Subgroups — #22558271

Order 8576.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16863798	C₂	1	1	yes	maximal, characteristic
2	#16863217	C₂	1	1	yes	maximal, characteristic
3	#364×C₆₇	C₂	1	1	yes	maximal, characteristic
4	#14524035	C₄	1	2	yes	—
5	#14523983	—	2	1	no	—
6	#14523977	—	2	1	no	—
7	#14523975	—	2	1	no	—
8	M₃₂×C₆₇	—	2	1	no	—
9	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₆₇	—	2	2	no	—
11	Dic₆₇⋊C₄	—	4	2	no	—
12	C₆₇⋊C₁₆	—	4	1	no	—
13	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
14	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
15	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₆₇	—	4	1	no	—
17	C₂×C₄×C₆₇	—	2	1	no	—
18	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
19	C₂×Dic₆₇	—	4	2	no	—
20	C₈×C₆₇	SD₁₆	1	1	yes	characteristic
21	C₈×C₆₇	SD₁₆	1	1	yes	characteristic
22	C₈×C₆₇	Q₁₆	1	1	yes	characteristic
23	C₈×C₆₇	Q₁₆	1	1	yes	characteristic
24	C₂²×C₆₇	#100	1	1	yes	characteristic
25	C₄×C₆₇	—	4	1	no	—
26	C₄×C₆₇	#103	1	1	yes	characteristic
27	C₄×C₆₇	#103	1	1	yes	characteristic
28	Dic₆₇	—	8	2	no	—
29	C₂×C₆₇	—	2	1	no	—
30	C₂×C₆₇	#333	1	1	yes	characteristic
31	#1366	—	67	1	no	maximal, Sylow
32	C₆₇	#1366	1	1	yes	characteristic, complemented, Sylow
33	#499	—	67	1	no	—
34	#364	—	67	1	no	—
35	#364	D₆₇	1	1	yes	characteristic
36	#125	—	67	2	no	—
37	M₃₂	—	2	1	no	—
38	M₃₂	—	134	1	no	—
39	#107	—	134	1	no	—
40	#106	—	134	1	no	—
41	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
42	C₂×C₈	D₂₆₈	1	1	yes	characteristic
43	C₂×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
44	C₄⋊C₄	—	268	2	no	—
45	C₄⋊C₄	—	134	2	no	—
46	C₄²	C₄×D₆₇	1	1	yes	characteristic
47	C₁₆	—	4	1	no	—
48	C₁₆	—	268	1	no	—
49	C₂×C₄	—	2	1	no	—
50	C₂×C₄	—	268	2	no	—
51	C₂×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
52	C₈	#11761616	1	1	yes	characteristic
53	C₈	#11761617	1	1	yes	characteristic
54	C₈	#11761618	1	1	yes	characteristic
55	C₈	Dic₂₆₈	1	1	yes	characteristic
56	C₂²	#14523959	1	1	yes	characteristic
57	C₄	—	4	1	no	—
58	C₄	—	536	2	no	—
59	C₄	#14523966	1	1	yes	characteristic
60	C₄	#14523968	1	1	yes	characteristic
61	C₂	—	2	1	no	—
62	C₂	#16863144	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16863144
commutator	C₄²×C₆₇	C₂×C₄
Frattini	C₄×C₈	C₂×D₆₇
Fitting	#364×C₆₇	C₂
socle	C₂×C₆₇	#333
hypercenter	#364	D₆₇
hypocenter	C₆₇	#1366
LCS	G ▷ C₄²×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
32. C₆₇	#1366	semidirect