Subgroups — C₃₁×#22572 #21479331

Order 7936.

1006 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#416	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2346	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2339	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1933	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1927	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1864	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1825	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1807	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1788	C₂	1	1	yes	maximal, characteristic
10	C₄×C₃₁×#102	C₂	1	2	yes	maximal, complemented
11	C₈×C₂²⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
15	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#106	C₂²	1	2	yes	—
17	C₃₁×#416	C₄	1	8	yes	complemented
18	C₂×C₃₁×#103	C₂²	1	2	yes	—
19	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#102	C₂²	1	2	yes	—
22	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
26	C₃₁×#391	C₂²	1	2	yes	—
27	C₃₁×#389	C₂²	1	2	yes	—
28	C₃₁×#388	—	2	2	no	—
29	C₃₁×#386	—	2	2	no	—
30	C₃₁×#382	—	2	1	no	—
31	C₃₁×#381	—	2	1	no	—
32	C₃₁×#380	C₂²	1	1	yes	characteristic
33	C₃₁×#380	C₂²	1	1	yes	characteristic
34	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
35	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
36	C₃₁×#376	—	2	1	no	—
37	C₂×C₃₁×#95	—	2	1	no	—
38	C₃₁×#340	C₂²	1	2	yes	—
39	C₃₁×#340	C₂²	1	2	yes	—
40	C₃₁×#340	C₂²	1	2	yes	—
41	C₃₁×#336	C₂²	1	2	yes	—
42	C₂×#41×C₃₁	—	2	2	no	—
43	C₂×#41×C₃₁	—	2	4	no	—
44	C₂×#41×C₃₁	—	4	2	no	—
45	C₂×#41×C₃₁	C₂×C₄	1	4	yes	—
46	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
47	C₂²×D₄×C₃₁	—	2	1	no	—
48	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
49	C₂³×C₄×C₃₁	—	2	1	no	—
50	C₂³×C₄×C₃₁	—	2	2	no	—
51	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₃₁	—	4	2	no	—
53	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	complemented
54	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
55	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
57	C₄×D₄×C₃₁	—	2	4	no	—
58	C₄×D₄×C₃₁	—	4	2	no	—
59	C₃₁×#117	—	2	2	no	—
60	C₃₁×#117	C₂×C₄	1	4	yes	complemented
61	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
62	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
63	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
64	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
66	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
67	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
68	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
69	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
70	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
71	C₂×C₄²×C₃₁	—	2	1	no	—
72	C₂×C₄²×C₃₁	—	2	2	no	—
73	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
74	C₃₁×#106	—	4	2	no	—
75	C₃₁×#103	—	4	2	no	—
76	C₃₁×#103	C₂×C₄	1	4	yes	complemented
77	C₃₁×#103	C₂×C₄	1	4	yes	complemented
78	C₃₁×#102	—	2	4	no	—
79	C₃₁×#102	C₂×C₄	1	4	yes	—
80	C₃₁×#102	C₂×C₄	1	4	yes	—
81	C₃₁×C₂²⋊C₈	—	2	2	no	—
82	C₄×C₈×C₃₁	—	4	2	no	—
83	C₃₁×#95	—	2	1	no	—
84	C₃₁×#95	—	2	2	no	—
85	C₃₁×#95	—	2	1	no	—
86	C₃₁×#95	—	2	2	no	—
87	C₃₁×#95	—	2	2	no	—
88	C₃₁×#95	—	2	2	no	—
89	C₃₁×#95	D₄	1	1	yes	characteristic
90	C₃₁×#95	D₄	1	1	yes	characteristic
91	C₂⁴×C₃₁	—	2	1	no	—
92	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
93	#41×C₃₁	—	2	8	no	—
94	#41×C₃₁	—	4	4	no	—
95	#41×C₃₁	—	4	8	no	—
96	C₂×Q₈×C₃₁	—	2	1	no	—
97	C₂×Q₈×C₃₁	—	2	2	no	—
98	C₂×Q₈×C₃₁	—	2	2	no	—
99	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
100	C₂×D₄×C₃₁	—	2	4	no	—
101	C₂×D₄×C₃₁	—	2	1	no	—
102	C₂×D₄×C₃₁	—	2	2	no	—
103	C₂×D₄×C₃₁	—	4	4	no	—
104	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
105	C₂×D₄×C₃₁	#41	1	4	yes	—
106	C₂²×C₄×C₃₁	—	2	2	no	—
107	C₂²×C₄×C₃₁	—	2	2	no	—
108	C₂²×C₄×C₃₁	—	2	1	no	—
109	C₂²×C₄×C₃₁	—	2	1	no	—
110	C₂²×C₄×C₃₁	—	2	1	no	—
111	C₂²×C₄×C₃₁	—	2	4	no	—
112	C₂²×C₄×C₃₁	—	2	2	no	—
113	C₂²×C₄×C₃₁	—	2	2	no	—
114	C₂²×C₄×C₃₁	—	4	4	no	—
115	C₂²×C₄×C₃₁	—	4	4	no	—
116	C₂²×C₄×C₃₁	—	4	4	no	—
117	C₂²×C₄×C₃₁	—	8	2	no	—
118	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
119	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
120	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
122	C₂²×C₄×C₃₁	#41	1	2	yes	—
123	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
124	C₂×C₈×C₃₁	—	2	2	no	—
125	C₂×C₈×C₃₁	—	2	2	no	—
126	C₂×C₈×C₃₁	—	2	2	no	—
127	C₂×C₈×C₃₁	—	2	2	no	—
128	C₂×C₈×C₃₁	—	2	2	no	—
129	C₂×C₈×C₃₁	—	4	4	no	—
130	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
131	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
132	C₄⋊C₄×C₃₁	—	2	1	no	—
133	C₄⋊C₄×C₃₁	—	2	1	no	—
134	C₄⋊C₄×C₃₁	—	2	2	no	—
135	C₄⋊C₄×C₃₁	—	2	2	no	—
136	C₄⋊C₄×C₃₁	—	2	2	no	—
137	C₄⋊C₄×C₃₁	—	2	2	no	—
138	C₄⋊C₄×C₃₁	—	8	2	no	—
139	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
140	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
141	C₂²⋊C₄×C₃₁	—	2	4	no	—
142	C₂²⋊C₄×C₃₁	—	2	4	no	—
143	C₂²⋊C₄×C₃₁	—	4	4	no	—
144	C₂²⋊C₄×C₃₁	—	8	1	no	—
145	C₂²⋊C₄×C₃₁	—	8	1	no	—
146	C₄²×C₃₁	—	2	4	no	—
147	C₄²×C₃₁	—	2	2	no	—
148	C₄²×C₃₁	—	2	2	no	—
149	C₄²×C₃₁	—	4	4	no	—
150	#22572	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
151	C₂³×C₃₁	—	2	2	no	—
152	C₂³×C₃₁	—	2	4	no	—
153	C₂³×C₃₁	—	2	2	no	—
154	C₂³×C₃₁	—	2	2	no	—
155	C₂³×C₃₁	—	2	1	no	—
156	C₂³×C₃₁	—	2	1	no	—
157	C₂³×C₃₁	—	4	2	no	—
158	C₂³×C₃₁	—	4	1	no	—
159	C₂³×C₃₁	—	4	1	no	—
160	C₂³×C₃₁	C₄×D₄	1	2	yes	—
161	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
162	Q₈×C₃₁	—	2	2	no	—
163	Q₈×C₃₁	—	2	2	no	—
164	Q₈×C₃₁	—	4	2	no	—
165	D₄×C₃₁	—	2	2	no	—
166	D₄×C₃₁	—	2	4	no	—
167	D₄×C₃₁	—	4	8	no	—
168	D₄×C₃₁	#117	1	4	yes	—
169	C₂×C₄×C₃₁	—	2	2	no	—
170	C₂×C₄×C₃₁	—	2	2	no	—
171	C₂×C₄×C₃₁	—	2	2	no	—
172	C₂×C₄×C₃₁	—	2	2	no	—
173	C₂×C₄×C₃₁	—	2	2	no	—
174	C₂×C₄×C₃₁	—	2	4	no	—
175	C₂×C₄×C₃₁	—	2	2	no	—
176	C₂×C₄×C₃₁	—	2	2	no	—
177	C₂×C₄×C₃₁	—	2	2	no	—
178	C₂×C₄×C₃₁	—	2	2	no	—
179	C₂×C₄×C₃₁	—	2	2	no	—
180	C₂×C₄×C₃₁	—	2	2	no	—
181	C₂×C₄×C₃₁	—	4	8	no	—
182	C₂×C₄×C₃₁	—	4	2	no	—
183	C₂×C₄×C₃₁	—	4	2	no	—
184	C₂×C₄×C₃₁	—	4	1	no	—
185	C₂×C₄×C₃₁	—	4	1	no	—
186	C₂×C₄×C₃₁	—	4	2	no	—
187	C₂×C₄×C₃₁	—	4	2	no	—
188	C₂×C₄×C₃₁	—	4	2	no	—
189	C₂×C₄×C₃₁	—	4	2	no	—
190	C₂×C₄×C₃₁	—	4	2	no	—
191	C₂×C₄×C₃₁	—	4	2	no	—
192	C₂×C₄×C₃₁	—	8	8	no	—
193	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
194	C₂×C₄×C₃₁	#135	1	4	yes	—
195	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
196	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
197	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
198	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
199	C₈×C₃₁	—	2	2	no	—
200	C₈×C₃₁	—	2	2	no	—
201	C₈×C₃₁	—	4	2	no	—
202	C₂×#416	C₂×C₃₁	1	1	yes	characteristic
203	#2346	C₂×C₃₁	1	1	yes	characteristic, complemented
204	#2339	C₂×C₃₁	1	1	yes	characteristic
205	#1933	C₂×C₃₁	1	2	yes	complemented
206	#1927	C₂×C₃₁	1	2	yes	complemented
207	#1864	C₂×C₃₁	1	2	yes	complemented
208	#1825	C₂×C₃₁	1	1	yes	characteristic, complemented
209	#1807	C₂×C₃₁	1	1	yes	characteristic, complemented
210	#1788	C₂×C₃₁	1	1	yes	characteristic
211	C₄×#102	C₂×C₃₁	1	2	yes	complemented
212	C₈×C₂²⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
213	C₂²×C₃₁	—	2	2	no	—
214	C₂²×C₃₁	—	2	4	no	—
215	C₂²×C₃₁	—	2	2	no	—
216	C₂²×C₃₁	—	2	2	no	—
217	C₂²×C₃₁	—	4	2	no	—
218	C₂²×C₃₁	—	4	2	no	—
219	C₂²×C₃₁	—	4	4	no	—
220	C₂²×C₃₁	—	4	2	no	—
221	C₂²×C₃₁	—	4	2	no	—
222	C₂²×C₃₁	—	4	4	no	—
223	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
224	C₂²×C₃₁	#546	1	2	yes	—
225	C₂²×C₃₁	C₂×#135	1	2	yes	—
226	C₂²×C₃₁	#577	1	1	yes	characteristic
227	C₂²×C₃₁	#577	1	1	yes	characteristic
228	C₄×C₃₁	—	2	2	no	—
229	C₄×C₃₁	—	2	4	no	—
230	C₄×C₃₁	—	4	2	no	—
231	C₄×C₃₁	—	4	2	no	—
232	C₄×C₃₁	—	4	2	no	—
233	C₄×C₃₁	—	4	2	no	—
234	C₄×C₃₁	—	8	2	no	—
235	C₄×C₃₁	—	8	2	no	—
236	C₄×C₃₁	C₂×#117	1	1	yes	characteristic
237	C₄×C₃₁	#518	1	1	yes	characteristic
238	C₄×C₃₁	#551	1	2	yes	—
239	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
240	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
241	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
242	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
243	C₂×#106	C₂²×C₃₁	1	2	yes	—
244	#416	C₄×C₃₁	1	8	yes	complemented
245	C₂×#103	C₂²×C₃₁	1	2	yes	—
246	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
247	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
248	C₂×#102	C₂²×C₃₁	1	2	yes	—
249	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
250	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
251	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
252	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
253	#391	C₂²×C₃₁	1	2	yes	—
254	#389	C₂²×C₃₁	1	2	yes	—
255	#388	—	2	2	no	—
256	#386	—	2	2	no	—
257	#382	—	2	1	no	—
258	#381	—	2	1	no	—
259	#380	C₂²×C₃₁	1	1	yes	characteristic
260	#380	C₂²×C₃₁	1	1	yes	characteristic
261	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
262	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
263	#376	—	2	1	no	—
264	C₂×#95	—	2	1	no	—
265	#340	C₂²×C₃₁	1	2	yes	—
266	#340	C₂²×C₃₁	1	2	yes	—
267	#340	C₂²×C₃₁	1	2	yes	—
268	#336	C₂²×C₃₁	1	2	yes	—
269	C₂×C₃₁	—	4	2	no	—
270	C₂×C₃₁	—	4	4	no	—
271	C₂×C₃₁	#2342	1	1	yes	characteristic
272	C₂×C₃₁	C₄×#135	1	1	yes	characteristic
273	C₂×C₃₁	#2952	1	1	yes	characteristic
274	C₂×C₃₁	#3296	1	2	yes	—
275	C₂×C₃₁	#3303	1	2	yes	—
276	C₂×#41	—	2	2	no	—
277	C₂×#41	—	2	4	no	—
278	C₂×#41	—	4	2	no	—
279	C₂×#41	C₂×C₄×C₃₁	1	4	yes	—
280	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
281	C₂²×D₄	—	2	1	no	—
282	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
283	C₂³×C₄	—	2	1	no	—
284	C₂³×C₄	—	2	2	no	—
285	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
286	C₂²×C₈	—	4	2	no	—
287	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	complemented
288	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
289	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
290	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
291	C₄×D₄	—	2	4	no	—
292	C₄×D₄	—	4	2	no	—
293	#117	—	2	2	no	—
294	#117	C₂×C₄×C₃₁	1	4	yes	complemented
295	C₂×C₄⋊C₄	—	2	2	no	—
296	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
297	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
298	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
299	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
300	C₂×C₂²⋊C₄	—	2	1	no	—
301	C₂×C₂²⋊C₄	—	2	2	no	—
302	C₂×C₂²⋊C₄	—	2	1	no	—
303	C₂×C₂²⋊C₄	—	2	1	no	—
304	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
305	C₂×C₄²	—	2	1	no	—
306	C₂×C₄²	—	2	2	no	—
307	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
308	#106	—	4	2	no	—
309	#103	—	4	2	no	—
310	#103	C₂×C₄×C₃₁	1	4	yes	complemented
311	#103	C₂×C₄×C₃₁	1	4	yes	complemented
312	#102	—	2	4	no	—
313	#102	C₂×C₄×C₃₁	1	4	yes	—
314	#102	C₂×C₄×C₃₁	1	4	yes	—
315	C₂²⋊C₈	—	2	2	no	—
316	C₄×C₈	—	4	2	no	—
317	#95	—	2	1	no	—
318	#95	—	2	2	no	—
319	#95	—	2	1	no	—
320	#95	—	2	2	no	—
321	#95	—	2	2	no	—
322	#95	—	2	2	no	—
323	#95	D₄×C₃₁	1	1	yes	characteristic
324	#95	D₄×C₃₁	1	1	yes	characteristic
325	C₃₁	#22572	1	1	yes	characteristic, direct product factor, Sylow
326	C₂⁴	—	2	1	no	—
327	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
328	#41	—	2	8	no	—
329	#41	—	4	4	no	—
330	#41	—	4	8	no	—
331	C₂×Q₈	—	2	1	no	—
332	C₂×Q₈	—	2	2	no	—
333	C₂×Q₈	—	2	2	no	—
334	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
335	C₂×D₄	—	2	4	no	—
336	C₂×D₄	—	2	1	no	—
337	C₂×D₄	—	2	2	no	—
338	C₂×D₄	—	4	4	no	—
339	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
340	C₂×D₄	#41×C₃₁	1	4	yes	—
341	C₂²×C₄	—	2	2	no	—
342	C₂²×C₄	—	2	2	no	—
343	C₂²×C₄	—	2	1	no	—
344	C₂²×C₄	—	2	1	no	—
345	C₂²×C₄	—	2	1	no	—
346	C₂²×C₄	—	2	4	no	—
347	C₂²×C₄	—	2	2	no	—
348	C₂²×C₄	—	2	2	no	—
349	C₂²×C₄	—	4	4	no	—
350	C₂²×C₄	—	4	4	no	—
351	C₂²×C₄	—	4	4	no	—
352	C₂²×C₄	—	8	2	no	—
353	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
354	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
355	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
356	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
357	C₂²×C₄	#41×C₃₁	1	2	yes	—
358	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
359	C₂×C₈	—	2	2	no	—
360	C₂×C₈	—	2	2	no	—
361	C₂×C₈	—	2	2	no	—
362	C₂×C₈	—	2	2	no	—
363	C₂×C₈	—	2	2	no	—
364	C₂×C₈	—	4	4	no	—
365	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
366	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
367	C₄⋊C₄	—	2	1	no	—
368	C₄⋊C₄	—	2	1	no	—
369	C₄⋊C₄	—	2	2	no	—
370	C₄⋊C₄	—	2	2	no	—
371	C₄⋊C₄	—	2	2	no	—
372	C₄⋊C₄	—	2	2	no	—
373	C₄⋊C₄	—	8	2	no	—
374	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
375	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
376	C₂²⋊C₄	—	2	4	no	—
377	C₂²⋊C₄	—	2	4	no	—
378	C₂²⋊C₄	—	4	4	no	—
379	C₂²⋊C₄	—	8	1	no	—
380	C₂²⋊C₄	—	8	1	no	—
381	C₄²	—	2	4	no	—
382	C₄²	—	2	2	no	—
383	C₄²	—	2	2	no	—
384	C₄²	—	4	4	no	—
385	C₂³	—	2	2	no	—
386	C₂³	—	2	4	no	—
387	C₂³	—	2	2	no	—
388	C₂³	—	2	2	no	—
389	C₂³	—	2	1	no	—
390	C₂³	—	2	1	no	—
391	C₂³	—	4	2	no	—
392	C₂³	—	4	1	no	—
393	C₂³	—	4	1	no	—
394	C₂³	C₄×D₄×C₃₁	1	2	yes	—
395	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
396	Q₈	—	2	2	no	—
397	Q₈	—	2	2	no	—
398	Q₈	—	4	2	no	—
399	D₄	—	2	2	no	—
400	D₄	—	2	4	no	—
401	D₄	—	4	8	no	—
402	D₄	C₃₁×#117	1	4	yes	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	2	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	2	2	no	—
408	C₂×C₄	—	2	4	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	2	2	no	—
412	C₂×C₄	—	2	2	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	4	8	no	—
416	C₂×C₄	—	4	2	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	—	4	1	no	—
419	C₂×C₄	—	4	1	no	—
420	C₂×C₄	—	4	2	no	—
421	C₂×C₄	—	4	2	no	—
422	C₂×C₄	—	4	2	no	—
423	C₂×C₄	—	4	2	no	—
424	C₂×C₄	—	4	2	no	—
425	C₂×C₄	—	4	2	no	—
426	C₂×C₄	—	8	8	no	—
427	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
428	C₂×C₄	C₃₁×#135	1	4	yes	—
429	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
430	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
431	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
432	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
433	C₈	—	2	2	no	—
434	C₈	—	2	2	no	—
435	C₈	—	4	2	no	—
436	C₂²	—	2	2	no	—
437	C₂²	—	2	4	no	—
438	C₂²	—	2	2	no	—
439	C₂²	—	2	2	no	—
440	C₂²	—	4	2	no	—
441	C₂²	—	4	2	no	—
442	C₂²	—	4	4	no	—
443	C₂²	—	4	2	no	—
444	C₂²	—	4	2	no	—
445	C₂²	—	4	4	no	—
446	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
447	C₂²	C₃₁×#546	1	2	yes	—
448	C₂²	C₂×C₃₁×#135	1	2	yes	—
449	C₂²	C₃₁×#577	1	1	yes	characteristic
450	C₂²	C₃₁×#577	1	1	yes	characteristic
451	C₄	—	2	2	no	—
452	C₄	—	2	4	no	—
453	C₄	—	4	2	no	—
454	C₄	—	4	2	no	—
455	C₄	—	4	2	no	—
456	C₄	—	4	2	no	—
457	C₄	—	8	2	no	—
458	C₄	—	8	2	no	—
459	C₄	C₂×C₃₁×#117	1	1	yes	characteristic
460	C₄	C₃₁×#518	1	1	yes	characteristic
461	C₄	C₃₁×#551	1	2	yes	—
462	C₂	—	4	2	no	—
463	C₂	—	4	4	no	—
464	C₂	C₃₁×#2342	1	1	yes	characteristic
465	C₂	C₄×C₃₁×#135	1	1	yes	characteristic
466	C₂	C₃₁×#2952	1	1	yes	characteristic
467	C₂	C₃₁×#3296	1	2	yes	—
468	C₂	C₃₁×#3303	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂²×#41×C₃₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄×C₂²⋊C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2346	C₂	semidirect
4. C₃₁×#1933	C₂	semidirect
5. C₃₁×#1927	C₂	semidirect
6. C₃₁×#1864	C₂	semidirect
7. C₃₁×#1825	C₂	semidirect
8. C₃₁×#1807	C₂	semidirect
10. C₄×C₃₁×#102	C₂	semidirect
11. C₈×C₂²⋊C₄×C₃₁	C₂	semidirect
17. C₃₁×#416	C₄	semidirect
53. C₂²×C₈×C₃₁	C₂×C₄	semidirect
60. C₃₁×#117	C₂×C₄	semidirect
76. C₃₁×#103	C₂×C₄	semidirect
77. C₃₁×#103	C₂×C₄	semidirect
150. #22572	C₃₁	direct (N × H)
203. #2346	C₂×C₃₁	semidirect
205. #1933	C₂×C₃₁	semidirect
206. #1927	C₂×C₃₁	semidirect
207. #1864	C₂×C₃₁	semidirect
208. #1825	C₂×C₃₁	semidirect
209. #1807	C₂×C₃₁	semidirect
211. C₄×#102	C₂×C₃₁	semidirect
212. C₈×C₂²⋊C₄	C₂×C₃₁	semidirect
244. #416	C₄×C₃₁	semidirect
287. C₂²×C₈	C₂×C₄×C₃₁	semidirect
294. #117	C₂×C₄×C₃₁	semidirect
310. #103	C₂×C₄×C₃₁	semidirect
311. #103	C₂×C₄×C₃₁	semidirect
325. C₃₁	#22572	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×#416	2
C₃₁×#2346	2
C₃₁×#2339	2
C₃₁×#1933	2
C₃₁×#1927	2
C₃₁×#1864	2
C₃₁×#1825	2
C₃₁×#1807	2
C₃₁×#1788	2
C₄×C₃₁×#102	2
C₈×C₂²⋊C₄×C₃₁	2
#22572	31

Subgroups — C31×#22572 #21479331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22572 #21479331