Subgroups — #35232562

Order 21152.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24902371	C₂	1	1	yes	maximal, characteristic
2	Q₈×D₆₆₁	C₂	1	1	yes	maximal, characteristic, complemented
3	Dic₂₆₄₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#24902362	C₂	1	1	yes	maximal, characteristic, complemented
5	#24902360	C₂	1	1	yes	maximal, characteristic, complemented
6	#24902355	C₂	1	1	yes	maximal, characteristic, complemented
7	SD₁₆×C₆₆₁	C₂	1	1	yes	maximal, characteristic, complemented
8	Dic₁₃₂₂	—	2	1	no	—
9	Dic₁₃₂₂	C₂²	1	1	yes	characteristic
10	Dic₁₃₂₂	C₂²	1	1	yes	characteristic
11	C₂²⋊D₆₆₁	—	2	1	no	—
12	C₂×Dic₆₆₁	—	2	1	no	—
13	C₄×D₆₆₁	—	2	1	no	—
14	C₄×D₆₆₁	C₂²	1	1	yes	characteristic
15	#19126971	C₂²	1	1	yes	characteristic
16	Q₈×C₆₆₁	C₂²	1	1	yes	characteristic
17	D₄×C₆₆₁	C₂²	1	1	yes	characteristic
18	C₈×C₆₆₁	C₂²	1	1	yes	characteristic
19	C₂²×C₆₆₁	—	2	1	no	—
20	C₂×D₆₆₁	D₄	1	1	yes	characteristic
21	C₄×C₆₆₁	—	2	1	no	—
22	C₄×C₆₆₁	C₂³	1	1	yes	characteristic
23	Dic₆₆₁	—	2	1	no	—
24	Dic₆₆₁	—	2	1	no	—
25	Dic₆₆₁	D₄	1	1	yes	characteristic
26	C₂×C₆₆₁	—	4	1	no	—
27	C₂×C₆₆₁	C₂×D₄	1	1	yes	characteristic
28	D₆₆₁	—	2	1	no	—
29	C₆₆₁	#137	1	1	yes	characteristic, complemented, Sylow
30	#137	—	661	1	no	maximal, Sylow
31	#41	—	661	1	no	—
32	C₂×Q₈	—	661	1	no	—
33	Q₁₆	—	661	1	no	—
34	Q₁₆	—	661	1	no	—
35	SD₁₆	—	661	1	no	—
36	SD₁₆	D₆₆₁	1	1	yes	characteristic, complemented
37	M₁₆	—	661	1	no	—
38	Q₈	—	1322	1	no	—
39	Q₈	—	661	1	no	—
40	Q₈	—	661	1	no	—
41	Q₈	C₂×D₆₆₁	1	1	yes	characteristic
42	D₄	—	1322	1	no	—
43	D₄	C₂×D₆₆₁	1	1	yes	characteristic
44	C₂×C₄	—	1322	1	no	—
45	C₂×C₄	—	1322	1	no	—
46	C₂×C₄	—	661	1	no	—
47	C₈	—	661	1	no	—
48	C₈	C₂×D₆₆₁	1	1	yes	characteristic
49	C₂²	—	2	1	no	—
50	C₂²	—	661	1	no	—
51	C₄	—	2	1	no	—
52	C₄	—	1322	1	no	—
53	C₄	—	1322	1	no	—
54	C₄	—	661	1	no	—
55	C₄	C₂²×D₆₆₁	1	1	yes	characteristic
56	C₂	—	4	1	no	—
57	C₂	—	1322	1	no	—
58	C₂	D₄×D₆₆₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₆₆₁
commutator, cosocle	C₄×C₆₆₁	C₂³
Frattini	C₄	C₂²×D₆₆₁
Fitting	SD₁₆×C₆₆₁	C₂
socle	C₂×C₆₆₁	C₂×D₄
prime radical	#24902371	C₂
hypercenter	SD₁₆	D₆₆₁
hypocenter	C₆₆₁	#137
LCS	G ▷ C₄×C₆₆₁ ▷ C₂×C₆₆₁ ▷ C₆₆₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ SD₁₆

Complements

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

N	H ≅ G/N	type
2. Q₈×D₆₆₁	C₂	semidirect
3. Dic₂₆₄₄	C₂	semidirect
4. #24902362	C₂	semidirect
5. #24902360	C₂	semidirect
6. #24902355	C₂	semidirect
7. SD₁₆×C₆₆₁	C₂	semidirect
29. C₆₆₁	#137	semidirect
36. SD₁₆	D₆₆₁	semidirect

Maximal subgroups

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

subgroup	index
#24902371	2
Q₈×D₆₆₁	2
Dic₂₆₄₄	2
#24902362	2
#24902360	2
#24902355	2
SD₁₆×C₆₆₁	2
#137	661