Subgroups — #16810682

Order 3392.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13133174	C₂	1	1	yes	maximal, characteristic
2	#13133146	C₂	1	2	yes	maximal, complemented
3	#13133129	C₂	1	2	yes	maximal
4	#13133128	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₅₃	C₂²	1	1	yes	characteristic
7	C₂×C₈×C₅₃	C₂²	1	2	yes	—
8	C₄²×C₅₃	C₂²	1	1	yes	characteristic
9	#11719921	C₄	1	2	yes	—
10	#11719919	C₄	1	2	yes	—
11	C₄×Dic₅₃	C₂²	1	1	yes	characteristic
12	C₂×#88322	C₂²	1	2	yes	—
13	C₂²×D₅₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₅₃	C₂×C₄	1	2	yes	—
15	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
16	C₂×Dic₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂×Dic₅₃	C₂×C₄	1	2	yes	—
18	C₄×D₅₃	—	2	2	no	—
19	C₈×C₅₃	—	2	2	no	—
20	#88322	—	2	2	no	—
21	C₂²×C₅₃	C₂²×C₄	1	1	yes	characteristic
22	C₂×D₅₃	—	2	1	no	—
23	C₂×D₅₃	—	2	2	no	—
24	C₄×C₅₃	—	2	2	no	—
25	C₄×C₅₃	#41	1	2	yes	—
26	Dic₅₃	—	2	1	no	—
27	Dic₅₃	—	2	2	no	—
28	C₂×C₅₃	#117	1	1	yes	characteristic
29	C₂×C₅₃	#131	1	2	yes	—
30	D₅₃	—	4	1	no	—
31	#433	—	53	1	no	maximal, Sylow
32	C₅₃	#433	1	1	yes	characteristic, complemented, Sylow
33	#117	—	53	1	no	—
34	C₄⋊C₈	—	53	2	no	—
35	C₂²⋊C₈	—	53	2	no	—
36	#97	—	53	1	no	—
37	C₄×C₈	D₅₃	1	1	yes	characteristic, complemented
38	C₂²×C₄	—	53	1	no	—
39	C₂×C₈	—	53	2	no	—
40	C₂×C₈	C₂×D₅₃	1	2	yes	—
41	C₄⋊C₄	—	53	2	no	—
42	C₂²⋊C₄	—	53	2	no	—
43	C₄²	—	53	1	no	—
44	C₄²	C₂×D₅₃	1	1	yes	characteristic
45	C₂³	—	53	1	no	—
46	C₂×C₄	—	53	2	no	—
47	C₂×C₄	—	53	1	no	—
48	C₂×C₄	—	106	2	no	—
49	C₂×C₄	C₄×D₅₃	1	2	yes	—
50	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
51	C₈	—	2	2	no	—
52	C₈	—	106	2	no	—
53	C₂²	—	106	1	no	—
54	C₂²	—	106	2	no	—
55	C₂²	C₂×C₄×D₅₃	1	1	yes	characteristic
56	C₄	—	2	2	no	—
57	C₄	—	106	1	no	—
58	C₄	—	106	2	no	—
59	C₄	#11719945	1	2	yes	—
60	C₂	—	212	1	no	—
61	C₂	#13133174	1	1	yes	characteristic
62	C₂	#13133223	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₅₃
commutator, socle	C₂²×C₅₃	C₂²×C₄
Fitting	C₄×C₈×C₅₃	C₂
prime radical	C₂²×D₅₃	C₂×C₄
cosocle	C₂×C₄×C₅₃	C₂³
hypercenter	C₄×C₈	D₅₃
hypocenter	C₅₃	#433
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #13133146	C₂	semidirect
4. #13133128	C₂	semidirect
5. C₄×C₈×C₅₃	C₂	semidirect
32. C₅₃	#433	semidirect
37. C₄×C₈	D₅₃	semidirect

Maximal subgroups

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

subgroup	index
#13133174	2
#13133146	2
#13133129	2
#13133128	2
C₄×C₈×C₅₃	2
#433	53