Subgroups — #22559819

Order 8576.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16863801	C₂	1	2	yes	maximal
2	#16863747	C₂	1	4	yes	maximal
3	C₈²×C₆₇	C₂	1	1	yes	maximal, characteristic
4	#14524043	C₂²	1	4	yes	—
5	#14523977	—	2	4	no	—
6	#14523968	—	2	8	no	—
7	C₄×C₈×C₆₇	C₂²	1	2	yes	—
8	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
9	C₂×Dic₁₃₄	—	2	4	no	—
10	C₄⋊Dic₆₇	—	2	4	no	—
11	C₄⋊Dic₆₇	—	2	4	no	—
12	C₂×C₈×C₆₇	D₄	1	4	yes	—
13	C₂×C₈×C₆₇	D₄	1	2	yes	—
14	C₄²×C₆₇	C₂³	1	1	yes	characteristic
15	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
16	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
17	C₂×Dic₆₇	—	4	4	no	—
18	Dic₁₃₄	—	4	8	no	—
19	C₈×C₆₇	—	2	4	no	—
20	C₈×C₆₇	Q₁₆	1	4	yes	—
21	C₂²×C₆₇	#127	1	1	yes	characteristic
22	C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
23	C₄×C₆₇	#135	1	4	yes	—
24	Dic₆₇	—	8	4	no	—
25	C₂×C₆₇	#494	1	1	yes	characteristic
26	C₂×C₆₇	#495	1	2	yes	—
27	#1710	—	67	1	no	maximal, Sylow
28	C₆₇	#1710	1	1	yes	characteristic, complemented, Sylow
29	#500	—	67	2	no	—
30	#487	—	67	4	no	—
31	C₈²	D₆₇	1	1	yes	characteristic
32	C₄⋊Q₈	—	67	4	no	—
33	#107	—	134	4	no	—
34	#103	—	134	8	no	—
35	C₄×C₈	C₂×D₆₇	1	2	yes	—
36	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
37	C₂×Q₈	—	134	4	no	—
38	C₂×C₈	D₂₆₈	1	4	yes	—
39	C₂×C₈	D₂₆₈	1	2	yes	—
40	C₄⋊C₄	—	134	4	no	—
41	C₄⋊C₄	—	134	4	no	—
42	C₄²	C₂²×D₆₇	1	1	yes	characteristic
43	Q₈	—	268	8	no	—
44	C₂×C₄	—	268	4	no	—
45	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
46	C₂×C₄	C₂×D₂₆₈	1	2	yes	—
47	C₈	—	2	4	no	—
48	C₈	Dic₂₆₈	1	4	yes	—
49	C₂²	#14524040	1	1	yes	characteristic
50	C₄	—	536	4	no	—
51	C₄	C₂×Dic₂₆₈	1	2	yes	—
52	C₄	#14524065	1	4	yes	—
53	C₂	#16863776	1	1	yes	characteristic
54	C₂	#16863780	1	2	yes	—

	subgroup	G/H
center	C₂²	#14524040
commutator	C₄²×C₆₇	C₂³
Frattini	C₄²	C₂²×D₆₇
Fitting	C₈²×C₆₇	C₂
socle	C₂²×C₆₇	#127
hypercenter	C₈²	D₆₇
hypocenter	C₆₇	#1710
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
28. C₆₇	#1710	semidirect