Subgroups — #16811024

Order 3392.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13133208	C₂	1	2	yes	maximal
2	#13133151	C₂	1	1	yes	maximal, characteristic
3	#13133148	C₂	1	2	yes	maximal
4	C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
5	#107×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₈×C₅₃	C₂²	1	1	yes	characteristic
7	C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
8	C₄⋊Dic₅₃	—	2	2	no	—
9	C₄⋊Dic₅₃	C₂²	1	2	yes	—
10	#11719919	—	2	2	no	—
11	C₄×Dic₅₃	C₂²	1	1	yes	characteristic
12	C₂×#88322	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₅₃	—	2	2	no	—
14	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
15	C₂×Dic₅₃	—	2	2	no	—
16	C₂×Dic₅₃	D₄	1	1	yes	characteristic
17	C₂×Dic₅₃	D₄	1	1	yes	characteristic
18	C₈×C₅₃	Q₈	1	2	yes	—
19	#88322	Q₈	1	2	yes	—
20	C₂²×C₅₃	C₂×D₄	1	1	yes	characteristic
21	C₄×C₅₃	—	4	2	no	—
22	C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
23	C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
24	Dic₅₃	—	2	1	no	—
25	Dic₅₃	—	2	1	no	—
26	Dic₅₃	—	4	2	no	—
27	C₂×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
28	C₂×C₅₃	#135	1	1	yes	characteristic
29	C₂×C₅₃	#135	1	1	yes	characteristic
30	#499	—	53	1	no	maximal, Sylow
31	C₅₃	#499	1	1	yes	characteristic, complemented, Sylow
32	#125	—	53	2	no	—
33	#107	—	53	1	no	—
34	#107	D₅₃	1	1	yes	characteristic
35	#106	—	53	2	no	—
36	C₄×C₈	—	53	1	no	—
37	C₂×C₈	—	53	1	no	—
38	C₂×C₈	C₂×D₅₃	1	1	yes	characteristic
39	C₄⋊C₄	—	53	2	no	—
40	C₄⋊C₄	—	106	2	no	—
41	C₄⋊C₄	—	106	2	no	—
42	C₄⋊C₄	C₂×D₅₃	1	2	yes	—
43	C₄²	—	53	1	no	—
44	C₂×C₄	—	2	2	no	—
45	C₂×C₄	—	53	1	no	—
46	C₂×C₄	—	53	1	no	—
47	C₂×C₄	—	106	2	no	—
48	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
49	C₈	—	53	2	no	—
50	C₈	Dic₁₀₆	1	2	yes	—
51	C₂²	D₄×D₅₃	1	1	yes	characteristic
52	C₄	—	4	2	no	—
53	C₄	—	106	1	no	—
54	C₄	—	106	1	no	—
55	C₄	—	212	2	no	—
56	C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
57	C₄	Q₈×D₅₃	1	1	yes	characteristic
58	C₂	#13133216	1	1	yes	characteristic
59	C₂	#13133236	1	1	yes	characteristic
60	C₂	#13133240	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₅₃
commutator, cosocle	C₂×C₄×C₅₃	C₂³
Frattini	C₂×C₄	C₂²×D₅₃
Fitting	#107×C₅₃	C₂
socle, prime radical	C₂²×C₅₃	C₂×D₄
hypercenter	#107	D₅₃
hypocenter	C₅₃	#499
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
31. C₅₃	#499	semidirect

Maximal subgroups

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

subgroup	index
#13133208	2
#13133151	2
#13133148	2
C₈×Dic₅₃	2
#107×C₅₃	2
#499	53