Subgroups — #35232557

Order 21152.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24902371	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×D₆₆₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#24902361	C₂	1	1	yes	maximal, characteristic, complemented
4	#24902360	C₂	1	1	yes	maximal, characteristic, complemented
5	#24902357	C₂	1	1	yes	maximal, characteristic, complemented
6	#24902355	C₂	1	1	yes	maximal, characteristic, complemented
7	D₈×C₆₆₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₆₆₁	—	2	1	no	—
9	Dic₁₃₂₂	C₂²	1	1	yes	characteristic, complemented
10	D₂₆₄₄	C₂²	1	1	yes	characteristic
11	C₂²⋊D₆₆₁	—	2	1	no	—
12	C₂²⋊D₆₆₁	—	2	1	no	—
13	C₂×Dic₆₆₁	—	2	1	no	—
14	C₄×D₆₆₁	C₂²	1	1	yes	characteristic
15	#19126971	C₂²	1	1	yes	characteristic, complemented
16	D₄×C₆₆₁	C₂²	1	1	yes	characteristic
17	D₄×C₆₆₁	C₂²	1	1	yes	characteristic, complemented
18	C₈×C₆₆₁	C₂²	1	1	yes	characteristic, complemented
19	C₂²×C₆₆₁	—	2	1	no	—
20	C₂²×C₆₆₁	—	2	1	no	—
21	C₂×D₆₆₁	—	2	1	no	—
22	C₂×D₆₆₁	—	4	2	no	—
23	C₂×D₆₆₁	D₄	1	1	yes	characteristic
24	C₄×C₆₆₁	C₂³	1	1	yes	characteristic
25	Dic₆₆₁	—	2	1	no	—
26	Dic₆₆₁	D₄	1	1	yes	characteristic
27	C₂×C₆₆₁	—	4	1	no	—
28	C₂×C₆₆₁	—	4	1	no	—
29	C₂×C₆₆₁	C₂×D₄	1	1	yes	characteristic
30	D₆₆₁	—	2	1	no	—
31	D₆₆₁	—	4	1	no	—
32	C₆₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
33	AGL(1,ℤ/8)	—	661	1	no	maximal, Sylow
34	#41	—	661	1	no	—
35	C₂×D₄	—	661	1	no	—
36	SD₁₆	—	661	1	no	—
37	SD₁₆	—	661	1	no	—
38	D₈	—	661	1	no	—
39	D₈	D₆₆₁	1	1	yes	characteristic, complemented
40	M₁₆	—	661	1	no	—
41	C₂³	—	1322	1	no	—
42	Q₈	—	661	1	no	—
43	D₄	—	1322	1	no	—
44	D₄	—	1322	1	no	—
45	D₄	—	661	1	no	—
46	D₄	C₂×D₆₆₁	1	1	yes	characteristic
47	D₄	C₂×D₆₆₁	1	1	yes	characteristic, complemented
48	C₂×C₄	—	1322	1	no	—
49	C₂×C₄	—	661	1	no	—
50	C₈	—	661	1	no	—
51	C₈	C₂×D₆₆₁	1	1	yes	characteristic, complemented
52	C₂²	—	2	1	no	—
53	C₂²	—	2	1	no	—
54	C₂²	—	1322	1	no	—
55	C₂²	—	2644	2	no	—
56	C₂²	—	661	1	no	—
57	C₄	—	1322	1	no	—
58	C₄	—	661	1	no	—
59	C₄	C₂²×D₆₆₁	1	1	yes	characteristic
60	C₂	—	4	1	no	—
61	C₂	—	4	1	no	—
62	C₂	—	1322	1	no	—
63	C₂	—	2644	1	no	—
64	C₂	D₄×D₆₆₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₆₆₁
commutator, cosocle	C₄×C₆₆₁	C₂³
Frattini	C₄	C₂²×D₆₆₁
Fitting	D₈×C₆₆₁	C₂
socle	C₂×C₆₆₁	C₂×D₄
hypercenter	D₈	D₆₆₁
hypocenter	C₆₆₁	AGL(1,ℤ/8)
LCS	G ▷ C₄×C₆₆₁ ▷ C₂×C₆₆₁ ▷ C₆₆₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₈

Complements

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

N	H ≅ G/N	type
1. #24902371	C₂	semidirect
2. D₄×D₆₆₁	C₂	semidirect
3. #24902361	C₂	semidirect
4. #24902360	C₂	semidirect
5. #24902357	C₂	semidirect
6. #24902355	C₂	semidirect
7. D₈×C₆₆₁	C₂	semidirect
9. Dic₁₃₂₂	C₂²	semidirect
15. #19126971	C₂²	semidirect
17. D₄×C₆₆₁	C₂²	semidirect
18. C₈×C₆₆₁	C₂²	semidirect
32. C₆₆₁	AGL(1,ℤ/8)	semidirect
39. D₈	D₆₆₁	semidirect
47. D₄	C₂×D₆₆₁	semidirect
51. C₈	C₂×D₆₆₁	semidirect

Maximal subgroups

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

subgroup	index
#24902371	2
D₄×D₆₆₁	2
#24902361	2
#24902360	2
#24902357	2
#24902355	2
D₈×C₆₆₁	2
AGL(1,ℤ/8)	661