Subgroups — #27305112

Order 13568.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276990	C₂	1	2	yes	maximal
2	#20272915	C₂	1	4	yes	maximal
3	C₄×C₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
4	#16811028	—	2	2	no	—
5	#16810449	—	2	2	no	—
6	#16810449	C₂²	1	4	yes	—
7	C₂×C₃₂×C₅₃	C₂²	1	2	yes	—
8	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
9	#13133217	—	4	2	no	—
10	#13133151	—	2	4	no	—
11	#13133151	—	4	2	no	—
12	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
13	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
14	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
15	C₃₂×C₅₃	Q₈	1	4	yes	complemented
16	C₂×Dic₁₀₆	—	8	2	no	—
17	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
18	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
19	C₄²×C₅₃	D₈	1	1	yes	characteristic
20	C₄⋊Dic₅₃	—	4	4	no	—
21	C₄⋊Dic₅₃	—	8	2	no	—
22	C₁₆×C₅₃	Q₁₆	1	2	yes	—
23	C₁₆×C₅₃	C₂×Q₈	1	2	yes	—
24	C₂×C₄×C₅₃	D₁₆	1	1	yes	characteristic
25	C₂×C₄×C₅₃	D₁₆	1	1	yes	characteristic
26	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
27	C₂×Dic₅₃	—	8	4	no	—
28	Dic₁₀₆	—	16	4	no	—
29	C₈×C₅₃	Q₃₂	1	2	yes	—
30	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
31	C₈×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
32	#13731	—	53	1	no	maximal, Sylow
33	C₂²×C₅₃	C₂×D₁₆	1	1	yes	characteristic
34	C₄×C₅₃	D₃₂	1	2	yes	—
35	C₄×C₅₃	Q₆₄	1	2	yes	—
36	C₄×C₅₃	#500	1	1	yes	characteristic
37	C₄×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
38	Dic₅₃	—	16	4	no	—
39	#2252	—	53	2	no	—
40	#1423	—	53	4	no	—
41	C₄×C₃₂	D₅₃	1	1	yes	characteristic
42	C₂×C₅₃	#2252	1	1	yes	characteristic
43	C₂×C₅₃	C₂×D₃₂	1	1	yes	characteristic
44	C₂×C₅₃	C₂×Q₆₄	1	1	yes	characteristic
45	#500	—	106	2	no	—
46	C₂×C₃₂	C₂×D₅₃	1	2	yes	—
47	#366	—	53	4	no	—
48	#366	—	106	2	no	—
49	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
50	C₅₃	#13731	1	1	yes	characteristic, complemented, Sylow
51	C₄⋊Q₈	—	212	2	no	—
52	C₂×C₁₆	D₂₁₂	1	1	yes	characteristic
53	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
54	#107	—	106	4	no	—
55	#107	—	212	2	no	—
56	C₄×C₈	D₂₁₂	1	1	yes	characteristic
57	C₃₂	Dic₁₀₆	1	4	yes	complemented
58	C₂×Q₈	—	424	2	no	—
59	C₂×C₈	D₄₂₄	1	1	yes	characteristic
60	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
61	C₄⋊C₄	—	424	2	no	—
62	C₄⋊C₄	—	212	4	no	—
63	C₄²	D₄₂₄	1	1	yes	characteristic
64	C₁₆	Dic₂₁₂	1	2	yes	—
65	C₁₆	C₂×Dic₁₀₆	1	2	yes	—
66	Q₈	—	848	4	no	—
67	C₂×C₄	—	424	4	no	—
68	C₂×C₄	D₈₄₈	1	1	yes	characteristic
69	C₂×C₄	D₈₄₈	1	1	yes	characteristic
70	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
71	C₈	Dic₄₂₄	1	2	yes	—
72	C₈	#13133217	1	1	yes	characteristic
73	C₈	C₂×Dic₂₁₂	1	1	yes	characteristic
74	C₂²	C₂×D₈₄₈	1	1	yes	characteristic
75	C₄	—	848	4	no	—
76	C₄	D₁₆₉₆	1	2	yes	—
77	C₄	Dic₈₄₈	1	2	yes	—
78	C₄	#16811028	1	1	yes	characteristic
79	C₄	C₂×Dic₄₂₄	1	1	yes	characteristic
80	C₂	#20276990	1	1	yes	characteristic
81	C₂	C₂×D₁₆₉₆	1	1	yes	characteristic
82	C₂	C₂×Dic₈₄₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈₄₈
commutator, cosocle	C₂×C₁₆×C₅₃	C₂³
Frattini	C₂×C₁₆	C₂²×D₅₃
Fitting	C₄×C₃₂×C₅₃	C₂
socle, prime radical	C₂²×C₅₃	C₂×D₁₆
hypercenter	C₄×C₃₂	D₅₃
hypocenter	C₅₃	#13731
LCS	G ▷ C₂×C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

Complements

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

N	H ≅ G/N	type
15. C₃₂×C₅₃	Q₈	semidirect
50. C₅₃	#13731	semidirect
57. C₃₂	Dic₁₀₆	semidirect

Maximal subgroups

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

subgroup	index
#20276990	2
#20272915	2
C₄×C₃₂×C₅₃	2
#13731	53