Subgroups — C₃₁×#34241 #21491000

Order 7936.

2330 conjugacy classes of subgroups.

The subgroup lattice has 640 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×#2373	C₂	1	4	yes	maximal, complemented
3	C₃₁×#2368	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2364	C₂	1	2	yes	maximal
5	C₃₁×#2355	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2352	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2352	C₂	1	2	yes	maximal, complemented
8	D₄×C₄⋊C₄×C₃₁	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2337	C₂	1	2	yes	maximal, complemented
10	C₃₁×#2336	C₂	1	2	yes	maximal, complemented
11	C₃₁×#2314	C₂	1	2	yes	maximal, complemented
12	C₄×C₃₁×#123	C₂	1	4	yes	maximal, complemented
13	C₃₁×#2283	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#2274	C₂	1	2	yes	maximal, complemented
15	C₂×C₄×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₃₁×#529	C₄	1	16	yes	complemented
17	C₂×C₃₁×#126	C₂²	1	2	yes	—
18	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#123	C₂²	1	2	yes	—
21	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#121	C₂²	1	1	yes	characteristic
23	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
24	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	—
25	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
26	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
27	C₂×C₃₁×#117	C₂²	1	2	yes	—
28	C₂×C₃₁×#117	C₂²	1	2	yes	—
29	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
30	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
31	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
32	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
33	C₃₁×#391	C₂²	1	4	yes	—
34	C₃₁×#389	C₂²	1	4	yes	—
35	C₃₁×#388	C₂²	1	4	yes	—
36	C₃₁×#388	C₂²	1	4	yes	—
37	C₃₁×#388	C₂²	1	4	yes	—
38	C₃₁×#388	C₂²	1	4	yes	—
39	C₃₁×#387	C₂²	1	4	yes	—
40	C₃₁×#387	C₂²	1	4	yes	—
41	C₃₁×#387	C₂²	1	4	yes	—
42	C₃₁×#387	C₂²	1	4	yes	—
43	C₃₁×#387	C₂²	1	4	yes	—
44	C₃₁×#387	C₂²	1	4	yes	—
45	C₃₁×#386	C₂²	1	4	yes	—
46	C₃₁×#386	C₂²	1	4	yes	—
47	C₃₁×#385	C₂²	1	4	yes	—
48	C₃₁×#385	C₂²	1	4	yes	—
49	C₃₁×#384	C₂²	1	2	yes	—
50	C₃₁×#383	C₂²	1	2	yes	—
51	C₃₁×#382	C₂²	1	2	yes	—
52	C₃₁×#382	C₂²	1	2	yes	—
53	C₃₁×#381	C₂²	1	2	yes	—
54	C₃₁×#380	C₂²	1	2	yes	—
55	C₃₁×#380	C₂²	1	2	yes	—
56	C₃₁×#380	C₂²	1	1	yes	characteristic
57	C₃₁×#380	C₂²	1	1	yes	characteristic
58	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
59	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
60	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
61	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
62	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
63	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
64	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
65	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
66	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
67	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
68	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
69	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
70	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
71	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
72	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
73	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
74	C₃₁×#376	C₂²	1	2	yes	—
75	C₂×C₃₁×#95	C₂²	1	2	yes	—
76	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
77	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
78	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
79	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
80	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
81	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
82	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
83	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
84	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
85	C₃₁×#126	C₂×C₄	1	16	yes	complemented
86	C₃₁×#125	C₂×C₄	1	8	yes	complemented
87	C₃₁×#124	C₂×C₄	1	8	yes	complemented
88	C₃₁×#123	C₂×C₄	1	16	yes	complemented
89	C₃₁×C₂²⋊Q₈	C₂×C₄	1	8	yes	complemented
90	C₃₁×#121	C₂×C₄	1	8	yes	—
91	C₄×Q₈×C₃₁	C₂×C₄	1	8	yes	complemented
92	C₄×D₄×C₃₁	—	4	8	no	—
93	C₄×D₄×C₃₁	C₂×C₄	1	8	yes	complemented
94	C₄×D₄×C₃₁	C₂×C₄	1	16	yes	complemented
95	C₃₁×#117	—	2	8	no	—
96	C₃₁×#117	C₂×C₄	1	16	yes	complemented
97	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
98	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
99	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
100	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
101	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
102	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
103	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
104	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
105	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
106	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
107	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
108	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
109	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
110	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
111	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
112	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
113	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
114	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
115	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
116	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
117	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
118	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
119	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
120	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
121	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
122	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
123	C₂×C₄²×C₃₁	—	4	4	no	—
124	C₂×C₄²×C₃₁	C₂×C₄	1	8	yes	complemented
125	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
126	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
127	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
128	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
129	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
130	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
131	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
132	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
133	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
134	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
135	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
136	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
137	C₃₁×#95	C₂³	1	2	yes	—
138	C₃₁×#95	C₂³	1	4	yes	—
139	C₃₁×#95	C₂³	1	2	yes	—
140	C₃₁×#95	C₂³	1	4	yes	—
141	C₃₁×#95	C₂³	1	2	yes	—
142	C₃₁×#95	C₂³	1	2	yes	—
143	C₃₁×#95	C₂³	1	4	yes	—
144	C₃₁×#95	C₂³	1	4	yes	—
145	C₃₁×#95	C₂³	1	4	yes	—
146	C₃₁×#95	C₂³	1	4	yes	—
147	C₃₁×#95	C₂³	1	4	yes	—
148	C₃₁×#95	C₂³	1	4	yes	—
149	C₃₁×#95	C₂³	1	4	yes	—
150	C₃₁×#95	C₂³	1	4	yes	—
151	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
152	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
153	C₂×Q₈×C₃₁	—	4	2	no	—
154	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
155	C₂×D₄×C₃₁	—	2	4	no	—
156	C₂×D₄×C₃₁	—	2	4	no	—
157	C₂×D₄×C₃₁	—	4	2	no	—
158	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
159	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
160	C₂²×C₄×C₃₁	—	2	2	no	—
161	C₂²×C₄×C₃₁	—	2	2	no	—
162	C₂²×C₄×C₃₁	—	2	4	no	—
163	C₂²×C₄×C₃₁	—	2	4	no	—
164	C₂²×C₄×C₃₁	—	2	4	no	—
165	C₂²×C₄×C₃₁	—	2	8	no	—
166	C₂²×C₄×C₃₁	—	2	8	no	—
167	C₂²×C₄×C₃₁	—	2	8	no	—
168	C₂²×C₄×C₃₁	—	4	2	no	—
169	C₂²×C₄×C₃₁	—	4	4	no	—
170	C₂²×C₄×C₃₁	—	4	4	no	—
171	C₂²×C₄×C₃₁	—	4	4	no	—
172	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
173	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
174	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
175	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
176	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
177	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
178	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
179	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
180	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
181	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
182	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
183	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
184	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
185	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
186	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
187	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
188	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
189	C₄⋊C₄×C₃₁	—	2	8	no	—
190	C₄⋊C₄×C₃₁	—	2	8	no	—
191	C₄⋊C₄×C₃₁	—	2	4	no	—
192	C₄⋊C₄×C₃₁	—	2	4	no	—
193	C₄⋊C₄×C₃₁	—	2	4	no	—
194	C₄⋊C₄×C₃₁	—	4	4	no	—
195	C₄⋊C₄×C₃₁	—	4	4	no	—
196	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
197	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
198	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
199	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
200	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
201	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
202	C₄⋊C₄×C₃₁	#41	1	8	yes	—
203	C₂²⋊C₄×C₃₁	—	2	4	no	—
204	C₂²⋊C₄×C₃₁	—	2	4	no	—
205	C₂²⋊C₄×C₃₁	—	2	8	no	—
206	C₂²⋊C₄×C₃₁	—	4	4	no	—
207	C₂²⋊C₄×C₃₁	—	4	4	no	—
208	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
209	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
210	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
211	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
212	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
213	C₄²×C₃₁	—	2	8	no	—
214	C₄²×C₃₁	—	4	2	no	—
215	C₄²×C₃₁	—	4	4	no	—
216	C₄²×C₃₁	—	4	2	no	—
217	C₄²×C₃₁	—	4	4	no	—
218	C₄²×C₃₁	—	4	4	no	—
219	C₄²×C₃₁	—	4	4	no	—
220	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
221	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
222	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
223	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
224	C₄²×C₃₁	C₂²×C₄	1	8	yes	—
225	#34241	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
226	C₂³×C₃₁	—	2	2	no	—
227	C₂³×C₃₁	—	2	2	no	—
228	C₂³×C₃₁	—	2	2	no	—
229	C₂³×C₃₁	—	2	2	no	—
230	C₂³×C₃₁	—	2	2	no	—
231	C₂³×C₃₁	—	2	2	no	—
232	C₂³×C₃₁	—	2	2	no	—
233	C₂³×C₃₁	—	2	2	no	—
234	C₂³×C₃₁	C₂³×C₄	1	4	yes	—
235	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
236	C₂³×C₃₁	H(5,𝔽₂)	1	2	yes	—
237	C₂³×C₃₁	2₋¹⁺⁴	1	2	yes	—
238	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
239	Q₈×C₃₁	—	4	4	no	—
240	D₄×C₃₁	—	2	16	no	—
241	D₄×C₃₁	—	4	4	no	—
242	C₂×C₄×C₃₁	—	2	8	no	—
243	C₂×C₄×C₃₁	—	2	2	no	—
244	C₂×C₄×C₃₁	—	2	2	no	—
245	C₂×C₄×C₃₁	—	2	2	no	—
246	C₂×C₄×C₃₁	—	2	1	no	—
247	C₂×C₄×C₃₁	—	2	1	no	—
248	C₂×C₄×C₃₁	—	2	4	no	—
249	C₂×C₄×C₃₁	—	2	2	no	—
250	C₂×C₄×C₃₁	—	2	1	no	—
251	C₂×C₄×C₃₁	—	2	1	no	—
252	C₂×C₄×C₃₁	—	2	2	no	—
253	C₂×C₄×C₃₁	—	2	2	no	—
254	C₂×C₄×C₃₁	—	2	2	no	—
255	C₂×C₄×C₃₁	—	2	4	no	—
256	C₂×C₄×C₃₁	—	2	2	no	—
257	C₂×C₄×C₃₁	—	2	2	no	—
258	C₂×C₄×C₃₁	—	2	8	no	—
259	C₂×C₄×C₃₁	—	2	16	no	—
260	C₂×C₄×C₃₁	—	2	4	no	—
261	C₂×C₄×C₃₁	—	2	4	no	—
262	C₂×C₄×C₃₁	—	2	4	no	—
263	C₂×C₄×C₃₁	—	2	4	no	—
264	C₂×C₄×C₃₁	—	2	4	no	—
265	C₂×C₄×C₃₁	—	2	4	no	—
266	C₂×C₄×C₃₁	—	2	4	no	—
267	C₂×C₄×C₃₁	—	2	4	no	—
268	C₂×C₄×C₃₁	—	2	4	no	—
269	C₂×C₄×C₃₁	—	4	4	no	—
270	C₂×C₄×C₃₁	—	4	16	no	—
271	C₂×C₄×C₃₁	—	8	8	no	—
272	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
273	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
274	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
275	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
276	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
277	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
278	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
279	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
280	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
281	C₂×C₄×C₃₁	C₂×#41	1	8	yes	—
282	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
283	C₂×#529	C₂×C₃₁	1	1	yes	characteristic
284	#2373	C₂×C₃₁	1	4	yes	complemented
285	#2368	C₂×C₃₁	1	2	yes	complemented
286	#2364	C₂×C₃₁	1	2	yes	—
287	#2355	C₂×C₃₁	1	2	yes	complemented
288	#2352	C₂×C₃₁	1	2	yes	complemented
289	#2352	C₂×C₃₁	1	2	yes	complemented
290	D₄×C₄⋊C₄	C₂×C₃₁	1	2	yes	complemented
291	#2337	C₂×C₃₁	1	2	yes	complemented
292	#2336	C₂×C₃₁	1	2	yes	complemented
293	#2314	C₂×C₃₁	1	2	yes	complemented
294	C₄×#123	C₂×C₃₁	1	4	yes	complemented
295	#2283	C₂×C₃₁	1	1	yes	characteristic, complemented
296	#2274	C₂×C₃₁	1	2	yes	complemented
297	C₂×C₄×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
298	C₂²×C₃₁	—	2	2	no	—
299	C₂²×C₃₁	—	2	2	no	—
300	C₂²×C₃₁	—	2	4	no	—
301	C₂²×C₃₁	—	2	4	no	—
302	C₂²×C₃₁	—	2	4	no	—
303	C₂²×C₃₁	—	2	4	no	—
304	C₂²×C₃₁	—	2	4	no	—
305	C₂²×C₃₁	—	4	2	no	—
306	C₂²×C₃₁	—	4	2	no	—
307	C₂²×C₃₁	—	4	2	no	—
308	C₂²×C₃₁	—	4	2	no	—
309	C₂²×C₃₁	#520	1	4	yes	—
310	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
311	C₂²×C₃₁	C₂²×#41	1	2	yes	—
312	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
313	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
314	C₂²×C₃₁	#585	1	2	yes	—
315	C₄×C₃₁	—	2	4	no	—
316	C₄×C₃₁	—	2	2	no	—
317	C₄×C₃₁	—	2	2	no	—
318	C₄×C₃₁	—	2	2	no	—
319	C₄×C₃₁	—	2	4	no	—
320	C₄×C₃₁	—	2	2	no	—
321	C₄×C₃₁	—	2	2	no	—
322	C₄×C₃₁	—	2	4	no	—
323	C₄×C₃₁	—	2	8	no	—
324	C₄×C₃₁	—	4	4	no	—
325	C₄×C₃₁	—	4	4	no	—
326	C₄×C₃₁	—	4	4	no	—
327	C₄×C₃₁	C₄×#41	1	4	yes	—
328	#529	C₄×C₃₁	1	16	yes	complemented
329	C₂×#126	C₂²×C₃₁	1	2	yes	—
330	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
331	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
332	C₂×#123	C₂²×C₃₁	1	2	yes	—
333	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
334	C₂×#121	C₂²×C₃₁	1	1	yes	characteristic
335	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
336	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	—
337	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
338	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
339	C₂×#117	C₂²×C₃₁	1	2	yes	—
340	C₂×#117	C₂²×C₃₁	1	2	yes	—
341	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
342	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
343	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
344	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
345	#391	C₂²×C₃₁	1	4	yes	—
346	#389	C₂²×C₃₁	1	4	yes	—
347	#388	C₂²×C₃₁	1	4	yes	—
348	#388	C₂²×C₃₁	1	4	yes	—
349	#388	C₂²×C₃₁	1	4	yes	—
350	#388	C₂²×C₃₁	1	4	yes	—
351	#387	C₂²×C₃₁	1	4	yes	—
352	#387	C₂²×C₃₁	1	4	yes	—
353	#387	C₂²×C₃₁	1	4	yes	—
354	#387	C₂²×C₃₁	1	4	yes	—
355	#387	C₂²×C₃₁	1	4	yes	—
356	#387	C₂²×C₃₁	1	4	yes	—
357	#386	C₂²×C₃₁	1	4	yes	—
358	#386	C₂²×C₃₁	1	4	yes	—
359	#385	C₂²×C₃₁	1	4	yes	—
360	#385	C₂²×C₃₁	1	4	yes	—
361	#384	C₂²×C₃₁	1	2	yes	—
362	#383	C₂²×C₃₁	1	2	yes	—
363	#382	C₂²×C₃₁	1	2	yes	—
364	#382	C₂²×C₃₁	1	2	yes	—
365	#381	C₂²×C₃₁	1	2	yes	—
366	#380	C₂²×C₃₁	1	2	yes	—
367	#380	C₂²×C₃₁	1	2	yes	—
368	#380	C₂²×C₃₁	1	1	yes	characteristic
369	#380	C₂²×C₃₁	1	1	yes	characteristic
370	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
371	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
372	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
373	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
374	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
375	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
376	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
377	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
378	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
379	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
380	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
381	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
382	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
383	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
384	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
385	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
386	#376	C₂²×C₃₁	1	2	yes	—
387	C₂×#95	C₂²×C₃₁	1	2	yes	—
388	C₂×C₃₁	—	2	4	no	—
389	C₂×C₃₁	—	4	4	no	—
390	C₂×C₃₁	C₂×C₄×#41	1	1	yes	characteristic
391	C₂×C₃₁	C₂×#520	1	1	yes	characteristic
392	C₂×C₃₁	#3428	1	1	yes	characteristic
393	C₂×C₃₁	#3513	1	2	yes	—
394	C₂×C₃₁	#3516	1	2	yes	—
395	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
396	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
397	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
398	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
399	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
400	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
401	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
402	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
403	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
404	#126	C₂×C₄×C₃₁	1	16	yes	complemented
405	#125	C₂×C₄×C₃₁	1	8	yes	complemented
406	#124	C₂×C₄×C₃₁	1	8	yes	complemented
407	#123	C₂×C₄×C₃₁	1	16	yes	complemented
408	C₂²⋊Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
409	#121	C₂×C₄×C₃₁	1	8	yes	—
410	C₄×Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
411	C₄×D₄	—	4	8	no	—
412	C₄×D₄	C₂×C₄×C₃₁	1	8	yes	complemented
413	C₄×D₄	C₂×C₄×C₃₁	1	16	yes	complemented
414	#117	—	2	8	no	—
415	#117	C₂×C₄×C₃₁	1	16	yes	complemented
416	C₂×C₄⋊C₄	—	2	8	no	—
417	C₂×C₄⋊C₄	—	2	8	no	—
418	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
419	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
420	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
421	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
422	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
423	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
424	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
425	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
426	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
427	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
428	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
429	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
430	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
431	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
432	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
433	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
434	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
435	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
436	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
437	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
438	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
439	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
440	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
441	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
442	C₂×C₄²	—	4	4	no	—
443	C₂×C₄²	C₂×C₄×C₃₁	1	8	yes	complemented
444	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
445	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
446	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
447	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
448	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
449	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
450	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
451	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
452	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
453	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
454	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
455	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
456	#95	C₂³×C₃₁	1	2	yes	—
457	#95	C₂³×C₃₁	1	4	yes	—
458	#95	C₂³×C₃₁	1	2	yes	—
459	#95	C₂³×C₃₁	1	4	yes	—
460	#95	C₂³×C₃₁	1	2	yes	—
461	#95	C₂³×C₃₁	1	2	yes	—
462	#95	C₂³×C₃₁	1	4	yes	—
463	#95	C₂³×C₃₁	1	4	yes	—
464	#95	C₂³×C₃₁	1	4	yes	—
465	#95	C₂³×C₃₁	1	4	yes	—
466	#95	C₂³×C₃₁	1	4	yes	—
467	#95	C₂³×C₃₁	1	4	yes	—
468	#95	C₂³×C₃₁	1	4	yes	—
469	#95	C₂³×C₃₁	1	4	yes	—
470	C₃₁	#34241	1	1	yes	characteristic, direct product factor, Sylow
471	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
472	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
473	C₂×Q₈	—	4	2	no	—
474	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
475	C₂×D₄	—	2	4	no	—
476	C₂×D₄	—	2	4	no	—
477	C₂×D₄	—	4	2	no	—
478	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
479	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
480	C₂²×C₄	—	2	2	no	—
481	C₂²×C₄	—	2	2	no	—
482	C₂²×C₄	—	2	4	no	—
483	C₂²×C₄	—	2	4	no	—
484	C₂²×C₄	—	2	4	no	—
485	C₂²×C₄	—	2	8	no	—
486	C₂²×C₄	—	2	8	no	—
487	C₂²×C₄	—	2	8	no	—
488	C₂²×C₄	—	4	2	no	—
489	C₂²×C₄	—	4	4	no	—
490	C₂²×C₄	—	4	4	no	—
491	C₂²×C₄	—	4	4	no	—
492	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
493	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
494	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
495	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
496	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
497	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
498	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
499	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
500	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
501	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
502	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
503	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
504	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
505	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
506	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
507	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
508	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
509	C₄⋊C₄	—	2	8	no	—
510	C₄⋊C₄	—	2	8	no	—
511	C₄⋊C₄	—	2	4	no	—
512	C₄⋊C₄	—	2	4	no	—
513	C₄⋊C₄	—	2	4	no	—
514	C₄⋊C₄	—	4	4	no	—
515	C₄⋊C₄	—	4	4	no	—
516	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
517	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
518	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
519	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
520	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
521	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
522	C₄⋊C₄	#41×C₃₁	1	8	yes	—
523	C₂²⋊C₄	—	2	4	no	—
524	C₂²⋊C₄	—	2	4	no	—
525	C₂²⋊C₄	—	2	8	no	—
526	C₂²⋊C₄	—	4	4	no	—
527	C₂²⋊C₄	—	4	4	no	—
528	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
529	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
530	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
531	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
532	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
533	C₄²	—	2	8	no	—
534	C₄²	—	4	2	no	—
535	C₄²	—	4	4	no	—
536	C₄²	—	4	2	no	—
537	C₄²	—	4	4	no	—
538	C₄²	—	4	4	no	—
539	C₄²	—	4	4	no	—
540	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
541	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
542	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
543	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
544	C₄²	C₂²×C₄×C₃₁	1	8	yes	—
545	C₂³	—	2	2	no	—
546	C₂³	—	2	2	no	—
547	C₂³	—	2	2	no	—
548	C₂³	—	2	2	no	—
549	C₂³	—	2	2	no	—
550	C₂³	—	2	2	no	—
551	C₂³	—	2	2	no	—
552	C₂³	—	2	2	no	—
553	C₂³	C₂³×C₄×C₃₁	1	4	yes	—
554	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
555	C₂³	C₃₁×H(5,𝔽₂)	1	2	yes	—
556	C₂³	C₃₁×2₋¹⁺⁴	1	2	yes	—
557	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
558	Q₈	—	4	4	no	—
559	D₄	—	2	16	no	—
560	D₄	—	4	4	no	—
561	C₂×C₄	—	2	8	no	—
562	C₂×C₄	—	2	2	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	2	2	no	—
565	C₂×C₄	—	2	1	no	—
566	C₂×C₄	—	2	1	no	—
567	C₂×C₄	—	2	4	no	—
568	C₂×C₄	—	2	2	no	—
569	C₂×C₄	—	2	1	no	—
570	C₂×C₄	—	2	1	no	—
571	C₂×C₄	—	2	2	no	—
572	C₂×C₄	—	2	2	no	—
573	C₂×C₄	—	2	2	no	—
574	C₂×C₄	—	2	4	no	—
575	C₂×C₄	—	2	2	no	—
576	C₂×C₄	—	2	2	no	—
577	C₂×C₄	—	2	8	no	—
578	C₂×C₄	—	2	16	no	—
579	C₂×C₄	—	2	4	no	—
580	C₂×C₄	—	2	4	no	—
581	C₂×C₄	—	2	4	no	—
582	C₂×C₄	—	2	4	no	—
583	C₂×C₄	—	2	4	no	—
584	C₂×C₄	—	2	4	no	—
585	C₂×C₄	—	2	4	no	—
586	C₂×C₄	—	2	4	no	—
587	C₂×C₄	—	2	4	no	—
588	C₂×C₄	—	4	4	no	—
589	C₂×C₄	—	4	16	no	—
590	C₂×C₄	—	8	8	no	—
591	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
592	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
593	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
594	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
595	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
596	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
597	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
598	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
599	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
600	C₂×C₄	C₂×#41×C₃₁	1	8	yes	—
601	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
602	C₂²	—	2	2	no	—
603	C₂²	—	2	2	no	—
604	C₂²	—	2	4	no	—
605	C₂²	—	2	4	no	—
606	C₂²	—	2	4	no	—
607	C₂²	—	2	4	no	—
608	C₂²	—	2	4	no	—
609	C₂²	—	4	2	no	—
610	C₂²	—	4	2	no	—
611	C₂²	—	4	2	no	—
612	C₂²	—	4	2	no	—
613	C₂²	C₃₁×#520	1	4	yes	—
614	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
615	C₂²	C₂²×#41×C₃₁	1	2	yes	—
616	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
617	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
618	C₂²	C₃₁×#585	1	2	yes	—
619	C₄	—	2	4	no	—
620	C₄	—	2	2	no	—
621	C₄	—	2	2	no	—
622	C₄	—	2	2	no	—
623	C₄	—	2	4	no	—
624	C₄	—	2	2	no	—
625	C₄	—	2	2	no	—
626	C₄	—	2	4	no	—
627	C₄	—	2	8	no	—
628	C₄	—	4	4	no	—
629	C₄	—	4	4	no	—
630	C₄	—	4	4	no	—
631	C₄	C₄×#41×C₃₁	1	4	yes	—
632	C₂	—	2	4	no	—
633	C₂	—	4	4	no	—
634	C₂	C₂×C₄×#41×C₃₁	1	1	yes	characteristic
635	C₂	C₂×C₃₁×#520	1	1	yes	characteristic
636	C₂	C₃₁×#3428	1	1	yes	characteristic
637	C₂	C₃₁×#3513	1	2	yes	—
638	C₂	C₃₁×#3516	1	2	yes	—

	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×#2373	C₂	semidirect
3. C₃₁×#2368	C₂	semidirect
5. C₃₁×#2355	C₂	semidirect
6. C₃₁×#2352	C₂	semidirect
7. C₃₁×#2352	C₂	semidirect
8. D₄×C₄⋊C₄×C₃₁	C₂	semidirect
9. C₃₁×#2337	C₂	semidirect
10. C₃₁×#2336	C₂	semidirect
11. C₃₁×#2314	C₂	semidirect
12. C₄×C₃₁×#123	C₂	semidirect
13. C₃₁×#2283	C₂	semidirect
14. C₃₁×#2274	C₂	semidirect
15. C₂×C₄×C₄⋊C₄×C₃₁	C₂	semidirect
16. C₃₁×#529	C₄	semidirect
85. C₃₁×#126	C₂×C₄	semidirect
86. C₃₁×#125	C₂×C₄	semidirect
87. C₃₁×#124	C₂×C₄	semidirect
88. C₃₁×#123	C₂×C₄	semidirect
89. C₃₁×C₂²⋊Q₈	C₂×C₄	semidirect
91. C₄×Q₈×C₃₁	C₂×C₄	semidirect
93. C₄×D₄×C₃₁	C₂×C₄	semidirect
94. C₄×D₄×C₃₁	C₂×C₄	semidirect
96. C₃₁×#117	C₂×C₄	semidirect
124. C₂×C₄²×C₃₁	C₂×C₄	semidirect
225. #34241	C₃₁	direct (N × H)
284. #2373	C₂×C₃₁	semidirect
285. #2368	C₂×C₃₁	semidirect
287. #2355	C₂×C₃₁	semidirect
288. #2352	C₂×C₃₁	semidirect
289. #2352	C₂×C₃₁	semidirect
290. D₄×C₄⋊C₄	C₂×C₃₁	semidirect
291. #2337	C₂×C₃₁	semidirect
292. #2336	C₂×C₃₁	semidirect
293. #2314	C₂×C₃₁	semidirect
294. C₄×#123	C₂×C₃₁	semidirect
295. #2283	C₂×C₃₁	semidirect
296. #2274	C₂×C₃₁	semidirect
297. C₂×C₄×C₄⋊C₄	C₂×C₃₁	semidirect
328. #529	C₄×C₃₁	semidirect
404. #126	C₂×C₄×C₃₁	semidirect
405. #125	C₂×C₄×C₃₁	semidirect
406. #124	C₂×C₄×C₃₁	semidirect
407. #123	C₂×C₄×C₃₁	semidirect
408. C₂²⋊Q₈	C₂×C₄×C₃₁	semidirect
410. C₄×Q₈	C₂×C₄×C₃₁	semidirect
412. C₄×D₄	C₂×C₄×C₃₁	semidirect
413. C₄×D₄	C₂×C₄×C₃₁	semidirect
415. #117	C₂×C₄×C₃₁	semidirect
443. C₂×C₄²	C₂×C₄×C₃₁	semidirect
470. C₃₁	#34241	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×#2373	2
C₃₁×#2368	2
C₃₁×#2364	2
C₃₁×#2355	2
C₃₁×#2352	2
D₄×C₄⋊C₄×C₃₁	2
C₃₁×#2337	2
C₃₁×#2336	2
C₃₁×#2314	2
C₄×C₃₁×#123	2
C₃₁×#2283	2
C₃₁×#2274	2
C₂×C₄×C₄⋊C₄×C₃₁	2
#34241	31

Subgroups — C31×#34241 #21491000

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#34241 #21491000