Subgroups — #21462233

Order 7808.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16840683	C₂	1	2	yes	maximal
2	C₄⋊#11758187	C₂	1	1	yes	maximal, characteristic
3	#16840648	C₂	1	2	yes	maximal
4	C₄×#11758187	C₂	1	1	yes	maximal, characteristic
5	#16839886	C₂	1	1	yes	maximal, characteristic
6	C₂×#11758187	C₂²	1	1	yes	characteristic
7	C₂×#11758187	C₂²	1	2	yes	—
8	C₂×#11758187	C₂²	1	1	yes	characteristic
9	C₄×Dic₁₂₂	C₄	1	1	yes	characteristic
10	#14519706	C₄	1	2	yes	—
11	C₄×#92418	C₂²	1	1	yes	characteristic
12	C₈×Dic₆₁	C₂²	1	2	yes	—
13	C₄⋊C₈×C₆₁	C₄	1	1	yes	characteristic
14	C₂×Dic₁₂₂	C₈	1	1	yes	characteristic
15	C₂×C₈×C₆₁	C₂×C₄	1	2	yes	—
16	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
17	C₄⋊Dic₆₁	C₈	1	1	yes	characteristic
18	#11758196	C₈	1	2	yes	—
19	C₄×Dic₆₁	C₂×C₄	1	2	yes	—
20	C₂×#92418	C₂×C₄	1	1	yes	characteristic
21	C₂×#92418	C₂³	1	1	yes	characteristic
22	#11758187	—	2	1	no	—
23	#11758187	—	2	2	no	—
24	#11758187	Q₈	1	2	yes	complemented
25	C₂×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
26	C₂×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
27	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
28	C₂×Dic₆₁	C₂×C₈	1	2	yes	—
29	C₂×Dic₆₁	C₂×C₈	1	2	yes	—
30	Dic₁₂₂	C₁₆	1	4	yes	complemented
31	C₈×C₆₁	—	2	2	no	—
32	#92418	—	2	1	no	—
33	#92418	C₂×Q₈	1	1	yes	characteristic
34	#92418	#41	1	1	yes	characteristic
35	C₂²×C₆₁	C₂²×C₈	1	1	yes	characteristic
36	C₄×C₆₁	—	2	1	no	—
37	C₄×C₆₁	C₂×C₁₆	1	2	yes	—
38	C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
39	C₄×C₆₁	#131	1	1	yes	characteristic
40	Dic₆₁	—	2	2	no	—
41	Dic₆₁	C₂×C₁₆	1	4	yes	—
42	Q₈×C₁₆	—	61	1	no	maximal, Sylow
43	C₂×C₆₁	C₈×Q₈	1	1	yes	characteristic
44	C₂×C₆₁	C₂²×C₁₆	1	1	yes	characteristic
45	C₂×C₆₁	#504	1	1	yes	characteristic
46	C₈×Q₈	—	61	1	no	—
47	C₄⋊C₁₆	—	61	1	no	—
48	C₄⋊C₁₆	—	61	2	no	—
49	C₄×C₁₆	—	61	1	no	—
50	C₄×C₁₆	—	61	2	no	—
51	C₆₁	Q₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
52	C₄×Q₈	—	61	1	no	—
53	C₂×C₁₆	—	61	1	no	—
54	C₂×C₁₆	—	61	1	no	—
55	C₂×C₁₆	—	61	2	no	—
56	C₄⋊C₈	—	61	2	no	—
57	C₄⋊C₈	#6923	1	1	yes	characteristic
58	C₄×C₈	—	61	2	no	—
59	C₄×C₈	—	61	1	no	—
60	C₂×Q₈	—	61	1	no	—
61	C₂×C₈	—	61	1	no	—
62	C₂×C₈	—	61	1	no	—
63	C₂×C₈	C₂×#6923	1	2	yes	—
64	C₄⋊C₄	—	61	2	no	—
65	C₄⋊C₄	—	61	1	no	—
66	C₄²	—	61	2	no	—
67	C₄²	C₂×#6923	1	1	yes	characteristic
68	C₁₆	—	61	2	no	—
69	C₁₆	—	122	1	no	—
70	C₁₆	—	122	2	no	—
71	Q₈	—	61	4	no	—
72	C₂×C₄	—	61	2	no	—
73	C₂×C₄	—	61	2	no	—
74	C₂×C₄	D₆₁⋊C₈	1	1	yes	characteristic
75	C₂×C₄	D₆₁⋊C₈	1	1	yes	characteristic
76	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
77	C₈	—	2	2	no	—
78	C₈	—	61	1	no	—
79	C₈	—	61	1	no	—
80	C₈	—	122	1	no	—
81	C₂²	C₂×D₆₁⋊C₈	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	61	4	no	—
84	C₄	—	122	2	no	—
85	C₄	D₆₁⋊C₁₆	1	2	yes	—
86	C₄	Q₈×#6923	1	1	yes	characteristic
87	C₄	#14519864	1	1	yes	characteristic
88	C₂	#16840811	1	1	yes	characteristic
89	C₂	C₂×D₆₁⋊C₁₆	1	1	yes	characteristic
90	C₂	#16840821	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#6923
commutator	C₂×C₆₁	C₂²×C₁₆
Fitting	C₄⋊C₈×C₆₁	C₄
socle	C₂²×C₆₁	C₂²×C₈
hypercenter	C₄⋊C₈	#6923
hypocenter	C₆₁	Q₈×C₁₆
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
24. #11758187	Q₈	semidirect
30. Dic₁₂₂	C₁₆	semidirect
51. C₆₁	Q₈×C₁₆	semidirect