Group #20276872 — subgroups

Subgroups — #20276872

Order 6784.

136 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 53 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	#16811025	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810450	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810448	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄⋊#11719908	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810424	C₂	1	1	yes	maximal, characteristic
6	#16810424	C₂	1	1	yes	maximal, characteristic
7	C₄×D₈×C₅₃	C₂	1	1	yes	maximal, characteristic
8	#13133209	—	2	1	no	—
9	C₂×#11719908	C₂²	1	1	yes	characteristic
10	C₂×#11719908	C₂²	1	1	yes	characteristic
11	#13133151	C₂²	1	1	yes	characteristic
12	#13133151	C₂²	1	1	yes	characteristic
13	#13133149	—	2	1	no	—
14	C₂×D₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₄×D₄×C₅₃	—	2	1	no	—
16	#107×C₅₃	C₂²	1	1	yes	characteristic
17	#102×C₅₃	—	2	1	no	—
18	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×D₄×C₅₃	—	2	1	no	—
20	C₂²×C₄×C₅₃	—	4	1	no	—
21	D₈×C₅₃	—	2	1	no	—
22	D₈×C₅₃	Q₈	1	2	yes	—
23	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
24	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
25	C₄⋊C₄×C₅₃	—	2	1	no	—
26	C₂²⋊C₄×C₅₃	—	4	1	no	—
27	C₄²×C₅₃	D₄	1	1	yes	characteristic
28	C₄⋊Dic₅₃	—	2	1	no	—
29	C₄⋊Dic₅₃	—	2	1	no	—
30	#11719919	—	4	1	no	—
31	#11719919	—	4	1	no	—
32	#11719908	—	2	1	no	—
33	#11719908	—	2	1	no	—
34	C₂³×C₅₃	—	4	1	no	—
35	D₄×C₅₃	—	2	2	no	—
36	D₄×C₅₃	—	4	1	no	—
37	C₂×C₄×C₅₃	—	4	1	no	—
38	C₂×C₄×C₅₃	—	8	2	no	—
39	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
40	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
41	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂×Dic₅₃	—	4	1	no	—
43	C₂×Dic₅₃	—	4	1	no	—
44	C₈×C₅₃	—	2	1	no	—
45	C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
46	C₈×C₅₃	#41	1	1	yes	characteristic
47	C₂²×C₅₃	—	4	2	no	—
48	C₂²×C₅₃	—	8	1	no	—
49	C₂²×C₅₃	—	8	1	no	—
50	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
51	C₄×C₅₃	—	2	1	no	—
52	C₄×C₅₃	—	2	1	no	—
53	C₄×C₅₃	—	8	1	no	—
54	C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
55	C₄×C₅₃	#137	1	1	yes	characteristic
56	Dic₅₃	—	8	1	no	—
57	Dic₅₃	—	8	1	no	—
58	#2228	—	53	1	no	maximal, Sylow
59	C₂×C₅₃	—	8	2	no	—
60	C₂×C₅₃	#474	1	1	yes	characteristic
61	C₂×C₅₃	#508	1	1	yes	characteristic
62	C₂×C₅₃	#509	1	1	yes	characteristic
63	#499	—	53	1	no	—
64	C₄×D₈	D₅₃	1	1	yes	characteristic
65	#367	—	53	1	no	—
66	#366	—	53	1	no	—
67	C₄⋊C₁₆	—	53	1	no	—
68	#357	—	53	1	no	—
69	#357	—	53	1	no	—
70	C₅₃	#2228	1	1	yes	characteristic, complemented, Sylow
71	C₂×D₈	C₂×D₅₃	1	1	yes	characteristic
72	#125	—	106	1	no	—
73	C₄×D₄	—	2	1	no	—
74	C₂×C₁₆	—	53	1	no	—
75	C₂×C₁₆	—	53	1	no	—
76	#107	—	53	1	no	—
77	#107	—	53	1	no	—
78	#107	C₂×D₅₃	1	1	yes	characteristic
79	#106	—	106	1	no	—
80	#102	—	2	1	no	—
81	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
82	C₂×D₄	—	2	1	no	—
83	C₂²×C₄	—	4	1	no	—
84	D₈	—	2	1	no	—
85	D₈	Dic₁₀₆	1	2	yes	—
86	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
87	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
88	C₄⋊C₄	—	2	1	no	—
89	C₄⋊C₄	—	106	1	no	—
90	C₄⋊C₄	—	106	1	no	—
91	C₄⋊C₄	—	212	1	no	—
92	C₄⋊C₄	—	212	1	no	—
93	C₂²⋊C₄	—	4	1	no	—
94	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
95	C₁₆	—	106	1	no	—
96	C₁₆	—	106	1	no	—
97	C₂³	—	4	1	no	—
98	D₄	—	2	2	no	—
99	D₄	—	4	1	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	—	8	2	no	—
102	C₂×C₄	—	212	1	no	—
103	C₂×C₄	—	212	1	no	—
104	C₂×C₄	#11719922	1	1	yes	characteristic
105	C₂×C₄	#11719922	1	1	yes	characteristic
106	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
107	C₈	—	2	1	no	—
108	C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
109	C₈	#11719945	1	1	yes	characteristic
110	C₂²	—	4	2	no	—
111	C₂²	—	8	1	no	—
112	C₂²	—	8	1	no	—
113	C₂²	C₂×#11719922	1	1	yes	characteristic
114	C₄	—	2	1	no	—
115	C₄	—	2	1	no	—
116	C₄	—	8	1	no	—
117	C₄	—	424	1	no	—
118	C₄	—	424	1	no	—
119	C₄	#13133196	1	1	yes	characteristic
120	C₄	#13133247	1	1	yes	characteristic
121	C₂	—	8	2	no	—
122	C₂	#16810903	1	1	yes	characteristic
123	C₂	#16811053	1	1	yes	characteristic
124	C₂	#16811059	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811025

C₂

yes

maximal, characteristic, complemented

#16810450

C₂

yes

maximal, characteristic, complemented

#16810448

C₂

yes

maximal, characteristic, complemented

C₄⋊#11719908

C₂

yes

maximal, characteristic, complemented

#16810424

C₂

yes

maximal, characteristic

#16810424

C₂

yes

maximal, characteristic

C₄×D₈×C₅₃

C₂

yes

maximal, characteristic

#13133209

—

C₂×#11719908

C₂²

yes

characteristic

C₂×#11719908

C₂²

yes

characteristic

#13133151

C₂²

yes

characteristic

#13133151

C₂²

yes

characteristic

#13133149

—

C₂×D₈×C₅₃

C₂²

yes

characteristic

C₄×D₄×C₅₃

—

#107×C₅₃

C₂²

yes

characteristic

#102×C₅₃

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×D₄×C₅₃

—

C₂²×C₄×C₅₃

—

D₈×C₅₃

—

D₈×C₅₃

Q₈

yes

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

D₄

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

—

—

—

—

C₂³×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

D₈

yes

characteristic

C₂×C₄×C₅₃

D₈

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

C₈×C₅₃

C₂×Q₈

yes

characteristic

C₈×C₅₃

#41

yes

characteristic

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×D₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₄×C₅₃

#137

yes

characteristic

Dic₅₃

—

Dic₅₃

—

#2228

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₄×D₈

D₅₃

yes

characteristic

#367

—

#366

—

C₄⋊C₁₆

—

#357

—

#357

—

C₅₃

#2228

yes

characteristic, complemented, Sylow

C₂×D₈

C₂×D₅₃

yes

characteristic

#125

—

106

—

C₄×D₄

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

—

#107

—

#107

C₂×D₅₃

yes

characteristic

#106

—

106

—

#102

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

D₈

—

D₈

Dic₁₀₆

yes

—

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

212

—

C₄⋊C₄

—

212

—

C₂²⋊C₄

—

C₄²

C₂²⋊D₅₃

yes

characteristic

C₁₆

—

106

—

C₁₆

—

106

—

C₂³

—

D₄

—

D₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

212

—

103

C₂×C₄

—

212

—

104

C₂×C₄

#11719922

yes

characteristic

105

C₂×C₄

#11719922

yes

characteristic

106

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

107

C₈

—

108

C₈

C₂×Dic₁₀₆

yes

characteristic

109

C₈

#11719945

yes

characteristic

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

C₂×#11719922

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

424

—

118

C₄

—

424

—

119

C₄

#13133196

yes

characteristic

120

C₄

#13133247

yes

characteristic

121

C₂

—

122

C₂

#16810903

yes

characteristic

123

C₂

#16811053

yes

characteristic

124

C₂

#16811059

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#11719922
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₄×D₈×C₅₃	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	C₄×D₈	D₅₃
hypocenter	C₅₃	#2228
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×D₈

subgroup

G/H

center

C₂²

C₂×#11719922

commutator

C₂×C₈×C₅₃

C₂³

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₄×D₈×C₅₃

C₂

socle

C₂²×C₅₃

C₂×D₈

hypercenter

C₄×D₈

D₅₃

hypocenter

C₅₃

#2228

LCS

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×D₈

N	H ≅ G/N	type
1. #16811025	C₂	semidirect
2. #16810450	C₂	semidirect
3. #16810448	C₂	semidirect
4. C₄⋊#11719908	C₂	semidirect
70. C₅₃	#2228	semidirect

H ≅ G/N

type

1. #16811025

C₂

semidirect

2. #16810450

C₂

semidirect

3. #16810448

C₂

semidirect

4. C₄⋊#11719908

C₂

semidirect

70. C₅₃

#2228

semidirect

Subgroups — #20276872

Conjugacy classes of subgroups

Complements