Subgroups — C₃₁×#7475 #21464234

Order 7936.

170 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×M₆₄	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2158	C₂	1	2	yes	maximal
3	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
4	C₃₁×#427	—	2	2	no	—
5	C₃₁×M₆₄	—	2	2	no	—
6	C₂×C₃₁×C₃₂	C₄	1	2	yes	complemented
7	C₃₁×#367	C₄	1	4	yes	complemented
8	C₃₁×#366	—	2	2	no	—
9	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
10	C₃₁×#117	—	4	2	no	—
11	C₂×C₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
12	C₂×C₁₆×C₃₁	C₂×C₄	1	2	yes	—
13	C₂×C₁₆×C₃₁	D₄	1	2	yes	—
14	C₂×C₁₆×C₃₁	Q₈	1	1	yes	characteristic
15	C₃₁×#107	—	2	4	no	—
16	C₃₁×#106	—	4	2	no	—
17	C₃₁×C₃₂	—	2	2	no	—
18	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
19	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
20	C₂×C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
21	C₂×C₈×C₃₁	SD₁₆	1	2	yes	—
22	C₂×C₈×C₃₁	Q₁₆	1	1	yes	characteristic
23	C₄⋊C₄×C₃₁	—	4	4	no	—
24	C₂²⋊C₄×C₃₁	—	8	2	no	—
25	C₄²×C₃₁	—	8	2	no	—
26	C₁₆×C₃₁	C₄²	1	1	yes	characteristic
27	C₁₆×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
28	C₁₆×C₃₁	C₄⋊C₄	1	2	yes	—
29	#7475	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂³×C₃₁	D₁₆	1	1	yes	characteristic
31	C₂×C₄×C₃₁	—	8	4	no	—
32	C₂×C₄×C₃₁	#102	1	2	yes	—
33	C₂×C₄×C₃₁	#107	1	1	yes	characteristic
34	C₂×C₄×C₃₁	SD₃₂	1	2	yes	—
35	C₂×C₄×C₃₁	Q₃₂	1	1	yes	characteristic
36	C₈×C₃₁	#95	1	1	yes	characteristic
37	C₈×C₃₁	#103	1	2	yes	—
38	C₈×C₃₁	#106	1	1	yes	characteristic
39	C₂×M₆₄	C₂×C₃₁	1	1	yes	characteristic
40	#2158	C₂×C₃₁	1	2	yes	—
41	C₂²×C₃₁	—	4	1	no	—
42	C₂²×C₃₁	#357	1	2	yes	—
43	C₂²×C₃₁	#366	1	1	yes	characteristic
44	C₄×C₃₁	—	16	4	no	—
45	C₄×C₃₁	#340	1	1	yes	characteristic
46	C₄×C₃₁	#358	1	2	yes	—
47	C₄×C₃₁	#367	1	1	yes	characteristic
48	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
49	#427	—	2	2	no	—
50	M₆₄	—	2	2	no	—
51	C₂×C₃₂	C₄×C₃₁	1	2	yes	complemented
52	#367	C₄×C₃₁	1	4	yes	complemented
53	#366	—	2	2	no	—
54	C₂×C₃₁	—	2	2	no	—
55	C₂×C₃₁	—	2	1	no	—
56	C₂×C₃₁	#1380	1	1	yes	characteristic
57	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
58	#117	—	4	2	no	—
59	C₂×C₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
60	C₂×C₁₆	C₂×C₄×C₃₁	1	2	yes	—
61	C₂×C₁₆	D₄×C₃₁	1	2	yes	—
62	C₂×C₁₆	Q₈×C₃₁	1	1	yes	characteristic
63	#107	—	2	4	no	—
64	#106	—	4	2	no	—
65	C₃₂	—	2	2	no	—
66	C₃₁	#7475	1	1	yes	characteristic, direct product factor, Sylow
67	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
68	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
69	C₂×C₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
70	C₂×C₈	SD₁₆×C₃₁	1	2	yes	—
71	C₂×C₈	Q₁₆×C₃₁	1	1	yes	characteristic
72	C₄⋊C₄	—	4	4	no	—
73	C₂²⋊C₄	—	8	2	no	—
74	C₄²	—	8	2	no	—
75	C₁₆	C₄²×C₃₁	1	1	yes	characteristic
76	C₁₆	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
77	C₁₆	C₄⋊C₄×C₃₁	1	2	yes	—
78	C₂³	C₃₁×D₁₆	1	1	yes	characteristic
79	C₂×C₄	—	8	4	no	—
80	C₂×C₄	C₃₁×#102	1	2	yes	—
81	C₂×C₄	C₃₁×#107	1	1	yes	characteristic
82	C₂×C₄	C₃₁×SD₃₂	1	2	yes	—
83	C₂×C₄	C₃₁×Q₃₂	1	1	yes	characteristic
84	C₈	C₃₁×#95	1	1	yes	characteristic
85	C₈	C₃₁×#103	1	2	yes	—
86	C₈	C₃₁×#106	1	1	yes	characteristic
87	C₂²	—	4	1	no	—
88	C₂²	C₃₁×#357	1	2	yes	—
89	C₂²	C₃₁×#366	1	1	yes	characteristic
90	C₄	—	16	4	no	—
91	C₄	C₃₁×#340	1	1	yes	characteristic
92	C₄	C₃₁×#358	1	2	yes	—
93	C₄	C₃₁×#367	1	1	yes	characteristic
94	C₂	—	2	2	no	—
95	C₂	—	2	1	no	—
96	C₂	C₃₁×#1380	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₃₁	#367
commutator	C₁₆	C₄²×C₃₁
Frattini, cosocle	C₂²×C₁₆	C₂²×C₃₁
socle	C₂×C₃₁	#1380
prime radical	C₂³×C₃₁	D₁₆
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ C₂²×C₈×C₃₁ ◁ C₂²×C₁₆×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₂×C₃₁×C₃₂	C₄	semidirect
7. C₃₁×#367	C₄	semidirect
29. #7475	C₃₁	direct (N × H)
51. C₂×C₃₂	C₄×C₃₁	semidirect
52. #367	C₄×C₃₁	semidirect
66. C₃₁	#7475	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×M₆₄	2
C₃₁×#2158	2
#7475	31

Subgroups — C31×#7475 #21464234

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#7475 #21464234