Group #20276754 — subgroups

Subgroups — #20276754

Order 6784.

174 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 174 conjugacy classes of subgroups — too many to draw. Only its 49 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×#13133162	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13133161	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810998	C₂	1	1	yes	maximal, characteristic
4	#16810717	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810441	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810423	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#357	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₂₁₂	C₂²	1	1	yes	characteristic
9	C₂×D₄₂₄	C₂²	1	1	yes	characteristic
10	C₂×#11719922	—	2	1	no	—
11	#13133213	—	2	1	no	—
12	#13133182	—	2	1	no	—
13	#13133162	—	2	2	no	—
14	#13133161	—	2	2	no	—
15	C₂×#11719908	C₂²	1	1	yes	characteristic
16	#13133140	—	2	1	no	—
17	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
18	C₂×D₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
20	#107×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×D₄×C₅₃	—	2	1	no	—
22	C₂×C₂²⋊D₅₃	—	4	1	no	—
23	C₂×C₂²⋊D₅₃	—	4	1	no	—
24	C₂×D₂₁₂	—	2	1	no	—
25	C₂×Dic₁₀₆	—	2	1	no	—
26	D₈×C₅₃	—	2	1	no	—
27	D₈×C₅₃	—	2	1	no	—
28	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₅₃	—	2	1	no	—
30	#11719922	—	4	2	no	—
31	#11719919	—	4	1	no	—
32	C₄×Dic₅₃	—	4	1	no	—
33	C₄×Dic₅₃	D₄	1	1	yes	characteristic
34	C₂×#88322	D₄	1	1	yes	characteristic
35	Dic₂₁₂	—	2	1	no	—
36	Dic₂₁₂	D₄	1	2	yes	complemented
37	D₄₂₄	—	2	1	no	—
38	D₄₂₄	—	2	1	no	—
39	C₁₆×C₅₃	—	2	1	no	—
40	#11719908	—	2	1	no	—
41	C₂³×C₅₃	—	4	1	no	—
42	C₂²×D₅₃	—	4	1	no	—
43	D₄×C₅₃	—	4	1	no	—
44	D₄×C₅₃	—	4	1	no	—
45	C₂×C₄×C₅₃	—	4	1	no	—
46	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂²⋊D₅₃	—	8	2	no	—
48	C₂²⋊D₅₃	—	8	2	no	—
49	C₂×Dic₅₃	—	4	1	no	—
50	C₂×Dic₅₃	D₈	1	1	yes	characteristic
51	C₂×Dic₅₃	D₈	1	1	yes	characteristic
52	D₂₁₂	—	4	1	no	—
53	D₂₁₂	—	4	1	no	—
54	Dic₁₀₆	—	2	2	no	—
55	Dic₁₀₆	—	4	1	no	—
56	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₈×C₅₃	#41	1	1	yes	characteristic
58	#88322	—	2	1	no	—
59	C₂²×C₅₃	—	8	1	no	—
60	C₂²×C₅₃	—	8	1	no	—
61	C₂²×C₅₃	—	8	1	no	—
62	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
63	C₂×D₅₃	—	8	1	no	—
64	C₂×D₅₃	—	8	1	no	—
65	C₂×D₅₃	—	8	1	no	—
66	C₄×C₅₃	—	8	1	no	—
67	C₄×C₅₃	#121	1	1	yes	characteristic
68	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
69	Dic₅₃	—	2	1	no	—
70	Dic₅₃	—	4	2	no	—
71	Dic₅₃	SD₃₂	1	2	yes	complemented
72	#2207	—	53	1	no	maximal, Sylow
73	C₂×C₅₃	—	16	1	no	—
74	C₂×C₅₃	#459	1	1	yes	characteristic
75	C₂×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
76	C₂×C₅₃	#509	1	1	yes	characteristic
77	D₅₃	—	16	1	no	—
78	C₂×SD₃₂	—	53	1	no	—
79	C₂×SD₃₂	—	53	1	no	—
80	#493	—	53	1	no	—
81	C₄×Q₁₆	—	53	1	no	—
82	C₄⋊C₁₆	—	53	1	no	—
83	#357	—	53	1	no	—
84	#357	D₅₃	1	1	yes	characteristic, complemented
85	C₅₃	#2207	1	1	yes	characteristic, complemented, Sylow
86	C₂×Q₁₆	—	53	1	no	—
87	C₂×D₈	—	53	1	no	—
88	C₂×D₈	—	106	1	no	—
89	C₂×D₈	C₂×D₅₃	1	1	yes	characteristic
90	#127	—	106	1	no	—
91	C₄×Q₈	—	106	1	no	—
92	SD₃₂	—	106	2	no	—
93	SD₃₂	—	106	2	no	—
94	C₂×C₁₆	—	53	1	no	—
95	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
96	#107	C₂×D₅₃	1	1	yes	characteristic
97	#103	—	106	1	no	—
98	C₄×C₈	—	53	1	no	—
99	C₂×Q₈	—	106	1	no	—
100	C₂×D₄	—	2	1	no	—
101	C₂×D₄	—	106	1	no	—
102	C₂×D₄	—	212	1	no	—
103	C₂×D₄	—	212	1	no	—
104	Q₁₆	—	53	2	no	—
105	Q₁₆	—	106	1	no	—
106	D₈	—	2	1	no	—
107	D₈	—	2	1	no	—
108	D₈	—	106	1	no	—
109	D₈	—	106	1	no	—
110	D₈	—	212	2	no	—
111	C₂×C₈	—	53	1	no	—
112	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	212	1	no	—
115	C₄²	—	53	1	no	—
116	C₄²	—	212	1	no	—
117	C₁₆	—	2	1	no	—
118	C₁₆	—	106	1	no	—
119	C₂³	—	4	1	no	—
120	C₂³	—	212	1	no	—
121	Q₈	—	106	2	no	—
122	Q₈	—	212	1	no	—
123	D₄	—	4	1	no	—
124	D₄	—	4	1	no	—
125	D₄	—	424	2	no	—
126	D₄	—	424	2	no	—
127	D₄	—	212	1	no	—
128	D₄	—	212	1	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	53	1	no	—
131	C₂×C₄	—	53	1	no	—
132	C₂×C₄	—	212	1	no	—
133	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
134	C₈	—	106	1	no	—
135	C₈	#11719945	1	1	yes	characteristic
136	C₈	D₄×D₅₃	1	1	yes	characteristic
137	C₂²	—	8	1	no	—
138	C₂²	—	8	1	no	—
139	C₂²	—	8	1	no	—
140	C₂²	—	424	1	no	—
141	C₂²	—	424	1	no	—
142	C₂²	—	424	1	no	—
143	C₂²	D₈×D₅₃	1	1	yes	characteristic
144	C₄	—	8	1	no	—
145	C₄	—	53	2	no	—
146	C₄	—	106	1	no	—
147	C₄	—	212	2	no	—
148	C₄	#13133190	1	1	yes	characteristic
149	C₄	#13133244	1	1	yes	characteristic
150	C₂	—	16	1	no	—
151	C₂	—	848	1	no	—
152	C₂	#16810817	1	1	yes	characteristic
153	C₂	SD₃₂×D₅₃	1	1	yes	characteristic
154	C₂	#16811057	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133162

C₂

yes

maximal, characteristic, complemented

C₂×#13133161

C₂

yes

maximal, characteristic, complemented

#16810998

C₂

yes

maximal, characteristic

#16810717

C₂

yes

maximal, characteristic, complemented

#16810441

C₂

yes

maximal, characteristic, complemented

#16810423

C₂

yes

maximal, characteristic, complemented

C₅₃×#357

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₁₂

C₂²

yes

characteristic

C₂×D₄₂₄

C₂²

yes

—

—

—

—

—

C₂²

yes

characteristic

#13133140

—

C₈×Dic₅₃

C₂²

yes

characteristic

C₂×D₈×C₅₃

C₂²

yes

characteristic

C₂×C₁₆×C₅₃

C₂²

yes

characteristic

#107×C₅₃

C₂²

yes

characteristic

C₂×D₄×C₅₃

—

C₂×C₂²⋊D₅₃

—

C₂×C₂²⋊D₅₃

—

C₂×D₂₁₂

—

C₂×Dic₁₀₆

—

D₈×C₅₃

—

D₈×C₅₃

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

#11719922

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

D₄

yes

characteristic

C₂×#88322

D₄

yes

characteristic

Dic₂₁₂

—

Dic₂₁₂

D₄

yes

complemented

D₄₂₄

—

D₄₂₄

—

C₁₆×C₅₃

—

#11719908

—

C₂³×C₅₃

—

C₂²×D₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²⋊D₅₃

—

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

D₈

yes

characteristic

C₂×Dic₅₃

D₈

yes

characteristic

D₂₁₂

—

D₂₁₂

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

C₂×D₄

yes

C₈×C₅₃

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×D₈

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

SD₃₂

yes

complemented

#2207

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#459

yes

characteristic

C₂×C₅₃

C₂×SD₃₂

yes

characteristic

C₂×C₅₃

#509

yes

characteristic

D₅₃

—

C₂×SD₃₂

—

C₂×SD₃₂

—

#493

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

#357

—

#357

D₅₃

yes

characteristic, complemented

C₅₃

#2207

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

—

106

—

C₂×D₈

C₂×D₅₃

yes

characteristic

#127

—

106

—

C₄×Q₈

—

106

—

SD₃₂

—

106

—

SD₃₂

—

106

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₅₃

yes

characteristic

#107

C₂×D₅₃

yes

characteristic

#103

—

106

—

C₄×C₈

—

C₂×Q₈

—

106

—

100

C₂×D₄

—

101

C₂×D₄

—

106

—

102

C₂×D₄

—

212

—

103

C₂×D₄

—

212

—

104

Q₁₆

—

105

Q₁₆

—

106

—

106

D₈

—

107

D₈

—

108

D₈

—

106

—

109

D₈

—

106

—

110

D₈

—

212

—

111

C₂×C₈

—

112

C₂×C₈

C₂²×D₅₃

yes

characteristic

113

C₄⋊C₄

—

114

C₄⋊C₄

—

212

—

115

C₄²

—

116

C₄²

—

212

—

117

C₁₆

—

118

C₁₆

—

106

—

119

C₂³

—

120

C₂³

—

212

—

121

Q₈

—

106

—

122

Q₈

—

212

—

123

D₄

—

124

D₄

—

125

D₄

—

424

—

126

D₄

—

424

—

127

D₄

—

212

—

128

D₄

—

212

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

212

—

133

C₂×C₄

D₄×D₅₃

yes

characteristic

134

C₈

—

106

—

135

C₈

#11719945

yes

characteristic

136

C₈

D₄×D₅₃

yes

characteristic

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

424

—

141

C₂²

—

424

—

142

C₂²

—

424

—

143

C₂²

D₈×D₅₃

yes

characteristic

144

C₄

—

145

C₄

—

146

C₄

—

106

—

147

C₄

—

212

—

148

C₄

#13133190

yes

characteristic

149

C₄

#13133244

yes

characteristic

150

C₂

—

151

C₂

—

848

—

152

C₂

#16810817

yes

characteristic

153

C₂

SD₃₂×D₅₃

yes

characteristic

154

C₂

#16811057

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₅₃
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#357	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	#357	D₅₃
hypocenter	C₅₃	#2207
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

subgroup

G/H

center

C₂²

D₈×D₅₃

commutator

C₂×C₈×C₅₃

C₂³

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₅₃×#357

C₂

socle

C₂²×C₅₃

C₂×D₈

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

N	H ≅ G/N	type
1. C₂×#13133162	C₂	semidirect
2. C₂×#13133161	C₂	semidirect
4. #16810717	C₂	semidirect
5. #16810441	C₂	semidirect
6. #16810423	C₂	semidirect
7. C₅₃×#357	C₂	semidirect
36. Dic₂₁₂	D₄	semidirect
71. Dic₅₃	SD₃₂	semidirect
84. #357	D₅₃	semidirect
85. C₅₃	#2207	semidirect

H ≅ G/N

type

1. C₂×#13133162

C₂

semidirect

2. C₂×#13133161

C₂

semidirect

4. #16810717

C₂

semidirect

5. #16810441

C₂

semidirect

6. #16810423

C₂

semidirect

7. C₅₃×#357

C₂

semidirect

36. Dic₂₁₂

D₄

semidirect

71. Dic₅₃

SD₃₂

semidirect

84. #357

D₅₃

semidirect

85. C₅₃

#2207

semidirect

Subgroups — #20276754

Conjugacy classes of subgroups

Complements