Subgroups — #16810423

Order 3392.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄₂₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#11719908	C₂	1	1	yes	maximal, characteristic, complemented
3	#107×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₂₁₂	—	2	1	no	—
5	C₂×C₈×C₅₃	C₂²	1	1	yes	characteristic
6	C₄⋊C₄×C₅₃	—	2	1	no	—
7	D₄₂₄	—	2	1	no	—
8	D₄₂₄	C₄	1	2	yes	complemented
9	#11719908	—	2	1	no	—
10	C₂²×D₅₃	—	4	1	no	—
11	C₂×C₄×C₅₃	—	4	1	no	—
12	C₂×C₄×C₅₃	D₄	1	1	yes	characteristic
13	D₂₁₂	—	2	2	no	—
14	D₂₁₂	—	4	1	no	—
15	C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₈×C₅₃	D₄	1	1	yes	characteristic
17	C₂²×C₅₃	D₈	1	1	yes	characteristic
18	C₂×D₅₃	—	4	2	no	—
19	C₂×D₅₃	—	8	1	no	—
20	C₂×D₅₃	—	8	1	no	—
21	C₄×C₅₃	—	8	1	no	—
22	C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₄×C₅₃	SD₁₆	1	1	yes	characteristic
24	C₂×C₅₃	#102	1	1	yes	characteristic
25	C₂×C₅₃	D₁₆	1	1	yes	characteristic
26	C₂×C₅₃	SD₃₂	1	1	yes	characteristic
27	D₅₃	—	8	2	no	—
28	#357	—	53	1	no	maximal, Sylow
29	C₅₃	#357	1	1	yes	characteristic, complemented, Sylow
30	C₂×D₈	—	53	1	no	—
31	C₂×C₁₆	—	53	1	no	—
32	#107	D₅₃	1	1	yes	characteristic, complemented
33	C₂×D₄	—	106	1	no	—
34	D₈	—	53	2	no	—
35	D₈	—	106	1	no	—
36	C₂×C₈	C₂×D₅₃	1	1	yes	characteristic
37	C₄⋊C₄	—	2	1	no	—
38	C₁₆	—	106	1	no	—
39	C₂³	—	212	1	no	—
40	D₄	—	106	2	no	—
41	D₄	—	212	1	no	—
42	C₂×C₄	—	4	1	no	—
43	C₂×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
44	C₈	C₄×D₅₃	1	1	yes	characteristic
45	C₈	D₂₁₂	1	1	yes	characteristic
46	C₂²	—	424	1	no	—
47	C₂²	—	424	1	no	—
48	C₂²	—	212	2	no	—
49	C₂²	#11719922	1	1	yes	characteristic
50	C₄	—	8	1	no	—
51	C₄	#11719921	1	1	yes	characteristic
52	C₄	#11719924	1	1	yes	characteristic
53	C₂	—	424	2	no	—
54	C₂	#13133137	1	1	yes	characteristic
55	C₂	#13133158	1	1	yes	characteristic
56	C₂	#13133160	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11719922
commutator	C₈×C₅₃	C₂×C₄
Frattini	C₂×C₈	C₂×D₅₃
Fitting	#107×C₅₃	C₂
socle	C₂²×C₅₃	D₈
prime radical	C₂×D₄₂₄	C₂
cosocle	C₂×C₈×C₅₃	C₂²
hypercenter	#107	D₅₃
hypocenter	C₅₃	#357
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #107

Complements

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

N	H ≅ G/N	type
2. C₂×#11719908	C₂	semidirect
3. #107×C₅₃	C₂	semidirect
8. D₄₂₄	C₄	semidirect
29. C₅₃	#357	semidirect
32. #107	D₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄₂₄	2
C₂×#11719908	2
#107×C₅₃	2
#357	53