Subgroups — C₃₁×#7537 #21464296

Order 7936.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×D₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×C₆₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#1423	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃₁×D₁₆	—	2	1	no	—
5	C₃₁×D₃₂	—	2	1	no	—
6	C₃₁×D₃₂	C₄	1	2	yes	complemented
7	C₂×C₃₁×C₃₂	C₂²	1	1	yes	characteristic
8	C₃₁×#366	—	2	1	no	—
9	C₃₁×C₆₄	—	2	1	no	—
10	C₂×D₈×C₃₁	—	4	1	no	—
11	C₃₁×D₁₆	—	2	2	no	—
12	C₃₁×D₁₆	—	4	1	no	—
13	C₂×C₁₆×C₃₁	D₄	1	1	yes	characteristic
14	C₃₁×#107	—	4	1	no	—
15	C₃₁×C₃₂	C₂×C₄	1	1	yes	characteristic
16	C₃₁×C₃₂	D₄	1	1	yes	characteristic
17	C₂×D₄×C₃₁	—	8	1	no	—
18	D₈×C₃₁	—	4	2	no	—
19	D₈×C₃₁	—	8	1	no	—
20	C₂×C₈×C₃₁	D₈	1	1	yes	characteristic
21	C₄⋊C₄×C₃₁	—	8	1	no	—
22	C₁₆×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
23	C₁₆×C₃₁	SD₁₆	1	1	yes	characteristic
24	#7537	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂³×C₃₁	—	16	1	no	—
26	D₄×C₃₁	—	8	2	no	—
27	D₄×C₃₁	—	16	1	no	—
28	C₂×C₄×C₃₁	—	16	1	no	—
29	C₂×C₄×C₃₁	D₁₆	1	1	yes	characteristic
30	C₈×C₃₁	#102	1	1	yes	characteristic
31	C₈×C₃₁	SD₃₂	1	1	yes	characteristic
32	C₂×D₃₂	C₂×C₃₁	1	1	yes	characteristic
33	C₂×C₆₄	C₂×C₃₁	1	1	yes	characteristic, complemented
34	#1423	C₂×C₃₁	1	1	yes	characteristic, complemented
35	C₂²×C₃₁	—	16	2	no	—
36	C₂²×C₃₁	—	32	1	no	—
37	C₂²×C₃₁	—	32	1	no	—
38	C₂²×C₃₁	D₃₂	1	1	yes	characteristic
39	C₄×C₃₁	—	32	1	no	—
40	C₄×C₃₁	#357	1	1	yes	characteristic
41	C₄×C₃₁	SD₆₄	1	1	yes	characteristic
42	C₂×D₁₆	—	2	1	no	—
43	D₃₂	—	2	1	no	—
44	D₃₂	C₄×C₃₁	1	2	yes	complemented
45	C₂×C₃₂	C₂²×C₃₁	1	1	yes	characteristic
46	#366	—	2	1	no	—
47	C₆₄	—	2	1	no	—
48	C₂×C₃₁	—	32	2	no	—
49	C₂×C₃₁	#1415	1	1	yes	characteristic
50	C₂×C₃₁	D₆₄	1	1	yes	characteristic
51	C₂×C₃₁	SD₁₂₈	1	1	yes	characteristic
52	C₂×D₈	—	4	1	no	—
53	D₁₆	—	2	2	no	—
54	D₁₆	—	4	1	no	—
55	C₂×C₁₆	D₄×C₃₁	1	1	yes	characteristic
56	#107	—	4	1	no	—
57	C₃₂	C₂×C₄×C₃₁	1	1	yes	characteristic
58	C₃₂	D₄×C₃₁	1	1	yes	characteristic
59	C₃₁	#7537	1	1	yes	characteristic, direct product factor, Sylow
60	C₂×D₄	—	8	1	no	—
61	D₈	—	4	2	no	—
62	D₈	—	8	1	no	—
63	C₂×C₈	D₈×C₃₁	1	1	yes	characteristic
64	C₄⋊C₄	—	8	1	no	—
65	C₁₆	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
66	C₁₆	SD₁₆×C₃₁	1	1	yes	characteristic
67	C₂³	—	16	1	no	—
68	D₄	—	8	2	no	—
69	D₄	—	16	1	no	—
70	C₂×C₄	—	16	1	no	—
71	C₂×C₄	C₃₁×D₁₆	1	1	yes	characteristic
72	C₈	C₃₁×#102	1	1	yes	characteristic
73	C₈	C₃₁×SD₃₂	1	1	yes	characteristic
74	C₂²	—	16	2	no	—
75	C₂²	—	32	1	no	—
76	C₂²	—	32	1	no	—
77	C₂²	C₃₁×D₃₂	1	1	yes	characteristic
78	C₄	—	32	1	no	—
79	C₄	C₃₁×#357	1	1	yes	characteristic
80	C₄	C₃₁×SD₆₄	1	1	yes	characteristic
81	C₂	—	32	2	no	—
82	C₂	C₃₁×#1415	1	1	yes	characteristic
83	C₂	C₃₁×D₆₄	1	1	yes	characteristic
84	C₂	C₃₁×SD₁₂₈	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	D₃₂
commutator	C₃₂	C₂×C₄×C₃₁
Frattini, cosocle	C₂×C₃₂	C₂²×C₃₁
prime radical	C₂×C₃₁×D₃₂	C₂
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₈×C₃₁ ◁ C₂×C₁₆×C₃₁ ◁ C₂×C₃₁×C₃₂ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×C₆₄	C₂	semidirect
3. C₃₁×#1423	C₂	semidirect
6. C₃₁×D₃₂	C₄	semidirect
24. #7537	C₃₁	direct (N × H)
33. C₂×C₆₄	C₂×C₃₁	semidirect
34. #1423	C₂×C₃₁	semidirect
44. D₃₂	C₄×C₃₁	semidirect
59. C₃₁	#7537	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×D₃₂	2
C₂×C₃₁×C₆₄	2
C₃₁×#1423	2
#7537	31

Subgroups — C31×#7537 #21464296

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#7537 #21464296