Subgroups — #16810980

Order 3392.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13133205	C₂	1	1	yes	maximal, characteristic
2	#13133140	C₂	1	2	yes	maximal, complemented
3	#13133137	C₂	1	2	yes	maximal
4	C₄×#88322	C₂	1	1	yes	maximal, characteristic, complemented
5	#125×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₂₁₂	C₂²	1	1	yes	characteristic
7	C₂×Dic₁₀₆	C₂²	1	1	yes	characteristic
8	C₄⋊C₄×C₅₃	—	2	2	no	—
9	C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
10	C₄²×C₅₃	C₂²	1	1	yes	characteristic
11	#11719921	—	2	2	no	—
12	C₂×#88322	C₂²	1	2	yes	—
13	C₂²×D₅₃	—	2	1	no	—
14	C₂×C₄×C₅₃	—	2	2	no	—
15	C₂×C₄×C₅₃	D₄	1	2	yes	—
16	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
17	C₂×Dic₅₃	—	2	1	no	—
18	D₂₁₂	—	2	2	no	—
19	Dic₁₀₆	—	2	2	no	—
20	#88322	—	2	2	no	—
21	C₂²×C₅₃	C₂×D₄	1	1	yes	characteristic
22	C₂×D₅₃	—	4	1	no	—
23	C₂×D₅₃	—	4	2	no	—
24	C₄×C₅₃	—	2	2	no	—
25	C₄×C₅₃	—	4	2	no	—
26	C₄×C₅₃	#41	1	2	yes	—
27	Dic₅₃	—	4	1	no	—
28	C₂×C₅₃	#124	1	1	yes	characteristic
29	C₂×C₅₃	#135	1	2	yes	—
30	D₅₃	—	8	1	no	—
31	#488	—	53	1	no	maximal, Sylow
32	C₅₃	#488	1	1	yes	characteristic, complemented, Sylow
33	#125	D₅₃	1	1	yes	characteristic, complemented
34	#124	—	53	1	no	—
35	#103	—	53	2	no	—
36	#102	—	53	2	no	—
37	C₄×C₈	—	53	1	no	—
38	C₂×Q₈	—	53	1	no	—
39	C₂×D₄	—	53	1	no	—
40	C₂×C₈	—	53	2	no	—
41	C₄⋊C₄	—	2	2	no	—
42	C₄⋊C₄	C₂×D₅₃	1	2	yes	—
43	C₂²⋊C₄	—	106	2	no	—
44	C₄²	C₂×D₅₃	1	1	yes	characteristic
45	C₂³	—	106	1	no	—
46	Q₈	—	106	2	no	—
47	D₄	—	106	2	no	—
48	C₂×C₄	—	2	2	no	—
49	C₂×C₄	—	106	1	no	—
50	C₂×C₄	C₂²⋊D₅₃	1	2	yes	—
51	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
52	C₈	—	106	2	no	—
53	C₂²	—	212	1	no	—
54	C₂²	—	212	2	no	—
55	C₂²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
56	C₄	—	2	2	no	—
57	C₄	—	4	2	no	—
58	C₄	—	212	1	no	—
59	C₄	#11719949	1	2	yes	—
60	C₂	—	424	1	no	—
61	C₂	#13133206	1	1	yes	characteristic
62	C₂	#13133237	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₅₃
commutator, cosocle	C₂×C₄×C₅₃	C₂³
Frattini	C₂×C₄	C₂²×D₅₃
Fitting	#125×C₅₃	C₂
socle	C₂²×C₅₃	C₂×D₄
prime radical	C₂×D₂₁₂	C₂²
hypercenter	#125	D₅₃
hypocenter	C₅₃	#488
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
2. #13133140	C₂	semidirect
4. C₄×#88322	C₂	semidirect
5. #125×C₅₃	C₂	semidirect
32. C₅₃	#488	semidirect
33. #125	D₅₃	semidirect

Maximal subgroups

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

subgroup	index
#13133205	2
#13133140	2
#13133137	2
C₄×#88322	2
#125×C₅₃	2
#488	53