Subgroups — C₃₁×#29287 #21486046

Order 7936.

756 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₄×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2088	C₂	1	4	yes	maximal
4	C₃₁×#1988	C₂	1	2	yes	maximal
5	C₃₁×#1987	C₂	1	2	yes	maximal
6	C₃₁×#1948	C₂	1	2	yes	maximal
7	C₃₁×#1948	C₂	1	2	yes	maximal
8	C₃₁×#1842	C₂	1	1	yes	maximal, characteristic
9	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
12	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
13	C₂×C₃₁×#106	C₂²	1	4	yes	—
14	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
17	C₃₁×#391	—	2	1	no	—
18	C₃₁×#391	—	2	1	no	—
19	C₃₁×#389	—	2	1	no	—
20	C₃₁×#389	—	2	1	no	—
21	C₃₁×#389	C₂²	1	2	yes	—
22	C₃₁×#389	C₂²	1	2	yes	—
23	C₃₁×#384	C₂²	1	1	yes	characteristic
24	C₃₁×#384	C₂²	1	1	yes	characteristic
25	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
26	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
27	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
28	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
29	C₄³×C₃₁	C₂²	1	1	yes	characteristic
30	C₃₁×#340	C₂²	1	4	yes	—
31	C₃₁×#340	C₂²	1	4	yes	—
32	C₃₁×#340	C₂²	1	4	yes	—
33	C₂²×Q₈×C₃₁	—	2	1	no	—
34	C₂²×Q₈×C₃₁	—	2	1	no	—
35	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
36	C₃₁×C₄⋊Q₈	—	2	2	no	—
37	C₃₁×C₄⋊Q₈	—	2	2	no	—
38	C₃₁×C₄⋊Q₈	Q₈	1	4	yes	complemented
39	C₃₁×C₄⋊Q₈	Q₈	1	4	yes	complemented
40	C₄×Q₈×C₃₁	—	4	4	no	—
41	C₄×Q₈×C₃₁	—	4	4	no	—
42	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
43	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
44	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
45	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
46	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
47	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
48	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
49	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
50	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
51	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
52	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
53	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
54	C₂×C₄²×C₃₁	—	2	2	no	—
55	C₂×C₄²×C₃₁	—	2	2	no	—
56	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
58	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
59	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
60	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
61	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
63	C₃₁×#106	—	4	4	no	—
64	C₃₁×C₄⋊C₈	—	2	4	no	—
65	C₃₁×C₄⋊C₈	—	2	4	no	—
66	C₃₁×C₄⋊C₈	—	4	2	no	—
67	C₃₁×#95	—	2	2	no	—
68	C₃₁×#95	—	2	2	no	—
69	C₃₁×#95	—	2	2	no	—
70	C₃₁×#95	—	2	2	no	—
71	C₂×Q₈×C₃₁	—	2	4	no	—
72	C₂×Q₈×C₃₁	—	2	4	no	—
73	C₂×Q₈×C₃₁	—	4	2	no	—
74	C₂×Q₈×C₃₁	—	4	2	no	—
75	C₂×Q₈×C₃₁	—	4	2	no	—
76	C₂×Q₈×C₃₁	—	4	2	no	—
77	C₂²×C₄×C₃₁	—	2	2	no	—
78	C₂²×C₄×C₃₁	—	2	2	no	—
79	C₂²×C₄×C₃₁	—	2	2	no	—
80	C₂²×C₄×C₃₁	—	2	2	no	—
81	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
88	C₂×C₈×C₃₁	—	2	4	no	—
89	C₂×C₈×C₃₁	—	2	4	no	—
90	C₂×C₈×C₃₁	—	2	4	no	—
91	C₄⋊C₄×C₃₁	—	2	4	no	—
92	C₄⋊C₄×C₃₁	—	2	4	no	—
93	C₄⋊C₄×C₃₁	—	2	2	no	—
94	C₄⋊C₄×C₃₁	—	2	2	no	—
95	C₄⋊C₄×C₃₁	—	2	2	no	—
96	C₄⋊C₄×C₃₁	—	2	2	no	—
97	C₄⋊C₄×C₃₁	—	2	2	no	—
98	C₄⋊C₄×C₃₁	—	2	2	no	—
99	C₄⋊C₄×C₃₁	—	4	2	no	—
100	C₄⋊C₄×C₃₁	—	4	2	no	—
101	C₄⋊C₄×C₃₁	—	4	4	no	—
102	C₄⋊C₄×C₃₁	—	4	4	no	—
103	C₄⋊C₄×C₃₁	—	4	2	no	—
104	C₄⋊C₄×C₃₁	—	4	2	no	—
105	C₄⋊C₄×C₃₁	C₂×Q₈	1	4	yes	—
106	C₄⋊C₄×C₃₁	C₂×Q₈	1	4	yes	—
107	C₄²×C₃₁	—	2	2	no	—
108	C₄²×C₃₁	—	2	2	no	—
109	C₄²×C₃₁	—	2	1	no	—
110	C₄²×C₃₁	—	2	1	no	—
111	C₄²×C₃₁	—	4	1	no	—
112	C₄²×C₃₁	—	4	4	no	—
113	C₄²×C₃₁	—	4	4	no	—
114	C₄²×C₃₁	SD₁₆	1	4	yes	—
115	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
116	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
117	C₄²×C₃₁	#41	1	2	yes	—
118	C₄²×C₃₁	#41	1	2	yes	—
119	#29287	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
121	Q₈×C₃₁	—	4	8	no	—
122	Q₈×C₃₁	—	4	8	no	—
123	C₂×C₄×C₃₁	—	2	1	no	—
124	C₂×C₄×C₃₁	—	2	1	no	—
125	C₂×C₄×C₃₁	—	2	1	no	—
126	C₂×C₄×C₃₁	—	2	1	no	—
127	C₂×C₄×C₃₁	—	2	1	no	—
128	C₂×C₄×C₃₁	—	2	1	no	—
129	C₂×C₄×C₃₁	—	2	4	no	—
130	C₂×C₄×C₃₁	—	2	4	no	—
131	C₂×C₄×C₃₁	—	2	1	no	—
132	C₂×C₄×C₃₁	—	2	1	no	—
133	C₂×C₄×C₃₁	—	2	1	no	—
134	C₂×C₄×C₃₁	—	4	2	no	—
135	C₂×C₄×C₃₁	—	4	2	no	—
136	C₂×C₄×C₃₁	—	4	2	no	—
137	C₂×C₄×C₃₁	—	4	2	no	—
138	C₂×C₄×C₃₁	—	4	2	no	—
139	C₂×C₄×C₃₁	—	4	2	no	—
140	C₂×C₄×C₃₁	—	4	2	no	—
141	C₂×C₄×C₃₁	—	4	2	no	—
142	C₂×C₄×C₃₁	—	4	2	no	—
143	C₂×C₄×C₃₁	—	4	2	no	—
144	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
145	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
146	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
147	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
148	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
149	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
150	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
151	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
152	C₂×C₄×C₃₁	#137	1	2	yes	—
153	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
154	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
155	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
156	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
157	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
158	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
159	C₈×C₃₁	—	4	4	no	—
160	C₄×C₄⋊Q₈	C₂×C₃₁	1	1	yes	characteristic
161	C₄×C₄⋊Q₈	C₂×C₃₁	1	1	yes	characteristic
162	#2088	C₂×C₃₁	1	4	yes	—
163	#1988	C₂×C₃₁	1	2	yes	—
164	#1987	C₂×C₃₁	1	2	yes	—
165	#1948	C₂×C₃₁	1	2	yes	—
166	#1948	C₂×C₃₁	1	2	yes	—
167	#1842	C₂×C₃₁	1	1	yes	characteristic
168	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
169	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
170	C₂²×C₃₁	#537	1	1	yes	characteristic
171	C₂²×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
172	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
173	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
174	C₂²×C₃₁	#575	1	1	yes	characteristic
175	C₄×C₃₁	—	2	2	no	—
176	C₄×C₃₁	—	2	2	no	—
177	C₄×C₃₁	—	2	2	no	—
178	C₄×C₃₁	—	4	4	no	—
179	C₄×C₃₁	—	4	1	no	—
180	C₄×C₃₁	—	4	4	no	—
181	C₄×C₃₁	—	8	2	no	—
182	C₄×C₃₁	—	8	2	no	—
183	C₄×C₃₁	#475	1	4	yes	—
184	C₄×C₃₁	#476	1	4	yes	—
185	C₄×C₃₁	#557	1	1	yes	characteristic
186	C₄×C₃₁	#557	1	1	yes	characteristic
187	C₄×C₃₁	Q₈²	1	1	yes	characteristic
188	C₄×C₃₁	#559	1	1	yes	characteristic
189	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
190	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
191	C₂×C₄×Q₈	—	2	1	no	—
192	C₂×C₄×Q₈	—	2	1	no	—
193	C₂×#106	C₂²×C₃₁	1	4	yes	—
194	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
195	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
196	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
197	#391	—	2	1	no	—
198	#391	—	2	1	no	—
199	#389	—	2	1	no	—
200	#389	—	2	1	no	—
201	#389	C₂²×C₃₁	1	2	yes	—
202	#389	C₂²×C₃₁	1	2	yes	—
203	#384	C₂²×C₃₁	1	1	yes	characteristic
204	#384	C₂²×C₃₁	1	1	yes	characteristic
205	C₄×C₄⋊C₄	—	2	1	no	—
206	C₄×C₄⋊C₄	—	2	1	no	—
207	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
208	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
209	C₄³	C₂²×C₃₁	1	1	yes	characteristic
210	#340	C₂²×C₃₁	1	4	yes	—
211	#340	C₂²×C₃₁	1	4	yes	—
212	#340	C₂²×C₃₁	1	4	yes	—
213	C₂×C₃₁	#2582	1	1	yes	characteristic
214	C₂×C₃₁	C₂×#476	1	1	yes	characteristic
215	C₂×C₃₁	C₂×#475	1	1	yes	characteristic
216	C₂×C₃₁	#3077	1	1	yes	characteristic
217	C₂×C₃₁	#3078	1	1	yes	characteristic
218	C₂×C₃₁	#3246	1	1	yes	characteristic
219	C₂×C₃₁	#3254	1	1	yes	characteristic
220	C₂²×Q₈	—	2	1	no	—
221	C₂²×Q₈	—	2	1	no	—
222	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
223	C₄⋊Q₈	—	2	2	no	—
224	C₄⋊Q₈	—	2	2	no	—
225	C₄⋊Q₈	Q₈×C₃₁	1	4	yes	complemented
226	C₄⋊Q₈	Q₈×C₃₁	1	4	yes	complemented
227	C₄×Q₈	—	4	4	no	—
228	C₄×Q₈	—	4	4	no	—
229	C₂×C₄⋊C₄	—	2	1	no	—
230	C₂×C₄⋊C₄	—	2	1	no	—
231	C₂×C₄⋊C₄	—	2	1	no	—
232	C₂×C₄⋊C₄	—	2	1	no	—
233	C₂×C₄⋊C₄	—	2	2	no	—
234	C₂×C₄⋊C₄	—	2	2	no	—
235	C₂×C₄⋊C₄	—	2	1	no	—
236	C₂×C₄⋊C₄	—	2	1	no	—
237	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
238	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
239	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
240	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
241	C₂×C₄²	—	2	2	no	—
242	C₂×C₄²	—	2	2	no	—
243	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
244	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
245	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
246	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
247	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
248	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
249	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
250	#106	—	4	4	no	—
251	C₄⋊C₈	—	2	4	no	—
252	C₄⋊C₈	—	2	4	no	—
253	C₄⋊C₈	—	4	2	no	—
254	#95	—	2	2	no	—
255	#95	—	2	2	no	—
256	#95	—	2	2	no	—
257	#95	—	2	2	no	—
258	C₃₁	#29287	1	1	yes	characteristic, direct product factor, Sylow
259	C₂×Q₈	—	2	4	no	—
260	C₂×Q₈	—	2	4	no	—
261	C₂×Q₈	—	4	2	no	—
262	C₂×Q₈	—	4	2	no	—
263	C₂×Q₈	—	4	2	no	—
264	C₂×Q₈	—	4	2	no	—
265	C₂²×C₄	—	2	2	no	—
266	C₂²×C₄	—	2	2	no	—
267	C₂²×C₄	—	2	2	no	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
270	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
271	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
272	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
273	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
274	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
275	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
276	C₂×C₈	—	2	4	no	—
277	C₂×C₈	—	2	4	no	—
278	C₂×C₈	—	2	4	no	—
279	C₄⋊C₄	—	2	4	no	—
280	C₄⋊C₄	—	2	4	no	—
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	4	2	no	—
288	C₄⋊C₄	—	4	2	no	—
289	C₄⋊C₄	—	4	4	no	—
290	C₄⋊C₄	—	4	4	no	—
291	C₄⋊C₄	—	4	2	no	—
292	C₄⋊C₄	—	4	2	no	—
293	C₄⋊C₄	C₂×Q₈×C₃₁	1	4	yes	—
294	C₄⋊C₄	C₂×Q₈×C₃₁	1	4	yes	—
295	C₄²	—	2	2	no	—
296	C₄²	—	2	2	no	—
297	C₄²	—	2	1	no	—
298	C₄²	—	2	1	no	—
299	C₄²	—	4	1	no	—
300	C₄²	—	4	4	no	—
301	C₄²	—	4	4	no	—
302	C₄²	SD₁₆×C₃₁	1	4	yes	—
303	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
304	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
305	C₄²	#41×C₃₁	1	2	yes	—
306	C₄²	#41×C₃₁	1	2	yes	—
307	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
308	Q₈	—	4	8	no	—
309	Q₈	—	4	8	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	4	no	—
317	C₂×C₄	—	2	4	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
332	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
333	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
334	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
335	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
336	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
337	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
338	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
339	C₂×C₄	C₃₁×#137	1	2	yes	—
340	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
341	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
342	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
343	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
344	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
345	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
346	C₈	—	4	4	no	—
347	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
348	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
349	C₂²	C₃₁×#537	1	1	yes	characteristic
350	C₂²	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
351	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
352	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
353	C₂²	C₃₁×#575	1	1	yes	characteristic
354	C₄	—	2	2	no	—
355	C₄	—	2	2	no	—
356	C₄	—	2	2	no	—
357	C₄	—	4	4	no	—
358	C₄	—	4	1	no	—
359	C₄	—	4	4	no	—
360	C₄	—	8	2	no	—
361	C₄	—	8	2	no	—
362	C₄	C₃₁×#475	1	4	yes	—
363	C₄	C₃₁×#476	1	4	yes	—
364	C₄	C₃₁×#557	1	1	yes	characteristic
365	C₄	C₃₁×#557	1	1	yes	characteristic
366	C₄	Q₈²×C₃₁	1	1	yes	characteristic
367	C₄	C₃₁×#559	1	1	yes	characteristic
368	C₂	C₃₁×#2582	1	1	yes	characteristic
369	C₂	C₂×C₃₁×#476	1	1	yes	characteristic
370	C₂	C₂×C₃₁×#475	1	1	yes	characteristic
371	C₂	C₃₁×#3077	1	1	yes	characteristic
372	C₂	C₃₁×#3078	1	1	yes	characteristic
373	C₂	C₃₁×#3246	1	1	yes	characteristic
374	C₂	C₃₁×#3254	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
38. C₃₁×C₄⋊Q₈	Q₈	semidirect
39. C₃₁×C₄⋊Q₈	Q₈	semidirect
119. #29287	C₃₁	direct (N × H)
225. C₄⋊Q₈	Q₈×C₃₁	semidirect
226. C₄⋊Q₈	Q₈×C₃₁	semidirect
258. C₃₁	#29287	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₃₁×C₄⋊Q₈	2
C₃₁×#2088	2
C₃₁×#1988	2
C₃₁×#1987	2
C₃₁×#1948	2
C₃₁×#1842	2
#29287	31

Subgroups — C31×#29287 #21486046

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#29287 #21486046