Subgroups — #14524070

Order 2144.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11761629	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×D₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#11761615	C₂	1	1	yes	maximal, characteristic, complemented
4	D₅₃₆	C₂	1	1	yes	maximal, characteristic, complemented
5	#11761613	C₂	1	1	yes	maximal, characteristic, complemented
6	#11761611	C₂	1	1	yes	maximal, characteristic, complemented
7	SD₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₆₇	—	2	1	no	—
9	Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
10	D₄×C₆₇	C₂²	1	1	yes	characteristic
11	C₂²⋊D₆₇	—	2	1	no	—
12	D₂₆₈	—	2	1	no	—
13	D₂₆₈	C₂²	1	1	yes	characteristic
14	D₂₆₈	C₂²	1	1	yes	characteristic, complemented
15	C₄×D₆₇	—	2	1	no	—
16	C₄×D₆₇	C₂²	1	1	yes	characteristic
17	C₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
18	C₆₇⋊C₈	C₂²	1	1	yes	characteristic, complemented
19	C₂²×C₆₇	—	2	1	no	—
20	C₂×D₆₇	—	2	1	no	—
21	C₂×D₆₇	—	2	1	no	—
22	C₂×D₆₇	—	4	2	no	—
23	C₂×D₆₇	D₄	1	1	yes	characteristic
24	C₄×C₆₇	—	2	1	no	—
25	C₄×C₆₇	C₂³	1	1	yes	characteristic
26	Dic₆₇	D₄	1	1	yes	characteristic
27	C₂×C₆₇	—	4	1	no	—
28	C₂×C₆₇	C₂×D₄	1	1	yes	characteristic
29	D₆₇	—	2	1	no	—
30	D₆₇	—	4	1	no	—
31	D₆₇	—	4	1	no	—
32	C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
33	AGL(1,ℤ/8)	—	67	1	no	maximal, Sylow
34	#41	—	67	1	no	—
35	C₂×D₄	—	67	1	no	—
36	SD₁₆	—	67	1	no	—
37	SD₁₆	D₆₇	1	1	yes	characteristic, complemented
38	D₈	—	67	1	no	—
39	D₈	—	67	1	no	—
40	M₁₆	—	67	1	no	—
41	C₂³	—	134	1	no	—
42	Q₈	C₂×D₆₇	1	1	yes	characteristic, complemented
43	D₄	—	67	1	no	—
44	D₄	—	67	1	no	—
45	D₄	—	134	1	no	—
46	D₄	—	134	1	no	—
47	D₄	C₂×D₆₇	1	1	yes	characteristic
48	C₂×C₄	—	67	1	no	—
49	C₂×C₄	—	134	1	no	—
50	C₈	—	67	1	no	—
51	C₈	C₂×D₆₇	1	1	yes	characteristic, complemented
52	C₂²	—	2	1	no	—
53	C₂²	—	268	2	no	—
54	C₂²	—	67	1	no	—
55	C₂²	—	134	1	no	—
56	C₂²	—	134	1	no	—
57	C₄	—	2	1	no	—
58	C₄	—	67	1	no	—
59	C₄	C₂²×D₆₇	1	1	yes	characteristic
60	C₂	—	4	1	no	—
61	C₂	—	268	1	no	—
62	C₂	—	268	1	no	—
63	C₂	—	134	1	no	—
64	C₂	D₄×D₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₆₇
commutator, cosocle	C₄×C₆₇	C₂³
Frattini	C₄	C₂²×D₆₇
Fitting	SD₁₆×C₆₇	C₂
socle	C₂×C₆₇	C₂×D₄
hypercenter	SD₁₆	D₆₇
hypocenter	C₆₇	AGL(1,ℤ/8)
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ SD₁₆

Complements

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

N	H ≅ G/N	type
1. #11761629	C₂	semidirect
2. D₄×D₆₇	C₂	semidirect
3. #11761615	C₂	semidirect
4. D₅₃₆	C₂	semidirect
5. #11761613	C₂	semidirect
6. #11761611	C₂	semidirect
7. SD₁₆×C₆₇	C₂	semidirect
9. Q₈×C₆₇	C₂²	semidirect
14. D₂₆₈	C₂²	semidirect
17. C₈×C₆₇	C₂²	semidirect
18. C₆₇⋊C₈	C₂²	semidirect
32. C₆₇	AGL(1,ℤ/8)	semidirect
37. SD₁₆	D₆₇	semidirect
42. Q₈	C₂×D₆₇	semidirect
51. C₈	C₂×D₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#11761629	2
D₄×D₆₇	2
#11761615	2
D₅₃₆	2
#11761613	2
#11761611	2
SD₁₆×C₆₇	2
AGL(1,ℤ/8)	67