Subgroups — #35232555

Order 21152.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24902372	C₂	1	2	yes	maximal, complemented
2	Dic₂₆₄₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#24902361	C₂	1	2	yes	maximal, complemented
4	D₅₂₈₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₈×C₆₆₁	C₂	1	1	yes	maximal, characteristic, complemented
6	Dic₁₃₂₂	C₂²	1	2	yes	—
7	D₂₆₄₄	C₂²	1	2	yes	—
8	C₂²⋊D₆₆₁	—	2	2	no	—
9	C₄×D₆₆₁	—	2	2	no	—
10	C₂×C₄×C₆₆₁	C₂²	1	1	yes	characteristic
11	C₈×C₆₆₁	C₂²	1	1	yes	characteristic
12	C₈×C₆₆₁	C₂²	1	1	yes	characteristic
13	C₂²×C₆₆₁	D₄	1	1	yes	characteristic
14	C₂×D₆₆₁	—	2	2	no	—
15	C₄×C₆₆₁	D₄	1	1	yes	characteristic
16	C₄×C₆₆₁	C₂³	1	1	yes	characteristic
17	Dic₆₆₁	—	2	2	no	—
18	C₂×C₆₆₁	—	2	1	no	—
19	C₂×C₆₆₁	C₂×D₄	1	1	yes	characteristic
20	D₆₆₁	—	4	2	no	—
21	C₆₆₁	#135	1	1	yes	characteristic, complemented, Sylow
22	#135	—	661	1	no	maximal, Sylow
23	#41	—	661	2	no	—
24	Q₁₆	—	661	1	no	—
25	SD₁₆	—	661	2	no	—
26	D₈	—	661	1	no	—
27	C₂×C₈	D₆₆₁	1	1	yes	characteristic, complemented
28	Q₈	—	661	2	no	—
29	D₄	—	1322	2	no	—
30	D₄	—	661	2	no	—
31	C₂×C₄	—	1322	2	no	—
32	C₂×C₄	C₂×D₆₆₁	1	1	yes	characteristic
33	C₈	C₂×D₆₆₁	1	1	yes	characteristic
34	C₈	C₂×D₆₆₁	1	1	yes	characteristic
35	C₂²	—	1322	2	no	—
36	C₂²	D₂₆₄₄	1	1	yes	characteristic
37	C₄	—	1322	2	no	—
38	C₄	D₂₆₄₄	1	1	yes	characteristic
39	C₄	C₂²×D₆₆₁	1	1	yes	characteristic
40	C₂	—	2	1	no	—
41	C₂	—	2644	2	no	—
42	C₂	C₂×D₂₆₄₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₂₆₄₄
commutator, cosocle	C₄×C₆₆₁	C₂³
Frattini	C₄	C₂²×D₆₆₁
Fitting	C₂×C₈×C₆₆₁	C₂
socle	C₂×C₆₆₁	C₂×D₄
hypercenter	C₂×C₈	D₆₆₁
hypocenter	C₆₆₁	#135
LCS	G ▷ C₄×C₆₆₁ ▷ C₂×C₆₆₁ ▷ C₆₆₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
1. #24902372	C₂	semidirect
2. Dic₂₆₄₄	C₂	semidirect
3. #24902361	C₂	semidirect
4. D₅₂₈₈	C₂	semidirect
5. C₂×C₈×C₆₆₁	C₂	semidirect
21. C₆₆₁	#135	semidirect
27. C₂×C₈	D₆₆₁	semidirect

Maximal subgroups

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

subgroup	index
#24902372	2
Dic₂₆₄₄	2
#24902361	2
D₅₂₈₈	2
C₂×C₈×C₆₆₁	2
#135	661