Subgroups — #21462216

Order 7808.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₁₆×#6923	C₂	1	2	yes	maximal
2	#16840689	C₂	1	1	yes	maximal, characteristic
3	#16840675	C₂	1	2	yes	maximal
4	#16840649	C₂	1	1	yes	maximal, characteristic
5	Q₃₂×D₆₁	C₂	1	1	yes	maximal, characteristic
6	Q₈×#6923	—	2	2	no	—
7	D₆₁⋊C₁₆	C₂²	1	1	yes	characteristic
8	#14519846	C₂²	1	2	yes	—
9	#14519840	—	2	2	no	—
10	C₈×#6923	C₂²	1	1	yes	characteristic
11	Q₁₆×D₆₁	C₂²	1	2	yes	—
12	Dic₄₈₈	C₄	1	1	yes	characteristic, complemented
13	#14519723	C₄	1	2	yes	complemented
14	C₁₆×D₆₁	C₂²	1	1	yes	characteristic
15	Q₃₂×C₆₁	C₄	1	1	yes	characteristic, complemented
16	Q₈×D₆₁	—	2	2	no	—
17	C₄⋊#6923	—	2	2	no	—
18	C₄⋊#6923	—	4	2	no	—
19	C₄×#6923	—	4	2	no	—
20	C₄×#6923	D₄	1	1	yes	characteristic
21	D₆₁⋊C₈	D₄	1	1	yes	characteristic
22	Q₁₆×C₆₁	C₂×C₄	1	2	yes	—
23	#11758202	—	2	2	no	—
24	Dic₂₄₄	C₂×C₄	1	2	yes	—
25	C₈×D₆₁	C₂³	1	1	yes	characteristic
26	#11758187	—	2	1	no	—
27	C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
28	#11758185	C₂×C₄	1	1	yes	characteristic
29	C₂×#6923	—	4	2	no	—
30	C₂×#6923	D₈	1	2	yes	—
31	Q₈×C₆₁	—	2	2	no	—
32	C₄×D₆₁	—	4	2	no	—
33	C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
34	Dic₁₂₂	—	2	2	no	—
35	Dic₁₂₂	—	4	2	no	—
36	#92420	—	2	1	no	—
37	C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
38	#92418	#41	1	1	yes	characteristic
39	C₂×D₆₁	C₂×D₈	1	1	yes	characteristic
40	#6923	—	2	2	no	—
41	#6923	—	8	2	no	—
42	C₄×C₆₁	—	4	2	no	—
43	C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
44	Dic₆₁	—	4	2	no	—
45	Dic₆₁	#135	1	1	yes	characteristic
46	#2176	—	61	1	no	maximal, Sylow
47	C₂×C₆₁	C₄×D₈	1	1	yes	characteristic
48	D₆₁	#510	1	2	yes	—
49	C₂×Q₃₂	—	61	1	no	—
50	C₄×Q₁₆	—	61	2	no	—
51	#367	—	61	1	no	—
52	#358	—	61	2	no	—
53	#346	—	61	1	no	—
54	C₆₁	#2176	1	1	yes	characteristic, complemented, Sylow
55	C₂×Q₁₆	—	61	2	no	—
56	C₄×Q₈	—	122	2	no	—
57	Q₃₂	—	61	2	no	—
58	Q₃₂	—	61	1	no	—
59	Q₃₂	#6923	1	1	yes	characteristic, complemented
60	C₂×C₁₆	—	61	1	no	—
61	C₂×C₁₆	—	61	1	no	—
62	#107	—	61	2	no	—
63	#103	—	122	2	no	—
64	C₄×C₈	—	61	1	no	—
65	C₂×Q₈	—	122	2	no	—
66	Q₁₆	—	61	2	no	—
67	Q₁₆	—	122	2	no	—
68	Q₁₆	C₂×#6923	1	2	yes	—
69	C₂×C₈	—	61	1	no	—
70	C₂×C₈	—	61	1	no	—
71	C₄⋊C₄	—	122	2	no	—
72	C₄⋊C₄	—	244	2	no	—
73	C₄²	—	61	1	no	—
74	C₄²	—	244	2	no	—
75	C₁₆	—	61	1	no	—
76	C₁₆	—	122	1	no	—
77	C₁₆	C₂×#6923	1	1	yes	characteristic
78	Q₈	—	2	2	no	—
79	Q₈	—	122	2	no	—
80	Q₈	—	244	2	no	—
81	C₂×C₄	—	61	1	no	—
82	C₂×C₄	—	61	2	no	—
83	C₂×C₄	—	244	2	no	—
84	C₂×C₄	—	244	2	no	—
85	C₈	—	61	1	no	—
86	C₈	—	122	1	no	—
87	C₈	C₂²×#6923	1	1	yes	characteristic
88	C₂²	—	61	1	no	—
89	C₄	—	4	2	no	—
90	C₄	—	61	1	no	—
91	C₄	—	122	2	no	—
92	C₄	—	488	2	no	—
93	C₄	—	244	2	no	—
94	C₄	D₄×#6923	1	1	yes	characteristic
95	C₂	—	61	2	no	—
96	C₂	D₈×#6923	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×#6923
commutator	C₈×C₆₁	C₂²×C₄
Frattini	C₈	C₂²×#6923
Fitting	Q₃₂×C₆₁	C₄
socle	C₂×C₆₁	C₄×D₈
hypercenter	Q₃₂	#6923
hypocenter	C₆₁	#2176
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

Complements

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

N	H ≅ G/N	type
12. Dic₄₈₈	C₄	semidirect
13. #14519723	C₄	semidirect
15. Q₃₂×C₆₁	C₄	semidirect
54. C₆₁	#2176	semidirect
59. Q₃₂	#6923	semidirect