Subgroups — C₃₁×#8240 #21464999

Order 7936.

522 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2929	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1752	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×C₂²⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1501	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1501	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1298	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1297	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×#131	C₄	1	1	yes	characteristic, complemented
9	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₃₁×#525	C₂²	1	1	yes	characteristic
11	C₃₁×#435	—	2	1	no	—
12	C₃₁×#435	—	2	1	no	—
13	C₈×D₄×C₃₁	—	2	1	no	—
14	C₈×D₄×C₃₁	—	2	1	no	—
15	C₃₁×#424	C₄	1	1	yes	characteristic, complemented
16	C₃₁×#408	C₄	1	1	yes	characteristic, complemented
17	C₃₁×#407	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
20	C₃₁×#405	C₄	1	1	yes	characteristic, complemented
21	C₂×C₃₁×#97	—	2	1	no	—
22	C₂×C₄×C₈×C₃₁	—	2	1	no	—
23	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
24	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
25	C₃₁×#336	—	2	1	no	—
26	C₃₁×#336	—	2	1	no	—
27	C₃₁×#328	C₄	1	2	yes	complemented
28	C₃₁×#328	C₄	1	2	yes	complemented
29	C₃₁×#327	C₄	1	2	yes	complemented
30	C₃₁×#327	C₄	1	2	yes	complemented
31	C₂×#41×C₃₁	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
33	C₃₁×#131	—	4	2	no	—
34	C₂×M₁₆×C₃₁	—	2	1	no	—
35	C₂×M₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
36	C₂×M₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₃₁	—	2	1	no	—
38	C₂²×C₈×C₃₁	—	2	1	no	—
39	C₂²×C₈×C₃₁	—	2	1	no	—
40	C₂²×C₈×C₃₁	—	4	2	no	—
41	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
47	C₃₁×#123	—	2	1	no	—
48	C₃₁×C₂²⋊Q₈	C₂×C₄	1	1	yes	characteristic
49	C₃₁×C₂²⋊Q₈	C₂×C₄	1	1	yes	characteristic
50	C₃₁×#121	C₂×C₄	1	1	yes	characteristic
51	C₃₁×#121	C₂×C₄	1	1	yes	characteristic
52	C₃₁×#120	—	2	1	no	—
53	C₄×D₄×C₃₁	—	2	1	no	—
54	C₄×D₄×C₃₁	—	2	1	no	—
55	C₃₁×#117	C₂×C₄	1	1	yes	characteristic
56	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
57	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
58	C₂×C₄²×C₃₁	—	2	1	no	—
59	C₂×C₄²×C₃₁	—	2	1	no	—
60	C₃₁×C₄⋊C₈	—	2	1	no	—
61	C₃₁×C₄⋊C₈	—	2	1	no	—
62	C₃₁×C₂²⋊C₈	—	2	1	no	—
63	C₃₁×C₂²⋊C₈	—	2	1	no	—
64	C₃₁×C₂²⋊C₈	—	2	1	no	—
65	C₃₁×C₂²⋊C₈	—	2	1	no	—
66	C₃₁×C₂²⋊C₈	—	2	1	no	—
67	C₃₁×C₂²⋊C₈	—	2	1	no	—
68	C₃₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
69	C₃₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
70	C₃₁×#97	—	2	1	no	—
71	C₃₁×#97	—	4	2	no	—
72	C₄×C₈×C₃₁	—	2	1	no	—
73	C₄×C₈×C₃₁	—	4	2	no	—
74	C₃₁×#95	C₂×C₄	1	2	yes	—
75	C₃₁×#95	C₂×C₄	1	2	yes	—
76	C₂⁴×C₃₁	#41	1	1	yes	characteristic
77	#41×C₃₁	—	4	1	no	—
78	#41×C₃₁	—	4	1	no	—
79	C₂×Q₈×C₃₁	C₄²	1	1	yes	characteristic
80	C₂×D₄×C₃₁	—	2	1	no	—
81	C₂×D₄×C₃₁	—	2	1	no	—
82	C₂×D₄×C₃₁	—	2	1	no	—
83	C₂×D₄×C₃₁	C₄²	1	1	yes	characteristic
84	C₂²×C₄×C₃₁	—	2	2	no	—
85	C₂²×C₄×C₃₁	—	2	1	no	—
86	C₂²×C₄×C₃₁	—	2	2	no	—
87	C₂²×C₄×C₃₁	—	4	1	no	—
88	C₂²×C₄×C₃₁	—	4	1	no	—
89	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
95	M₁₆×C₃₁	—	4	2	no	—
96	M₁₆×C₃₁	—	4	1	no	—
97	M₁₆×C₃₁	—	4	1	no	—
98	C₂×C₈×C₃₁	—	2	2	no	—
99	C₂×C₈×C₃₁	—	2	2	no	—
100	C₂×C₈×C₃₁	—	2	2	no	—
101	C₂×C₈×C₃₁	—	2	1	no	—
102	C₂×C₈×C₃₁	—	2	1	no	—
103	C₂×C₈×C₃₁	—	2	2	no	—
104	C₂×C₈×C₃₁	—	2	2	no	—
105	C₂×C₈×C₃₁	—	4	1	no	—
106	C₂×C₈×C₃₁	—	4	1	no	—
107	C₂×C₈×C₃₁	—	4	1	no	—
108	C₂×C₈×C₃₁	—	4	2	no	—
109	C₂×C₈×C₃₁	—	4	1	no	—
110	C₂×C₈×C₃₁	—	4	1	no	—
111	C₂×C₈×C₃₁	—	4	1	no	—
112	C₂×C₈×C₃₁	—	4	1	no	—
113	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
117	C₄⋊C₄×C₃₁	—	2	1	no	—
118	C₄⋊C₄×C₃₁	—	2	1	no	—
119	C₄⋊C₄×C₃₁	C₄²	1	1	yes	characteristic
120	C₄⋊C₄×C₃₁	C₄²	1	1	yes	characteristic
121	C₂²⋊C₄×C₃₁	—	2	4	no	—
122	C₂²⋊C₄×C₃₁	—	2	1	no	—
123	C₂²⋊C₄×C₃₁	—	2	1	no	—
124	C₂²⋊C₄×C₃₁	—	2	1	no	—
125	C₂²⋊C₄×C₃₁	—	2	1	no	—
126	C₂²⋊C₄×C₃₁	—	2	1	no	—
127	C₂²⋊C₄×C₃₁	—	4	2	no	—
128	C₄²×C₃₁	—	2	1	no	—
129	C₄²×C₃₁	—	4	1	no	—
130	C₄²×C₃₁	—	4	1	no	—
131	C₄²×C₃₁	—	4	1	no	—
132	C₄²×C₃₁	—	4	1	no	—
133	#8240	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
134	C₂³×C₃₁	—	2	2	no	—
135	C₂³×C₃₁	—	2	2	no	—
136	C₂³×C₃₁	—	2	1	no	—
137	C₂³×C₃₁	—	4	1	no	—
138	C₂³×C₃₁	#117	1	1	yes	characteristic
139	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
140	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
141	Q₈×C₃₁	—	4	1	no	—
142	D₄×C₃₁	—	4	1	no	—
143	D₄×C₃₁	—	4	2	no	—
144	D₄×C₃₁	—	4	2	no	—
145	D₄×C₃₁	—	4	2	no	—
146	C₂×C₄×C₃₁	—	2	1	no	—
147	C₂×C₄×C₃₁	—	2	1	no	—
148	C₂×C₄×C₃₁	—	2	1	no	—
149	C₂×C₄×C₃₁	—	2	1	no	—
150	C₂×C₄×C₃₁	—	2	1	no	—
151	C₂×C₄×C₃₁	—	2	1	no	—
152	C₂×C₄×C₃₁	—	2	4	no	—
153	C₂×C₄×C₃₁	—	2	1	no	—
154	C₂×C₄×C₃₁	—	2	1	no	—
155	C₂×C₄×C₃₁	—	2	1	no	—
156	C₂×C₄×C₃₁	—	4	1	no	—
157	C₂×C₄×C₃₁	—	4	1	no	—
158	C₂×C₄×C₃₁	—	4	1	no	—
159	C₂×C₄×C₃₁	—	4	2	no	—
160	C₂×C₄×C₃₁	—	4	2	no	—
161	C₂×C₄×C₃₁	—	4	1	no	—
162	C₂×C₄×C₃₁	—	4	1	no	—
163	C₂×C₄×C₃₁	—	4	2	no	—
164	C₂×C₄×C₃₁	—	4	2	no	—
165	C₂×C₄×C₃₁	—	4	2	no	—
166	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
167	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
168	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
169	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
170	C₈×C₃₁	—	2	1	no	—
171	C₈×C₃₁	—	2	1	no	—
172	C₈×C₃₁	—	2	1	no	—
173	C₈×C₃₁	—	4	2	no	—
174	C₈×C₃₁	—	4	2	no	—
175	C₈×C₃₁	—	4	1	no	—
176	C₈×C₃₁	—	4	1	no	—
177	C₈×C₃₁	#99	1	2	yes	—
178	#2929	C₂×C₃₁	1	1	yes	characteristic
179	#1752	C₂×C₃₁	1	1	yes	characteristic, complemented
180	C₈×C₂²⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
181	#1501	C₂×C₃₁	1	1	yes	characteristic
182	#1501	C₂×C₃₁	1	1	yes	characteristic
183	#1298	C₂×C₃₁	1	1	yes	characteristic, complemented
184	#1297	C₂×C₃₁	1	1	yes	characteristic, complemented
185	C₂²×C₃₁	—	2	1	no	—
186	C₂²×C₃₁	—	2	1	no	—
187	C₂²×C₃₁	—	2	1	no	—
188	C₂²×C₃₁	—	2	1	no	—
189	C₂²×C₃₁	—	2	1	no	—
190	C₂²×C₃₁	—	2	1	no	—
191	C₂²×C₃₁	—	2	1	no	—
192	C₂²×C₃₁	—	4	4	no	—
193	C₂²×C₃₁	—	4	1	no	—
194	C₂²×C₃₁	—	4	1	no	—
195	C₂²×C₃₁	—	4	1	no	—
196	C₂²×C₃₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
197	C₂²×C₃₁	#443	1	2	yes	—
198	C₂²×C₃₁	#444	1	2	yes	—
199	C₄×C₃₁	—	2	1	no	—
200	C₄×C₃₁	—	2	1	no	—
201	C₄×C₃₁	—	2	1	no	—
202	C₄×C₃₁	—	4	4	no	—
203	C₄×C₃₁	—	4	1	no	—
204	C₄×C₃₁	—	4	1	no	—
205	C₄×C₃₁	—	4	1	no	—
206	C₄×C₃₁	—	8	2	no	—
207	C₄×C₃₁	C₂×#99	1	1	yes	characteristic
208	C₄×C₃₁	#410	1	1	yes	characteristic
209	C₂×#131	C₄×C₃₁	1	1	yes	characteristic, complemented
210	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
211	#525	C₂²×C₃₁	1	1	yes	characteristic
212	#435	—	2	1	no	—
213	#435	—	2	1	no	—
214	C₈×D₄	—	2	1	no	—
215	C₈×D₄	—	2	1	no	—
216	#424	C₄×C₃₁	1	1	yes	characteristic, complemented
217	#408	C₄×C₃₁	1	1	yes	characteristic, complemented
218	#407	C₂²×C₃₁	1	1	yes	characteristic
219	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
220	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
221	#405	C₄×C₃₁	1	1	yes	characteristic, complemented
222	C₂×#97	—	2	1	no	—
223	C₂×C₄×C₈	—	2	1	no	—
224	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
225	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
226	#336	—	2	1	no	—
227	#336	—	2	1	no	—
228	#328	C₄×C₃₁	1	2	yes	complemented
229	#328	C₄×C₃₁	1	2	yes	complemented
230	#327	C₄×C₃₁	1	2	yes	complemented
231	#327	C₄×C₃₁	1	2	yes	complemented
232	C₂×C₃₁	—	2	2	no	—
233	C₂×C₃₁	—	2	2	no	—
234	C₂×C₃₁	—	4	1	no	—
235	C₂×C₃₁	—	8	1	no	—
236	C₂×C₃₁	C₄×#99	1	1	yes	characteristic
237	C₂×C₃₁	#1764	1	1	yes	characteristic
238	C₂×C₃₁	#1765	1	1	yes	characteristic
239	C₂×#41	C₂×C₄×C₃₁	1	1	yes	characteristic
240	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
241	#131	—	4	2	no	—
242	C₂×M₁₆	—	2	1	no	—
243	C₂×M₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
244	C₂×M₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
245	C₂²×C₈	—	2	1	no	—
246	C₂²×C₈	—	2	1	no	—
247	C₂²×C₈	—	2	1	no	—
248	C₂²×C₈	—	4	2	no	—
249	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
250	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
251	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
252	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
253	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
254	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
255	#123	—	2	1	no	—
256	C₂²⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
257	C₂²⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
258	#121	C₂×C₄×C₃₁	1	1	yes	characteristic
259	#121	C₂×C₄×C₃₁	1	1	yes	characteristic
260	#120	—	2	1	no	—
261	C₄×D₄	—	2	1	no	—
262	C₄×D₄	—	2	1	no	—
263	#117	C₂×C₄×C₃₁	1	1	yes	characteristic
264	C₂×C₂²⋊C₄	—	2	1	no	—
265	C₂×C₂²⋊C₄	—	2	1	no	—
266	C₂×C₄²	—	2	1	no	—
267	C₂×C₄²	—	2	1	no	—
268	C₄⋊C₈	—	2	1	no	—
269	C₄⋊C₈	—	2	1	no	—
270	C₂²⋊C₈	—	2	1	no	—
271	C₂²⋊C₈	—	2	1	no	—
272	C₂²⋊C₈	—	2	1	no	—
273	C₂²⋊C₈	—	2	1	no	—
274	C₂²⋊C₈	—	2	1	no	—
275	C₂²⋊C₈	—	2	1	no	—
276	C₂²⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
277	C₂²⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
278	#97	—	2	1	no	—
279	#97	—	4	2	no	—
280	C₄×C₈	—	2	1	no	—
281	C₄×C₈	—	4	2	no	—
282	#95	C₂×C₄×C₃₁	1	2	yes	—
283	#95	C₂×C₄×C₃₁	1	2	yes	—
284	C₃₁	#8240	1	1	yes	characteristic, direct product factor, Sylow
285	C₂⁴	#41×C₃₁	1	1	yes	characteristic
286	#41	—	4	1	no	—
287	#41	—	4	1	no	—
288	C₂×Q₈	C₄²×C₃₁	1	1	yes	characteristic
289	C₂×D₄	—	2	1	no	—
290	C₂×D₄	—	2	1	no	—
291	C₂×D₄	—	2	1	no	—
292	C₂×D₄	C₄²×C₃₁	1	1	yes	characteristic
293	C₂²×C₄	—	2	2	no	—
294	C₂²×C₄	—	2	1	no	—
295	C₂²×C₄	—	2	2	no	—
296	C₂²×C₄	—	4	1	no	—
297	C₂²×C₄	—	4	1	no	—
298	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
299	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
300	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
301	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
302	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
303	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
304	M₁₆	—	4	2	no	—
305	M₁₆	—	4	1	no	—
306	M₁₆	—	4	1	no	—
307	C₂×C₈	—	2	2	no	—
308	C₂×C₈	—	2	2	no	—
309	C₂×C₈	—	2	2	no	—
310	C₂×C₈	—	2	1	no	—
311	C₂×C₈	—	2	1	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	—	4	1	no	—
315	C₂×C₈	—	4	1	no	—
316	C₂×C₈	—	4	1	no	—
317	C₂×C₈	—	4	2	no	—
318	C₂×C₈	—	4	1	no	—
319	C₂×C₈	—	4	1	no	—
320	C₂×C₈	—	4	1	no	—
321	C₂×C₈	—	4	1	no	—
322	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
323	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
324	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
325	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
326	C₄⋊C₄	—	2	1	no	—
327	C₄⋊C₄	—	2	1	no	—
328	C₄⋊C₄	C₄²×C₃₁	1	1	yes	characteristic
329	C₄⋊C₄	C₄²×C₃₁	1	1	yes	characteristic
330	C₂²⋊C₄	—	2	4	no	—
331	C₂²⋊C₄	—	2	1	no	—
332	C₂²⋊C₄	—	2	1	no	—
333	C₂²⋊C₄	—	2	1	no	—
334	C₂²⋊C₄	—	2	1	no	—
335	C₂²⋊C₄	—	2	1	no	—
336	C₂²⋊C₄	—	4	2	no	—
337	C₄²	—	2	1	no	—
338	C₄²	—	4	1	no	—
339	C₄²	—	4	1	no	—
340	C₄²	—	4	1	no	—
341	C₄²	—	4	1	no	—
342	C₂³	—	2	2	no	—
343	C₂³	—	2	2	no	—
344	C₂³	—	2	1	no	—
345	C₂³	—	4	1	no	—
346	C₂³	C₃₁×#117	1	1	yes	characteristic
347	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
348	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
349	Q₈	—	4	1	no	—
350	D₄	—	4	1	no	—
351	D₄	—	4	2	no	—
352	D₄	—	4	2	no	—
353	D₄	—	4	2	no	—
354	C₂×C₄	—	2	1	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	2	1	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	2	1	no	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	2	4	no	—
361	C₂×C₄	—	2	1	no	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	2	1	no	—
364	C₂×C₄	—	4	1	no	—
365	C₂×C₄	—	4	1	no	—
366	C₂×C₄	—	4	1	no	—
367	C₂×C₄	—	4	2	no	—
368	C₂×C₄	—	4	2	no	—
369	C₂×C₄	—	4	1	no	—
370	C₂×C₄	—	4	1	no	—
371	C₂×C₄	—	4	2	no	—
372	C₂×C₄	—	4	2	no	—
373	C₂×C₄	—	4	2	no	—
374	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
375	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
376	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
377	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
378	C₈	—	2	1	no	—
379	C₈	—	2	1	no	—
380	C₈	—	2	1	no	—
381	C₈	—	4	2	no	—
382	C₈	—	4	2	no	—
383	C₈	—	4	1	no	—
384	C₈	—	4	1	no	—
385	C₈	C₃₁×#99	1	2	yes	—
386	C₂²	—	2	1	no	—
387	C₂²	—	2	1	no	—
388	C₂²	—	2	1	no	—
389	C₂²	—	2	1	no	—
390	C₂²	—	2	1	no	—
391	C₂²	—	2	1	no	—
392	C₂²	—	2	1	no	—
393	C₂²	—	4	4	no	—
394	C₂²	—	4	1	no	—
395	C₂²	—	4	1	no	—
396	C₂²	—	4	1	no	—
397	C₂²	C₄×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
398	C₂²	C₃₁×#443	1	2	yes	—
399	C₂²	C₃₁×#444	1	2	yes	—
400	C₄	—	2	1	no	—
401	C₄	—	2	1	no	—
402	C₄	—	2	1	no	—
403	C₄	—	4	4	no	—
404	C₄	—	4	1	no	—
405	C₄	—	4	1	no	—
406	C₄	—	4	1	no	—
407	C₄	—	8	2	no	—
408	C₄	C₂×C₃₁×#99	1	1	yes	characteristic
409	C₄	C₃₁×#410	1	1	yes	characteristic
410	C₂	—	2	2	no	—
411	C₂	—	2	2	no	—
412	C₂	—	4	1	no	—
413	C₂	—	8	1	no	—
414	C₂	C₄×C₃₁×#99	1	1	yes	characteristic
415	C₂	C₃₁×#1764	1	1	yes	characteristic
416	C₂	C₃₁×#1765	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₃₁
Frattini, cosocle	C₂³×C₄	C₂³×C₃₁
socle	C₂²×C₃₁	C₄×C₂²⋊C₄
prime radical	C₃₁×#525	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂³×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#1752	C₂	semidirect
3. C₈×C₂²⋊C₄×C₃₁	C₂	semidirect
6. C₃₁×#1298	C₂	semidirect
7. C₃₁×#1297	C₂	semidirect
8. C₂×C₃₁×#131	C₄	semidirect
15. C₃₁×#424	C₄	semidirect
16. C₃₁×#408	C₄	semidirect
20. C₃₁×#405	C₄	semidirect
27. C₃₁×#328	C₄	semidirect
28. C₃₁×#328	C₄	semidirect
29. C₃₁×#327	C₄	semidirect
30. C₃₁×#327	C₄	semidirect
133. #8240	C₃₁	direct (N × H)
179. #1752	C₂×C₃₁	semidirect
180. C₈×C₂²⋊C₄	C₂×C₃₁	semidirect
183. #1298	C₂×C₃₁	semidirect
184. #1297	C₂×C₃₁	semidirect
209. C₂×#131	C₄×C₃₁	semidirect
216. #424	C₄×C₃₁	semidirect
217. #408	C₄×C₃₁	semidirect
221. #405	C₄×C₃₁	semidirect
228. #328	C₄×C₃₁	semidirect
229. #328	C₄×C₃₁	semidirect
230. #327	C₄×C₃₁	semidirect
231. #327	C₄×C₃₁	semidirect
284. C₃₁	#8240	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#2929	2
C₃₁×#1752	2
C₈×C₂²⋊C₄×C₃₁	2
C₃₁×#1501	2
C₃₁×#1298	2
C₃₁×#1297	2
#8240	31

Subgroups — C31×#8240 #21464999

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8240 #21464999