Subgroups — #14521660

Order 1984.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#11758523	C₂	1	2	yes	maximal
3	Q₈×Dic₃₁	C₂	1	2	yes	maximal, complemented
4	#11758475	C₂	1	4	yes	maximal
5	#11758466	C₂	1	4	yes	maximal, complemented
6	#11758449	C₂	1	1	yes	maximal, characteristic
7	#11758447	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×C₃₁	C₂²	1	2	yes	—
9	C₂×C₄×D₃₁	C₂²	1	1	yes	characteristic
10	C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
11	C₄²×C₃₁	C₂²	1	1	yes	characteristic
12	(C₂×D₃₁)⋊C₄	C₂²	1	2	yes	—
13	(C₂×D₃₁)⋊C₄	C₂²	1	4	yes	—
14	C₄⋊Dic₃₁	C₂²	1	2	yes	—
15	C₄⋊Dic₃₁	C₂²	1	4	yes	—
16	Dic₃₁⋊C₄	C₂²	1	4	yes	—
17	Dic₃₁⋊C₄	C₂²	1	2	yes	—
18	Dic₃₁⋊C₄	C₂²	1	4	yes	—
19	C₄×Dic₃₁	C₂²	1	1	yes	characteristic
20	C₄×Dic₃₁	C₂²	1	4	yes	—
21	C₂²×D₃₁	C₂³	1	1	yes	characteristic
22	Q₈×C₃₁	—	2	4	no	—
23	C₂×C₄×C₃₁	C₂³	1	4	yes	—
24	C₂×C₄×C₃₁	C₂³	1	2	yes	—
25	C₂×C₄×C₃₁	C₂³	1	1	yes	characteristic
26	C₂×Dic₃₁	C₂³	1	2	yes	—
27	C₂×Dic₃₁	C₂³	1	4	yes	—
28	C₂×Dic₃₁	C₂³	1	1	yes	characteristic
29	C₄×D₃₁	—	2	2	no	—
30	C₂²×C₃₁	C₂⁴	1	1	yes	characteristic
31	C₂×D₃₁	—	2	1	no	—
32	C₂×D₃₁	—	2	2	no	—
33	C₄×C₃₁	—	2	4	no	—
34	C₄×C₃₁	—	2	2	no	—
35	C₄×C₃₁	#41	1	2	yes	—
36	Dic₃₁	—	2	1	no	—
37	Dic₃₁	—	2	2	no	—
38	Dic₃₁	—	2	4	no	—
39	#541	—	31	1	no	maximal, Sylow
40	C₂×C₃₁	C₂×#41	1	1	yes	characteristic
41	C₂×C₃₁	2₋¹⁺⁴	1	2	yes	—
42	D₃₁	—	4	1	no	—
43	C₄⋊Q₈	D₃₁	1	1	yes	characteristic, complemented
44	#126	—	31	4	no	—
45	#125	—	31	4	no	—
46	#125	—	31	1	no	—
47	C₂²⋊Q₈	—	31	2	no	—
48	C₄×Q₈	—	31	2	no	—
49	#117	—	31	1	no	—
50	C₃₁	#541	1	1	yes	characteristic, complemented, Sylow
51	C₂×Q₈	C₂×D₃₁	1	2	yes	—
52	C₂²×C₄	—	31	1	no	—
53	C₄⋊C₄	—	31	2	no	—
54	C₄⋊C₄	—	31	4	no	—
55	C₄⋊C₄	—	31	4	no	—
56	C₄⋊C₄	—	31	2	no	—
57	C₄⋊C₄	—	31	4	no	—
58	C₄⋊C₄	C₂×D₃₁	1	4	yes	—
59	C₂²⋊C₄	—	31	2	no	—
60	C₂²⋊C₄	—	31	4	no	—
61	C₄²	—	31	1	no	—
62	C₄²	—	31	4	no	—
63	C₄²	C₂×D₃₁	1	1	yes	characteristic
64	C₂³	—	31	1	no	—
65	Q₈	—	2	4	no	—
66	C₂×C₄	—	31	2	no	—
67	C₂×C₄	—	31	4	no	—
68	C₂×C₄	—	31	1	no	—
69	C₂×C₄	—	62	2	no	—
70	C₂×C₄	C₂²×D₃₁	1	4	yes	—
71	C₂×C₄	C₂²×D₃₁	1	2	yes	—
72	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
73	C₂²	—	62	1	no	—
74	C₂²	—	62	2	no	—
75	C₂²	C₂³×D₃₁	1	1	yes	characteristic
76	C₄	—	2	4	no	—
77	C₄	—	2	2	no	—
78	C₄	—	62	1	no	—
79	C₄	—	62	2	no	—
80	C₄	—	62	4	no	—
81	C₄	#92497	1	2	yes	—
82	C₂	—	124	1	no	—
83	C₂	C₂×#92497	1	1	yes	characteristic
84	C₂	#11758564	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₁
commutator, socle, cosocle	C₂²×C₃₁	C₂⁴
Fitting	C₃₁×C₄⋊Q₈	C₂
prime radical	C₂²×D₃₁	C₂³
hypercenter	C₄⋊Q₈	D₃₁
hypocenter	C₃₁	#541
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₃₁×C₄⋊Q₈	C₂	semidirect
3. Q₈×Dic₃₁	C₂	semidirect
5. #11758466	C₂	semidirect
7. #11758447	C₂	semidirect
43. C₄⋊Q₈	D₃₁	semidirect
50. C₃₁	#541	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×C₄⋊Q₈	2
#11758523	2
Q₈×Dic₃₁	2
#11758475	2
#11758466	2
#11758449	2
#11758447	2
#541	31