Subgroups — #14524064

Order 2144.

62 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	#11761627	C₂	1	1	yes	maximal, characteristic, complemented
3	#11761618	C₂	1	1	yes	maximal, characteristic, complemented
4	#11761617	C₂	1	1	yes	maximal, characteristic, complemented
5	#11761613	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×D₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	SD₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	Q₈×C₆₇	C₂²	1	1	yes	characteristic
9	D₄×C₆₇	C₂²	1	1	yes	characteristic
10	C₂²⋊D₆₇	—	2	1	no	—
11	C₂×Dic₆₇	—	2	1	no	—
12	D₂₆₈	—	2	1	no	—
13	D₂₆₈	C₂²	1	1	yes	characteristic
14	C₄×D₆₇	—	2	1	no	—
15	C₄×D₆₇	C₂²	1	1	yes	characteristic
16	Dic₁₃₄	C₂²	1	1	yes	characteristic
17	C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₆₇⋊C₈	C₂²	1	1	yes	characteristic
19	C₂²×C₆₇	—	2	1	no	—
20	C₂×D₆₇	—	2	1	no	—
21	C₂×D₆₇	D₄	1	1	yes	characteristic
22	C₄×C₆₇	—	2	1	no	—
23	C₄×C₆₇	C₂³	1	1	yes	characteristic
24	Dic₆₇	—	2	1	no	—
25	Dic₆₇	D₄	1	1	yes	characteristic
26	C₂×C₆₇	—	4	1	no	—
27	C₂×C₆₇	C₂×D₄	1	1	yes	characteristic
28	D₆₇	—	2	1	no	—
29	D₆₇	—	4	1	no	—
30	C₆₇	#135	1	1	yes	characteristic, complemented, Sylow
31	#135	—	67	1	no	maximal, Sylow
32	#41	—	67	1	no	—
33	#41	—	67	1	no	—
34	Q₁₆	—	67	1	no	—
35	SD₁₆	—	67	1	no	—
36	SD₁₆	D₆₇	1	1	yes	characteristic, complemented
37	D₈	—	67	1	no	—
38	C₂×C₈	—	67	1	no	—
39	Q₈	—	67	1	no	—
40	Q₈	C₂×D₆₇	1	1	yes	characteristic
41	D₄	—	67	1	no	—
42	D₄	—	134	1	no	—
43	D₄	—	134	1	no	—
44	D₄	C₂×D₆₇	1	1	yes	characteristic
45	C₂×C₄	—	67	1	no	—
46	C₂×C₄	—	134	1	no	—
47	C₂×C₄	—	134	1	no	—
48	C₈	—	67	1	no	—
49	C₈	C₂×D₆₇	1	1	yes	characteristic
50	C₂²	—	2	1	no	—
51	C₂²	—	67	1	no	—
52	C₂²	—	134	1	no	—
53	C₄	—	2	1	no	—
54	C₄	—	67	1	no	—
55	C₄	—	134	1	no	—
56	C₄	C₂²×D₆₇	1	1	yes	characteristic
57	C₂	—	4	1	no	—
58	C₂	—	268	1	no	—
59	C₂	—	134	1	no	—
60	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	#135
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. #11761627	C₂	semidirect
3. #11761618	C₂	semidirect
4. #11761617	C₂	semidirect
5. #11761613	C₂	semidirect
6. C₈×D₆₇	C₂	semidirect
7. SD₁₆×C₆₇	C₂	semidirect
30. C₆₇	#135	semidirect
36. SD₁₆	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
#11761627	2
#11761618	2
#11761617	2
#11761613	2
C₈×D₆₇	2
SD₁₆×C₆₇	2
#135	67