Subgroups — #16811366

Order 3392.

260 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13133204	C₂	1	3	yes	maximal, complemented
2	#13133203	C₂	1	6	yes	maximal, complemented
3	#13133186	C₂	1	2	yes	maximal, complemented
4	#13133185	C₂	1	3	yes	maximal, complemented
5	#120×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂⁴×C₅₃	C₂²	1	1	yes	characteristic, complemented
7	C₂³×D₅₃	C₂²	1	1	yes	characteristic, complemented
8	C₂×D₄×C₅₃	C₂²	1	3	yes	—
9	C₂×C₂²⋊D₅₃	C₂²	1	6	yes	—
10	C₂×Dic₁₀₆	C₂²	1	3	yes	complemented
11	C₂²⋊C₄×C₅₃	C₂²	1	3	yes	complemented
12	C₂²⋊Dic₅₃	C₂²	1	3	yes	complemented
13	C₂²⋊Dic₅₃	C₂²	1	6	yes	complemented
14	#11719921	C₂²	1	6	yes	complemented
15	C₄×Dic₅₃	C₂²	1	3	yes	complemented
16	C₂³×C₅₃	—	4	3	no	—
17	C₂³×C₅₃	C₂³	1	3	yes	—
18	C₂³×C₅₃	C₂³	1	1	yes	characteristic
19	C₂²×D₅₃	—	4	3	no	—
20	C₂²×D₅₃	C₂³	1	2	yes	—
21	D₄×C₅₃	—	4	3	no	—
22	C₂×C₄×C₅₃	C₂³	1	3	yes	—
23	C₂²⋊D₅₃	—	4	6	no	—
24	C₂×Dic₅₃	C₂³	1	6	yes	—
25	Dic₁₀₆	—	4	3	no	—
26	C₂²×C₅₃	—	2	6	no	—
27	C₂²×C₅₃	—	2	3	no	—
28	C₂²×C₅₃	—	2	3	no	—
29	C₂²×C₅₃	—	4	4	no	—
30	C₂²×C₅₃	C₂⁴	1	1	yes	characteristic
31	C₂×D₅₃	—	2	6	no	—
32	C₂×D₅₃	—	4	4	no	—
33	C₄×C₅₃	—	2	3	no	—
34	Dic₅₃	—	2	6	no	—
35	C₂×C₅₃	—	4	1	no	—
36	C₂×C₅₃	—	4	3	no	—
37	C₂×C₅₃	H(5,𝔽₂)	1	3	yes	—
38	D₅₃	—	4	2	no	—
39	H(3,𝔽₄)	—	53	1	no	maximal, Sylow
40	C₅₃	H(3,𝔽₄)	1	1	yes	characteristic, complemented, Sylow
41	#124	—	53	3	no	—
42	#124	—	53	6	no	—
43	#120	—	53	2	no	—
44	#120	—	53	3	no	—
45	#120	D₅₃	1	1	yes	characteristic, complemented
46	C₂⁴	—	53	1	no	—
47	C₂⁴	C₂×D₅₃	1	1	yes	characteristic, complemented
48	C₂×Q₈	—	53	3	no	—
49	C₂×D₄	—	53	6	no	—
50	C₂×D₄	C₂×D₅₃	1	3	yes	—
51	C₂²⋊C₄	—	53	3	no	—
52	C₂²⋊C₄	—	53	6	no	—
53	C₂²⋊C₄	—	53	6	no	—
54	C₂²⋊C₄	C₂×D₅₃	1	3	yes	complemented
55	C₄²	—	53	3	no	—
56	C₂³	—	4	3	no	—
57	C₂³	—	53	2	no	—
58	C₂³	—	212	3	no	—
59	C₂³	C₂²×D₅₃	1	1	yes	characteristic
60	C₂³	C₂²×D₅₃	1	3	yes	—
61	Q₈	—	212	3	no	—
62	D₄	—	4	3	no	—
63	D₄	—	212	6	no	—
64	C₂×C₄	—	53	6	no	—
65	C₂×C₄	C₂²×D₅₃	1	3	yes	—
66	C₂²	—	2	3	no	—
67	C₂²	—	2	3	no	—
68	C₂²	—	2	6	no	—
69	C₂²	—	4	4	no	—
70	C₂²	—	106	6	no	—
71	C₂²	—	212	4	no	—
72	C₂²	C₂³×D₅₃	1	1	yes	characteristic
73	C₄	—	2	3	no	—
74	C₄	—	106	6	no	—
75	C₂	—	4	3	no	—
76	C₂	—	4	1	no	—
77	C₂	—	212	2	no	—
78	C₂	#13133262	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅₃
commutator, socle, cosocle	C₂²×C₅₃	C₂⁴
Fitting	#120×C₅₃	C₂
hypercenter	#120	D₅₃
hypocenter	C₅₃	H(3,𝔽₄)
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #120

Complements

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

N	H ≅ G/N	type
1. #13133204	C₂	semidirect
2. #13133203	C₂	semidirect
3. #13133186	C₂	semidirect
4. #13133185	C₂	semidirect
5. #120×C₅₃	C₂	semidirect
6. C₂⁴×C₅₃	C₂²	semidirect
7. C₂³×D₅₃	C₂²	semidirect
10. C₂×Dic₁₀₆	C₂²	semidirect
11. C₂²⋊C₄×C₅₃	C₂²	semidirect
12. C₂²⋊Dic₅₃	C₂²	semidirect
13. C₂²⋊Dic₅₃	C₂²	semidirect
14. #11719921	C₂²	semidirect
15. C₄×Dic₅₃	C₂²	semidirect
40. C₅₃	H(3,𝔽₄)	semidirect
45. #120	D₅₃	semidirect
47. C₂⁴	C₂×D₅₃	semidirect
54. C₂²⋊C₄	C₂×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
#13133204	2
#13133203	2
#13133186	2
#13133185	2
#120×C₅₃	2
H(3,𝔽₄)	53