Subgroups — #22559821

Order 8576.

178 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16863747	C₂	1	3	yes	maximal, complemented
2	#16863743	C₂	1	3	yes	maximal
3	C₈²×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#14524043	C₂²	1	3	yes	—
5	#14524040	C₂²	1	1	yes	characteristic
6	#14523968	—	2	6	no	—
7	#14523965	—	2	6	no	—
8	C₄×C₈×C₆₇	C₂²	1	3	yes	—
9	C₂×Dic₁₃₄	—	2	3	no	—
10	C₂×D₂₆₈	—	2	3	no	—
11	C₄⋊Dic₆₇	—	2	3	no	—
12	C₄⋊Dic₆₇	—	2	3	no	—
13	C₂×C₈×C₆₇	D₄	1	6	yes	—
14	C₄²×C₆₇	C₂³	1	1	yes	characteristic
15	C₂²×D₆₇	—	4	1	no	—
16	C₂×C₄×C₆₇	C₂×D₄	1	3	yes	—
17	C₂×Dic₆₇	—	4	3	no	—
18	D₂₆₈	—	4	6	no	—
19	Dic₁₃₄	—	4	6	no	—
20	C₈×C₆₇	—	2	6	no	—
21	C₂²×C₆₇	#127	1	1	yes	characteristic
22	C₂×D₆₇	—	8	3	no	—
23	C₄×C₆₇	#135	1	6	yes	—
24	Dic₆₇	—	8	3	no	—
25	C₂×C₆₇	#495	1	3	yes	—
26	D₆₇	—	16	1	no	—
27	#1711	—	67	1	no	maximal, Sylow
28	C₆₇	#1711	1	1	yes	characteristic, complemented, Sylow
29	#487	—	67	3	no	—
30	#486	—	67	3	no	—
31	C₈²	D₆₇	1	1	yes	characteristic, complemented
32	C₄⋊Q₈	—	67	3	no	—
33	#127	—	67	1	no	—
34	#103	—	134	6	no	—
35	#102	—	134	6	no	—
36	C₄×C₈	C₂×D₆₇	1	3	yes	—
37	C₂×Q₈	—	134	3	no	—
38	C₂×D₄	—	134	3	no	—
39	C₂×C₈	D₂₆₈	1	6	yes	—
40	C₄⋊C₄	—	134	3	no	—
41	C₄⋊C₄	—	134	3	no	—
42	C₄²	C₂²×D₆₇	1	1	yes	characteristic
43	C₂³	—	268	1	no	—
44	Q₈	—	268	6	no	—
45	D₄	—	268	6	no	—
46	C₂×C₄	—	268	3	no	—
47	C₂×C₄	C₂×D₂₆₈	1	3	yes	—
48	C₈	—	2	6	no	—
49	C₂²	—	536	3	no	—
50	C₂²	#14524040	1	1	yes	characteristic
51	C₄	—	536	3	no	—
52	C₄	#14524065	1	6	yes	—
53	C₂	—	1072	1	no	—
54	C₂	#16863780	1	3	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	#1711
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
1. #16863747	C₂	semidirect
3. C₈²×C₆₇	C₂	semidirect
28. C₆₇	#1711	semidirect
31. C₈²	D₆₇	semidirect