Subgroups — #27273464

Order 13568.

486 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274460	C₂	1	3	yes	maximal
2	C₂×#16810345	C₂	1	3	yes	maximal
3	C₂×C₈²×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×#88322	C₂²	1	3	yes	—
5	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
6	#16810377	C₄	1	12	yes	—
7	#16810345	C₄	1	12	yes	complemented
8	C₂×C₄×C₈×C₅₃	C₂²	1	3	yes	—
9	C₈²×C₅₃	C₄	1	4	yes	complemented
10	C₂²×#88322	C₂×C₄	1	6	yes	—
11	C₂×C₄×Dic₅₃	C₂×C₄	1	6	yes	—
12	C₄×#88322	C₂×C₄	1	12	yes	—
13	C₂²×C₈×C₅₃	C₂×C₄	1	6	yes	—
14	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
15	C₄×C₈×C₅₃	C₂×C₄	1	6	yes	—
16	C₄×C₈×C₅₃	C₂×C₄	1	6	yes	—
17	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
18	C₂²×Dic₅₃	C₄²	1	4	yes	—
19	C₂×C₈×C₅₃	—	2	12	no	—
20	C₂×C₈×C₅₃	C₄²	1	12	yes	—
21	C₄²×C₅₃	C₂²×C₄	1	3	yes	—
22	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
23	C₄×Dic₅₃	—	2	12	no	—
24	C₂×#88322	—	2	12	no	—
25	C₂×#88322	C₄²	1	12	yes	—
26	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
27	C₂×C₄×C₅₃	C₂×C₄²	1	3	yes	—
28	C₂×C₄×C₅₃	C₂×C₄²	1	3	yes	—
29	C₂×C₄×C₅₃	#131	1	12	yes	—
30	C₂×Dic₅₃	—	2	12	no	—
31	C₈×C₅₃	—	2	12	no	—
32	#88322	—	2	12	no	—
33	#7845	—	53	1	no	maximal, Sylow
34	C₂²×C₅₃	C₄³	1	1	yes	characteristic
35	C₂²×C₅₃	#405	1	6	yes	—
36	C₄×C₅₃	#405	1	6	yes	—
37	C₄×C₅₃	#405	1	6	yes	—
38	Dic₅₃	—	4	4	no	—
39	#1744	—	53	3	no	—
40	C₂×#322	—	53	3	no	—
41	C₂×C₈²	D₅₃	1	1	yes	characteristic
42	C₂×C₅₃	#1454	1	4	yes	—
43	C₂×C₅₃	#1726	1	3	yes	—
44	C₂×C₄×C₈	—	53	3	no	—
45	C₂×C₄×C₈	C₂×D₅₃	1	3	yes	—
46	C₄³	—	53	1	no	—
47	#336	—	53	12	no	—
48	#322	—	53	12	no	—
49	C₈²	Dic₅₃	1	4	yes	complemented
50	C₅₃	#7845	1	1	yes	characteristic, complemented, Sylow
51	C₂²×C₈	—	53	6	no	—
52	C₂²×C₈	C₄×D₅₃	1	6	yes	—
53	C₂×C₄²	—	53	6	no	—
54	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
55	C₄×C₈	—	53	12	no	—
56	C₄×C₈	C₄×D₅₃	1	6	yes	—
57	C₄×C₈	C₂×Dic₅₃	1	6	yes	—
58	C₂²×C₄	—	53	4	no	—
59	C₂²×C₄	C₂×C₄×D₅₃	1	3	yes	—
60	C₂×C₈	—	2	12	no	—
61	C₂×C₈	—	53	12	no	—
62	C₂×C₈	—	106	12	no	—
63	C₂×C₈	C₄×Dic₅₃	1	12	yes	—
64	C₄²	—	106	12	no	—
65	C₄²	C₂×C₄×D₅₃	1	3	yes	—
66	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
67	C₂³	C₄²×D₅₃	1	1	yes	characteristic
68	C₂×C₄	—	106	12	no	—
69	C₂×C₄	C₄²×D₅₃	1	3	yes	—
70	C₂×C₄	C₂×C₄×Dic₅₃	1	3	yes	—
71	C₂×C₄	#13133223	1	12	yes	—
72	C₈	—	2	12	no	—
73	C₈	—	106	12	no	—
74	C₂²	C₄²×Dic₅₃	1	1	yes	characteristic
75	C₂²	#16810577	1	6	yes	—
76	C₄	—	212	4	no	—
77	C₄	#16810577	1	6	yes	—
78	C₄	#16810578	1	6	yes	—
79	C₂	#20273014	1	4	yes	—
80	C₂	#20274401	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₅₃
commutator	C₂²×C₅₃	C₄³
Fitting	C₂×C₈²×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×C₈²	D₅₃
hypocenter	C₅₃	#7845
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
7. #16810345	C₄	semidirect
9. C₈²×C₅₃	C₄	semidirect
49. C₈²	Dic₅₃	semidirect
50. C₅₃	#7845	semidirect

Maximal subgroups

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

subgroup	index
#20274460	2
C₂×#16810345	2
C₂×C₈²×C₅₃	2
#7845	53