Subgroups — #27415081

Order 13568.

604 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃⋊#2104	C₂	1	7	yes	maximal
2	#20274446	C₂	1	7	yes	maximal
3	C₅₃×#2104	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133173	C₂²	1	7	yes	—
5	#16810390	C₂²	1	7	yes	—
6	#16810389	C₂²	1	14	yes	—
7	C₅₃×#342	C₂²	1	7	yes	—
8	C₂³×Dic₅₃	C₂³	1	1	yes	characteristic
9	#13133173	—	2	14	no	—
10	#13133173	—	2	14	no	—
11	C₂×#11719919	—	2	7	no	—
12	C₂×C₂²⋊Dic₅₃	C₂³	1	7	yes	—
13	C₂×C₄×Dic₅₃	—	2	7	no	—
14	C₂×C₂²⋊C₄×C₅₃	C₂³	1	7	yes	—
15	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
16	C₂²×C₄×C₅₃	—	2	7	no	—
17	C₂²×Dic₅₃	—	2	7	no	—
18	C₂²×Dic₅₃	#41	1	7	yes	—
19	C₂²×Dic₅₃	#41	1	7	yes	—
20	C₂²⋊C₄×C₅₃	—	2	14	no	—
21	C₂²⋊Dic₅₃	—	2	14	no	—
22	#11719919	—	4	14	no	—
23	C₄×Dic₅₃	—	4	7	no	—
24	C₄×Dic₅₃	—	4	7	no	—
25	C₂³×C₅₃	—	8	1	no	—
26	C₂³×C₅₃	C₂×#41	1	7	yes	—
27	C₂×C₄×C₅₃	—	4	7	no	—
28	C₂×C₄×C₅₃	—	8	7	no	—
29	C₂×Dic₅₃	—	4	7	no	—
30	C₂×Dic₅₃	—	8	7	no	—
31	C₂×Dic₅₃	#126	1	7	yes	—
32	C₂×Dic₅₃	#126	1	7	yes	—
33	C₂×Dic₅₃	#126	1	14	yes	—
34	#23847	—	53	1	no	maximal, Sylow
35	C₂²×C₅₃	—	4	7	no	—
36	C₂²×C₅₃	C₂×#126	1	7	yes	—
37	C₄×C₅₃	—	8	7	no	—
38	Dic₅₃	—	8	7	no	—
39	Dic₅₃	#401	1	1	yes	characteristic
40	Dic₅₃	#401	1	7	yes	—
41	#2104	—	53	7	no	—
42	#2104	D₅₃	1	1	yes	characteristic
43	#1740	—	53	7	no	—
44	C₂×C₅₃	—	2	7	no	—
45	C₂×C₅₃	C₂×#401	1	1	yes	characteristic
46	C₂×#117	—	53	7	no	—
47	#342	—	53	14	no	—
48	#342	—	53	7	no	—
49	#342	C₂×D₅₃	1	7	yes	—
50	C₅₃	#23847	1	1	yes	characteristic, complemented, Sylow
51	C₂³×C₄	—	53	1	no	—
52	#117	—	106	14	no	—
53	#117	—	106	14	no	—
54	C₂×C₄⋊C₄	—	106	7	no	—
55	C₂×C₂²⋊C₄	—	53	7	no	—
56	C₂×C₂²⋊C₄	C₂²×D₅₃	1	7	yes	—
57	C₂×C₄²	—	106	7	no	—
58	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
59	C₂²×C₄	—	2	7	no	—
60	C₂²×C₄	—	53	7	no	—
61	C₂²×C₄	—	53	7	no	—
62	C₂²×C₄	—	106	7	no	—
63	C₄⋊C₄	—	212	14	no	—
64	C₂²⋊C₄	—	2	14	no	—
65	C₂²⋊C₄	—	106	14	no	—
66	C₄²	—	212	7	no	—
67	C₄²	—	212	7	no	—
68	C₂³	—	8	1	no	—
69	C₂³	#41×D₅₃	1	7	yes	—
70	C₂×C₄	—	4	7	no	—
71	C₂×C₄	—	8	7	no	—
72	C₂×C₄	—	53	7	no	—
73	C₂×C₄	—	53	7	no	—
74	C₂×C₄	—	53	14	no	—
75	C₂×C₄	—	424	7	no	—
76	C₂×C₄	—	212	7	no	—
77	C₂²	—	4	7	no	—
78	C₂²	#126×D₅₃	1	7	yes	—
79	C₄	—	8	7	no	—
80	C₄	—	53	1	no	—
81	C₄	—	53	7	no	—
82	C₄	—	424	7	no	—
83	C₂	—	2	7	no	—
84	C₂	#401×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#401×D₅₃
commutator, prime radical, cosocle	C₂⁴×C₅₃	C₂⁴
Frattini	C₂⁴	C₂³×D₅₃
Fitting	C₅₃×#2104	C₂
socle	C₂×C₅₃	C₂×#401
hypercenter	#2104	D₅₃
hypocenter	C₅₃	#23847
LCS	G ▷ C₂⁴×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ #2104

Complements

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

N	H ≅ G/N	type
50. C₅₃	#23847	semidirect

Maximal subgroups

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

subgroup	index
C₅₃⋊#2104	2
#20274446	2
C₅₃×#2104	2
#23847	53