Subgroups — #35232564

Order 21152.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24902372	C₂	1	1	yes	maximal, characteristic
2	#24902362	C₂	1	2	yes	maximal, complemented
3	#24902359	C₂	1	2	yes	maximal, complemented
4	#24902356	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×Q₈×C₆₆₁	C₂	1	1	yes	maximal, characteristic, complemented
6	Dic₁₃₂₂	C₂²	1	1	yes	characteristic
7	D₂₆₄₄	C₂²	1	1	yes	characteristic
8	C₂²⋊D₆₆₁	—	2	1	no	—
9	C₄×D₆₆₁	—	2	1	no	—
10	#19126971	C₂²	1	2	yes	—
11	Q₈×C₆₆₁	—	2	1	no	—
12	Q₈×C₆₆₁	C₂²	1	2	yes	—
13	C₂×C₄×C₆₆₁	—	2	1	no	—
14	C₂×C₄×C₆₆₁	C₂²	1	1	yes	characteristic
15	C₂²×C₆₆₁	D₄	1	1	yes	characteristic
16	C₂×D₆₆₁	—	2	1	no	—
17	C₄×C₆₆₁	—	2	2	no	—
18	C₄×C₆₆₁	D₄	1	1	yes	characteristic
19	C₄×C₆₆₁	C₂³	1	1	yes	characteristic
20	Dic₆₆₁	—	2	1	no	—
21	C₂×C₆₆₁	—	2	1	no	—
22	C₂×C₆₆₁	C₂×D₄	1	1	yes	characteristic
23	D₆₆₁	—	4	1	no	—
24	C₆₆₁	#137	1	1	yes	characteristic, complemented, Sylow
25	#137	—	661	1	no	maximal, Sylow
26	#41	—	661	1	no	—
27	C₂×Q₈	D₆₆₁	1	1	yes	characteristic, complemented
28	Q₁₆	—	661	2	no	—
29	SD₁₆	—	661	2	no	—
30	M₁₆	—	661	1	no	—
31	Q₈	—	2	1	no	—
32	Q₈	—	661	1	no	—
33	Q₈	C₂×D₆₆₁	1	2	yes	—
34	D₄	—	1322	1	no	—
35	D₄	—	661	1	no	—
36	C₂×C₄	—	2	1	no	—
37	C₂×C₄	—	1322	1	no	—
38	C₂×C₄	C₂×D₆₆₁	1	1	yes	characteristic
39	C₈	—	661	2	no	—
40	C₂²	—	1322	1	no	—
41	C₂²	C₂²⋊D₆₆₁	1	1	yes	characteristic
42	C₄	—	2	2	no	—
43	C₄	—	1322	1	no	—
44	C₄	C₂²⋊D₆₆₁	1	1	yes	characteristic
45	C₄	C₂²×D₆₆₁	1	1	yes	characteristic
46	C₂	—	2	1	no	—
47	C₂	—	2644	1	no	—
48	C₂	C₂×C₂²⋊D₆₆₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×C₂²⋊D₆₆₁
commutator, cosocle	C₄×C₆₆₁	C₂³
Frattini	C₄	C₂²×D₆₆₁
Fitting	C₂×Q₈×C₆₆₁	C₂
socle	C₂×C₆₆₁	C₂×D₄
prime radical	#24902372	C₂
hypercenter	C₂×Q₈	D₆₆₁
hypocenter	C₆₆₁	#137
LCS	G ▷ C₄×C₆₆₁ ▷ C₂×C₆₆₁ ▷ C₆₆₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
2. #24902362	C₂	semidirect
3. #24902359	C₂	semidirect
4. #24902356	C₂	semidirect
5. C₂×Q₈×C₆₆₁	C₂	semidirect
24. C₆₆₁	#137	semidirect
27. C₂×Q₈	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
#24902362	2
#24902359	2
#24902356	2
C₂×Q₈×C₆₆₁	2
#137	661