Subgroups — #16811360

Order 3392.

252 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13133211	C₂	1	2	yes	maximal, complemented
2	#13133198	C₂	1	3	yes	maximal, complemented
3	#13133189	C₂	1	6	yes	maximal, complemented
4	#13133185	C₂	1	3	yes	maximal, complemented
5	#127×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₅₃	C₂²	1	1	yes	characteristic
7	C₂×D₄×C₅₃	C₂²	1	3	yes	—
8	C₂×D₄×C₅₃	C₂²	1	3	yes	complemented
9	C₂×C₂²⋊D₅₃	C₂²	1	6	yes	—
10	C₂²×Dic₅₃	C₂²	1	3	yes	complemented
11	C₄²×C₅₃	C₂²	1	1	yes	characteristic, complemented
12	C₂²⋊Dic₅₃	C₂²	1	3	yes	—
13	C₂²⋊Dic₅₃	C₂²	1	3	yes	complemented
14	#11719921	C₂²	1	6	yes	—
15	#11719919	C₂²	1	6	yes	complemented
16	C₂³×C₅₃	C₂³	1	3	yes	—
17	C₂³×C₅₃	C₂³	1	1	yes	characteristic
18	C₂²×D₅₃	—	4	3	no	—
19	C₂²×D₅₃	C₂³	1	2	yes	—
20	D₄×C₅₃	—	4	3	no	—
21	D₄×C₅₃	—	4	3	no	—
22	C₂×C₄×C₅₃	C₂³	1	3	yes	—
23	C₂²⋊D₅₃	—	4	6	no	—
24	C₂×Dic₅₃	—	4	3	no	—
25	C₂×Dic₅₃	C₂³	1	6	yes	—
26	C₂²×C₅₃	—	2	3	no	—
27	C₂²×C₅₃	—	2	6	no	—
28	C₂²×C₅₃	—	2	3	no	—
29	C₂²×C₅₃	C₂⁴	1	1	yes	characteristic
30	C₂×D₅₃	—	2	6	no	—
31	C₂×D₅₃	—	4	4	no	—
32	C₄×C₅₃	—	2	3	no	—
33	Dic₅₃	—	2	6	no	—
34	C₂×C₅₃	—	4	1	no	—
35	C₂×C₅₃	—	4	3	no	—
36	C₂×C₅₃	H(5,𝔽₂)	1	3	yes	—
37	D₅₃	—	4	2	no	—
38	#560	—	53	1	no	maximal, Sylow
39	C₅₃	#560	1	1	yes	characteristic, complemented, Sylow
40	#127	D₅₃	1	1	yes	characteristic, complemented
41	#126	—	53	2	no	—
42	#123	—	53	3	no	—
43	#121	—	53	6	no	—
44	#120	—	53	3	no	—
45	C₂⁴	—	53	1	no	—
46	C₂×D₄	—	53	6	no	—
47	C₂×D₄	C₂×D₅₃	1	3	yes	—
48	C₂×D₄	C₂×D₅₃	1	3	yes	complemented
49	C₂²×C₄	—	53	3	no	—
50	C₄⋊C₄	—	53	6	no	—
51	C₂²⋊C₄	—	53	3	no	—
52	C₂²⋊C₄	—	53	3	no	—
53	C₂²⋊C₄	—	53	6	no	—
54	C₄²	C₂×D₅₃	1	1	yes	characteristic, complemented
55	C₂³	—	53	2	no	—
56	C₂³	—	212	3	no	—
57	C₂³	C₂²×D₅₃	1	1	yes	characteristic
58	C₂³	C₂²×D₅₃	1	3	yes	—
59	D₄	—	4	3	no	—
60	D₄	—	4	3	no	—
61	D₄	—	212	6	no	—
62	C₂×C₄	—	53	6	no	—
63	C₂×C₄	—	212	3	no	—
64	C₂×C₄	C₂²×D₅₃	1	3	yes	—
65	C₂²	—	2	3	no	—
66	C₂²	—	2	6	no	—
67	C₂²	—	2	3	no	—
68	C₂²	—	106	6	no	—
69	C₂²	—	212	4	no	—
70	C₂²	C₂³×D₅₃	1	1	yes	characteristic
71	C₄	—	2	3	no	—
72	C₄	—	106	6	no	—
73	C₂	—	4	1	no	—
74	C₂	—	4	3	no	—
75	C₂	—	212	2	no	—
76	C₂	#13133262	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅₃
commutator, socle, cosocle	C₂²×C₅₃	C₂⁴
Fitting	#127×C₅₃	C₂
hypercenter	#127	D₅₃
hypocenter	C₅₃	#560
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #127

Complements

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

N	H ≅ G/N	type
1. #13133211	C₂	semidirect
2. #13133198	C₂	semidirect
3. #13133189	C₂	semidirect
4. #13133185	C₂	semidirect
5. #127×C₅₃	C₂	semidirect
8. C₂×D₄×C₅₃	C₂²	semidirect
10. C₂²×Dic₅₃	C₂²	semidirect
11. C₄²×C₅₃	C₂²	semidirect
13. C₂²⋊Dic₅₃	C₂²	semidirect
15. #11719919	C₂²	semidirect
39. C₅₃	#560	semidirect
40. #127	D₅₃	semidirect
48. C₂×D₄	C₂×D₅₃	semidirect
54. C₄²	C₂×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
#13133211	2
#13133198	2
#13133189	2
#13133185	2
#127×C₅₃	2
#560	53