Subgroups — #16810956

Order 3392.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic
2	#13133150	C₂	1	2	yes	maximal, complemented
3	#13133141	C₂	1	2	yes	maximal, complemented
4	C₂²⋊#88322	C₂	1	1	yes	maximal, characteristic
5	C₂²⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₈×C₅₃	C₂²	1	1	yes	characteristic
7	C₂²×C₄×C₅₃	C₂²	1	1	yes	characteristic
8	C₂²×Dic₅₃	—	2	1	no	—
9	C₄⋊C₄×C₅₃	—	2	1	no	—
10	C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
11	C₂²⋊C₄×C₅₃	—	2	1	no	—
12	C₄⋊Dic₅₃	—	2	1	no	—
13	C₄⋊Dic₅₃	C₂²	1	2	yes	—
14	C₂×#88322	C₂²	1	2	yes	—
15	C₂³×C₅₃	D₄	1	1	yes	characteristic
16	Q₈×C₅₃	—	2	2	no	—
17	C₂×C₄×C₅₃	—	2	2	no	—
18	C₂×C₄×C₅₃	—	2	1	no	—
19	C₂×C₄×C₅₃	—	2	1	no	—
20	C₂×C₄×C₅₃	D₄	1	1	yes	characteristic
21	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
22	C₂×Dic₅₃	—	2	2	no	—
23	C₂×Dic₅₃	—	4	2	no	—
24	#88322	—	2	2	no	—
25	C₂²×C₅₃	—	2	1	no	—
26	C₂²×C₅₃	—	2	1	no	—
27	C₂²×C₅₃	Q₁₆	1	2	yes	—
28	C₂²×C₅₃	C₂×D₄	1	1	yes	characteristic
29	C₄×C₅₃	—	2	1	no	—
30	C₄×C₅₃	—	4	1	no	—
31	C₄×C₅₃	—	4	1	no	—
32	C₄×C₅₃	#41	1	2	yes	—
33	Dic₅₃	—	4	2	no	—
34	C₂×C₅₃	—	2	2	no	—
35	C₂×C₅₃	#123	1	1	yes	characteristic
36	C₂×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
37	C₂×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
38	#484	—	53	1	no	maximal, Sylow
39	C₅₃	#484	1	1	yes	characteristic, complemented, Sylow
40	C₂²⋊Q₈	D₅₃	1	1	yes	characteristic
41	C₂×C₄⋊C₄	—	53	1	no	—
42	#107	—	53	2	no	—
43	#103	—	53	2	no	—
44	C₂²⋊C₈	—	53	1	no	—
45	C₂×Q₈	C₂×D₅₃	1	1	yes	characteristic
46	C₂²×C₄	—	106	1	no	—
47	C₂²×C₄	C₂×D₅₃	1	1	yes	characteristic
48	C₂×C₈	—	53	2	no	—
49	C₄⋊C₄	—	2	1	no	—
50	C₄⋊C₄	—	53	2	no	—
51	C₄⋊C₄	—	106	1	no	—
52	C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
53	C₂²⋊C₄	—	2	1	no	—
54	C₂³	C₂²⋊D₅₃	1	1	yes	characteristic
55	Q₈	—	2	2	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	106	2	no	—
60	C₂×C₄	—	212	2	no	—
61	C₂×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
63	C₈	—	106	2	no	—
64	C₂²	—	2	1	no	—
65	C₂²	—	2	1	no	—
66	C₂²	#11719925	1	2	yes	—
67	C₂²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	4	1	no	—
70	C₄	—	4	1	no	—
71	C₄	—	212	2	no	—
72	C₄	#11719947	1	2	yes	—
73	C₂	—	2	2	no	—
74	C₂	#13133198	1	1	yes	characteristic
75	C₂	C₂×#11719925	1	1	yes	characteristic
76	C₂	#13133242	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₅₃
commutator, cosocle	C₂×C₄×C₅₃	C₂³
Frattini	C₂×C₄	C₂²×D₅₃
Fitting	C₂²⋊Q₈×C₅₃	C₂
socle	C₂²×C₅₃	C₂×D₄
prime radical	C₂³×C₅₃	D₄
hypercenter	C₂²⋊Q₈	D₅₃
hypocenter	C₅₃	#484
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #13133150	C₂	semidirect
3. #13133141	C₂	semidirect
39. C₅₃	#484	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊Dic₅₃	2
#13133150	2
#13133141	2
C₂²⋊#88322	2
C₂²⋊Q₈×C₅₃	2
#484	53