Subgroups — #35232579

Order 21152.

166 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24902373	C₂	1	2	yes	maximal, complemented
2	#24902372	C₂	1	3	yes	maximal, complemented
3	C₂×D₂₆₄₄	C₂	1	3	yes	maximal, complemented
4	D₄×D₆₆₁	C₂	1	6	yes	maximal, complemented
5	#41×C₆₆₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₆₆₁	C₂²	1	6	yes	complemented
7	Dic₁₃₂₂	C₂²	1	1	yes	characteristic, complemented
8	D₂₆₄₄	C₂²	1	6	yes	complemented
9	D₂₆₄₄	C₂²	1	3	yes	complemented
10	C₂²⋊D₆₆₁	C₂²	1	6	yes	complemented
11	C₄×D₆₆₁	C₂²	1	6	yes	complemented
12	Q₈×C₆₆₁	C₂²	1	1	yes	characteristic, complemented
13	D₄×C₆₆₁	C₂²	1	3	yes	complemented
14	C₂×C₄×C₆₆₁	C₂²	1	3	yes	complemented
15	C₂²×C₆₆₁	C₂³	1	3	yes	—
16	C₂×D₆₆₁	—	4	6	no	—
17	C₂×D₆₆₁	C₂³	1	6	yes	—
18	C₄×C₆₆₁	C₂³	1	1	yes	characteristic
19	C₄×C₆₆₁	C₂³	1	3	yes	—
20	Dic₆₆₁	C₂³	1	2	yes	—
21	C₂×C₆₆₁	—	2	3	no	—
22	C₂×C₆₆₁	C₂⁴	1	1	yes	characteristic
23	D₆₆₁	—	2	6	no	—
24	C₆₆₁	H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
25	H(5,𝔽₂)	—	661	1	no	maximal, Sylow
26	#41	—	661	2	no	—
27	#41	—	661	3	no	—
28	#41	D₆₆₁	1	1	yes	characteristic, complemented
29	C₂×D₄	—	661	6	no	—
30	C₂×D₄	—	661	3	no	—
31	C₂³	—	661	6	no	—
32	Q₈	—	661	1	no	—
33	Q₈	C₂×D₆₆₁	1	1	yes	characteristic, complemented
34	D₄	—	661	6	no	—
35	D₄	—	661	6	no	—
36	D₄	—	661	3	no	—
37	D₄	C₂×D₆₆₁	1	3	yes	complemented
38	C₂×C₄	—	661	6	no	—
39	C₂×C₄	C₂×D₆₆₁	1	3	yes	complemented
40	C₂²	—	2644	6	no	—
41	C₂²	—	661	6	no	—
42	C₂²	C₂²×D₆₆₁	1	3	yes	—
43	C₄	—	661	2	no	—
44	C₄	C₂²×D₆₆₁	1	3	yes	—
45	C₄	C₂²×D₆₆₁	1	1	yes	characteristic
46	C₂	—	2	3	no	—
47	C₂	—	1322	6	no	—
48	C₂	C₂³×D₆₆₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×D₆₆₁
commutator, socle, cosocle	C₂×C₆₆₁	C₂⁴
Fitting	#41×C₆₆₁	C₂
hypercenter	#41	D₆₆₁
hypocenter	C₆₆₁	H(5,𝔽₂)
LCS	G ▷ C₂×C₆₆₁ ▷ C₆₆₁
UCS	C₁ ◁ C₂ ◁ #41

Complements

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

N	H ≅ G/N	type
1. #24902373	C₂	semidirect
2. #24902372	C₂	semidirect
3. C₂×D₂₆₄₄	C₂	semidirect
4. D₄×D₆₆₁	C₂	semidirect
5. #41×C₆₆₁	C₂	semidirect
6. C₂²×D₆₆₁	C₂²	semidirect
7. Dic₁₃₂₂	C₂²	semidirect
8. D₂₆₄₄	C₂²	semidirect
9. D₂₆₄₄	C₂²	semidirect
10. C₂²⋊D₆₆₁	C₂²	semidirect
11. C₄×D₆₆₁	C₂²	semidirect
12. Q₈×C₆₆₁	C₂²	semidirect
13. D₄×C₆₆₁	C₂²	semidirect
14. C₂×C₄×C₆₆₁	C₂²	semidirect
24. C₆₆₁	H(5,𝔽₂)	semidirect
28. #41	D₆₆₁	semidirect
33. Q₈	C₂×D₆₆₁	semidirect
37. D₄	C₂×D₆₆₁	semidirect
39. 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
#24902373	2
#24902372	2
C₂×D₂₆₄₄	2
D₄×D₆₆₁	2
#41×C₆₆₁	2
H(5,𝔽₂)	661