	2	no	—
186	C₂²×C₄×C₃₁	—	2	4	no	—
187	C₂²×C₄×C₃₁	—	2	8	no	—
188	C₂²×C₄×C₃₁	—	4	4	no	—
189	C₂²×C₄×C₃₁	—	4	4	no	—
190	C₂²×C₄×C₃₁	—	4	4	no	—
191	C₂²×C₄×C₃₁	—	4	4	no	—
192	C₂²×C₄×C₃₁	—	4	4	no	—
193	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
194	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
195	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
196	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
197	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
198	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
199	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
200	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
201	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
202	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
203	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
204	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
205	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
206	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
207	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
208	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
209	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
210	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
211	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
212	C₄⋊C₄×C₃₁	—	2	2	no	—
213	C₄⋊C₄×C₃₁	—	2	4	no	—
214	C₄⋊C₄×C₃₁	—	2	4	no	—
215	C₄⋊C₄×C₃₁	—	2	4	no	—
216	C₄⋊C₄×C₃₁	—	2	4	no	—
217	C₄⋊C₄×C₃₁	—	2	2	no	—
218	C₄⋊C₄×C₃₁	—	2	2	no	—
219	C₄⋊C₄×C₃₁	—	2	2	no	—
220	C₄⋊C₄×C₃₁	—	4	4	no	—
221	C₄⋊C₄×C₃₁	—	4	4	no	—
222	C₄⋊C₄×C₃₁	—	4	4	no	—
223	C₄⋊C₄×C₃₁	—	4	2	no	—
224	C₄⋊C₄×C₃₁	—	4	4	no	—
225	C₄⋊C₄×C₃₁	—	4	2	no	—
226	C₄⋊C₄×C₃₁	—	4	4	no	—
227	C₄⋊C₄×C₃₁	—	4	4	no	—
228	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
229	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
230	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
231	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
232	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
233	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
234	C₂²⋊C₄×C₃₁	—	2	4	no	—
235	C₂²⋊C₄×C₃₁	—	2	4	no	—
236	C₂²⋊C₄×C₃₁	—	4	4	no	—
237	C₂²⋊C₄×C₃₁	—	4	4	no	—
238	C₂²⋊C₄×C₃₁	—	4	4	no	—
239	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
240	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
241	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
242	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
243	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
244	C₄²×C₃₁	—	2	4	no	—
245	C₄²×C₃₁	—	2	4	no	—
246	C₄²×C₃₁	—	4	4	no	—
247	C₄²×C₃₁	—	4	4	no	—
248	C₄²×C₃₁	—	4	4	no	—
249	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
250	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
251	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
252	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
253	C₄²×C₃₁	C₂²×C₄	1	8	yes	—
254	#34384	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
255	C₂³×C₃₁	—	2	1	no	—
256	C₂³×C₃₁	—	2	1	no	—
257	C₂³×C₃₁	—	2	1	no	—
258	C₂³×C₃₁	—	2	1	no	—
259	C₂³×C₃₁	—	2	2	no	—
260	C₂³×C₃₁	—	2	2	no	—
261	C₂³×C₃₁	—	2	2	no	—
262	C₂³×C₃₁	—	2	2	no	—
263	C₂³×C₃₁	—	2	2	no	—
264	C₂³×C₃₁	—	2	2	no	—
265	C₂³×C₃₁	C₂³×C₄	1	4	yes	—
266	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
267	C₂³×C₃₁	H(5,𝔽₂)	1	2	yes	—
268	C₂³×C₃₁	2₋¹⁺⁴	1	2	yes	—
269	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
270	Q₈×C₃₁	—	2	8	no	—
271	D₄×C₃₁	—	2	16	no	—
272	D₄×C₃₁	—	2	8	no	—
273	C₂×C₄×C₃₁	—	2	8	no	—
274	C₂×C₄×C₃₁	—	2	2	no	—
275	C₂×C₄×C₃₁	—	2	2	no	—
276	C₂×C₄×C₃₁	—	2	2	no	—
277	C₂×C₄×C₃₁	—	2	1	no	—
278	C₂×C₄×C₃₁	—	2	1	no	—
279	C₂×C₄×C₃₁	—	2	1	no	—
280	C₂×C₄×C₃₁	—	2	1	no	—
281	C₂×C₄×C₃₁	—	2	2	no	—
282	C₂×C₄×C₃₁	—	2	1	no	—
283	C₂×C₄×C₃₁	—	2	1	no	—
284	C₂×C₄×C₃₁	—	2	1	no	—
285	C₂×C₄×C₃₁	—	2	1	no	—
286	C₂×C₄×C₃₁	—	2	2	no	—
287	C₂×C₄×C₃₁	—	2	2	no	—
288	C₂×C₄×C₃₁	—	2	8	no	—
289	C₂×C₄×C₃₁	—	2	16	no	—
290	C₂×C₄×C₃₁	—	2	8	no	—
291	C₂×C₄×C₃₁	—	2	2	no	—
292	C₂×C₄×C₃₁	—	2	2	no	—
293	C₂×C₄×C₃₁	—	2	2	no	—
294	C₂×C₄×C₃₁	—	2	2	no	—
295	C₂×C₄×C₃₁	—	2	4	no	—
296	C₂×C₄×C₃₁	—	2	4	no	—
297	C₂×C₄×C₃₁	—	2	4	no	—
298	C₂×C₄×C₃₁	—	2	2	no	—
299	C₂×C₄×C₃₁	—	2	4	no	—
300	C₂×C₄×C₃₁	—	2	4	no	—
301	C₂×C₄×C₃₁	—	2	2	no	—
302	C₂×C₄×C₃₁	—	2	4	no	—
303	C₂×C₄×C₃₁	—	2	2	no	—
304	C₂×C₄×C₃₁	—	2	2	no	—
305	C₂×C₄×C₃₁	—	2	2	no	—
306	C₂×C₄×C₃₁	—	2	2	no	—
307	C₂×C₄×C₃₁	—	2	2	no	—
308	C₂×C₄×C₃₁	—	2	2	no	—
309	C₂×C₄×C₃₁	—	8	8	no	—
310	C₂×C₄×C₃₁	—	8	8	no	—
311	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
312	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
313	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
314	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
315	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
316	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
317	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
318	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
319	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
320	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
321	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
322	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
323	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
324	C₂×#529	C₂×C₃₁	1	1	yes	characteristic
325	#2383	C₂×C₃₁	1	4	yes	complemented
326	#2361	C₂×C₃₁	1	2	yes	—
327	#2360	C₂×C₃₁	1	2	yes	complemented
328	#2355	C₂×C₃₁	1	1	yes	characteristic, complemented
329	#2354	C₂×C₃₁	1	1	yes	characteristic, complemented
330	#2352	C₂×C₃₁	1	2	yes	complemented
331	Q₈×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
332	D₄×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
333	D₄×C₄⋊C₄	C₂×C₃₁	1	2	yes	complemented
334	#2330	C₂×C₃₁	1	2	yes	complemented
335	#2326	C₂×C₃₁	1	2	yes	complemented
336	#2322	C₂×C₃₁	1	4	yes	complemented
337	#2317	C₂×C₃₁	1	2	yes	complemented
338	#2314	C₂×C₃₁	1	2	yes	complemented
339	C₂×#384	C₂×C₃₁	1	1	yes	characteristic, complemented
340	#2283	C₂×C₃₁	1	1	yes	characteristic, complemented
341	C₂²×C₃₁	—	2	2	no	—
342	C₂²×C₃₁	—	2	2	no	—
343	C₂²×C₃₁	—	2	2	no	—
344	C₂²×C₃₁	—	2	2	no	—
345	C₂²×C₃₁	—	2	2	no	—
346	C₂²×C₃₁	—	2	2	no	—
347	C₂²×C₃₁	—	2	4	no	—
348	C₂²×C₃₁	—	2	4	no	—
349	C₂²×C₃₁	—	2	4	no	—
350	C₂²×C₃₁	—	4	2	no	—
351	C₂²×C₃₁	—	4	2	no	—
352	C₂²×C₃₁	—	4	2	no	—
353	C₂²×C₃₁	—	4	2	no	—
354	C₂²×C₃₁	#520	1	4	yes	—
355	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
356	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
357	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
358	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
359	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
360	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
361	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
362	C₄×C₃₁	—	2	2	no	—
363	C₄×C₃₁	—	2	2	no	—
364	C₄×C₃₁	—	2	4	no	—
365	C₄×C₃₁	—	2	2	no	—
366	C₄×C₃₁	—	2	2	no	—
367	C₄×C₃₁	—	2	4	no	—
368	C₄×C₃₁	—	2	4	no	—
369	C₄×C₃₁	—	4	4	no	—
370	C₄×C₃₁	—	4	2	no	—
371	C₄×C₃₁	—	4	4	no	—
372	C₄×C₃₁	—	4	2	no	—
373	C₄×C₃₁	—	4	2	no	—
374	C₄×C₃₁	—	4	2	no	—
375	C₄×C₃₁	#520	1	4	yes	—
376	C₄×C₃₁	#520	1	4	yes	—
377	#529	C₄×C₃₁	1	16	yes	complemented
378	C₂×#126	C₂²×C₃₁	1	2	yes	—
379	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
380	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
381	C₂×#123	C₂²×C₃₁	1	2	yes	—
382	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
383	C₂×#121	C₂²×C₃₁	1	1	yes	characteristic
384	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
385	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
386	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	—
387	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
388	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
389	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
390	C₂×#117	C₂²×C₃₁	1	2	yes	—
391	C₂×#117	C₂²×C₃₁	1	2	yes	—
392	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
393	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
394	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
395	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
396	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
397	#391	C₂²×C₃₁	1	2	yes	—
398	#390	C₂²×C₃₁	1	2	yes	—
399	#389	C₂²×C₃₁	1	4	yes	—
400	#389	C₂²×C₃₁	1	4	yes	—
401	#389	C₂²×C₃₁	1	4	yes	—
402	#389	C₂²×C₃₁	1	2	yes	—
403	#389	C₂²×C₃₁	1	4	yes	—
404	#389	C₂²×C₃₁	1	2	yes	—
405	#388	C₂²×C₃₁	1	4	yes	—
406	#388	C₂²×C₃₁	1	4	yes	—
407	#388	C₂²×C₃₁	1	4	yes	—
408	#388	C₂²×C₃₁	1	4	yes	—
409	#387	C₂²×C₃₁	1	4	yes	—
410	#387	C₂²×C₃₁	1	4	yes	—
411	#387	C₂²×C₃₁	1	4	yes	—
412	#387	C₂²×C₃₁	1	4	yes	—
413	#386	C₂²×C₃₁	1	4	yes	—
414	#385	C₂²×C₃₁	1	4	yes	—
415	#385	C₂²×C₃₁	1	4	yes	—
416	#385	C₂²×C₃₁	1	4	yes	—
417	#385	C₂²×C₃₁	1	4	yes	—
418	#385	C₂²×C₃₁	1	4	yes	—
419	#384	C₂²×C₃₁	1	2	yes	—
420	#384	C₂²×C₃₁	1	2	yes	—
421	#384	C₂²×C₃₁	1	2	yes	—
422	#384	C₂²×C₃₁	1	1	yes	characteristic
423	#384	C₂²×C₃₁	1	1	yes	characteristic
424	#384	C₂²×C₃₁	1	2	yes	—
425	#383	C₂²×C₃₁	1	2	yes	—
426	#382	C₂²×C₃₁	1	1	yes	characteristic
427	#382	C₂²×C₃₁	1	1	yes	characteristic
428	#382	C₂²×C₃₁	1	2	yes	—
429	#381	C₂²×C₃₁	1	2	yes	—
430	#380	C₂²×C₃₁	1	2	yes	—
431	#380	C₂²×C₃₁	1	2	yes	—
432	#380	C₂²×C₃₁	1	1	yes	characteristic
433	#380	C₂²×C₃₁	1	1	yes	characteristic
434	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
435	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
436	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
437	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
438	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
439	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
440	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
441	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
442	#376	C₂²×C₃₁	1	2	yes	—
443	C₂×C₃₁	—	2	4	no	—
444	C₂×C₃₁	—	4	4	no	—
445	C₂×C₃₁	C₂×#520	1	1	yes	characteristic
446	C₂×C₃₁	C₂×#520	1	1	yes	characteristic
447	C₂×C₃₁	C₂×#520	1	1	yes	characteristic
448	C₂×C₃₁	#3543	1	1	yes	characteristic
449	C₂×C₃₁	#3544	1	1	yes	characteristic
450	C₂×C₃₁	#3552	1	1	yes	characteristic
451	C₂×C₃₁	#3552	1	1	yes	characteristic
452	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
453	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
454	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
455	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
456	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
457	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
458	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
459	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
460	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
461	#126	C₂×C₄×C₃₁	1	16	yes	complemented
462	#125	C₂×C₄×C₃₁	1	8	yes	complemented
463	#124	C₂×C₄×C₃₁	1	8	yes	complemented
464	#123	C₂×C₄×C₃₁	1	16	yes	complemented
465	C₂²⋊Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
466	#121	C₂×C₄×C₃₁	1	8	yes	—
467	C₄×Q₈	—	4	4	no	—
468	C₄×Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
469	C₄×D₄	—	4	4	no	—
470	C₄×D₄	—	4	8	no	—
471	C₄×D₄	C₂×C₄×C₃₁	1	8	yes	complemented
472	C₄×D₄	C₂×C₄×C₃₁	1	16	yes	complemented
473	#117	—	4	4	no	—
474	#117	C₂×C₄×C₃₁	1	16	yes	complemented
475	C₂×C₄⋊C₄	—	4	4	no	—
476	C₂×C₄⋊C₄	—	4	4	no	—
477	C₂×C₄⋊C₄	—	4	4	no	—
478	C₂×C₄⋊C₄	—	4	4	no	—
479	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
480	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
481	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
482	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
483	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
484	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
485	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
486	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
487	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
488	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
489	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
490	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
491	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
492	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
493	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
494	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
495	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
496	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
497	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
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	2	yes	—
501	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
502	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
503	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
504	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
505	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
506	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
507	C₂×C₂²⋊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×C₄²	C₂×C₄×C₃₁	1	8	yes	complemented
512	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
513	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
514	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
515	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
516	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
517	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
518	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
519	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
520	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
521	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
522	#95	C₂³×C₃₁	1	2	yes	—
523	#95	C₂³×C₃₁	1	2	yes	—
524	#95	C₂³×C₃₁	1	2	yes	—
525	#95	C₂³×C₃₁	1	2	yes	—
526	#95	C₂³×C₃₁	1	4	yes	—
527	#95	C₂³×C₃₁	1	2	yes	—
528	#95	C₂³×C₃₁	1	2	yes	—
529	#95	C₂³×C₃₁	1	4	yes	—
530	#95	C₂³×C₃₁	1	4	yes	—
531	#95	C₂³×C₃₁	1	2	yes	—
532	#95	C₂³×C₃₁	1	2	yes	—
533	#95	C₂³×C₃₁	1	4	yes	—
534	#95	C₂³×C₃₁	1	4	yes	—
535	#95	C₂³×C₃₁	1	4	yes	—
536	C₃₁	#34384	1	1	yes	characteristic, direct product factor, Sylow
537	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
538	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
539	C₂×Q₈	—	2	2	no	—
540	C₂×Q₈	—	2	2	no	—
541	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
542	C₂×D₄	—	2	4	no	—
543	C₂×D₄	—	2	4	no	—
544	C₂×D₄	—	2	2	no	—
545	C₂×D₄	—	2	2	no	—
546	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
547	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
548	C₂²×C₄	—	2	2	no	—
549	C₂²×C₄	—	2	2	no	—
550	C₂²×C₄	—	2	2	no	—
551	C₂²×C₄	—	2	2	no	—
552	C₂²×C₄	—	2	4	no	—
553	C₂²×C₄	—	2	2	no	—
554	C₂²×C₄	—	2	2	no	—
555	C₂²×C₄	—	2	4	no	—
556	C₂²×C₄	—	2	8	no	—
557	C₂²×C₄	—	4	4	no	—
558	C₂²×C₄	—	4	4	no	—
559	C₂²×C₄	—	4	4	no	—
560	C₂²×C₄	—	4	4	no	—
561	C₂²×C₄	—	4	4	no	—
562	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
563	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
564	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
565	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
566	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
567	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
568	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
569	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
570	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
571	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
572	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
573	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
574	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
575	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
576	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
577	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
578	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
579	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
580	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
581	C₄⋊C₄	—	2	2	no	—
582	C₄⋊C₄	—	2	4	no	—
583	C₄⋊C₄	—	2	4	no	—
584	C₄⋊C₄	—	2	4	no	—
585	C₄⋊C₄	—	2	4	no	—
586	C₄⋊C₄	—	2	2	no	—
587	C₄⋊C₄	—	2	2	no	—
588	C₄⋊C₄	—	2	2	no	—
589	C₄⋊C₄	—	4	4	no	—
590	C₄⋊C₄	—	4	4	no	—
591	C₄⋊C₄	—	4	4	no	—
592	C₄⋊C₄	—	4	2	no	—
593	C₄⋊C₄	—	4	4	no	—
594	C₄⋊C₄	—	4	2	no	—
595	C₄⋊C₄	—	4	4	no	—
596	C₄⋊C₄	—	4	4	no	—
597	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
598	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
599	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
600	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
601	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
602	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
603	C₂²⋊C₄	—	2	4	no	—
604	C₂²⋊C₄	—	2	4	no	—
605	C₂²⋊C₄	—	4	4	no	—
606	C₂²⋊C₄	—	4	4	no	—
607	C₂²⋊C₄	—	4	4	no	—
608	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
609	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
610	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
611	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
612	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
613	C₄²	—	2	4	no	—
614	C₄²	—	2	4	no	—
615	C₄²	—	4	4	no	—
616	C₄²	—	4	4	no	—
617	C₄²	—	4	4	no	—
618	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
619	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
620	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
621	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
622	C₄²	C₂²×C₄×C₃₁	1	8	yes	—
623	C₂³	—	2	1	no	—
624	C₂³	—	2	1	no	—
625	C₂³	—	2	1	no	—
626	C₂³	—	2	1	no	—
627	C₂³	—	2	2	no	—
628	C₂³	—	2	2	no	—
629	C₂³	—	2	2	no	—
630	C₂³	—	2	2	no	—
631	C₂³	—	2	2	no	—
632	C₂³	—	2	2	no	—
633	C₂³	C₂³×C₄×C₃₁	1	4	yes	—
634	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
635	C₂³	C₃₁×H(5,𝔽₂)	1	2	yes	—
636	C₂³	C₃₁×2₋¹⁺⁴	1	2	yes	—
637	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
638	Q₈	—	2	8	no	—
639	D₄	—	2	16	no	—
640	D₄	—	2	8	no	—
641	C₂×C₄	—	2	8	no	—
642	C₂×C₄	—	2	2	no	—
643	C₂×C₄	—	2	2	no	—
644	C₂×C₄	—	2	2	no	—
645	C₂×C₄	—	2	1	no	—
646	C₂×C₄	—	2	1	no	—
647	C₂×C₄	—	2	1	no	—
648	C₂×C₄	—	2	1	no	—
649	C₂×C₄	—	2	2	no	—
650	C₂×C₄	—	2	1	no	—
651	C₂×C₄	—	2	1	no	—
652	C₂×C₄	—	2	1	no	—
653	C₂×C₄	—	2	1	no	—
654	C₂×C₄	—	2	2	no	—
655	C₂×C₄	—	2	2	no	—
656	C₂×C₄	—	2	8	no	—
657	C₂×C₄	—	2	16	no	—
658	C₂×C₄	—	2	8	no	—
659	C₂×C₄	—	2	2	no	—
660	C₂×C₄	—	2	2	no	—
661	C₂×C₄	—	2	2	no	—
662	C₂×C₄	—	2	2	no	—
663	C₂×C₄	—	2	4	no	—
664	C₂×C₄	—	2	4	no	—
665	C₂×C₄	—	2	4	no	—
666	C₂×C₄	—	2	2	no	—
667	C₂×C₄	—	2	4	no	—
668	C₂×C₄	—	2	4	no	—
669	C₂×C₄	—	2	2	no	—
670	C₂×C₄	—	2	4	no	—
671	C₂×C₄	—	2	2	no	—
672	C₂×C₄	—	2	2	no	—
673	C₂×C₄	—	2	2	no	—
674	C₂×C₄	—	2	2	no	—
675	C₂×C₄	—	2	2	no	—
676	C₂×C₄	—	2	2	no	—
677	C₂×C₄	—	8	8	no	—
678	C₂×C₄	—	8	8	no	—
679	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
680	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
681	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
682	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
683	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
684	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
685	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
686	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
687	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
688	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
689	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
690	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
691	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
692	C₂²	—	2	2	no	—
693	C₂²	—	2	2	no	—
694	C₂²	—	2	2	no	—
695	C₂²	—	2	2	no	—
696	C₂²	—	2	2	no	—
697	C₂²	—	2	2	no	—
698	C₂²	—	2	4	no	—
699	C₂²	—	2	4	no	—
700	C₂²	—	2	4	no	—
701	C₂²	—	4	2	no	—
702	C₂²	—	4	2	no	—
703	C₂²	—	4	2	no	—
704	C₂²	—	4	2	no	—
705	C₂²	C₃₁×#520	1	4	yes	—
706	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
707	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
708	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
709	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
710	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
711	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
712	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
713	C₄	—	2	2	no	—
714	C₄	—	2	2	no	—
715	C₄	—	2	4	no	—
716	C₄	—	2	2	no	—
717	C₄	—	2	2	no	—
718	C₄	—	2	4	no	—
719	C₄	—	2	4	no	—
720	C₄	—	4	2	no	—
721	C₄	—	4	4	no	—
722	C₄	—	4	4	no	—
723	C₄	—	4	2	no	—
724	C₄	—	4	2	no	—
725	C₄	—	4	2	no	—
726	C₄	C₃₁×#520	1	4	yes	—
727	C₄	C₃₁×#520	1	4	yes	—
728	C₂	—	2	4	no	—
729	C₂	—	4	4	no	—
730	C₂	C₂×C₃₁×#520	1	1	yes	characteristic
731	C₂	C₂×C₃₁×#520	1	1	yes	characteristic
732	C₂	C₂×C₃₁×#520	1	1	yes	characteristic
733	C₂	C₃₁×#3543	1	1	yes	characteristic
734	C₂	C₃₁×#3544	1	1	yes	characteristic
735	C₂	C₃₁×#3552	1	1	yes	characteristic
736	C₂	C₃₁×#3552	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×#121	C₂²
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×#2383	C₂	semidirect
4. C₃₁×#2360	C₂	semidirect
5. C₃₁×#2355	C₂	semidirect
6. C₃₁×#2354	C₂	semidirect
7. C₃₁×#2352	C₂	semidirect
8. Q₈×C₄⋊C₄×C₃₁	C₂	semidirect
9. D₄×C₄⋊C₄×C₃₁	C₂	semidirect
10. D₄×C₄⋊C₄×C₃₁	C₂	semidirect
11. C₃₁×#2330	C₂	semidirect
12. C₃₁×#2326	C₂	semidirect
13. C₃₁×#2322	C₂	semidirect
14. C₃₁×#2317	C₂	semidirect
15. C₃₁×#2314	C₂	semidirect
16. C₂×C₃₁×#384	C₂	semidirect
17. C₃₁×#2283	C₂	semidirect
18. C₃₁×#529	C₄	semidirect
93. C₃₁×#126	C₂×C₄	semidirect
94. C₃₁×#125	C₂×C₄	semidirect
95. C₃₁×#124	C₂×C₄	semidirect
96. C₃₁×#123	C₂×C₄	semidirect
97. C₃₁×C₂²⋊Q₈	C₂×C₄	semidirect
100. C₄×Q₈×C₃₁	C₂×C₄	semidirect
103. C₄×D₄×C₃₁	C₂×C₄	semidirect
104. C₄×D₄×C₃₁	C₂×C₄	semidirect
106. C₃₁×#117	C₂×C₄	semidirect
143. C₂×C₄²×C₃₁	C₂×C₄	semidirect
254. #34384	C₃₁	direct (N × H)
325. #2383	C₂×C₃₁	semidirect
327. #2360	C₂×C₃₁	semidirect
328. #2355	C₂×C₃₁	semidirect
329. #2354	C₂×C₃₁	semidirect
330. #2352	C₂×C₃₁	semidirect
331. Q₈×C₄⋊C₄	C₂×C₃₁	semidirect
332. D₄×C₄⋊C₄	C₂×C₃₁	semidirect
333. D₄×C₄⋊C₄	C₂×C₃₁	semidirect
334. #2330	C₂×C₃₁	semidirect
335. #2326	C₂×C₃₁	semidirect
336. #2322	C₂×C₃₁	semidirect
337. #2317	C₂×C₃₁	semidirect
338. #2314	C₂×C₃₁	semidirect
339. C₂×#384	C₂×C₃₁	semidirect
340. #2283	C₂×C₃₁	semidirect
377. #529	C₄×C₃₁	semidirect
461. #126	C₂×C₄×C₃₁	semidirect
462. #125	C₂×C₄×C₃₁	semidirect
463. #124	C₂×C₄×C₃₁	semidirect
464. #123	C₂×C₄×C₃₁	semidirect
465. C₂²⋊Q₈	C₂×C₄×C₃₁	semidirect
468. C₄×Q₈	C₂×C₄×C₃₁	semidirect
471. C₄×D₄	C₂×C₄×C₃₁	semidirect
472. C₄×D₄	C₂×C₄×C₃₁	semidirect
474. #117	C₂×C₄×C₃₁	semidirect
511. C₂×C₄²	C₂×C₄×C₃₁	semidirect
536. C₃₁	#34384	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×#529	2
C₃₁×#2383	2
C₃₁×#2361	2
C₃₁×#2360	2
C₃₁×#2355	2
C₃₁×#2354	2
C₃₁×#2352	2
Q₈×C₄⋊C₄×C₃₁	2
D₄×C₄⋊C₄×C₃₁	2
C₃₁×#2330	2
C₃₁×#2326	2
C₃₁×#2322	2
C₃₁×#2317	2
C₃₁×#2314	2
C₂×C₃₁×#384	2
C₃₁×#2283	2
#34384	31

Subgroups — C31×#34384 #21491143

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#34384 #21491143