Subgroups — C₃₁×#8672 #21465431

Order 7936.

386 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2965	C₂	1	1	yes	maximal, characteristic
2	C₄×C₃₁×#108	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#1566	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1489	C₂	1	2	yes	maximal
5	C₃₁×#1299	C₂	1	2	yes	maximal, complemented
6	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
7	C₃₁×#446	C₄	1	1	yes	characteristic
8	C₃₁×#446	C₄	1	1	yes	characteristic
9	C₃₁×#436	C₄	1	1	yes	characteristic
10	C₃₁×#436	C₄	1	1	yes	characteristic
11	C₃₁×#435	—	2	1	no	—
12	C₃₁×#435	—	2	1	no	—
13	C₃₁×#431	—	2	1	no	—
14	C₃₁×#431	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#108	—	2	1	no	—
16	C₃₁×#423	C₂²	1	2	yes	—
17	C₄×M₁₆×C₃₁	—	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×#334	—	2	1	no	—
22	C₃₁×#334	—	2	1	no	—
23	C₃₁×Q₈⋊C₈	—	2	2	no	—
24	C₃₁×D₄⋊C₈	—	2	2	no	—
25	C₂×#41×C₃₁	Q₈	1	1	yes	characteristic
26	C₂×M₁₆×C₃₁	—	2	1	no	—
27	C₂×M₁₆×C₃₁	—	2	1	no	—
28	C₂×M₁₆×C₃₁	—	2	2	no	—
29	C₂×M₁₆×C₃₁	—	2	2	no	—
30	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
32	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
33	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
34	C₄×D₄×C₃₁	—	2	1	no	—
35	C₄×D₄×C₃₁	—	2	1	no	—
36	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
37	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
38	C₃₁×#117	—	2	1	no	—
39	C₃₁×#117	Q₈	1	1	yes	characteristic
40	C₂×C₄²×C₃₁	—	2	1	no	—
41	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
42	C₃₁×#108	—	2	4	no	—
43	C₃₁×C₄⋊C₈	—	2	1	no	—
44	C₃₁×C₄⋊C₈	—	2	1	no	—
45	C₃₁×C₄⋊C₈	—	2	2	no	—
46	C₃₁×C₄⋊C₈	—	2	2	no	—
47	C₃₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
48	C₃₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
49	C₃₁×C₂²⋊C₈	—	2	1	no	—
50	C₃₁×C₂²⋊C₈	—	2	1	no	—
51	C₃₁×C₂²⋊C₈	—	2	1	no	—
52	C₃₁×#97	—	2	1	no	—
53	C₃₁×#97	—	2	1	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	#41×C₃₁	—	4	2	no	—
62	C₂×Q₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
63	C₂×D₄×C₃₁	—	2	1	no	—
64	C₂×D₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₃₁	—	2	1	no	—
66	C₂²×C₄×C₃₁	—	2	1	no	—
67	C₂²×C₄×C₃₁	—	2	1	no	—
68	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
70	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
71	M₁₆×C₃₁	—	2	4	no	—
72	M₁₆×C₃₁	—	4	2	no	—
73	M₁₆×C₃₁	—	4	2	no	—
74	M₁₆×C₃₁	—	4	2	no	—
75	M₁₆×C₃₁	—	8	2	no	—
76	C₂×C₈×C₃₁	—	2	2	no	—
77	C₂×C₈×C₃₁	—	2	1	no	—
78	C₂×C₈×C₃₁	—	2	1	no	—
79	C₂×C₈×C₃₁	—	4	1	no	—
80	C₂×C₈×C₃₁	—	4	2	no	—
81	C₂×C₈×C₃₁	—	4	2	no	—
82	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
86	C₄⋊C₄×C₃₁	—	2	1	no	—
87	C₄⋊C₄×C₃₁	—	2	1	no	—
88	C₄⋊C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
89	C₄⋊C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
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	—	2	1	no	—
95	C₄²×C₃₁	—	2	1	no	—
96	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₃₁	#41	1	1	yes	characteristic
100	#8672	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂³×C₃₁	—	2	1	no	—
102	C₂³×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
103	Q₈×C₃₁	—	2	2	no	—
104	D₄×C₃₁	—	2	2	no	—
105	D₄×C₃₁	—	4	2	no	—
106	C₂×C₄×C₃₁	—	2	1	no	—
107	C₂×C₄×C₃₁	—	2	1	no	—
108	C₂×C₄×C₃₁	—	2	2	no	—
109	C₂×C₄×C₃₁	—	2	2	no	—
110	C₂×C₄×C₃₁	—	2	1	no	—
111	C₂×C₄×C₃₁	—	2	1	no	—
112	C₂×C₄×C₃₁	—	2	1	no	—
113	C₂×C₄×C₃₁	—	4	2	no	—
114	C₂×C₄×C₃₁	—	4	1	no	—
115	C₂×C₄×C₃₁	—	4	1	no	—
116	C₂×C₄×C₃₁	—	4	2	no	—
117	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
120	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
121	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
122	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
123	C₈×C₃₁	—	2	1	no	—
124	C₈×C₃₁	—	2	1	no	—
125	C₈×C₃₁	—	2	1	no	—
126	C₈×C₃₁	—	4	4	no	—
127	C₈×C₃₁	—	4	1	no	—
128	C₈×C₃₁	—	4	1	no	—
129	C₈×C₃₁	—	8	2	no	—
130	C₈×C₃₁	#104	1	2	yes	—
131	#2965	C₂×C₃₁	1	1	yes	characteristic
132	C₄×#108	C₂×C₃₁	1	1	yes	characteristic, complemented
133	#1566	C₂×C₃₁	1	1	yes	characteristic, complemented
134	#1489	C₂×C₃₁	1	2	yes	—
135	#1299	C₂×C₃₁	1	2	yes	complemented
136	C₂²×C₃₁	—	2	1	no	—
137	C₂²×C₃₁	—	2	1	no	—
138	C₂²×C₃₁	—	2	1	no	—
139	C₂²×C₃₁	—	4	1	no	—
140	C₂²×C₃₁	—	4	1	no	—
141	C₂²×C₃₁	—	4	1	no	—
142	C₂²×C₃₁	#380	1	1	yes	characteristic
143	C₄×C₃₁	—	2	1	no	—
144	C₄×C₃₁	—	2	1	no	—
145	C₄×C₃₁	—	2	1	no	—
146	C₄×C₃₁	—	4	1	no	—
147	C₄×C₃₁	—	4	1	no	—
148	C₄×C₃₁	—	4	1	no	—
149	C₄×C₃₁	C₂×#104	1	1	yes	characteristic
150	C₄×C₃₁	#421	1	1	yes	characteristic
151	C₄×C₃₁	#430	1	2	yes	—
152	C₄×C₃₁	#471	1	2	yes	—
153	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
154	#446	C₄×C₃₁	1	1	yes	characteristic
155	#446	C₄×C₃₁	1	1	yes	characteristic
156	#436	C₄×C₃₁	1	1	yes	characteristic
157	#436	C₄×C₃₁	1	1	yes	characteristic
158	#435	—	2	1	no	—
159	#435	—	2	1	no	—
160	#431	—	2	1	no	—
161	#431	C₂²×C₃₁	1	1	yes	characteristic
162	C₂×#108	—	2	1	no	—
163	#423	C₂²×C₃₁	1	2	yes	—
164	C₄×M₁₆	—	2	1	no	—
165	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
166	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
167	#341	—	2	1	no	—
168	#334	—	2	1	no	—
169	#334	—	2	1	no	—
170	Q₈⋊C₈	—	2	2	no	—
171	D₄⋊C₈	—	2	2	no	—
172	C₂×C₃₁	—	4	1	no	—
173	C₂×C₃₁	—	8	1	no	—
174	C₂×C₃₁	#1806	1	1	yes	characteristic
175	C₂×C₃₁	#1811	1	1	yes	characteristic
176	C₂×C₃₁	#1815	1	1	yes	characteristic
177	C₂×#41	Q₈×C₃₁	1	1	yes	characteristic
178	C₂×M₁₆	—	2	1	no	—
179	C₂×M₁₆	—	2	1	no	—
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	—	2	2	no	—
182	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
183	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
184	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
185	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
186	C₄×D₄	—	2	1	no	—
187	C₄×D₄	—	2	1	no	—
188	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
189	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
190	#117	—	2	1	no	—
191	#117	Q₈×C₃₁	1	1	yes	characteristic
192	C₂×C₄²	—	2	1	no	—
193	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
194	#108	—	2	4	no	—
195	C₄⋊C₈	—	2	1	no	—
196	C₄⋊C₈	—	2	1	no	—
197	C₄⋊C₈	—	2	2	no	—
198	C₄⋊C₈	—	2	2	no	—
199	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
200	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
201	C₂²⋊C₈	—	2	1	no	—
202	C₂²⋊C₈	—	2	1	no	—
203	C₂²⋊C₈	—	2	1	no	—
204	#97	—	2	1	no	—
205	#97	—	2	1	no	—
206	#97	—	2	2	no	—
207	C₄×C₈	—	2	1	no	—
208	C₄×C₈	—	2	2	no	—
209	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
210	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
211	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
212	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
213	C₃₁	#8672	1	1	yes	characteristic, direct product factor, Sylow
214	#41	—	4	2	no	—
215	C₂×Q₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
216	C₂×D₄	—	2	1	no	—
217	C₂×D₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
222	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
223	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
224	M₁₆	—	2	4	no	—
225	M₁₆	—	4	2	no	—
226	M₁₆	—	4	2	no	—
227	M₁₆	—	4	2	no	—
228	M₁₆	—	8	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	4	1	no	—
233	C₂×C₈	—	4	2	no	—
234	C₂×C₈	—	4	2	no	—
235	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
236	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
237	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
238	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	—	2	1	no	—
241	C₄⋊C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
242	C₄⋊C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
243	C₂²⋊C₄	—	2	1	no	—
244	C₂²⋊C₄	—	2	1	no	—
245	C₂²⋊C₄	—	2	1	no	—
246	C₄²	—	2	1	no	—
247	C₄²	—	2	1	no	—
248	C₄²	—	2	1	no	—
249	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
250	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
251	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
252	C₄²	#41×C₃₁	1	1	yes	characteristic
253	C₂³	—	2	1	no	—
254	C₂³	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
255	Q₈	—	2	2	no	—
256	D₄	—	2	2	no	—
257	D₄	—	4	2	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
270	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
271	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
272	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
273	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
274	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
275	C₈	—	2	1	no	—
276	C₈	—	2	1	no	—
277	C₈	—	2	1	no	—
278	C₈	—	4	4	no	—
279	C₈	—	4	1	no	—
280	C₈	—	4	1	no	—
281	C₈	—	8	2	no	—
282	C₈	C₃₁×#104	1	2	yes	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	4	1	no	—
287	C₂²	—	4	1	no	—
288	C₂²	—	4	1	no	—
289	C₂²	C₃₁×#380	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	4	1	no	—
294	C₄	—	4	1	no	—
295	C₄	—	4	1	no	—
296	C₄	C₂×C₃₁×#104	1	1	yes	characteristic
297	C₄	C₃₁×#421	1	1	yes	characteristic
298	C₄	C₃₁×#430	1	2	yes	—
299	C₄	C₃₁×#471	1	2	yes	—
300	C₂	—	4	1	no	—
301	C₂	—	8	1	no	—
302	C₂	C₃₁×#1806	1	1	yes	characteristic
303	C₂	C₃₁×#1811	1	1	yes	characteristic
304	C₂	C₃₁×#1815	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	#380
prime radical	C₂×#41×C₃₁	Q₈
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. C₄×C₃₁×#108	C₂	semidirect
3. C₃₁×#1566	C₂	semidirect
5. C₃₁×#1299	C₂	semidirect
100. #8672	C₃₁	direct (N × H)
132. C₄×#108	C₂×C₃₁	semidirect
133. #1566	C₂×C₃₁	semidirect
135. #1299	C₂×C₃₁	semidirect
213. C₃₁	#8672	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×#2965	2
C₄×C₃₁×#108	2
C₃₁×#1566	2
C₃₁×#1489	2
C₃₁×#1299	2
#8672	31

Subgroups — C31×#8672 #21465431

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8672 #21465431