Subgroups — #35232580

Order 21152.

146 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24902372	C₂	1	3	yes	maximal, complemented
2	#24902371	C₂	1	6	yes	maximal, complemented
3	C₂×Dic₁₃₂₂	C₂	1	3	yes	maximal, complemented
4	Q₈×D₆₆₁	C₂	1	2	yes	maximal, complemented
5	#41×C₆₆₁	C₂	1	1	yes	maximal, characteristic, complemented
6	Dic₁₃₂₂	C₂²	1	6	yes	—
7	Dic₁₃₂₂	C₂²	1	3	yes	—
8	D₂₆₄₄	C₂²	1	1	yes	characteristic
9	C₂²⋊D₆₆₁	C₂²	1	6	yes	—
10	C₂×Dic₆₆₁	C₂²	1	6	yes	—
11	C₄×D₆₆₁	C₂²	1	6	yes	—
12	Q₈×C₆₆₁	C₂²	1	1	yes	characteristic
13	D₄×C₆₆₁	C₂²	1	3	yes	—
14	C₂×C₄×C₆₆₁	C₂²	1	3	yes	—
15	C₂²×C₆₆₁	C₂³	1	3	yes	—
16	C₂×D₆₆₁	C₂³	1	2	yes	—
17	C₄×C₆₆₁	C₂³	1	1	yes	characteristic
18	C₄×C₆₆₁	C₂³	1	3	yes	—
19	Dic₆₆₁	C₂³	1	6	yes	—
20	C₂×C₆₆₁	—	2	3	no	—
21	C₂×C₆₆₁	C₂⁴	1	1	yes	characteristic
22	D₆₆₁	—	2	2	no	—
23	C₆₆₁	2₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
24	2₋¹⁺⁴	—	661	1	no	maximal, Sylow
25	#41	—	661	6	no	—
26	#41	—	661	3	no	—
27	#41	D₆₆₁	1	1	yes	characteristic, complemented
28	C₂×Q₈	—	661	2	no	—
29	C₂×Q₈	—	661	3	no	—
30	Q₈	—	661	6	no	—
31	Q₈	—	661	3	no	—
32	Q₈	C₂×D₆₆₁	1	1	yes	characteristic
33	D₄	—	661	6	no	—
34	D₄	—	661	1	no	—
35	D₄	C₂×D₆₆₁	1	3	yes	—
36	C₂×C₄	—	661	6	no	—
37	C₂×C₄	—	661	6	no	—
38	C₂×C₄	C₂×D₆₆₁	1	3	yes	—
39	C₂²	—	661	2	no	—
40	C₂²	C₂²×D₆₆₁	1	3	yes	—
41	C₄	—	661	6	no	—
42	C₄	C₂²×D₆₆₁	1	3	yes	—
43	C₄	C₂²×D₆₆₁	1	1	yes	characteristic
44	C₂	—	2	3	no	—
45	C₂	—	1322	2	no	—
46	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	2₋¹⁺⁴
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. #24902372	C₂	semidirect
2. #24902371	C₂	semidirect
3. C₂×Dic₁₃₂₂	C₂	semidirect
4. Q₈×D₆₆₁	C₂	semidirect
5. #41×C₆₆₁	C₂	semidirect
23. C₆₆₁	2₋¹⁺⁴	semidirect
27. #41	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×Dic₁₃₂₂	2
Q₈×D₆₆₁	2
#41×C₆₆₁	2
2₋¹⁺⁴	661