Subgroups — #35232595

Order 21152.

50 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₆₆₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄⋊#14581558	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×D₆₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#14581558	—	2	1	no	—
5	#19126978	—	2	1	no	—
6	C₂²×D₆₆₁	—	2	1	no	—
7	D₂₆₄₄	C₄	1	1	yes	characteristic, complemented
8	C₂²⋊D₆₆₁	—	2	1	no	—
9	C₄×D₆₆₁	C₂²	1	1	yes	characteristic
10	D₄×C₆₆₁	C₄	1	1	yes	characteristic, complemented
11	C₂²×C₆₆₁	—	2	1	no	—
12	C₂×D₆₆₁	—	2	1	no	—
13	C₂×D₆₆₁	—	4	1	no	—
14	C₂×D₆₆₁	—	4	1	no	—
15	C₂×D₆₆₁	D₄	1	1	yes	characteristic
16	#14581558	—	4	1	no	—
17	C₄×C₆₆₁	C₂×C₄	1	1	yes	characteristic
18	Dic₆₆₁	D₄	1	1	yes	characteristic
19	C₂×C₆₆₁	—	4	1	no	—
20	C₂×C₆₆₁	C₂²⋊C₄	1	1	yes	characteristic
21	D₆₆₁	—	4	1	no	—
22	D₆₆₁	D₈	1	1	yes	characteristic
23	D₆₆₁	SD₁₆	1	1	yes	characteristic
24	C₆₆₁	#102	1	1	yes	characteristic, complemented, Sylow
25	#102	—	661	1	no	maximal, Sylow
26	C₂×D₄	—	661	1	no	—
27	C₂×C₈	—	661	1	no	—
28	C₄⋊C₄	—	661	1	no	—
29	C₂³	—	1322	1	no	—
30	D₄	—	1322	1	no	—
31	D₄	—	661	1	no	—
32	D₄	#14581558	1	1	yes	characteristic, complemented
33	C₂×C₄	—	1322	1	no	—
34	C₂×C₄	—	661	1	no	—
35	C₈	—	1322	1	no	—
36	C₂²	—	2	1	no	—
37	C₂²	—	1322	1	no	—
38	C₂²	—	2644	1	no	—
39	C₂²	—	2644	1	no	—
40	C₂²	—	661	1	no	—
41	C₄	—	2644	1	no	—
42	C₄	—	661	1	no	—
43	C₄	C₂×#14581558	1	1	yes	characteristic
44	C₂	—	4	1	no	—
45	C₂	—	2644	1	no	—
46	C₂	—	661	1	no	—
47	C₂	—	661	1	no	—
48	C₂	C₂²⋊#14581558	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²⋊#14581558
commutator	C₄×C₆₆₁	C₂×C₄
Frattini	C₄	C₂×#14581558
Fitting	D₄×C₆₆₁	C₄
socle	C₂×C₆₆₁	C₂²⋊C₄
prime radical	D₄×D₆₆₁	C₂
cosocle	C₄×D₆₆₁	C₂²
hypercenter	D₄	#14581558
hypocenter	C₆₆₁	#102
LCS	G ▷ C₄×C₆₆₁ ▷ C₂×C₆₆₁ ▷ C₆₆₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₄

Complements

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

N	H ≅ G/N	type
1. D₆₆₁⋊C₈	C₂	semidirect
2. C₄⋊#14581558	C₂	semidirect
7. D₂₆₄₄	C₄	semidirect
10. D₄×C₆₆₁	C₄	semidirect
24. C₆₆₁	#102	semidirect
32. D₄	#14581558	semidirect

Maximal subgroups

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

subgroup	index
D₆₆₁⋊C₈	2
C₄⋊#14581558	2
D₄×D₆₆₁	2
#102	661