Subgroups — #35232541

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
2	#24902355	C₂	1	2	yes	maximal, complemented
3	C₂×#19126971	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×D₆₆₁	C₂	1	2	yes	maximal, complemented
5	M₁₆×C₆₆₁	C₂	1	1	yes	maximal, characteristic, complemented
6	Dic₁₃₂₂	C₄	1	1	yes	characteristic
7	D₂₆₄₄	C₄	1	1	yes	characteristic
8	C₂²⋊D₆₆₁	C₄	1	2	yes	—
9	C₄×D₆₆₁	C₂²	1	2	yes	—
10	#19126971	C₂²	1	1	yes	characteristic
11	#19126971	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₆₆₁	C₂²	1	1	yes	characteristic
13	C₈×C₆₆₁	C₂²	1	2	yes	—
14	C₂²×C₆₆₁	C₂×C₄	1	1	yes	characteristic
15	C₂×D₆₆₁	C₂×C₄	1	2	yes	—
16	C₄×C₆₆₁	C₂×C₄	1	1	yes	characteristic
17	C₄×C₆₆₁	C₂³	1	1	yes	characteristic
18	Dic₆₆₁	C₂×C₄	1	2	yes	—
19	C₂×C₆₆₁	—	2	1	no	—
20	C₂×C₆₆₁	C₂²×C₄	1	1	yes	characteristic
21	D₆₆₁	—	2	2	no	—
22	C₆₆₁	#131	1	1	yes	characteristic, complemented, Sylow
23	#131	—	661	1	no	maximal, Sylow
24	#41	—	661	1	no	—
25	M₁₆	—	661	2	no	—
26	M₁₆	D₆₆₁	1	1	yes	characteristic, complemented
27	C₂×C₈	—	661	1	no	—
28	C₂×C₈	—	661	2	no	—
29	Q₈	—	661	1	no	—
30	D₄	—	661	2	no	—
31	D₄	—	661	1	no	—
32	C₂×C₄	—	661	2	no	—
33	C₂×C₄	C₂×D₆₆₁	1	1	yes	characteristic
34	C₈	—	661	1	no	—
35	C₈	—	661	1	no	—
36	C₈	C₂×D₆₆₁	1	2	yes	—
37	C₂²	—	661	2	no	—
38	C₂²	C₄×D₆₆₁	1	1	yes	characteristic
39	C₄	—	661	2	no	—
40	C₄	C₄×D₆₆₁	1	1	yes	characteristic
41	C₄	C₂²×D₆₆₁	1	1	yes	characteristic
42	C₂	—	2	1	no	—
43	C₂	—	1322	2	no	—
44	C₂	C₂×C₄×D₆₆₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂²×D₆₆₁
commutator, socle	C₂×C₆₆₁	C₂²×C₄
Fitting	M₁₆×C₆₆₁	C₂
prime radical	#24902372	C₂
cosocle	C₄×C₆₆₁	C₂³
hypercenter	M₁₆	D₆₆₁
hypocenter	C₆₆₁	#131
LCS	G ▷ C₂×C₆₆₁ ▷ C₆₆₁
UCS	C₁ ◁ C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
2. #24902355	C₂	semidirect
3. C₂×#19126971	C₂	semidirect
4. C₈×D₆₆₁	C₂	semidirect
5. M₁₆×C₆₆₁	C₂	semidirect
22. C₆₆₁	#131	semidirect
26. M₁₆	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
#24902355	2
C₂×#19126971	2
C₈×D₆₆₁	2
M₁₆×C₆₆₁	2
#131	661