Subgroups — #35232565

Order 21152.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24902373	C₂	1	1	yes	maximal, characteristic
2	Q₈×D₆₆₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#24902362	C₂	1	1	yes	maximal, characteristic, complemented
4	#24902361	C₂	1	1	yes	maximal, characteristic, complemented
5	#24902359	C₂	1	1	yes	maximal, characteristic, complemented
6	#24902355	C₂	1	1	yes	maximal, characteristic, complemented
7	Q₁₆×C₆₆₁	C₂	1	1	yes	maximal, characteristic, complemented
8	Dic₁₃₂₂	—	2	1	no	—
9	Dic₁₃₂₂	C₂²	1	1	yes	characteristic
10	D₂₆₄₄	—	2	1	no	—
11	D₂₆₄₄	C₂²	1	1	yes	characteristic
12	C₄×D₆₆₁	—	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	Q₈×C₆₆₁	C₂²	1	1	yes	characteristic
18	C₈×C₆₆₁	C₂²	1	1	yes	characteristic
19	C₂×D₆₆₁	—	2	1	no	—
20	C₂×D₆₆₁	D₄	1	1	yes	characteristic
21	C₄×C₆₆₁	—	2	1	no	—
22	C₄×C₆₆₁	—	2	1	no	—
23	C₄×C₆₆₁	C₂³	1	1	yes	characteristic
24	Dic₆₆₁	—	2	1	no	—
25	Dic₆₆₁	D₄	1	1	yes	characteristic
26	C₂×C₆₆₁	C₂×D₄	1	1	yes	characteristic
27	D₆₆₁	—	2	1	no	—
28	D₆₆₁	—	4	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	D₆₆₁	1	1	yes	characteristic, complemented
35	SD₁₆	—	661	1	no	—
36	SD₁₆	—	661	1	no	—
37	M₁₆	—	661	1	no	—
38	Q₈	—	1322	1	no	—
39	Q₈	—	661	1	no	—
40	Q₈	C₂×D₆₆₁	1	1	yes	characteristic
41	Q₈	C₂×D₆₆₁	1	1	yes	characteristic
42	D₄	—	1322	1	no	—
43	D₄	—	661	1	no	—
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	—	1322	1	no	—
50	C₂²	—	661	1	no	—
51	C₄	—	2	1	no	—
52	C₄	—	2	1	no	—
53	C₄	—	1322	1	no	—
54	C₄	—	661	1	no	—
55	C₄	C₂²×D₆₆₁	1	1	yes	characteristic
56	C₂	—	1322	1	no	—
57	C₂	—	2644	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	Q₁₆×C₆₆₁	C₂
socle	C₂×C₆₆₁	C₂×D₄
prime radical	#24902373	C₂
hypercenter	Q₁₆	D₆₆₁
hypocenter	C₆₆₁	#137
LCS	G ▷ C₄×C₆₆₁ ▷ C₂×C₆₆₁ ▷ C₆₆₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
2. Q₈×D₆₆₁	C₂	semidirect
3. #24902362	C₂	semidirect
4. #24902361	C₂	semidirect
5. #24902359	C₂	semidirect
6. #24902355	C₂	semidirect
7. Q₁₆×C₆₆₁	C₂	semidirect
29. C₆₆₁	#137	semidirect
34. Q₁₆	D₆₆₁	semidirect

Maximal subgroups

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

subgroup	index
#24902373	2
Q₈×D₆₆₁	2
#24902362	2
#24902361	2
#24902359	2
#24902355	2
Q₁₆×C₆₆₁	2
#137	661