Subgroups — C₃₁×#7539 #21464298

Order 7936.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2262	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×C₆₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#1425	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#508	—	2	1	no	—
5	C₃₁×Q₆₄	C₄	1	1	yes	characteristic
6	C₃₁×SD₆₄	—	2	1	no	—
7	C₃₁×D₃₂	C₄	1	1	yes	characteristic
8	C₂×C₃₁×C₃₂	C₂²	1	1	yes	characteristic
9	C₃₁×#368	—	2	1	no	—
10	C₃₁×C₆₄	—	2	1	no	—
11	C₃₁×#135	—	4	1	no	—
12	C₃₁×Q₃₂	—	2	1	no	—
13	C₃₁×SD₃₂	—	4	1	no	—
14	C₃₁×D₁₆	—	2	1	no	—
15	C₂×C₁₆×C₃₁	D₄	1	1	yes	characteristic
16	C₃₁×#108	—	4	1	no	—
17	C₃₁×C₃₂	C₂×C₄	1	1	yes	characteristic
18	C₃₁×C₃₂	D₄	1	1	yes	characteristic
19	#41×C₃₁	—	8	1	no	—
20	Q₁₆×C₃₁	—	4	1	no	—
21	SD₁₆×C₃₁	—	8	1	no	—
22	D₈×C₃₁	—	4	1	no	—
23	M₁₆×C₃₁	—	8	1	no	—
24	C₂×C₈×C₃₁	D₈	1	1	yes	characteristic
25	C₁₆×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
26	C₁₆×C₃₁	SD₁₆	1	1	yes	characteristic
27	#7539	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	Q₈×C₃₁	—	8	1	no	—
29	D₄×C₃₁	—	8	1	no	—
30	D₄×C₃₁	—	16	1	no	—
31	C₂×C₄×C₃₁	—	16	1	no	—
32	C₂×C₄×C₃₁	D₁₆	1	1	yes	characteristic
33	C₈×C₃₁	—	16	1	no	—
34	C₈×C₃₁	#102	1	1	yes	characteristic
35	C₈×C₃₁	SD₃₂	1	1	yes	characteristic
36	#2262	C₂×C₃₁	1	1	yes	characteristic
37	C₂×C₆₄	C₂×C₃₁	1	1	yes	characteristic, complemented
38	#1425	C₂×C₃₁	1	1	yes	characteristic, complemented
39	C₂²×C₃₁	—	16	1	no	—
40	C₂²×C₃₁	SD₆₄	1	1	yes	characteristic
41	C₄×C₃₁	—	16	1	no	—
42	C₄×C₃₁	#357	1	1	yes	characteristic
43	C₄×C₃₁	D₃₂	1	1	yes	characteristic
44	#508	—	2	1	no	—
45	Q₆₄	C₄×C₃₁	1	1	yes	characteristic
46	SD₆₄	—	2	1	no	—
47	D₃₂	C₄×C₃₁	1	1	yes	characteristic
48	C₂×C₃₂	C₂²×C₃₁	1	1	yes	characteristic
49	#368	—	2	1	no	—
50	C₆₄	—	2	1	no	—
51	C₂×C₃₁	—	2	1	no	—
52	C₂×C₃₁	—	32	1	no	—
53	C₂×C₃₁	#1415	1	1	yes	characteristic
54	#135	—	4	1	no	—
55	Q₃₂	—	2	1	no	—
56	SD₃₂	—	4	1	no	—
57	D₁₆	—	2	1	no	—
58	C₂×C₁₆	D₄×C₃₁	1	1	yes	characteristic
59	#108	—	4	1	no	—
60	C₃₂	C₂×C₄×C₃₁	1	1	yes	characteristic
61	C₃₂	D₄×C₃₁	1	1	yes	characteristic
62	C₃₁	#7539	1	1	yes	characteristic, direct product factor, Sylow
63	#41	—	8	1	no	—
64	Q₁₆	—	4	1	no	—
65	SD₁₆	—	8	1	no	—
66	D₈	—	4	1	no	—
67	M₁₆	—	8	1	no	—
68	C₂×C₈	D₈×C₃₁	1	1	yes	characteristic
69	C₁₆	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
70	C₁₆	SD₁₆×C₃₁	1	1	yes	characteristic
71	Q₈	—	8	1	no	—
72	D₄	—	8	1	no	—
73	D₄	—	16	1	no	—
74	C₂×C₄	—	16	1	no	—
75	C₂×C₄	C₃₁×D₁₆	1	1	yes	characteristic
76	C₈	—	16	1	no	—
77	C₈	C₃₁×#102	1	1	yes	characteristic
78	C₈	C₃₁×SD₃₂	1	1	yes	characteristic
79	C₂²	—	16	1	no	—
80	C₂²	C₃₁×SD₆₄	1	1	yes	characteristic
81	C₄	—	16	1	no	—
82	C₄	C₃₁×#357	1	1	yes	characteristic
83	C₄	C₃₁×D₃₂	1	1	yes	characteristic
84	C₂	—	2	1	no	—
85	C₂	—	32	1	no	—
86	C₂	C₃₁×#1415	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₃₁	D₃₂
commutator	C₃₂	C₂×C₄×C₃₁
Frattini, cosocle	C₂×C₃₂	C₂²×C₃₁
socle	C₂×C₃₁	#1415
prime radical	C₃₁×#2262	C₂
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×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×#1425	C₂	semidirect
27. #7539	C₃₁	direct (N × H)
37. C₂×C₆₄	C₂×C₃₁	semidirect
38. #1425	C₂×C₃₁	semidirect
62. C₃₁	#7539	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×#2262	2
C₂×C₃₁×C₆₄	2
C₃₁×#1425	2
#7539	31

Subgroups — C31×#7539 #21464298

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#7539 #21464298