Subgroups — #22558344

Order 8576.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16863230	C₂	1	1	yes	maximal, characteristic
2	#16863171	C₂	1	1	yes	maximal, characteristic
3	M₆₄×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₆₇⋊C₁₆	C₄	1	1	yes	characteristic
5	C₈×Dic₆₇	C₄	1	1	yes	characteristic
6	C₆₇⋊C₃₂	C₄	1	2	yes	complemented
7	C₂×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
8	C₃₂×C₆₇	C₄	1	2	yes	complemented
9	C₂×C₆₇⋊C₈	C₈	1	1	yes	characteristic
10	C₄×Dic₆₇	C₈	1	1	yes	characteristic
11	C₆₇⋊C₁₆	C₈	1	2	yes	—
12	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
13	C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
14	C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
16	C₂×Dic₆₇	—	2	1	no	—
17	C₈×C₆₇	C₄²	1	1	yes	characteristic
18	C₈×C₆₇	C₂×C₈	1	1	yes	characteristic
19	C₆₇⋊C₈	—	2	1	no	—
20	C₂²×C₆₇	M₃₂	1	1	yes	characteristic
21	C₄×C₆₇	C₄×C₈	1	1	yes	characteristic
22	C₄×C₆₇	M₃₂	1	1	yes	characteristic
23	Dic₆₇	—	4	1	no	—
24	C₂×C₆₇	—	2	1	no	—
25	C₂×C₆₇	#346	1	1	yes	characteristic
26	#1398	—	67	1	no	maximal, Sylow
27	C₆₇	#1398	1	1	yes	characteristic, complemented, Sylow
28	M₆₄	—	67	1	no	—
29	M₆₄	D₆₇	1	1	yes	characteristic
30	#346	—	67	1	no	—
31	C₂×C₁₆	—	67	1	no	—
32	C₂×C₁₆	C₂×D₆₇	1	1	yes	characteristic
33	C₄×C₈	—	67	1	no	—
34	C₃₂	—	67	2	no	—
35	C₃₂	Dic₆₇	1	2	yes	complemented
36	C₂×C₈	—	67	1	no	—
37	C₂×C₈	C₄×D₆₇	1	1	yes	characteristic
38	C₄²	—	67	1	no	—
39	C₁₆	—	67	2	no	—
40	C₁₆	C₄×D₆₇	1	1	yes	characteristic
41	C₁₆	C₂×Dic₆₇	1	1	yes	characteristic
42	C₂×C₄	—	134	1	no	—
43	C₂×C₄	C₈×D₆₇	1	1	yes	characteristic
44	C₈	—	134	1	no	—
45	C₈	C₄×Dic₆₇	1	1	yes	characteristic
46	C₈	C₈×D₆₇	1	1	yes	characteristic
47	C₂²	#14523982	1	1	yes	characteristic
48	C₄	—	268	1	no	—
49	C₄	C₈×Dic₆₇	1	1	yes	characteristic
50	C₄	#14523982	1	1	yes	characteristic
51	C₂	—	2	1	no	—
52	C₂	#16863171	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₄×Dic₆₇
commutator	C₄×C₆₇	C₄×C₈
Frattini	C₂×C₁₆	C₂×D₆₇
Fitting	M₆₄×C₆₇	C₂
socle	C₂×C₆₇	#346
hypercenter	M₆₄	D₆₇
hypocenter	C₆₇	#1398
LCS	G ▷ C₄×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₈ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
6. C₆₇⋊C₃₂	C₄	semidirect
8. C₃₂×C₆₇	C₄	semidirect
27. C₆₇	#1398	semidirect
35. C₃₂	Dic₆₇	semidirect