Group #20276644 — subgroups

Subgroups — #20276644

Order 6784.

256 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 61 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	C₂×#13133161	C₂	1	2	yes	maximal, complemented
2	#16810864	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810424	C₂	1	2	yes	maximal, complemented
4	C₂²⋊#11719908	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₁₂	C₂²	1	1	yes	characteristic, complemented
7	#13133196	—	2	1	no	—
8	#13133161	—	2	4	no	—
9	C₂×#11719908	C₂²	1	2	yes	complemented
10	#13133151	C₂²	1	1	yes	characteristic, complemented
11	#13133142	—	2	1	no	—
12	C₂²×D₄×C₅₃	—	2	1	no	—
13	C₂×D₈×C₅₃	—	2	4	no	—
14	C₂×D₈×C₅₃	—	2	1	no	—
15	C₂×D₈×C₅₃	C₂²	1	2	yes	—
16	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂⁴×C₅₃	—	4	1	no	—
18	C₂×D₄×C₅₃	—	2	4	no	—
19	C₂×D₄×C₅₃	—	2	2	no	—
20	C₂×D₄×C₅₃	—	4	2	no	—
21	C₂×D₄×C₅₃	—	4	1	no	—
22	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
23	C₂×Dic₁₀₆	—	2	1	no	—
24	D₈×C₅₃	—	2	4	no	—
25	D₈×C₅₃	—	2	2	no	—
26	D₈×C₅₃	D₄	1	4	yes	—
27	C₂×C₈×C₅₃	—	2	2	no	—
28	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
29	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
30	C₂²⋊Dic₅₃	—	4	1	no	—
31	C₄⋊Dic₅₃	—	2	1	no	—
32	#11719919	—	4	1	no	—
33	Dic₂₁₂	—	2	2	no	—
34	#11719908	—	2	2	no	—
35	C₂³×C₅₃	—	4	4	no	—
36	C₂³×C₅₃	—	4	2	no	—
37	C₂³×C₅₃	—	8	2	no	—
38	C₂³×C₅₃	—	8	2	no	—
39	C₂³×C₅₃	D₈	1	1	yes	characteristic
40	D₄×C₅₃	—	2	4	no	—
41	D₄×C₅₃	—	4	4	no	—
42	D₄×C₅₃	—	4	2	no	—
43	C₂×C₄×C₅₃	—	2	1	no	—
44	C₂×C₄×C₅₃	—	2	1	no	—
45	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
46	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂×Dic₅₃	—	4	1	no	—
48	C₂×Dic₅₃	—	4	1	no	—
49	Dic₁₀₆	—	4	1	no	—
50	Dic₁₀₆	—	4	1	no	—
51	C₈×C₅₃	—	2	1	no	—
52	C₈×C₅₃	C₂×D₄	1	2	yes	—
53	C₂²×C₅₃	—	2	1	no	—
54	C₂²×C₅₃	—	2	1	no	—
55	C₂²×C₅₃	—	4	4	no	—
56	C₂²×C₅₃	—	8	8	no	—
57	C₂²×C₅₃	—	8	2	no	—
58	C₂²×C₅₃	—	8	2	no	—
59	C₂²×C₅₃	SD₃₂	1	2	yes	complemented
60	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
61	C₄×C₅₃	—	2	1	no	—
62	C₄×C₅₃	#120	1	1	yes	characteristic
63	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
64	Dic₅₃	—	8	1	no	—
65	Dic₅₃	—	8	1	no	—
66	#2187	—	53	1	no	maximal, Sylow
67	C₂×C₅₃	—	2	2	no	—
68	C₂×C₅₃	—	8	4	no	—
69	C₂×C₅₃	#447	1	1	yes	characteristic
70	C₂×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
71	C₂×C₅₃	#509	1	1	yes	characteristic
72	C₂²×D₈	D₅₃	1	1	yes	characteristic
73	C₂×SD₃₂	—	53	2	no	—
74	#467	—	53	1	no	—
75	#357	—	53	2	no	—
76	C₂²⋊C₁₆	—	53	1	no	—
77	C₅₃	#2187	1	1	yes	characteristic, complemented, Sylow
78	C₂²×D₄	—	2	1	no	—
79	C₂×Q₁₆	—	53	1	no	—
80	C₂×D₈	—	2	4	no	—
81	C₂×D₈	—	2	1	no	—
82	C₂×D₈	C₂×D₅₃	1	2	yes	—
83	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
84	C₂²⋊Q₈	—	106	1	no	—
85	SD₃₂	—	106	4	no	—
86	C₂×C₁₆	—	53	2	no	—
87	#107	—	53	1	no	—
88	#103	—	106	1	no	—
89	C₂⁴	—	4	1	no	—
90	C₂×Q₈	—	106	1	no	—
91	C₂×D₄	—	2	4	no	—
92	C₂×D₄	—	2	2	no	—
93	C₂×D₄	—	4	2	no	—
94	C₂×D₄	—	4	1	no	—
95	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
96	Q₁₆	—	106	2	no	—
97	D₈	—	2	4	no	—
98	D₈	—	2	2	no	—
99	D₈	C₂²⋊D₅₃	1	4	yes	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
102	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
103	C₄⋊C₄	—	106	1	no	—
104	C₄⋊C₄	—	212	1	no	—
105	C₂²⋊C₄	—	212	1	no	—
106	C₁₆	—	106	2	no	—
107	C₂³	—	4	4	no	—
108	C₂³	—	4	2	no	—
109	C₂³	—	8	2	no	—
110	C₂³	—	8	2	no	—
111	C₂³	#11719922	1	1	yes	characteristic
112	Q₈	—	212	1	no	—
113	Q₈	—	212	1	no	—
114	D₄	—	2	4	no	—
115	D₄	—	4	4	no	—
116	D₄	—	4	2	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	212	1	no	—
120	C₂×C₄	—	212	1	no	—
121	C₂×C₄	#11719922	1	1	yes	characteristic
122	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
125	C₂²	—	2	1	no	—
126	C₂²	—	2	1	no	—
127	C₂²	—	4	4	no	—
128	C₂²	—	8	8	no	—
129	C₂²	—	8	2	no	—
130	C₂²	—	8	2	no	—
131	C₂²	#13133161	1	2	yes	complemented
132	C₂²	C₂×#11719922	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	424	1	no	—
135	C₄	—	424	1	no	—
136	C₄	#13133186	1	1	yes	characteristic
137	C₄	#13133243	1	1	yes	characteristic
138	C₂	—	2	2	no	—
139	C₂	—	8	4	no	—
140	C₂	#16810757	1	1	yes	characteristic
141	C₂	C₂×#13133161	1	1	yes	characteristic
142	C₂	#16811059	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133161

C₂

yes

maximal, complemented

#16810864

C₂

yes

maximal, characteristic, complemented

#16810424

C₂

yes

maximal, complemented

C₂²⋊#11719908

C₂

yes

maximal, characteristic, complemented

C₂²×D₈×C₅₃

C₂

yes

maximal, characteristic

C₂×Dic₂₁₂

C₂²

yes

characteristic, complemented

#13133196

—

#13133161

—

C₂×#11719908

C₂²

yes

complemented

#13133151

C₂²

yes

characteristic, complemented

#13133142

—

C₂²×D₄×C₅₃

—

C₂×D₈×C₅₃

—

C₂×D₈×C₅₃

—

C₂×D₈×C₅₃

C₂²

yes

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂⁴×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂×Dic₁₀₆

—

D₈×C₅₃

—

D₈×C₅₃

—

D₈×C₅₃

D₄

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719919

—

Dic₂₁₂

—

#11719908

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

D₈

yes

characteristic

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

D₈

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

C₂×D₄

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

SD₃₂

yes

complemented

C₂²×C₅₃

C₂×D₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

#2187

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#447

yes

characteristic

C₂×C₅₃

C₂×SD₃₂

yes

characteristic

C₂×C₅₃

#509

yes

characteristic

C₂²×D₈

D₅₃

yes

characteristic

C₂×SD₃₂

—

#467

—

#357

—

C₂²⋊C₁₆

—

C₅₃

#2187

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

C₂×D₅₃

yes

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₂²⋊Q₈

—

106

—

SD₃₂

—

106

—

C₂×C₁₆

—

#107

—

#103

—

106

—

C₂⁴

—

C₂×Q₈

—

106

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

Q₁₆

—

106

—

D₈

—

D₈

—

D₈

C₂²⋊D₅₃

yes

—

100

C₂×C₈

—

101

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

102

C₂×C₈

C₂²×D₅₃

yes

characteristic

103

C₄⋊C₄

—

106

—

104

C₄⋊C₄

—

212

—

105

C₂²⋊C₄

—

212

—

106

C₁₆

—

106

—

107

C₂³

—

108

C₂³

—

109

C₂³

—

110

C₂³

—

111

C₂³

#11719922

yes

characteristic

112

Q₈

—

212

—

113

Q₈

—

212

—

114

D₄

—

115

D₄

—

116

D₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

212

—

120

C₂×C₄

—

212

—

121

C₂×C₄

#11719922

yes

characteristic

122

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

123

C₈

—

124

C₈

C₂×C₂²⋊D₅₃

yes

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

#13133161

yes

complemented

132

C₂²

C₂×#11719922

yes

characteristic

133

C₄

—

134

C₄

—

424

—

135

C₄

—

424

—

136

C₄

#13133186

yes

characteristic

137

C₄

#13133243

yes

characteristic

138

C₂

—

139

C₂

—

140

C₂

#16810757

yes

characteristic

141

C₂

C₂×#13133161

yes

characteristic

142

C₂

#16811059

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#11719922
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₂²×D₈×C₅₃	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	C₂²×D₈	D₅₃
hypocenter	C₅₃	#2187
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×D₈

subgroup

G/H

center

C₂²

C₂×#11719922

commutator

C₂×C₈×C₅₃

C₂³

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₂²×D₈×C₅₃

C₂

socle

C₂²×C₅₃

C₂×D₈

hypercenter

C₂²×D₈

D₅₃

hypocenter

C₅₃

#2187

LCS

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×D₈

N	H ≅ G/N	type
1. C₂×#13133161	C₂	semidirect
2. #16810864	C₂	semidirect
3. #16810424	C₂	semidirect
4. C₂²⋊#11719908	C₂	semidirect
6. C₂×Dic₂₁₂	C₂²	semidirect
9. C₂×#11719908	C₂²	semidirect
10. #13133151	C₂²	semidirect
59. C₂²×C₅₃	SD₃₂	semidirect
77. C₅₃	#2187	semidirect
131. C₂²	#13133161	semidirect

H ≅ G/N

type

1. C₂×#13133161

C₂

semidirect

2. #16810864

C₂

semidirect

3. #16810424

C₂

semidirect

4. C₂²⋊#11719908

C₂

semidirect

6. C₂×Dic₂₁₂

C₂²

semidirect

9. C₂×#11719908

C₂²

semidirect

10. #13133151

C₂²

semidirect

59. C₂²×C₅₃

SD₃₂

semidirect

77. C₅₃

#2187

semidirect

131. C₂²

#13133161

semidirect

Subgroups — #20276644

Conjugacy classes of subgroups

Complements