Subgroups — #30628366

Order 15712.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21463327	C₂	1	1	yes	maximal, characteristic
2	Q₈×D₄₉₁	C₂	1	1	yes	maximal, characteristic, complemented
3	Dic₁₉₆₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#21463318	C₂	1	1	yes	maximal, characteristic, complemented
5	#21463316	C₂	1	1	yes	maximal, characteristic, complemented
6	#21463311	C₂	1	1	yes	maximal, characteristic, complemented
7	SD₁₆×C₄₉₁	C₂	1	1	yes	maximal, characteristic, complemented
8	Dic₉₈₂	—	2	1	no	—
9	Dic₉₈₂	C₂²	1	1	yes	characteristic
10	Dic₉₈₂	C₂²	1	1	yes	characteristic
11	C₂²⋊D₄₉₁	—	2	1	no	—
12	C₂×Dic₄₉₁	—	2	1	no	—
13	C₄×D₄₉₁	—	2	1	no	—
14	C₄×D₄₉₁	C₂²	1	1	yes	characteristic
15	C₄₉₁⋊C₈	C₂²	1	1	yes	characteristic
16	Q₈×C₄₉₁	C₂²	1	1	yes	characteristic
17	D₄×C₄₉₁	C₂²	1	1	yes	characteristic
18	C₈×C₄₉₁	C₂²	1	1	yes	characteristic
19	C₂²×C₄₉₁	—	2	1	no	—
20	C₂×D₄₉₁	D₄	1	1	yes	characteristic
21	C₄×C₄₉₁	—	2	1	no	—
22	C₄×C₄₉₁	C₂³	1	1	yes	characteristic
23	Dic₄₉₁	—	2	1	no	—
24	Dic₄₉₁	—	2	1	no	—
25	Dic₄₉₁	D₄	1	1	yes	characteristic
26	C₂×C₄₉₁	—	4	1	no	—
27	C₂×C₄₉₁	C₂×D₄	1	1	yes	characteristic
28	D₄₉₁	—	2	1	no	—
29	C₄₉₁	#137	1	1	yes	characteristic, complemented, Sylow
30	#137	—	491	1	no	maximal, Sylow
31	#41	—	491	1	no	—
32	C₂×Q₈	—	491	1	no	—
33	Q₁₆	—	491	1	no	—
34	Q₁₆	—	491	1	no	—
35	SD₁₆	—	491	1	no	—
36	SD₁₆	D₄₉₁	1	1	yes	characteristic, complemented
37	M₁₆	—	491	1	no	—
38	Q₈	—	982	1	no	—
39	Q₈	—	491	1	no	—
40	Q₈	—	491	1	no	—
41	Q₈	C₂×D₄₉₁	1	1	yes	characteristic
42	D₄	—	982	1	no	—
43	D₄	C₂×D₄₉₁	1	1	yes	characteristic
44	C₂×C₄	—	982	1	no	—
45	C₂×C₄	—	982	1	no	—
46	C₂×C₄	—	491	1	no	—
47	C₈	—	491	1	no	—
48	C₈	C₂×D₄₉₁	1	1	yes	characteristic
49	C₂²	—	2	1	no	—
50	C₂²	—	491	1	no	—
51	C₄	—	2	1	no	—
52	C₄	—	982	1	no	—
53	C₄	—	982	1	no	—
54	C₄	—	491	1	no	—
55	C₄	C₂²×D₄₉₁	1	1	yes	characteristic
56	C₂	—	4	1	no	—
57	C₂	—	982	1	no	—
58	C₂	D₄×D₄₉₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₄₉₁
commutator, cosocle	C₄×C₄₉₁	C₂³
Frattini	C₄	C₂²×D₄₉₁
Fitting	SD₁₆×C₄₉₁	C₂
socle	C₂×C₄₉₁	C₂×D₄
prime radical	#21463327	C₂
hypercenter	SD₁₆	D₄₉₁
hypocenter	C₄₉₁	#137
LCS	G ▷ C₄×C₄₉₁ ▷ C₂×C₄₉₁ ▷ C₄₉₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ SD₁₆

Complements

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

N	H ≅ G/N	type
2. Q₈×D₄₉₁	C₂	semidirect
3. Dic₁₉₆₄	C₂	semidirect
4. #21463318	C₂	semidirect
5. #21463316	C₂	semidirect
6. #21463311	C₂	semidirect
7. SD₁₆×C₄₉₁	C₂	semidirect
29. C₄₉₁	#137	semidirect
36. SD₁₆	D₄₉₁	semidirect

Maximal subgroups

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

subgroup	index
#21463327	2
Q₈×D₄₉₁	2
Dic₁₉₆₄	2
#21463318	2
#21463316	2
#21463311	2
SD₁₆×C₄₉₁	2
#137	491