Subgroups — C₃₁×#9601 #21466360

Order 7936.

404 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#1842	C₂	1	1	yes	maximal, characteristic
2	C₄×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#335	C₂	1	2	yes	maximal
5	C₃₁×#1277	C₂	1	2	yes	maximal
6	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
7	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
8	C₂×C₃₁×C₄⋊C₈	C₄	1	1	yes	characteristic
9	C₂×C₃₁×C₄⋊C₈	C₄	1	1	yes	characteristic
10	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₃₁	—	2	1	no	—
14	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
15	C₄³×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#336	—	2	1	no	—
17	C₃₁×#336	C₄	1	2	yes	—
18	C₃₁×#335	—	2	4	no	—
19	C₂²×C₈×C₃₁	—	2	1	no	—
20	C₂²×C₈×C₃₁	—	2	1	no	—
21	C₂²×C₈×C₃₁	—	2	2	no	—
22	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₃₁	Q₈	1	1	yes	characteristic
29	C₂×C₄²×C₃₁	Q₈	1	1	yes	characteristic
30	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
31	C₃₁×C₄⋊C₈	—	2	2	no	—
32	C₃₁×C₄⋊C₈	—	2	2	no	—
33	C₃₁×C₄⋊C₈	—	2	1	no	—
34	C₃₁×C₄⋊C₈	—	2	1	no	—
35	C₃₁×C₄⋊C₈	—	2	2	no	—
36	C₃₁×C₄⋊C₈	—	4	2	no	—
37	C₃₁×C₄⋊C₈	—	4	2	no	—
38	C₃₁×C₄⋊C₈	C₈	1	4	yes	complemented
39	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
40	C₃₁×C₄⋊C₈	Q₈	1	2	yes	—
41	C₄×C₈×C₃₁	—	2	2	no	—
42	C₄×C₈×C₃₁	—	2	2	no	—
43	C₄×C₈×C₃₁	—	4	2	no	—
44	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₃₁	—	2	2	no	—
52	C₂×C₈×C₃₁	—	2	2	no	—
53	C₂×C₈×C₃₁	—	2	2	no	—
54	C₂×C₈×C₃₁	—	2	2	no	—
55	C₂×C₈×C₃₁	—	2	2	no	—
56	C₂×C₈×C₃₁	—	2	2	no	—
57	C₂×C₈×C₃₁	—	2	2	no	—
58	C₂×C₈×C₃₁	—	4	1	no	—
59	C₂×C₈×C₃₁	—	4	1	no	—
60	C₂×C₈×C₃₁	—	4	1	no	—
61	C₂×C₈×C₃₁	—	4	1	no	—
62	C₂×C₈×C₃₁	—	4	2	no	—
63	C₂×C₈×C₃₁	—	4	2	no	—
64	C₂×C₈×C₃₁	—	4	2	no	—
65	C₂×C₈×C₃₁	C₂×C₈	1	4	yes	—
66	C₄²×C₃₁	—	2	2	no	—
67	C₄²×C₃₁	—	2	1	no	—
68	C₄²×C₃₁	—	2	1	no	—
69	C₄²×C₃₁	—	2	1	no	—
70	C₄²×C₃₁	—	2	1	no	—
71	C₄²×C₃₁	—	4	1	no	—
72	C₄²×C₃₁	C₂×C₈	1	2	yes	—
73	C₄²×C₃₁	M₁₆	1	2	yes	—
74	C₄²×C₃₁	D₈	1	2	yes	—
75	C₄²×C₃₁	Q₁₆	1	2	yes	—
76	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
78	C₄²×C₃₁	#41	1	1	yes	characteristic
79	C₄²×C₃₁	#41	1	1	yes	characteristic
80	#9601	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₃₁	—	2	1	no	—
83	C₂×C₄×C₃₁	—	2	1	no	—
84	C₂×C₄×C₃₁	—	2	1	no	—
85	C₂×C₄×C₃₁	—	2	1	no	—
86	C₂×C₄×C₃₁	—	2	1	no	—
87	C₂×C₄×C₃₁	—	2	1	no	—
88	C₂×C₄×C₃₁	—	2	1	no	—
89	C₂×C₄×C₃₁	—	2	1	no	—
90	C₂×C₄×C₃₁	—	2	1	no	—
91	C₂×C₄×C₃₁	C₄⋊C₈	1	2	yes	—
92	C₂×C₄×C₃₁	C₄⋊C₈	1	2	yes	—
93	C₂×C₄×C₃₁	#107	1	2	yes	—
94	C₂×C₄×C₃₁	#107	1	2	yes	—
95	C₂×C₄×C₃₁	#108	1	2	yes	—
96	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₃₁	C₄×Q₈	1	1	yes	characteristic
98	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
99	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
100	C₂×C₄×C₃₁	#125	1	1	yes	characteristic
101	C₂×C₄×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
102	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
103	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
104	C₂×C₄×C₃₁	#131	1	1	yes	characteristic
105	C₂×C₄×C₃₁	#131	1	1	yes	characteristic
106	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
107	C₂×C₄×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
108	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
109	C₂×C₄×C₃₁	#137	1	1	yes	characteristic
110	C₈×C₃₁	—	2	4	no	—
111	C₈×C₃₁	—	4	2	no	—
112	C₈×C₃₁	—	4	2	no	—
113	C₈×C₃₁	—	4	2	no	—
114	C₈×C₃₁	—	8	2	no	—
115	#1842	C₂×C₃₁	1	1	yes	characteristic
116	C₄×C₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic
117	C₄×C₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic
118	C₂×#335	C₂×C₃₁	1	2	yes	—
119	#1277	C₂×C₃₁	1	2	yes	—
120	C₂²×C₃₁	#389	1	1	yes	characteristic
121	C₂²×C₃₁	C₂×C₄⋊C₈	1	1	yes	characteristic
122	C₂²×C₃₁	#424	1	1	yes	characteristic
123	C₂²×C₃₁	C₂×#107	1	1	yes	characteristic
124	C₂²×C₃₁	#428	1	1	yes	characteristic
125	C₂²×C₃₁	C₂×#108	1	1	yes	characteristic
126	C₂²×C₃₁	#430	1	1	yes	characteristic
127	C₄×C₃₁	—	2	2	no	—
128	C₄×C₃₁	—	2	2	no	—
129	C₄×C₃₁	—	2	1	no	—
130	C₄×C₃₁	—	2	1	no	—
131	C₄×C₃₁	—	4	1	no	—
132	C₄×C₃₁	#335	1	4	yes	complemented
133	C₄×C₃₁	C₈×D₄	1	1	yes	characteristic
134	C₄×C₃₁	#436	1	1	yes	characteristic
135	C₄×C₃₁	C₈×Q₈	1	1	yes	characteristic
136	C₄×C₃₁	#446	1	1	yes	characteristic
137	C₄×C₃₁	#459	1	1	yes	characteristic
138	C₄×C₃₁	#462	1	1	yes	characteristic
139	C₄×C₃₁	#474	1	1	yes	characteristic
140	C₄×C₃₁	#477	1	1	yes	characteristic
141	C₂×C₄⋊C₈	—	2	1	no	—
142	C₂×C₄⋊C₈	—	2	1	no	—
143	C₂×C₄⋊C₈	C₄×C₃₁	1	1	yes	characteristic
144	C₂×C₄⋊C₈	C₄×C₃₁	1	1	yes	characteristic
145	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
146	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
147	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
148	C₂×C₄×C₈	—	2	1	no	—
149	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
150	C₄³	C₂²×C₃₁	1	1	yes	characteristic
151	#336	—	2	1	no	—
152	#336	C₄×C₃₁	1	2	yes	—
153	#335	—	2	4	no	—
154	C₂×C₃₁	C₂×#335	1	1	yes	characteristic
155	C₂×C₃₁	#1565	1	1	yes	characteristic
156	C₂×C₃₁	#1571	1	1	yes	characteristic
157	C₂×C₃₁	#1574	1	1	yes	characteristic
158	C₂×C₃₁	#1939	1	1	yes	characteristic
159	C₂×C₃₁	#1947	1	1	yes	characteristic
160	C₂×C₃₁	#1950	1	1	yes	characteristic
161	C₂²×C₈	—	2	1	no	—
162	C₂²×C₈	—	2	1	no	—
163	C₂²×C₈	—	2	2	no	—
164	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
165	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
166	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
167	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
168	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
169	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
170	C₂×C₄²	Q₈×C₃₁	1	1	yes	characteristic
171	C₂×C₄²	Q₈×C₃₁	1	1	yes	characteristic
172	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
173	C₄⋊C₈	—	2	2	no	—
174	C₄⋊C₈	—	2	2	no	—
175	C₄⋊C₈	—	2	1	no	—
176	C₄⋊C₈	—	2	1	no	—
177	C₄⋊C₈	—	2	2	no	—
178	C₄⋊C₈	—	4	2	no	—
179	C₄⋊C₈	—	4	2	no	—
180	C₄⋊C₈	C₈×C₃₁	1	4	yes	complemented
181	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
182	C₄⋊C₈	Q₈×C₃₁	1	2	yes	—
183	C₄×C₈	—	2	2	no	—
184	C₄×C₈	—	2	2	no	—
185	C₄×C₈	—	4	2	no	—
186	C₃₁	#9601	1	1	yes	characteristic, direct product factor, Sylow
187	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
188	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
189	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
190	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
191	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
193	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	4	1	no	—
202	C₂×C₈	—	4	1	no	—
203	C₂×C₈	—	4	1	no	—
204	C₂×C₈	—	4	1	no	—
205	C₂×C₈	—	4	2	no	—
206	C₂×C₈	—	4	2	no	—
207	C₂×C₈	—	4	2	no	—
208	C₂×C₈	C₂×C₈×C₃₁	1	4	yes	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	2	1	no	—
212	C₄²	—	2	1	no	—
213	C₄²	—	2	1	no	—
214	C₄²	—	4	1	no	—
215	C₄²	C₂×C₈×C₃₁	1	2	yes	—
216	C₄²	M₁₆×C₃₁	1	2	yes	—
217	C₄²	D₈×C₃₁	1	2	yes	—
218	C₄²	Q₁₆×C₃₁	1	2	yes	—
219	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
220	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
221	C₄²	#41×C₃₁	1	1	yes	characteristic
222	C₄²	#41×C₃₁	1	1	yes	characteristic
223	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	C₃₁×C₄⋊C₈	1	2	yes	—
234	C₂×C₄	C₃₁×C₄⋊C₈	1	2	yes	—
235	C₂×C₄	C₃₁×#107	1	2	yes	—
236	C₂×C₄	C₃₁×#107	1	2	yes	—
237	C₂×C₄	C₃₁×#108	1	2	yes	—
238	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
239	C₂×C₄	C₄×Q₈×C₃₁	1	1	yes	characteristic
240	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
241	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
242	C₂×C₄	C₃₁×#125	1	1	yes	characteristic
243	C₂×C₄	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
244	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
245	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
246	C₂×C₄	C₃₁×#131	1	1	yes	characteristic
247	C₂×C₄	C₃₁×#131	1	1	yes	characteristic
248	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
249	C₂×C₄	C₂×Q₁₆×C₃₁	1	1	yes	characteristic
250	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
251	C₂×C₄	C₃₁×#137	1	1	yes	characteristic
252	C₈	—	2	4	no	—
253	C₈	—	4	2	no	—
254	C₈	—	4	2	no	—
255	C₈	—	4	2	no	—
256	C₈	—	8	2	no	—
257	C₂²	C₃₁×#389	1	1	yes	characteristic
258	C₂²	C₂×C₃₁×C₄⋊C₈	1	1	yes	characteristic
259	C₂²	C₃₁×#424	1	1	yes	characteristic
260	C₂²	C₂×C₃₁×#107	1	1	yes	characteristic
261	C₂²	C₃₁×#428	1	1	yes	characteristic
262	C₂²	C₂×C₃₁×#108	1	1	yes	characteristic
263	C₂²	C₃₁×#430	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	2	2	no	—
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	4	1	no	—
269	C₄	C₃₁×#335	1	4	yes	complemented
270	C₄	C₈×D₄×C₃₁	1	1	yes	characteristic
271	C₄	C₃₁×#436	1	1	yes	characteristic
272	C₄	C₈×Q₈×C₃₁	1	1	yes	characteristic
273	C₄	C₃₁×#446	1	1	yes	characteristic
274	C₄	C₃₁×#459	1	1	yes	characteristic
275	C₄	C₃₁×#462	1	1	yes	characteristic
276	C₄	C₃₁×#474	1	1	yes	characteristic
277	C₄	C₃₁×#477	1	1	yes	characteristic
278	C₂	C₂×C₃₁×#335	1	1	yes	characteristic
279	C₂	C₃₁×#1565	1	1	yes	characteristic
280	C₂	C₃₁×#1571	1	1	yes	characteristic
281	C₂	C₃₁×#1574	1	1	yes	characteristic
282	C₂	C₃₁×#1939	1	1	yes	characteristic
283	C₂	C₃₁×#1947	1	1	yes	characteristic
284	C₂	C₃₁×#1950	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle, prime radical	C₂³×C₃₁	C₂×C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
38. C₃₁×C₄⋊C₈	C₈	semidirect
80. #9601	C₃₁	direct (N × H)
132. C₄×C₃₁	#335	semidirect
180. C₄⋊C₈	C₈×C₃₁	semidirect
186. C₃₁	#9601	direct (N × H)
269. C₄	C₃₁×#335	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#1842	2
C₄×C₃₁×C₄⋊C₈	2
C₂×C₃₁×#335	2
C₃₁×#1277	2
#9601	31

Subgroups — C31×#9601 #21466360

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9601 #21466360