Subgroups — #21447721

Order 7808.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×Dic₁₂₂	C₂	1	1	yes	maximal, characteristic
2	#16839580	C₂	1	2	yes	maximal
3	C₄×#11758185	C₂	1	1	yes	maximal, characteristic
4	C₁₆×Dic₆₁	C₂	1	2	yes	maximal
5	C₆₁×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₁₂₂	C₄	1	1	yes	characteristic
7	C₂×#11758185	C₂²	1	1	yes	characteristic
8	C₂×#11758185	C₂²	1	1	yes	characteristic
9	C₄⋊#92418	C₄	1	1	yes	characteristic
10	#14519706	C₄	1	2	yes	—
11	C₈×Dic₆₁	C₂²	1	2	yes	—
12	C₂×C₁₆×C₆₁	C₂²	1	2	yes	—
13	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×Dic₁₂₂	C₈	1	1	yes	characteristic
15	C₂×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
17	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
18	C₄⋊Dic₆₁	C₈	1	1	yes	characteristic
19	#11758196	C₈	1	2	yes	—
20	C₄×Dic₆₁	C₂×C₄	1	2	yes	—
21	C₂×#92418	C₂×C₄	1	2	yes	—
22	C₁₆×C₆₁	—	2	2	no	—
23	#11758185	—	2	1	no	—
24	#11758185	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	1	no	—
32	C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
33	C₈×C₆₁	#41	1	1	yes	characteristic
34	#92418	—	2	2	no	—
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	2	no	—
48	C₄⋊C₁₆	D₆₁	1	1	yes	characteristic
49	C₄×C₁₆	—	61	2	no	—
50	C₄×C₁₆	—	61	1	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	C₂×D₆₁	1	2	yes	—
56	C₄⋊C₈	—	61	2	no	—
57	C₄⋊C₈	—	61	1	no	—
58	C₄×C₈	—	61	2	no	—
59	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
60	C₂×Q₈	—	61	1	no	—
61	C₂×C₈	—	61	2	no	—
62	C₂×C₈	C₄×D₆₁	1	1	yes	characteristic
63	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
64	C₄⋊C₄	—	61	2	no	—
65	C₄⋊C₄	—	61	1	no	—
66	C₄²	—	61	2	no	—
67	C₄²	C₄×D₆₁	1	1	yes	characteristic
68	C₁₆	—	2	2	no	—
69	C₁₆	—	61	2	no	—
70	C₁₆	—	122	1	no	—
71	Q₈	—	61	4	no	—
72	C₂×C₄	—	61	2	no	—
73	C₂×C₄	—	61	2	no	—
74	C₂×C₄	C₈×D₆₁	1	1	yes	characteristic
75	C₂×C₄	C₈×D₆₁	1	1	yes	characteristic
76	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
77	C₈	—	2	1	no	—
78	C₈	—	122	2	no	—
79	C₈	#11758224	1	1	yes	characteristic
80	C₈	Q₈×D₆₁	1	1	yes	characteristic
81	C₂²	C₂×C₈×D₆₁	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	61	4	no	—
84	C₄	—	122	2	no	—
85	C₄	C₁₆×D₆₁	1	2	yes	—
86	C₄	#14519742	1	1	yes	characteristic
87	C₄	#14519785	1	1	yes	characteristic
88	C₂	#16839886	1	1	yes	characteristic
89	C₂	C₂×C₁₆×D₆₁	1	1	yes	characteristic
90	C₂	#16840180	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₆₁
commutator	C₂×C₆₁	C₂²×C₁₆
Fitting	C₆₁×C₄⋊C₁₆	C₂
socle	C₂²×C₆₁	C₂²×C₈
hypercenter	C₄⋊C₁₆	D₆₁
hypocenter	C₆₁	Q₈×C₁₆
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
24. #11758185	Q₈	semidirect
30. Dic₁₂₂	C₁₆	semidirect
51. C₆₁	Q₈×C₁₆	semidirect