Subgroups — C₃₁×#12523 #21469282

Order 7936.

262 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2254	C₂	1	4	yes	maximal
2	C₃₁×#1713	C₂	1	2	yes	maximal
3	C₈×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#500	—	2	2	no	—
5	C₃₁×#500	C₂²	1	4	yes	—
6	C₃₁×#494	—	2	2	no	—
7	C₃₁×#367	—	2	8	no	—
8	C₄×C₁₆×C₃₁	C₂²	1	2	yes	—
9	C₈²×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×Q₁₆×C₃₁	—	4	4	no	—
11	C₃₁×C₄⋊Q₈	—	2	4	no	—
12	C₂×C₁₆×C₃₁	D₄	1	4	yes	—
13	C₃₁×#107	—	2	8	no	—
14	C₃₁×#107	—	4	4	no	—
15	C₄×C₈×C₃₁	D₄	1	2	yes	—
16	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
17	C₂×Q₈×C₃₁	—	4	4	no	—
18	Q₁₆×C₃₁	—	8	8	no	—
19	C₂×C₈×C₃₁	D₈	1	4	yes	—
20	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
21	C₄⋊C₄×C₃₁	—	4	4	no	—
22	C₄⋊C₄×C₃₁	—	4	4	no	—
23	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
24	C₁₆×C₃₁	Q₁₆	1	8	yes	complemented
25	#12523	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	Q₈×C₃₁	—	8	8	no	—
27	C₂×C₄×C₃₁	—	8	4	no	—
28	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
29	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
30	C₈×C₃₁	SD₃₂	1	8	yes	—
31	C₈×C₃₁	C₂×Q₁₆	1	4	yes	—
32	#2254	C₂×C₃₁	1	4	yes	—
33	#1713	C₂×C₃₁	1	2	yes	—
34	C₈×C₁₆	C₂×C₃₁	1	1	yes	characteristic
35	C₂²×C₃₁	#493	1	1	yes	characteristic
36	C₄×C₃₁	—	16	4	no	—
37	C₄×C₃₁	#494	1	2	yes	—
38	C₄×C₃₁	C₂×SD₃₂	1	4	yes	—
39	#500	—	2	2	no	—
40	#500	C₂²×C₃₁	1	4	yes	—
41	#494	—	2	2	no	—
42	#367	—	2	8	no	—
43	C₄×C₁₆	C₂²×C₃₁	1	2	yes	—
44	C₈²	C₂²×C₃₁	1	1	yes	characteristic
45	C₂×C₃₁	#1713	1	1	yes	characteristic
46	C₂×C₃₁	#2248	1	2	yes	—
47	C₂×Q₁₆	—	4	4	no	—
48	C₄⋊Q₈	—	2	4	no	—
49	C₂×C₁₆	D₄×C₃₁	1	4	yes	—
50	#107	—	2	8	no	—
51	#107	—	4	4	no	—
52	C₄×C₈	D₄×C₃₁	1	2	yes	—
53	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
54	C₃₁	#12523	1	1	yes	characteristic, direct product factor, Sylow
55	C₂×Q₈	—	4	4	no	—
56	Q₁₆	—	8	8	no	—
57	C₂×C₈	D₈×C₃₁	1	4	yes	—
58	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
59	C₄⋊C₄	—	4	4	no	—
60	C₄⋊C₄	—	4	4	no	—
61	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
62	C₁₆	Q₁₆×C₃₁	1	8	yes	complemented
63	Q₈	—	8	8	no	—
64	C₂×C₄	—	8	4	no	—
65	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
66	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
67	C₈	C₃₁×SD₃₂	1	8	yes	—
68	C₈	C₂×Q₁₆×C₃₁	1	4	yes	—
69	C₂²	C₃₁×#493	1	1	yes	characteristic
70	C₄	—	16	4	no	—
71	C₄	C₃₁×#494	1	2	yes	—
72	C₄	C₂×C₃₁×SD₃₂	1	4	yes	—
73	C₂	C₃₁×#1713	1	1	yes	characteristic
74	C₂	C₃₁×#2248	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂²×C₃₁	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₃₁
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄²×C₃₁ ◁ C₈²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
24. C₁₆×C₃₁	Q₁₆	semidirect
25. #12523	C₃₁	direct (N × H)
54. C₃₁	#12523	direct (N × H)
62. C₁₆	Q₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#2254	2
C₃₁×#1713	2
C₈×C₁₆×C₃₁	2
#12523	31

Subgroups — C31×#12523 #21469282

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#12523 #21469282