Subgroups — #35232553

Order 21152.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24902372	C₂	1	1	yes	maximal, characteristic, complemented
2	#24902362	C₂	1	1	yes	maximal, characteristic, complemented
3	#24902360	C₂	1	1	yes	maximal, characteristic, complemented
4	#24902359	C₂	1	1	yes	maximal, characteristic, complemented
5	#24902357	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#19126971	C₂	1	1	yes	maximal, characteristic, complemented
7	#41×C₆₆₁	C₂	1	1	yes	maximal, characteristic, complemented
8	Dic₁₃₂₂	C₂²	1	1	yes	characteristic
9	D₂₆₄₄	C₂²	1	1	yes	characteristic
10	C₂²⋊D₆₆₁	—	2	1	no	—
11	C₄×D₆₆₁	—	2	1	no	—
12	#19126971	C₂²	1	1	yes	characteristic
13	#19126971	C₂²	1	1	yes	characteristic
14	Q₈×C₆₆₁	C₂²	1	1	yes	characteristic
15	D₄×C₆₆₁	—	2	1	no	—
16	D₄×C₆₆₁	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₆₆₁	—	2	1	no	—
18	C₂×C₄×C₆₆₁	C₂²	1	1	yes	characteristic
19	C₂²×C₆₆₁	—	2	1	no	—
20	C₂²×C₆₆₁	D₄	1	1	yes	characteristic
21	C₂×D₆₆₁	—	2	1	no	—
22	C₄×C₆₆₁	—	2	1	no	—
23	C₄×C₆₆₁	D₄	1	1	yes	characteristic
24	C₄×C₆₆₁	C₂³	1	1	yes	characteristic
25	Dic₆₆₁	—	2	1	no	—
26	C₂×C₆₆₁	—	2	1	no	—
27	C₂×C₆₆₁	—	4	1	no	—
28	C₂×C₆₆₁	C₂×D₄	1	1	yes	characteristic
29	D₆₆₁	—	4	1	no	—
30	C₆₆₁	#135	1	1	yes	characteristic, complemented, Sylow
31	#135	—	661	1	no	maximal, Sylow
32	#41	—	661	1	no	—
33	#41	D₆₆₁	1	1	yes	characteristic, complemented
34	Q₁₆	—	661	1	no	—
35	SD₁₆	—	661	1	no	—
36	SD₁₆	—	661	1	no	—
37	D₈	—	661	1	no	—
38	C₂×C₈	—	661	1	no	—
39	Q₈	—	661	1	no	—
40	Q₈	C₂×D₆₆₁	1	1	yes	characteristic
41	D₄	—	2	1	no	—
42	D₄	—	1322	1	no	—
43	D₄	—	661	1	no	—
44	D₄	C₂×D₆₆₁	1	1	yes	characteristic
45	C₂×C₄	—	2	1	no	—
46	C₂×C₄	—	1322	1	no	—
47	C₂×C₄	C₂×D₆₆₁	1	1	yes	characteristic
48	C₈	—	661	1	no	—
49	C₈	—	661	1	no	—
50	C₂²	—	2	1	no	—
51	C₂²	—	1322	1	no	—
52	C₂²	C₂²⋊D₆₆₁	1	1	yes	characteristic
53	C₄	—	2	1	no	—
54	C₄	—	1322	1	no	—
55	C₄	C₂²⋊D₆₆₁	1	1	yes	characteristic
56	C₄	C₂²×D₆₆₁	1	1	yes	characteristic
57	C₂	—	2	1	no	—
58	C₂	—	4	1	no	—
59	C₂	—	2644	1	no	—
60	C₂	C₂×C₂²⋊D₆₆₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²⋊D₆₆₁
commutator, cosocle	C₄×C₆₆₁	C₂³
Frattini	C₄	C₂²×D₆₆₁
Fitting	#41×C₆₆₁	C₂
socle	C₂×C₆₆₁	C₂×D₄
hypercenter	#41	D₆₆₁
hypocenter	C₆₆₁	#135
LCS	G ▷ C₄×C₆₆₁ ▷ C₂×C₆₆₁ ▷ C₆₆₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #41

Complements

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

N	H ≅ G/N	type
1. #24902372	C₂	semidirect
2. #24902362	C₂	semidirect
3. #24902360	C₂	semidirect
4. #24902359	C₂	semidirect
5. #24902357	C₂	semidirect
6. C₂×#19126971	C₂	semidirect
7. #41×C₆₆₁	C₂	semidirect
30. C₆₆₁	#135	semidirect
33. #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
#24902362	2
#24902360	2
#24902359	2
#24902357	2
C₂×#19126971	2
#41×C₆₆₁	2
#135	661