Subgroups — #35232578

Order 21152.

166 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	#24902371	C₂	1	4	yes	maximal, complemented
3	C₂×C₂²⋊D₆₆₁	C₂	1	4	yes	maximal, complemented
4	D₄×D₆₆₁	C₂	1	4	yes	maximal, complemented
5	C₂×D₄×C₆₆₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₆₆₁	C₂²	1	4	yes	complemented
7	Dic₁₃₂₂	C₂²	1	2	yes	complemented
8	D₂₆₄₄	C₂²	1	2	yes	complemented
9	C₂²⋊D₆₆₁	C₂²	1	8	yes	complemented
10	C₂²⋊D₆₆₁	C₂²	1	4	yes	complemented
11	C₂×Dic₆₆₁	C₂²	1	4	yes	complemented
12	C₄×D₆₆₁	C₂²	1	4	yes	complemented
13	C₂³×C₆₆₁	C₂²	1	2	yes	complemented
14	D₄×C₆₆₁	C₂²	1	4	yes	complemented
15	C₂×C₄×C₆₆₁	C₂²	1	1	yes	characteristic, complemented
16	C₂²×C₆₆₁	—	4	2	no	—
17	C₂²×C₆₆₁	C₂³	1	1	yes	characteristic
18	C₂²×C₆₆₁	C₂³	1	4	yes	—
19	C₂×D₆₆₁	—	4	4	no	—
20	C₂×D₆₆₁	C₂³	1	4	yes	—
21	C₄×C₆₆₁	C₂³	1	2	yes	—
22	Dic₆₆₁	C₂³	1	4	yes	—
23	C₂×C₆₆₁	—	2	4	no	—
24	C₂×C₆₆₁	—	2	1	no	—
25	C₂×C₆₆₁	C₂⁴	1	1	yes	characteristic
26	D₆₆₁	—	2	4	no	—
27	C₆₆₁	H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
28	H(5,𝔽₂)	—	661	1	no	maximal, Sylow
29	#41	—	661	4	no	—
30	#41	—	661	2	no	—
31	C₂×D₄	—	661	4	no	—
32	C₂×D₄	—	661	4	no	—
33	C₂×D₄	D₆₆₁	1	1	yes	characteristic, complemented
34	C₂³	—	661	4	no	—
35	C₂³	C₂×D₆₆₁	1	2	yes	complemented
36	Q₈	—	661	2	no	—
37	D₄	—	661	8	no	—
38	D₄	—	661	4	no	—
39	D₄	—	661	2	no	—
40	D₄	C₂×D₆₆₁	1	4	yes	complemented
41	C₂×C₄	—	661	4	no	—
42	C₂×C₄	—	661	4	no	—
43	C₂×C₄	C₂×D₆₆₁	1	1	yes	characteristic, complemented
44	C₂²	—	4	2	no	—
45	C₂²	—	2644	4	no	—
46	C₂²	—	661	4	no	—
47	C₂²	C₂²×D₆₆₁	1	4	yes	—
48	C₂²	C₂²×D₆₆₁	1	1	yes	characteristic
49	C₄	—	661	4	no	—
50	C₄	C₂²×D₆₆₁	1	2	yes	—
51	C₂	—	2	1	no	—
52	C₂	—	2	4	no	—
53	C₂	—	1322	4	no	—
54	C₂	C₂³×D₆₆₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×D₆₆₁
commutator, socle, cosocle	C₂×C₆₆₁	C₂⁴
Fitting	C₂×D₄×C₆₆₁	C₂
hypercenter	C₂×D₄	D₆₆₁
hypocenter	C₆₆₁	H(5,𝔽₂)
LCS	G ▷ C₂×C₆₆₁ ▷ C₆₆₁
UCS	C₁ ◁ C₂ ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
1. #24902372	C₂	semidirect
2. #24902371	C₂	semidirect
3. C₂×C₂²⋊D₆₆₁	C₂	semidirect
4. D₄×D₆₆₁	C₂	semidirect
5. C₂×D₄×C₆₆₁	C₂	semidirect
6. C₂²×D₆₆₁	C₂²	semidirect
7. Dic₁₃₂₂	C₂²	semidirect
8. D₂₆₄₄	C₂²	semidirect
9. C₂²⋊D₆₆₁	C₂²	semidirect
10. C₂²⋊D₆₆₁	C₂²	semidirect
11. C₂×Dic₆₆₁	C₂²	semidirect
12. C₄×D₆₆₁	C₂²	semidirect
13. C₂³×C₆₆₁	C₂²	semidirect
14. D₄×C₆₆₁	C₂²	semidirect
15. C₂×C₄×C₆₆₁	C₂²	semidirect
27. C₆₆₁	H(5,𝔽₂)	semidirect
33. C₂×D₄	D₆₆₁	semidirect
35. C₂³	C₂×D₆₆₁	semidirect
40. D₄	C₂×D₆₆₁	semidirect
43. C₂×C₄	C₂×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
#24902371	2
C₂×C₂²⋊D₆₆₁	2
D₄×D₆₆₁	2
C₂×D₄×C₆₆₁	2
H(5,𝔽₂)	661