Subgroups — C₃₁×#8692 #21465451

Order 7936.

358 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2966	C₂	1	1	yes	maximal, characteristic
2	C₄×C₃₁×#108	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#1568	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1529	C₂	1	2	yes	maximal
5	C₃₁×#1291	C₂	1	2	yes	maximal, complemented
6	C₃₁×#529	C₂²	1	1	yes	characteristic
7	C₃₁×#446	C₄	1	1	yes	characteristic
8	C₈×Q₈×C₃₁	C₄	1	1	yes	characteristic
9	C₃₁×#436	C₄	1	1	yes	characteristic
10	C₃₁×#435	—	2	1	no	—
11	C₈×D₄×C₃₁	—	2	1	no	—
12	C₈×D₄×C₃₁	C₄	1	1	yes	characteristic
13	C₃₁×#433	—	2	1	no	—
14	C₃₁×#432	C₂²	1	2	yes	—
15	C₃₁×#432	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#108	—	2	1	no	—
17	C₃₁×#405	—	2	1	no	—
18	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
20	C₃₁×#341	—	2	1	no	—
21	C₃₁×#335	—	2	1	no	—
22	C₃₁×#334	—	2	1	no	—
23	C₃₁×#330	—	2	2	no	—
24	C₃₁×#329	—	2	2	no	—
25	C₂×M₁₆×C₃₁	—	2	1	no	—
26	C₂×M₁₆×C₃₁	—	2	2	no	—
27	C₂²×C₈×C₃₁	—	2	1	no	—
28	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
30	C₃₁×#126	—	2	1	no	—
31	C₃₁×#125	D₄	1	1	yes	characteristic
32	C₃₁×#124	D₄	1	1	yes	characteristic
33	C₃₁×#123	—	2	1	no	—
34	C₃₁×C₂²⋊Q₈	Q₈	1	1	yes	characteristic
35	C₃₁×#121	Q₈	1	1	yes	characteristic
36	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
37	C₄×D₄×C₃₁	—	2	1	no	—
38	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
39	C₃₁×#117	—	2	1	no	—
40	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
41	C₃₁×#108	—	2	4	no	—
42	C₃₁×C₄⋊C₈	—	2	1	no	—
43	C₃₁×C₄⋊C₈	—	2	1	no	—
44	C₃₁×C₄⋊C₈	—	2	2	no	—
45	C₃₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
46	C₃₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
47	C₃₁×C₂²⋊C₈	—	2	1	no	—
48	C₃₁×C₂²⋊C₈	—	2	1	no	—
49	C₃₁×C₂²⋊C₈	—	2	1	no	—
50	C₃₁×C₂²⋊C₈	—	2	2	no	—
51	C₃₁×#97	—	2	1	no	—
52	C₃₁×#97	—	2	1	no	—
53	C₃₁×#97	—	2	2	no	—
54	C₃₁×#97	—	2	2	no	—
55	C₄×C₈×C₃₁	—	2	1	no	—
56	C₄×C₈×C₃₁	—	2	2	no	—
57	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
60	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
61	C₂×Q₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
62	C₂×D₄×C₃₁	—	2	1	no	—
63	C₂×D₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₃₁	—	2	1	no	—
65	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
68	M₁₆×C₃₁	—	2	4	no	—
69	M₁₆×C₃₁	—	4	2	no	—
70	M₁₆×C₃₁	—	4	2	no	—
71	C₂×C₈×C₃₁	—	2	2	no	—
72	C₂×C₈×C₃₁	—	2	1	no	—
73	C₂×C₈×C₃₁	—	2	1	no	—
74	C₂×C₈×C₃₁	—	4	1	no	—
75	C₂×C₈×C₃₁	—	4	2	no	—
76	C₂×C₈×C₃₁	—	4	2	no	—
77	C₂×C₈×C₃₁	—	4	2	no	—
78	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
82	C₄⋊C₄×C₃₁	—	2	1	no	—
83	C₄⋊C₄×C₃₁	—	2	1	no	—
84	C₄⋊C₄×C₃₁	—	2	1	no	—
85	C₄⋊C₄×C₃₁	—	2	1	no	—
86	C₄⋊C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
87	C₄⋊C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
88	C₂²⋊C₄×C₃₁	—	2	1	no	—
89	C₂²⋊C₄×C₃₁	—	2	1	no	—
90	C₂²⋊C₄×C₃₁	—	2	1	no	—
91	C₂²⋊C₄×C₃₁	—	2	1	no	—
92	C₂²⋊C₄×C₃₁	—	2	1	no	—
93	C₄²×C₃₁	—	2	1	no	—
94	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₃₁	#41	1	1	yes	characteristic
98	#8692	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂³×C₃₁	—	2	1	no	—
100	C₂³×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
101	Q₈×C₃₁	—	4	1	no	—
102	D₄×C₃₁	—	4	1	no	—
103	D₄×C₃₁	—	4	2	no	—
104	C₂×C₄×C₃₁	—	2	1	no	—
105	C₂×C₄×C₃₁	—	2	1	no	—
106	C₂×C₄×C₃₁	—	2	1	no	—
107	C₂×C₄×C₃₁	—	2	1	no	—
108	C₂×C₄×C₃₁	—	2	1	no	—
109	C₂×C₄×C₃₁	—	4	1	no	—
110	C₂×C₄×C₃₁	—	4	1	no	—
111	C₂×C₄×C₃₁	—	4	1	no	—
112	C₂×C₄×C₃₁	—	4	1	no	—
113	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
116	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
117	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
118	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
119	C₈×C₃₁	—	2	1	no	—
120	C₈×C₃₁	—	2	1	no	—
121	C₈×C₃₁	—	2	1	no	—
122	C₈×C₃₁	—	4	4	no	—
123	C₈×C₃₁	—	4	1	no	—
124	C₈×C₃₁	—	4	1	no	—
125	C₈×C₃₁	—	8	2	no	—
126	C₈×C₃₁	#104	1	2	yes	—
127	#2966	C₂×C₃₁	1	1	yes	characteristic
128	C₄×#108	C₂×C₃₁	1	1	yes	characteristic, complemented
129	#1568	C₂×C₃₁	1	1	yes	characteristic, complemented
130	#1529	C₂×C₃₁	1	2	yes	—
131	#1291	C₂×C₃₁	1	2	yes	complemented
132	C₂²×C₃₁	—	2	1	no	—
133	C₂²×C₃₁	—	2	1	no	—
134	C₂²×C₃₁	—	2	1	no	—
135	C₂²×C₃₁	—	4	1	no	—
136	C₂²×C₃₁	—	4	1	no	—
137	C₂²×C₃₁	—	4	1	no	—
138	C₂²×C₃₁	#380	1	1	yes	characteristic
139	C₄×C₃₁	—	2	1	no	—
140	C₄×C₃₁	—	2	1	no	—
141	C₄×C₃₁	—	2	1	no	—
142	C₄×C₃₁	—	2	1	no	—
143	C₄×C₃₁	—	2	1	no	—
144	C₄×C₃₁	—	4	1	no	—
145	C₄×C₃₁	—	4	1	no	—
146	C₄×C₃₁	—	4	1	no	—
147	C₄×C₃₁	C₂×#104	1	1	yes	characteristic
148	C₄×C₃₁	#421	1	1	yes	characteristic
149	#529	C₂²×C₃₁	1	1	yes	characteristic
150	#446	C₄×C₃₁	1	1	yes	characteristic
151	C₈×Q₈	C₄×C₃₁	1	1	yes	characteristic
152	#436	C₄×C₃₁	1	1	yes	characteristic
153	#435	—	2	1	no	—
154	C₈×D₄	—	2	1	no	—
155	C₈×D₄	C₄×C₃₁	1	1	yes	characteristic
156	#433	—	2	1	no	—
157	#432	C₂²×C₃₁	1	2	yes	—
158	#432	C₂²×C₃₁	1	1	yes	characteristic
159	C₂×#108	—	2	1	no	—
160	#405	—	2	1	no	—
161	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
162	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
163	#341	—	2	1	no	—
164	#335	—	2	1	no	—
165	#334	—	2	1	no	—
166	#330	—	2	2	no	—
167	#329	—	2	2	no	—
168	C₂×C₃₁	—	4	1	no	—
169	C₂×C₃₁	—	8	1	no	—
170	C₂×C₃₁	#1806	1	1	yes	characteristic
171	C₂×C₃₁	#1813	1	1	yes	characteristic
172	C₂×C₃₁	#1814	1	1	yes	characteristic
173	C₂×M₁₆	—	2	1	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂²×C₈	—	2	1	no	—
176	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
177	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
178	#126	—	2	1	no	—
179	#125	D₄×C₃₁	1	1	yes	characteristic
180	#124	D₄×C₃₁	1	1	yes	characteristic
181	#123	—	2	1	no	—
182	C₂²⋊Q₈	Q₈×C₃₁	1	1	yes	characteristic
183	#121	Q₈×C₃₁	1	1	yes	characteristic
184	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
185	C₄×D₄	—	2	1	no	—
186	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
187	#117	—	2	1	no	—
188	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
189	#108	—	2	4	no	—
190	C₄⋊C₈	—	2	1	no	—
191	C₄⋊C₈	—	2	1	no	—
192	C₄⋊C₈	—	2	2	no	—
193	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
194	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
195	C₂²⋊C₈	—	2	1	no	—
196	C₂²⋊C₈	—	2	1	no	—
197	C₂²⋊C₈	—	2	1	no	—
198	C₂²⋊C₈	—	2	2	no	—
199	#97	—	2	1	no	—
200	#97	—	2	1	no	—
201	#97	—	2	2	no	—
202	#97	—	2	2	no	—
203	C₄×C₈	—	2	1	no	—
204	C₄×C₈	—	2	2	no	—
205	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
206	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
207	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
208	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
209	C₃₁	#8692	1	1	yes	characteristic, direct product factor, Sylow
210	C₂×Q₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
211	C₂×D₄	—	2	1	no	—
212	C₂×D₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
215	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
216	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
217	M₁₆	—	2	4	no	—
218	M₁₆	—	4	2	no	—
219	M₁₆	—	4	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
228	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
229	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
230	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
231	C₄⋊C₄	—	2	1	no	—
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
236	C₄⋊C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
237	C₂²⋊C₄	—	2	1	no	—
238	C₂²⋊C₄	—	2	1	no	—
239	C₂²⋊C₄	—	2	1	no	—
240	C₂²⋊C₄	—	2	1	no	—
241	C₂²⋊C₄	—	2	1	no	—
242	C₄²	—	2	1	no	—
243	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
244	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
245	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
246	C₄²	#41×C₃₁	1	1	yes	characteristic
247	C₂³	—	2	1	no	—
248	C₂³	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
249	Q₈	—	4	1	no	—
250	D₄	—	4	1	no	—
251	D₄	—	4	2	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	1	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
262	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
263	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
264	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
265	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
266	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
267	C₈	—	2	1	no	—
268	C₈	—	2	1	no	—
269	C₈	—	2	1	no	—
270	C₈	—	4	4	no	—
271	C₈	—	4	1	no	—
272	C₈	—	4	1	no	—
273	C₈	—	8	2	no	—
274	C₈	C₃₁×#104	1	2	yes	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	4	1	no	—
279	C₂²	—	4	1	no	—
280	C₂²	—	4	1	no	—
281	C₂²	C₃₁×#380	1	1	yes	characteristic
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	4	1	no	—
288	C₄	—	4	1	no	—
289	C₄	—	4	1	no	—
290	C₄	C₂×C₃₁×#104	1	1	yes	characteristic
291	C₄	C₃₁×#421	1	1	yes	characteristic
292	C₂	—	4	1	no	—
293	C₂	—	8	1	no	—
294	C₂	C₃₁×#1806	1	1	yes	characteristic
295	C₂	C₃₁×#1813	1	1	yes	characteristic
296	C₂	C₃₁×#1814	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	#380
commutator	C₄²	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
prime radical	C₃₁×#121	Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₃₁×#108	C₂	semidirect
3. C₃₁×#1568	C₂	semidirect
5. C₃₁×#1291	C₂	semidirect
98. #8692	C₃₁	direct (N × H)
128. C₄×#108	C₂×C₃₁	semidirect
129. #1568	C₂×C₃₁	semidirect
131. #1291	C₂×C₃₁	semidirect
209. C₃₁	#8692	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×#2966	2
C₄×C₃₁×#108	2
C₃₁×#1568	2
C₃₁×#1529	2
C₃₁×#1291	2
#8692	31

Subgroups — C31×#8692 #21465451

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8692 #21465451