Group #20274018 — subgroups

Subgroups — #20274018

Order 6784.

182 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 45 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃⋊#564	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810989	C₂	1	1	yes	maximal, characteristic
3	#16810988	C₂	1	2	yes	maximal
4	#16810364	C₂	1	2	yes	maximal, complemented
5	C₅₃×#330	C₂	1	1	yes	maximal, characteristic, complemented
6	#13133214	C₂²	1	1	yes	characteristic
7	#13133209	C₂²	1	2	yes	—
8	#13133208	—	2	2	no	—
9	#13133207	—	2	2	no	—
10	#13133207	—	2	2	no	—
11	#13133139	—	2	2	no	—
12	#13133139	—	2	2	no	—
13	#13133137	—	2	2	no	—
14	#13133128	C₂²	1	1	yes	characteristic
15	#125×C₅₃	C₂²	1	1	yes	characteristic
16	#97×C₅₃	C₂²	1	2	yes	—
17	C₂×D₂₁₂	—	2	1	no	—
18	C₂×D₂₁₂	—	2	2	no	—
19	C₂×C₈×C₅₃	—	2	2	no	—
20	C₄⋊C₄×C₅₃	—	2	2	no	—
21	C₄⋊C₄×C₅₃	D₄	1	2	yes	—
22	C₄²×C₅₃	C₂³	1	1	yes	characteristic
23	C₄⋊Dic₅₃	—	2	2	no	—
24	C₄⋊Dic₅₃	D₄	1	2	yes	—
25	C₄⋊Dic₅₃	D₄	1	2	yes	—
26	#11719919	—	2	2	no	—
27	#11719919	—	2	2	no	—
28	#11719919	—	2	2	no	—
29	#11719919	—	2	2	no	—
30	C₄×Dic₅₃	—	2	2	no	—
31	C₂×#88322	—	2	1	no	—
32	C₂²×D₅₃	—	4	1	no	—
33	C₂×C₄×C₅₃	—	2	2	no	—
34	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
36	C₂×Dic₅₃	—	2	2	no	—
37	C₂×Dic₅₃	—	2	2	no	—
38	D₂₁₂	—	4	2	no	—
39	D₂₁₂	—	4	4	no	—
40	C₈×C₅₃	—	4	2	no	—
41	#88322	—	4	1	no	—
42	C₂²×C₅₃	#120	1	1	yes	characteristic
43	C₂×D₅₃	—	8	2	no	—
44	C₂×D₅₃	—	8	1	no	—
45	C₄×C₅₃	—	2	1	no	—
46	C₄×C₅₃	—	2	2	no	—
47	C₄×C₅₃	—	4	2	no	—
48	Dic₅₃	—	4	2	no	—
49	Dic₅₃	—	4	2	no	—
50	#1658	—	53	1	no	maximal, Sylow
51	C₂×C₅₃	#454	1	2	yes	—
52	C₂×C₅₃	#454	1	1	yes	characteristic
53	D₅₃	—	16	1	no	—
54	#564	—	53	1	no	—
55	#490	—	53	1	no	—
56	#490	—	53	2	no	—
57	#330	—	53	2	no	—
58	#330	D₅₃	1	1	yes	characteristic, complemented
59	C₅₃	#1658	1	1	yes	characteristic, complemented, Sylow
60	#127	—	53	1	no	—
61	#125	—	53	2	no	—
62	#125	—	106	2	no	—
63	#125	—	106	2	no	—
64	#125	—	106	2	no	—
65	#125	C₂×D₅₃	1	1	yes	characteristic
66	#102	—	106	2	no	—
67	#102	—	106	2	no	—
68	#102	—	106	2	no	—
69	#97	—	53	1	no	—
70	#97	C₂×D₅₃	1	2	yes	—
71	C₂×D₄	—	106	1	no	—
72	C₂×D₄	—	106	2	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	106	1	no	—
75	C₄⋊C₄	—	2	2	no	—
76	C₄⋊C₄	—	53	2	no	—
77	C₄⋊C₄	—	53	2	no	—
78	C₄⋊C₄	—	106	2	no	—
79	C₄⋊C₄	—	106	2	no	—
80	C₄⋊C₄	—	106	2	no	—
81	C₄⋊C₄	—	106	2	no	—
82	C₄⋊C₄	—	106	2	no	—
83	C₄⋊C₄	D₂₁₂	1	2	yes	—
84	C₄²	—	106	2	no	—
85	C₄²	C₂²×D₅₃	1	1	yes	characteristic
86	C₂³	—	212	1	no	—
87	D₄	—	212	4	no	—
88	D₄	—	212	2	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	—	106	2	no	—
91	C₂×C₄	—	106	2	no	—
92	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
93	C₂×C₄	D₄×D₅₃	1	2	yes	—
94	C₈	—	4	2	no	—
95	C₈	—	212	1	no	—
96	C₂²	—	424	2	no	—
97	C₂²	—	424	1	no	—
98	C₂²	#13133187	1	1	yes	characteristic
99	C₄	—	2	2	no	—
100	C₄	—	2	1	no	—
101	C₄	—	4	2	no	—
102	C₄	—	212	2	no	—
103	C₄	—	212	2	no	—
104	C₂	—	848	1	no	—
105	C₂	#16810797	1	2	yes	—
106	C₂	#16810798	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃⋊#564

C₂

yes

maximal, characteristic, complemented

#16810989

C₂

yes

maximal, characteristic

#16810988

C₂

yes

maximal

#16810364

C₂

yes

maximal, complemented

C₅₃×#330

C₂

yes

maximal, characteristic, complemented

#13133214

C₂²

yes

characteristic

#13133209

C₂²

yes

—

—

—

—

—

—

—

C₂²

yes

characteristic

#125×C₅₃

C₂²

yes

characteristic

#97×C₅₃

C₂²

yes

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

D₄

yes

—

C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

D₄

yes

—

C₄⋊Dic₅₃

D₄

yes

—

—

—

—

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂²×D₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

#120

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

Dic₅₃

—

Dic₅₃

—

#1658

—

maximal, Sylow

C₂×C₅₃

#454

yes

—

C₂×C₅₃

#454

yes

characteristic

D₅₃

—

—

—

—

—

D₅₃

yes

characteristic, complemented

C₅₃

#1658

yes

characteristic, complemented, Sylow

#127

—

#125

—

#125

—

106

—

#125

—

106

—

#125

—

106

—

#125

C₂×D₅₃

yes

characteristic

#102

—

106

—

#102

—

106

—

#102

—

106

—

#97

—

#97

C₂×D₅₃

yes

—

C₂×D₄

—

106

—

C₂×D₄

—

106

—

C₂×C₈

—

C₂×C₈

—

106

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

D₂₁₂

yes

—

C₄²

—

106

—

C₄²

C₂²×D₅₃

yes

characteristic

C₂³

—

212

—

D₄

—

212

—

D₄

—

212

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

C₂×D₂₁₂

yes

characteristic

C₂×C₄

D₄×D₅₃

yes

—

C₈

—

C₈

—

212

—

C₂²

—

424

—

C₂²

—

424

—

C₂²

#13133187

yes

characteristic

C₄

—

100

C₄

—

101

C₄

—

102

C₄

—

212

—

103

C₄

—

212

—

104

C₂

—

848

—

105

C₂

#16810797

yes

—

106

C₂

#16810798

yes

characteristic

	subgroup	G/H
center	C₂²	#13133187
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#330	C₂
socle	C₂²×C₅₃	#120
hypercenter	#330	D₅₃
hypocenter	C₅₃	#1658
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #330

subgroup

G/H

center

C₂²

#13133187

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#330

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #330

N	H ≅ G/N	type
1. C₅₃⋊#564	C₂	semidirect
4. #16810364	C₂	semidirect
5. C₅₃×#330	C₂	semidirect
58. #330	D₅₃	semidirect
59. C₅₃	#1658	semidirect

H ≅ G/N

type

1. C₅₃⋊#564

C₂

semidirect

4. #16810364

C₂

semidirect

5. C₅₃×#330

C₂

semidirect

58. #330

D₅₃

semidirect

59. C₅₃

#1658

semidirect

Subgroups — #20274018

Conjugacy classes of subgroups

Complements