Subgroups — #34092127

Order 19904.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23755731	C₂	1	2	yes	maximal, complemented
2	#23755651	C₂	1	2	yes	maximal, complemented
3	D₄₉₇₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₆×D₃₁₁	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₃₂×C₃₁₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#19097316	—	2	2	no	—
7	#19097302	—	2	2	no	—
8	D₂₄₈₈	C₂²	1	2	yes	—
9	C₈×D₃₁₁	C₂²	1	1	yes	characteristic
10	C₃₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
11	Q₁₆×C₃₁₁	C₂²	1	2	yes	—
12	C₁₆×C₃₁₁	C₂²	1	1	yes	characteristic
13	D₁₂₄₄	—	2	2	no	—
14	D₁₂₄₄	—	4	2	no	—
15	C₄×D₃₁₁	—	4	2	no	—
16	C₄×D₃₁₁	D₄	1	1	yes	characteristic
17	C₃₁₁⋊C₈	D₄	1	1	yes	characteristic
18	Q₈×C₃₁₁	—	2	2	no	—
19	C₈×C₃₁₁	C₂³	1	1	yes	characteristic
20	C₂×D₃₁₁	—	4	2	no	—
21	C₂×D₃₁₁	D₈	1	1	yes	characteristic
22	C₄×C₃₁₁	—	4	2	no	—
23	C₄×C₃₁₁	C₂×D₄	1	1	yes	characteristic
24	Dic₃₁₁	D₈	1	1	yes	characteristic
25	C₂×C₃₁₁	C₂×D₈	1	1	yes	characteristic
26	D₃₁₁	—	2	1	no	—
27	D₃₁₁	—	8	2	no	—
28	C₃₁₁	#508	1	1	yes	characteristic, complemented, Sylow
29	#508	—	311	1	no	maximal, Sylow
30	#135	—	311	2	no	—
31	Q₃₂	D₃₁₁	1	1	yes	characteristic, complemented
32	SD₃₂	—	311	2	no	—
33	D₁₆	—	311	1	no	—
34	C₂×C₁₆	—	311	1	no	—
35	#41	—	622	2	no	—
36	Q₁₆	C₂×D₃₁₁	1	2	yes	—
37	SD₁₆	—	622	2	no	—
38	D₈	—	311	2	no	—
39	C₂×C₈	—	311	1	no	—
40	C₁₆	—	311	1	no	—
41	C₁₆	C₂×D₃₁₁	1	1	yes	characteristic
42	Q₈	—	2	2	no	—
43	D₄	—	622	2	no	—
44	D₄	—	1244	2	no	—
45	C₂×C₄	—	311	1	no	—
46	C₂×C₄	—	1244	2	no	—
47	C₈	—	311	1	no	—
48	C₈	C₂²×D₃₁₁	1	1	yes	characteristic
49	C₂²	—	311	1	no	—
50	C₂²	—	1244	2	no	—
51	C₄	—	4	2	no	—
52	C₄	—	311	1	no	—
53	C₄	D₄×D₃₁₁	1	1	yes	characteristic
54	C₂	—	622	1	no	—
55	C₂	—	2488	2	no	—
56	C₂	D₈×D₃₁₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×D₃₁₁
commutator, cosocle	C₈×C₃₁₁	C₂³
Frattini	C₈	C₂²×D₃₁₁
Fitting	Q₃₂×C₃₁₁	C₂
socle	C₂×C₃₁₁	C₂×D₈
hypercenter	Q₃₂	D₃₁₁
hypocenter	C₃₁₁	#508
LCS	G ▷ C₈×C₃₁₁ ▷ C₄×C₃₁₁ ▷ C₂×C₃₁₁ ▷ C₃₁₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

Complements

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

N	H ≅ G/N	type
1. #23755731	C₂	semidirect
2. #23755651	C₂	semidirect
3. D₄₉₇₆	C₂	semidirect
4. C₁₆×D₃₁₁	C₂	semidirect
5. Q₃₂×C₃₁₁	C₂	semidirect
28. C₃₁₁	#508	semidirect
31. Q₃₂	D₃₁₁	semidirect

Maximal subgroups

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

subgroup	index
#23755731	2
#23755651	2
D₄₉₇₆	2
C₁₆×D₃₁₁	2
Q₃₂×C₃₁₁	2
#508	311