Subgroups — #14521775

Order 1984.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₃₁	C₂	1	1	yes	maximal, characteristic
3	Q₈×Dic₃₁	C₂	1	1	yes	maximal, characteristic
4	#11758467	C₂	1	3	yes	maximal
5	#11758465	C₂	1	3	yes	maximal
6	#11758447	C₂	1	3	yes	maximal
7	C₄×Dic₆₂	C₂	1	3	yes	maximal
8	C₂×Q₈×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×Dic₆₂	C₂²	1	3	yes	—
10	C₄⋊C₄×C₃₁	C₂²	1	3	yes	—
11	C₄²×C₃₁	C₂²	1	3	yes	—
12	C₄⋊Dic₃₁	C₂²	1	3	yes	—
13	C₄⋊Dic₃₁	C₂²	1	3	yes	—
14	C₄⋊Dic₃₁	C₂²	1	3	yes	—
15	Dic₃₁⋊C₄	C₂²	1	1	yes	characteristic
16	Dic₃₁⋊C₄	C₂²	1	3	yes	—
17	Dic₃₁⋊C₄	C₂²	1	6	yes	—
18	C₄×Dic₃₁	C₂²	1	3	yes	—
19	C₄×Dic₃₁	C₂²	1	3	yes	—
20	Q₈×C₃₁	Q₈	1	4	yes	complemented
21	C₂×C₄×C₃₁	C₂³	1	1	yes	characteristic
22	C₂×C₄×C₃₁	C₂³	1	3	yes	—
23	C₂×C₄×C₃₁	C₂³	1	3	yes	—
24	C₂×Dic₃₁	C₂³	1	1	yes	characteristic
25	C₂×Dic₃₁	C₂³	1	3	yes	—
26	C₂×Dic₃₁	C₂³	1	3	yes	—
27	C₂×Dic₃₁	C₂³	1	1	yes	characteristic
28	Dic₆₂	—	2	6	no	—
29	C₂²×C₃₁	C₂⁴	1	1	yes	characteristic
30	C₄×C₃₁	—	2	3	no	—
31	C₄×C₃₁	—	2	1	no	—
32	C₄×C₃₁	C₂×Q₈	1	6	yes	—
33	Dic₃₁	—	2	3	no	—
34	Dic₃₁	—	2	1	no	—
35	Dic₃₁	—	2	3	no	—
36	Dic₃₁	#41	1	2	yes	—
37	#557	—	31	1	no	maximal, Sylow
38	C₂×C₃₁	C₂²×Q₈	1	1	yes	characteristic
39	C₂×C₃₁	C₂×#41	1	1	yes	characteristic
40	C₂×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
41	C₄⋊Q₈	—	31	3	no	—
42	#125	—	31	3	no	—
43	#125	—	31	3	no	—
44	C₄×Q₈	—	31	1	no	—
45	C₄×Q₈	—	31	1	no	—
46	C₄×Q₈	—	31	3	no	—
47	C₄×Q₈	D₃₁	1	1	yes	characteristic
48	C₃₁	#557	1	1	yes	characteristic, complemented, Sylow
49	C₂×Q₈	—	31	3	no	—
50	C₂×Q₈	C₂×D₃₁	1	1	yes	characteristic
51	C₄⋊C₄	—	31	1	no	—
52	C₄⋊C₄	—	31	3	no	—
53	C₄⋊C₄	—	31	6	no	—
54	C₄⋊C₄	—	31	3	no	—
55	C₄⋊C₄	—	31	3	no	—
56	C₄⋊C₄	—	31	3	no	—
57	C₄⋊C₄	C₂×D₃₁	1	3	yes	—
58	C₄²	—	31	3	no	—
59	C₄²	—	31	3	no	—
60	C₄²	C₂×D₃₁	1	3	yes	—
61	Q₈	—	62	6	no	—
62	Q₈	Dic₆₂	1	4	yes	complemented
63	C₂×C₄	—	31	1	no	—
64	C₂×C₄	—	31	1	no	—
65	C₂×C₄	—	31	3	no	—
66	C₂×C₄	—	31	3	no	—
67	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
68	C₂×C₄	C₂²×D₃₁	1	3	yes	—
69	C₂×C₄	C₂²×D₃₁	1	3	yes	—
70	C₂²	C₂³×D₃₁	1	1	yes	characteristic
71	C₄	—	2	1	no	—
72	C₄	—	2	3	no	—
73	C₄	—	31	2	no	—
74	C₄	—	62	1	no	—
75	C₄	—	62	3	no	—
76	C₄	—	62	3	no	—
77	C₄	C₂×Dic₆₂	1	6	yes	—
78	C₂	C₂²×Dic₆₂	1	1	yes	characteristic
79	C₂	#11758564	1	1	yes	characteristic
80	C₂	#41×D₃₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₁
commutator, socle, prime radical, cosocle	C₂²×C₃₁	C₂⁴
Fitting	C₄×Q₈×C₃₁	C₂
hypercenter	C₄×Q₈	D₃₁
hypocenter	C₃₁	#557
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
20. Q₈×C₃₁	Q₈	semidirect
48. C₃₁	#557	semidirect
62. Q₈	Dic₆₂	semidirect

Maximal subgroups

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

subgroup	index
C₄×Q₈×C₃₁	2
Q₈×Dic₃₁	2
#11758467	2
#11758465	2
#11758447	2
C₄×Dic₆₂	2
#557	31