Subgroups — C₃₁×#8236 #21464995

Order 7936.

528 conjugacy classes of subgroups.

The subgroup lattice has 304 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	C₃₁×#1767	C₂	1	1	yes	maximal, characteristic
3	C₄×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#332	C₂	1	2	yes	maximal
5	C₃₁×#1296	C₂	1	2	yes	maximal
6	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
7	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
8	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₃₁×#97	—	2	1	no	—
11	C₂×C₄×C₈×C₃₁	—	2	1	no	—
12	C₃₁×#391	—	2	1	no	—
13	C₃₁×#389	—	2	1	no	—
14	C₃₁×#389	C₄	1	2	yes	—
15	C₃₁×#384	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
17	C₄×C₄⋊C₄×C₃₁	C₄	1	2	yes	—
18	C₄³×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#336	—	2	1	no	—
20	C₃₁×#336	—	2	1	no	—
21	C₃₁×#332	C₄	1	8	yes	complemented
22	C₂²×Q₈×C₃₁	—	2	1	no	—
23	C₂²×C₈×C₃₁	—	2	2	no	—
24	C₂²×C₈×C₃₁	—	2	2	no	—
25	C₃₁×C₄⋊Q₈	—	2	2	no	—
26	C₃₁×C₄⋊Q₈	C₂×C₄	1	4	yes	—
27	C₄×Q₈×C₃₁	—	4	2	no	—
28	C₄×Q₈×C₃₁	—	4	2	no	—
29	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
30	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
31	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
32	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
33	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
34	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₃₁	—	2	2	no	—
36	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
43	C₃₁×C₄⋊C₈	—	2	2	no	—
44	C₃₁×C₄⋊C₈	—	2	2	no	—
45	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
46	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
47	C₃₁×#97	—	4	2	no	—
48	C₄×C₈×C₃₁	—	4	2	no	—
49	C₃₁×#95	—	2	2	no	—
50	C₃₁×#95	—	2	2	no	—
51	C₂×Q₈×C₃₁	—	2	4	no	—
52	C₂×Q₈×C₃₁	—	4	1	no	—
53	C₂×Q₈×C₃₁	—	4	1	no	—
54	C₂×Q₈×C₃₁	—	4	2	no	—
55	C₂²×C₄×C₃₁	—	2	2	no	—
56	C₂²×C₄×C₃₁	—	2	2	no	—
57	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₃₁	—	2	4	no	—
65	C₂×C₈×C₃₁	—	2	4	no	—
66	C₂×C₈×C₃₁	—	4	2	no	—
67	C₂×C₈×C₃₁	—	4	2	no	—
68	C₂×C₈×C₃₁	—	4	2	no	—
69	C₂×C₈×C₃₁	—	4	2	no	—
70	C₄⋊C₄×C₃₁	—	2	2	no	—
71	C₄⋊C₄×C₃₁	—	2	4	no	—
72	C₄⋊C₄×C₃₁	—	4	2	no	—
73	C₄⋊C₄×C₃₁	—	4	4	no	—
74	C₄⋊C₄×C₃₁	—	4	1	no	—
75	C₄⋊C₄×C₃₁	—	4	1	no	—
76	C₄⋊C₄×C₃₁	—	4	1	no	—
77	C₄⋊C₄×C₃₁	—	4	1	no	—
78	C₄⋊C₄×C₃₁	C₄²	1	4	yes	—
79	C₄²×C₃₁	—	2	2	no	—
80	C₄²×C₃₁	—	2	2	no	—
81	C₄²×C₃₁	—	2	1	no	—
82	C₄²×C₃₁	—	2	1	no	—
83	C₄²×C₃₁	—	2	2	no	—
84	C₄²×C₃₁	—	4	1	no	—
85	C₄²×C₃₁	—	4	4	no	—
86	C₄²×C₃₁	D₈	1	2	yes	—
87	C₄²×C₃₁	SD₁₆	1	2	yes	—
88	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
89	C₄²×C₃₁	#41	1	2	yes	—
90	#8236	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	Q₈×C₃₁	—	4	4	no	—
93	Q₈×C₃₁	—	4	4	no	—
94	C₂×C₄×C₃₁	—	2	1	no	—
95	C₂×C₄×C₃₁	—	2	1	no	—
96	C₂×C₄×C₃₁	—	2	1	no	—
97	C₂×C₄×C₃₁	—	2	1	no	—
98	C₂×C₄×C₃₁	—	2	1	no	—
99	C₂×C₄×C₃₁	—	2	1	no	—
100	C₂×C₄×C₃₁	—	2	1	no	—
101	C₂×C₄×C₃₁	—	2	1	no	—
102	C₂×C₄×C₃₁	—	2	4	no	—
103	C₂×C₄×C₃₁	—	2	1	no	—
104	C₂×C₄×C₃₁	—	2	1	no	—
105	C₂×C₄×C₃₁	—	2	1	no	—
106	C₂×C₄×C₃₁	—	4	2	no	—
107	C₂×C₄×C₃₁	—	4	2	no	—
108	C₂×C₄×C₃₁	—	4	2	no	—
109	C₂×C₄×C₃₁	—	4	2	no	—
110	C₂×C₄×C₃₁	—	4	2	no	—
111	C₂×C₄×C₃₁	#101	1	2	yes	—
112	C₂×C₄×C₃₁	#102	1	2	yes	—
113	C₂×C₄×C₃₁	#102	1	2	yes	—
114	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
115	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
116	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
120	C₂×C₄×C₃₁	#135	1	2	yes	—
121	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
122	C₂×C₄×C₃₁	#137	1	2	yes	—
123	C₂×C₄×C₃₁	#137	1	1	yes	characteristic
124	C₈×C₃₁	—	4	4	no	—
125	C₈×C₃₁	—	4	4	no	—
126	C₄×C₄⋊Q₈	C₂×C₃₁	1	1	yes	characteristic
127	#1767	C₂×C₃₁	1	1	yes	characteristic
128	C₄×C₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic
129	C₂×#332	C₂×C₃₁	1	2	yes	—
130	#1296	C₂×C₃₁	1	2	yes	—
131	C₂²×C₃₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
132	C₂²×C₃₁	C₂×#101	1	1	yes	characteristic
133	C₂²×C₃₁	#413	1	1	yes	characteristic
134	C₂²×C₃₁	C₂×#102	1	1	yes	characteristic
135	C₂²×C₃₁	#416	1	1	yes	characteristic
136	C₂²×C₃₁	#417	1	1	yes	characteristic
137	C₂²×C₃₁	#419	1	1	yes	characteristic
138	C₄×C₃₁	—	2	2	no	—
139	C₄×C₃₁	—	2	2	no	—
140	C₄×C₃₁	—	2	2	no	—
141	C₄×C₃₁	—	2	1	no	—
142	C₄×C₃₁	—	2	1	no	—
143	C₄×C₃₁	—	4	1	no	—
144	C₄×C₃₁	—	4	4	no	—
145	C₄×C₃₁	—	8	2	no	—
146	C₄×C₃₁	C₄×D₈	1	2	yes	—
147	C₄×C₃₁	C₄×SD₁₆	1	2	yes	—
148	C₄×C₃₁	#440	1	2	yes	—
149	C₄×C₃₁	#441	1	2	yes	—
150	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
151	C₂×C₄×Q₈	—	2	1	no	—
152	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
153	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
154	C₂×#97	—	2	1	no	—
155	C₂×C₄×C₈	—	2	1	no	—
156	#391	—	2	1	no	—
157	#389	—	2	1	no	—
158	#389	C₄×C₃₁	1	2	yes	—
159	#384	C₂²×C₃₁	1	1	yes	characteristic
160	C₄×C₄⋊C₄	—	2	1	no	—
161	C₄×C₄⋊C₄	C₄×C₃₁	1	2	yes	—
162	C₄³	C₂²×C₃₁	1	1	yes	characteristic
163	#336	—	2	1	no	—
164	#336	—	2	1	no	—
165	#332	C₄×C₃₁	1	8	yes	complemented
166	C₂×C₃₁	#1546	1	1	yes	characteristic
167	C₂×C₃₁	#1547	1	1	yes	characteristic
168	C₂×C₃₁	#1552	1	1	yes	characteristic
169	C₂×C₃₁	#1554	1	1	yes	characteristic
170	C₂×C₃₁	C₄×#101	1	1	yes	characteristic
171	C₂×C₃₁	C₄×#102	1	1	yes	characteristic
172	C₂×C₃₁	#1763	1	1	yes	characteristic
173	C₂²×Q₈	—	2	1	no	—
174	C₂²×C₈	—	2	2	no	—
175	C₂²×C₈	—	2	2	no	—
176	C₄⋊Q₈	—	2	2	no	—
177	C₄⋊Q₈	C₂×C₄×C₃₁	1	4	yes	—
178	C₄×Q₈	—	4	2	no	—
179	C₄×Q₈	—	4	2	no	—
180	C₂×C₄⋊C₄	—	2	1	no	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄⋊C₄	—	2	1	no	—
183	C₂×C₄⋊C₄	—	2	1	no	—
184	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
185	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
186	C₂×C₄²	—	2	2	no	—
187	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
188	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
189	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
190	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
191	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
192	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
193	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
194	C₄⋊C₈	—	2	2	no	—
195	C₄⋊C₈	—	2	2	no	—
196	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
197	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
198	#97	—	4	2	no	—
199	C₄×C₈	—	4	2	no	—
200	#95	—	2	2	no	—
201	#95	—	2	2	no	—
202	C₃₁	#8236	1	1	yes	characteristic, direct product factor, Sylow
203	C₂×Q₈	—	2	4	no	—
204	C₂×Q₈	—	4	1	no	—
205	C₂×Q₈	—	4	1	no	—
206	C₂×Q₈	—	4	2	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
210	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
211	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
212	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
213	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
214	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	4	2	no	—
219	C₂×C₈	—	4	2	no	—
220	C₂×C₈	—	4	2	no	—
221	C₂×C₈	—	4	2	no	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	2	4	no	—
224	C₄⋊C₄	—	4	2	no	—
225	C₄⋊C₄	—	4	4	no	—
226	C₄⋊C₄	—	4	1	no	—
227	C₄⋊C₄	—	4	1	no	—
228	C₄⋊C₄	—	4	1	no	—
229	C₄⋊C₄	—	4	1	no	—
230	C₄⋊C₄	C₄²×C₃₁	1	4	yes	—
231	C₄²	—	2	2	no	—
232	C₄²	—	2	2	no	—
233	C₄²	—	2	1	no	—
234	C₄²	—	2	1	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	4	1	no	—
237	C₄²	—	4	4	no	—
238	C₄²	D₈×C₃₁	1	2	yes	—
239	C₄²	SD₁₆×C₃₁	1	2	yes	—
240	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
241	C₄²	#41×C₃₁	1	2	yes	—
242	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
243	Q₈	—	4	4	no	—
244	Q₈	—	4	4	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	C₃₁×#101	1	2	yes	—
263	C₂×C₄	C₃₁×#102	1	2	yes	—
264	C₂×C₄	C₃₁×#102	1	2	yes	—
265	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
266	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
267	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
268	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
269	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
270	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
271	C₂×C₄	C₃₁×#135	1	2	yes	—
272	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
273	C₂×C₄	C₃₁×#137	1	2	yes	—
274	C₂×C₄	C₃₁×#137	1	1	yes	characteristic
275	C₈	—	4	4	no	—
276	C₈	—	4	4	no	—
277	C₂²	C₄×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
278	C₂²	C₂×C₃₁×#101	1	1	yes	characteristic
279	C₂²	C₃₁×#413	1	1	yes	characteristic
280	C₂²	C₂×C₃₁×#102	1	1	yes	characteristic
281	C₂²	C₃₁×#416	1	1	yes	characteristic
282	C₂²	C₃₁×#417	1	1	yes	characteristic
283	C₂²	C₃₁×#419	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	2	2	no	—
286	C₄	—	2	2	no	—
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	4	1	no	—
290	C₄	—	4	4	no	—
291	C₄	—	8	2	no	—
292	C₄	C₄×D₈×C₃₁	1	2	yes	—
293	C₄	C₄×SD₁₆×C₃₁	1	2	yes	—
294	C₄	C₃₁×#440	1	2	yes	—
295	C₄	C₃₁×#441	1	2	yes	—
296	C₂	C₃₁×#1546	1	1	yes	characteristic
297	C₂	C₃₁×#1547	1	1	yes	characteristic
298	C₂	C₃₁×#1552	1	1	yes	characteristic
299	C₂	C₃₁×#1554	1	1	yes	characteristic
300	C₂	C₄×C₃₁×#101	1	1	yes	characteristic
301	C₂	C₄×C₃₁×#102	1	1	yes	characteristic
302	C₂	C₃₁×#1763	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
21. C₃₁×#332	C₄	semidirect
90. #8236	C₃₁	direct (N × H)
165. #332	C₄×C₃₁	semidirect
202. C₃₁	#8236	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×#1767	2
C₄×C₃₁×C₄⋊C₈	2
C₂×C₃₁×#332	2
C₃₁×#1296	2
#8236	31

Subgroups — C31×#8236 #21464995

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8236 #21464995