Subgroups — C₃₁×#9753 #21466512

Order 7936.

502 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3147	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1770	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#1564	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1502	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1480	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#135	—	2	1	no	—
7	C₂×C₃₁×#135	—	2	1	no	—
8	C₃₁×#532	C₂²	1	2	yes	—
9	C₃₁×#495	—	2	1	no	—
10	C₃₁×#495	—	2	1	no	—
11	C₃₁×#494	C₄	1	1	yes	characteristic, complemented
12	C₃₁×#493	C₄	1	1	yes	characteristic, complemented
13	C₃₁×#492	C₄	1	2	yes	complemented
14	C₃₁×#431	C₂²	1	2	yes	—
15	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#97	—	2	1	no	—
17	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#336	—	2	1	no	—
20	C₃₁×#335	—	2	1	no	—
21	C₃₁×#334	—	2	1	no	—
22	C₃₁×#327	—	2	4	no	—
23	C₃₁×D₄⋊C₈	—	2	4	no	—
24	C₂×#41×C₃₁	—	2	2	no	—
25	C₃₁×#135	—	4	2	no	—
26	C₃₁×#135	—	4	2	no	—
27	C₃₁×#135	—	4	2	no	—
28	C₃₁×#135	—	4	2	no	—
29	C₂×Q₁₆×C₃₁	—	2	1	no	—
30	C₂×Q₁₆×C₃₁	—	2	1	no	—
31	C₂×SD₁₆×C₃₁	—	2	2	no	—
32	C₂×SD₁₆×C₃₁	—	2	2	no	—
33	C₂×D₈×C₃₁	—	2	1	no	—
34	C₂×D₈×C₃₁	—	2	1	no	—
35	C₂²×C₈×C₃₁	—	2	1	no	—
36	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
38	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
39	C₃₁×#127	C₂×C₄	1	2	yes	—
40	C₃₁×#124	—	2	2	no	—
41	C₃₁×#121	—	2	2	no	—
42	C₃₁×#121	—	2	2	no	—
43	C₄×D₄×C₃₁	—	2	2	no	—
44	C₄×D₄×C₃₁	—	2	2	no	—
45	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
46	C₂×C₄²×C₃₁	—	2	1	no	—
47	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
48	C₃₁×C₄⋊C₈	—	2	2	no	—
49	C₃₁×C₄⋊C₈	—	2	1	no	—
50	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
51	C₃₁×C₂²⋊C₈	—	2	2	no	—
52	C₃₁×#97	—	4	2	no	—
53	C₄×C₈×C₃₁	—	2	2	no	—
54	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
55	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
57	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
58	C₃₁×#95	D₄	1	2	yes	—
59	#41×C₃₁	—	4	2	no	—
60	#41×C₃₁	—	4	2	no	—
61	#41×C₃₁	—	4	2	no	—
62	#41×C₃₁	—	4	2	no	—
63	C₂×Q₈×C₃₁	—	2	2	no	—
64	C₂×D₄×C₃₁	—	2	2	no	—
65	C₂×D₄×C₃₁	—	2	2	no	—
66	C₂×D₄×C₃₁	—	2	2	no	—
67	C₂×D₄×C₃₁	—	4	2	no	—
68	C₂²×C₄×C₃₁	—	2	2	no	—
69	C₂²×C₄×C₃₁	—	4	2	no	—
70	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
73	Q₁₆×C₃₁	—	4	2	no	—
74	Q₁₆×C₃₁	—	4	2	no	—
75	SD₁₆×C₃₁	—	4	4	no	—
76	SD₁₆×C₃₁	—	4	4	no	—
77	D₈×C₃₁	—	4	2	no	—
78	D₈×C₃₁	—	4	2	no	—
79	C₂×C₈×C₃₁	—	2	1	no	—
80	C₂×C₈×C₃₁	—	2	1	no	—
81	C₂×C₈×C₃₁	—	2	2	no	—
82	C₂×C₈×C₃₁	—	4	1	no	—
83	C₂×C₈×C₃₁	—	4	1	no	—
84	C₂×C₈×C₃₁	—	4	2	no	—
85	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₈×C₃₁	D₈	1	2	yes	—
90	C₂×C₈×C₃₁	SD₁₆	1	2	yes	—
91	C₄⋊C₄×C₃₁	—	2	2	no	—
92	C₄⋊C₄×C₃₁	—	2	2	no	—
93	C₄⋊C₄×C₃₁	—	4	2	no	—
94	C₂²⋊C₄×C₃₁	—	4	2	no	—
95	C₂²⋊C₄×C₃₁	—	4	2	no	—
96	C₄²×C₃₁	—	4	1	no	—
97	C₄²×C₃₁	—	4	1	no	—
98	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
99	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₃₁	#41	1	1	yes	characteristic
102	#9753	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂³×C₃₁	—	4	2	no	—
104	C₂³×C₃₁	#121	1	1	yes	characteristic
105	Q₈×C₃₁	—	4	2	no	—
106	Q₈×C₃₁	—	4	2	no	—
107	D₄×C₃₁	—	4	2	no	—
108	D₄×C₃₁	—	4	2	no	—
109	D₄×C₃₁	—	4	4	no	—
110	D₄×C₃₁	—	8	2	no	—
111	D₄×C₃₁	—	8	2	no	—
112	D₄×C₃₁	—	8	2	no	—
113	C₂×C₄×C₃₁	—	2	2	no	—
114	C₂×C₄×C₃₁	—	4	1	no	—
115	C₂×C₄×C₃₁	—	4	2	no	—
116	C₂×C₄×C₃₁	—	4	2	no	—
117	C₂×C₄×C₃₁	—	4	2	no	—
118	C₂×C₄×C₃₁	—	4	2	no	—
119	C₂×C₄×C₃₁	—	8	2	no	—
120	C₂×C₄×C₃₁	—	8	2	no	—
121	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
125	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
126	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
127	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
129	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
130	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
131	C₈×C₃₁	—	2	1	no	—
132	C₈×C₃₁	—	2	1	no	—
133	C₈×C₃₁	—	4	2	no	—
134	C₈×C₃₁	—	8	2	no	—
135	C₈×C₃₁	#102	1	2	yes	—
136	C₈×C₃₁	#102	1	2	yes	—
137	#3147	C₂×C₃₁	1	1	yes	characteristic
138	#1770	C₂×C₃₁	1	1	yes	characteristic, complemented
139	#1564	C₂×C₃₁	1	1	yes	characteristic, complemented
140	#1502	C₂×C₃₁	1	2	yes	complemented
141	#1480	C₂×C₃₁	1	2	yes	complemented
142	C₂²×C₃₁	—	2	1	no	—
143	C₂²×C₃₁	—	2	1	no	—
144	C₂²×C₃₁	—	8	2	no	—
145	C₂²×C₃₁	—	8	2	no	—
146	C₂²×C₃₁	—	8	2	no	—
147	C₂²×C₃₁	#390	1	1	yes	characteristic
148	C₂²×C₃₁	#459	1	1	yes	characteristic
149	C₂²×C₃₁	#460	1	1	yes	characteristic
150	C₄×C₃₁	—	2	1	no	—
151	C₄×C₃₁	—	2	1	no	—
152	C₄×C₃₁	—	2	1	no	—
153	C₄×C₃₁	—	8	2	no	—
154	C₄×C₃₁	—	8	2	no	—
155	C₄×C₃₁	C₂×#102	1	1	yes	characteristic
156	C₄×C₃₁	#418	1	1	yes	characteristic
157	C₄×C₃₁	#444	1	2	yes	—
158	C₄×C₃₁	#459	1	1	yes	characteristic
159	C₄×C₃₁	#460	1	1	yes	characteristic
160	C₂×#135	—	2	1	no	—
161	C₂×#135	—	2	1	no	—
162	#532	C₂²×C₃₁	1	2	yes	—
163	#495	—	2	1	no	—
164	#495	—	2	1	no	—
165	#494	C₄×C₃₁	1	1	yes	characteristic, complemented
166	#493	C₄×C₃₁	1	1	yes	characteristic, complemented
167	#492	C₄×C₃₁	1	2	yes	complemented
168	#431	C₂²×C₃₁	1	2	yes	—
169	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
170	C₂×#97	—	2	1	no	—
171	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
172	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
173	#336	—	2	1	no	—
174	#335	—	2	1	no	—
175	#334	—	2	1	no	—
176	#327	—	2	4	no	—
177	D₄⋊C₈	—	2	4	no	—
178	C₂×C₃₁	—	2	1	no	—
179	C₂×C₃₁	—	2	1	no	—
180	C₂×C₃₁	—	16	2	no	—
181	C₂×C₃₁	#1975	1	1	yes	characteristic
182	C₂×C₃₁	#1976	1	1	yes	characteristic
183	C₂×C₃₁	#1982	1	1	yes	characteristic
184	C₂×#41	—	2	2	no	—
185	#135	—	4	2	no	—
186	#135	—	4	2	no	—
187	#135	—	4	2	no	—
188	#135	—	4	2	no	—
189	C₂×Q₁₆	—	2	1	no	—
190	C₂×Q₁₆	—	2	1	no	—
191	C₂×SD₁₆	—	2	2	no	—
192	C₂×SD₁₆	—	2	2	no	—
193	C₂×D₈	—	2	1	no	—
194	C₂×D₈	—	2	1	no	—
195	C₂²×C₈	—	2	1	no	—
196	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
197	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
198	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
199	#127	C₂×C₄×C₃₁	1	2	yes	—
200	#124	—	2	2	no	—
201	#121	—	2	2	no	—
202	#121	—	2	2	no	—
203	C₄×D₄	—	2	2	no	—
204	C₄×D₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	1	no	—
206	C₂×C₄²	—	2	1	no	—
207	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
208	C₄⋊C₈	—	2	2	no	—
209	C₄⋊C₈	—	2	1	no	—
210	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
211	C₂²⋊C₈	—	2	2	no	—
212	#97	—	4	2	no	—
213	C₄×C₈	—	2	2	no	—
214	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
215	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
216	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
217	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
218	#95	D₄×C₃₁	1	2	yes	—
219	C₃₁	#9753	1	1	yes	characteristic, direct product factor, Sylow
220	#41	—	4	2	no	—
221	#41	—	4	2	no	—
222	#41	—	4	2	no	—
223	#41	—	4	2	no	—
224	C₂×Q₈	—	2	2	no	—
225	C₂×D₄	—	2	2	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	4	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	4	2	no	—
231	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
233	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
234	Q₁₆	—	4	2	no	—
235	Q₁₆	—	4	2	no	—
236	SD₁₆	—	4	4	no	—
237	SD₁₆	—	4	4	no	—
238	D₈	—	4	2	no	—
239	D₈	—	4	2	no	—
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	4	1	no	—
244	C₂×C₈	—	4	1	no	—
245	C₂×C₈	—	4	2	no	—
246	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
247	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
248	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
249	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
250	C₂×C₈	D₈×C₃₁	1	2	yes	—
251	C₂×C₈	SD₁₆×C₃₁	1	2	yes	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₂²⋊C₄	—	4	2	no	—
256	C₂²⋊C₄	—	4	2	no	—
257	C₄²	—	4	1	no	—
258	C₄²	—	4	1	no	—
259	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
260	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
261	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
262	C₄²	#41×C₃₁	1	1	yes	characteristic
263	C₂³	—	4	2	no	—
264	C₂³	C₃₁×#121	1	1	yes	characteristic
265	Q₈	—	4	2	no	—
266	Q₈	—	4	2	no	—
267	D₄	—	4	2	no	—
268	D₄	—	4	2	no	—
269	D₄	—	4	4	no	—
270	D₄	—	8	2	no	—
271	D₄	—	8	2	no	—
272	D₄	—	8	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	8	2	no	—
280	C₂×C₄	—	8	2	no	—
281	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
282	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
283	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
284	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
285	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
286	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
287	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
288	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
289	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
290	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
291	C₈	—	2	1	no	—
292	C₈	—	2	1	no	—
293	C₈	—	4	2	no	—
294	C₈	—	8	2	no	—
295	C₈	C₃₁×#102	1	2	yes	—
296	C₈	C₃₁×#102	1	2	yes	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	8	2	no	—
300	C₂²	—	8	2	no	—
301	C₂²	—	8	2	no	—
302	C₂²	C₃₁×#390	1	1	yes	characteristic
303	C₂²	C₃₁×#459	1	1	yes	characteristic
304	C₂²	C₃₁×#460	1	1	yes	characteristic
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	8	2	no	—
309	C₄	—	8	2	no	—
310	C₄	C₂×C₃₁×#102	1	1	yes	characteristic
311	C₄	C₃₁×#418	1	1	yes	characteristic
312	C₄	C₃₁×#444	1	2	yes	—
313	C₄	C₃₁×#459	1	1	yes	characteristic
314	C₄	C₃₁×#460	1	1	yes	characteristic
315	C₂	—	2	1	no	—
316	C₂	—	2	1	no	—
317	C₂	—	16	2	no	—
318	C₂	C₃₁×#1975	1	1	yes	characteristic
319	C₂	C₃₁×#1976	1	1	yes	characteristic
320	C₂	C₃₁×#1982	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	#121
commutator	C₄²	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂²×C₃₁	#390
prime radical	C₃₁×#3147	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#1770	C₂	semidirect
3. C₃₁×#1564	C₂	semidirect
4. C₃₁×#1502	C₂	semidirect
5. C₃₁×#1480	C₂	semidirect
11. C₃₁×#494	C₄	semidirect
12. C₃₁×#493	C₄	semidirect
13. C₃₁×#492	C₄	semidirect
102. #9753	C₃₁	direct (N × H)
138. #1770	C₂×C₃₁	semidirect
139. #1564	C₂×C₃₁	semidirect
140. #1502	C₂×C₃₁	semidirect
141. #1480	C₂×C₃₁	semidirect
165. #494	C₄×C₃₁	semidirect
166. #493	C₄×C₃₁	semidirect
167. #492	C₄×C₃₁	semidirect
219. C₃₁	#9753	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×#3147	2
C₃₁×#1770	2
C₃₁×#1564	2
C₃₁×#1502	2
C₃₁×#1480	2
#9753	31

Subgroups — C31×#9753 #21466512

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9753 #21466512