Subgroups — #37551570

Order 23264.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26042740	C₂	1	2	yes	maximal, complemented
2	Dic₂₉₀₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#26042729	C₂	1	2	yes	maximal, complemented
4	D₅₈₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₈×C₇₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	Dic₁₄₅₄	C₂²	1	2	yes	—
7	D₂₉₀₈	C₂²	1	2	yes	—
8	C₂²⋊D₇₂₇	—	2	2	no	—
9	C₄×D₇₂₇	—	2	2	no	—
10	C₂×C₄×C₇₂₇	C₂²	1	1	yes	characteristic
11	C₈×C₇₂₇	C₂²	1	1	yes	characteristic
12	C₈×C₇₂₇	C₂²	1	1	yes	characteristic
13	C₂²×C₇₂₇	D₄	1	1	yes	characteristic
14	C₂×D₇₂₇	—	2	2	no	—
15	C₄×C₇₂₇	D₄	1	1	yes	characteristic
16	C₄×C₇₂₇	C₂³	1	1	yes	characteristic
17	Dic₇₂₇	—	2	2	no	—
18	C₂×C₇₂₇	—	2	1	no	—
19	C₂×C₇₂₇	C₂×D₄	1	1	yes	characteristic
20	D₇₂₇	—	4	2	no	—
21	C₇₂₇	#135	1	1	yes	characteristic, complemented, Sylow
22	#135	—	727	1	no	maximal, Sylow
23	#41	—	727	2	no	—
24	Q₁₆	—	727	1	no	—
25	SD₁₆	—	727	2	no	—
26	D₈	—	727	1	no	—
27	C₂×C₈	D₇₂₇	1	1	yes	characteristic, complemented
28	Q₈	—	727	2	no	—
29	D₄	—	1454	2	no	—
30	D₄	—	727	2	no	—
31	C₂×C₄	—	1454	2	no	—
32	C₂×C₄	C₂×D₇₂₇	1	1	yes	characteristic
33	C₈	C₂×D₇₂₇	1	1	yes	characteristic
34	C₈	C₂×D₇₂₇	1	1	yes	characteristic
35	C₂²	—	1454	2	no	—
36	C₂²	D₂₉₀₈	1	1	yes	characteristic
37	C₄	—	1454	2	no	—
38	C₄	D₂₉₀₈	1	1	yes	characteristic
39	C₄	C₂²×D₇₂₇	1	1	yes	characteristic
40	C₂	—	2	1	no	—
41	C₂	—	2908	2	no	—
42	C₂	C₂×D₂₉₀₈	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₂₉₀₈
commutator, cosocle	C₄×C₇₂₇	C₂³
Frattini	C₄	C₂²×D₇₂₇
Fitting	C₂×C₈×C₇₂₇	C₂
socle	C₂×C₇₂₇	C₂×D₄
hypercenter	C₂×C₈	D₇₂₇
hypocenter	C₇₂₇	#135
LCS	G ▷ C₄×C₇₂₇ ▷ C₂×C₇₂₇ ▷ C₇₂₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
1. #26042740	C₂	semidirect
2. Dic₂₉₀₈	C₂	semidirect
3. #26042729	C₂	semidirect
4. D₅₈₁₆	C₂	semidirect
5. C₂×C₈×C₇₂₇	C₂	semidirect
21. C₇₂₇	#135	semidirect
27. C₂×C₈	D₇₂₇	semidirect

Maximal subgroups

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

subgroup	index
#26042740	2
Dic₂₉₀₈	2
#26042729	2
D₅₈₁₆	2
C₂×C₈×C₇₂₇	2
#135	727