Group #20273939 — subgroups

Subgroups — #20273939

Order 6784.

200 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 59 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	Q₈×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
2	#16810974	C₂	1	1	yes	maximal, characteristic
3	#16810908	C₂	1	1	yes	maximal, characteristic
4	#16810834	C₂	1	1	yes	maximal, characteristic
5	#16810371	C₂	1	1	yes	maximal, characteristic
6	#16810355	C₂	1	1	yes	maximal, characteristic
7	C₅₃×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₁₂	—	2	1	no	—
9	#13133217	—	2	1	no	—
10	#13133217	C₂²	1	1	yes	characteristic
11	#13133217	C₂²	1	1	yes	characteristic
12	#13133216	—	2	1	no	—
13	#13133216	—	2	2	no	—
14	Q₈×Dic₅₃	—	2	1	no	—
15	C₄×Dic₁₀₆	—	2	1	no	—
16	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
17	#13133148	—	2	1	no	—
18	C₄⋊#88322	C₂²	1	1	yes	characteristic
19	#13133142	—	2	1	no	—
20	#13133142	—	2	2	no	—
21	#13133141	—	2	1	no	—
22	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×Q₈×C₅₃	D₄	1	1	yes	characteristic
26	C₂×Dic₁₀₆	—	2	1	no	—
27	C₂×Dic₁₀₆	—	2	2	no	—
28	C₂×Dic₁₀₆	D₄	1	1	yes	characteristic
29	C₂×Dic₁₀₆	D₄	1	2	yes	—
30	C₂×C₈×C₅₃	—	2	1	no	—
31	C₂×C₈×C₅₃	—	2	1	no	—
32	C₄⋊C₄×C₅₃	—	2	1	no	—
33	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
34	C₄²×C₅₃	—	2	1	no	—
35	C₄²×C₅₃	C₂³	1	1	yes	characteristic
36	C₄⋊Dic₅₃	—	2	2	no	—
37	C₄⋊Dic₅₃	—	2	1	no	—
38	C₄⋊Dic₅₃	—	2	1	no	—
39	C₄⋊Dic₅₃	—	2	1	no	—
40	C₄⋊Dic₅₃	—	2	1	no	—
41	C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
42	#11719919	—	2	2	no	—
43	#11719919	—	2	1	no	—
44	C₄×Dic₅₃	—	2	2	no	—
45	C₄×Dic₅₃	—	2	1	no	—
46	C₂×#88322	—	2	1	no	—
47	Dic₂₁₂	—	4	2	no	—
48	Q₈×C₅₃	—	2	1	no	—
49	Q₈×C₅₃	Q₁₆	1	2	yes	complemented
50	C₂×C₄×C₅₃	—	2	1	no	—
51	C₂×C₄×C₅₃	—	2	1	no	—
52	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂×Dic₅₃	—	2	1	no	—
56	C₂×Dic₅₃	—	2	1	no	—
57	C₂×Dic₅₃	—	2	2	no	—
58	C₂×Dic₅₃	—	4	1	no	—
59	Dic₁₀₆	—	2	1	no	—
60	Dic₁₀₆	—	4	2	no	—
61	Dic₁₀₆	—	4	1	no	—
62	Dic₁₀₆	—	4	1	no	—
63	Dic₁₀₆	—	8	2	no	—
64	Dic₁₀₆	SD₁₆	1	2	yes	—
65	C₈×C₅₃	—	4	1	no	—
66	C₈×C₅₃	—	4	1	no	—
67	#88322	—	4	1	no	—
68	C₂²×C₅₃	#120	1	1	yes	characteristic
69	C₄×C₅₃	—	2	2	no	—
70	C₄×C₅₃	—	2	1	no	—
71	C₄×C₅₃	—	4	1	no	—
72	C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
73	C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
74	C₄×C₅₃	#137	1	1	yes	characteristic
75	C₄×C₅₃	#137	1	1	yes	characteristic
76	Dic₅₃	—	2	2	no	—
77	Dic₅₃	—	4	2	no	—
78	Dic₅₃	—	4	1	no	—
79	Dic₅₃	—	8	1	no	—
80	#1645	—	53	1	no	maximal, Sylow
81	C₂×C₅₃	#448	1	1	yes	characteristic
82	C₂×C₅₃	#451	1	1	yes	characteristic
83	C₂×C₅₃	#455	1	1	yes	characteristic
84	Q₈²	—	53	1	no	—
85	#487	—	53	1	no	—
86	#475	—	53	1	no	—
87	#462	—	53	1	no	—
88	#333	—	53	1	no	—
89	Q₈⋊C₈	—	53	1	no	—
90	Q₈⋊C₈	D₅₃	1	1	yes	characteristic
91	C₅₃	#1645	1	1	yes	characteristic, complemented, Sylow
92	C₂×Q₁₆	—	106	1	no	—
93	C₄⋊Q₈	—	53	1	no	—
94	C₄⋊Q₈	—	53	1	no	—
95	C₄⋊Q₈	—	106	1	no	—
96	C₄⋊Q₈	—	106	2	no	—
97	C₄⋊Q₈	—	106	1	no	—
98	C₄×Q₈	—	53	1	no	—
99	C₄×Q₈	—	106	1	no	—
100	C₄×Q₈	—	106	1	no	—
101	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
102	#106	—	106	1	no	—
103	C₄⋊C₈	—	53	1	no	—
104	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
105	#103	—	106	1	no	—
106	#103	—	106	2	no	—
107	#103	—	106	1	no	—
108	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
109	C₂×Q₈	—	53	1	no	—
110	C₂×Q₈	—	53	2	no	—
111	C₂×Q₈	—	106	2	no	—
112	C₂×Q₈	—	106	1	no	—
113	C₂×Q₈	D₂₁₂	1	1	yes	characteristic
114	Q₁₆	—	212	2	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	106	1	no	—
118	C₄⋊C₄	—	2	1	no	—
119	C₄⋊C₄	—	53	1	no	—
120	C₄⋊C₄	—	106	2	no	—
121	C₄⋊C₄	—	106	2	no	—
122	C₄⋊C₄	—	106	1	no	—
123	C₄⋊C₄	—	106	1	no	—
124	C₄⋊C₄	—	106	1	no	—
125	C₄⋊C₄	—	106	1	no	—
126	C₄⋊C₄	—	106	1	no	—
127	C₄⋊C₄	D₂₁₂	1	1	yes	characteristic
128	C₄²	—	2	1	no	—
129	C₄²	—	106	2	no	—
130	C₄²	—	106	1	no	—
131	C₄²	C₂²×D₅₃	1	1	yes	characteristic
132	Q₈	—	2	1	no	—
133	Q₈	—	53	2	no	—
134	Q₈	—	106	1	no	—
135	Q₈	—	424	2	no	—
136	Q₈	—	212	2	no	—
137	Q₈	—	212	1	no	—
138	Q₈	—	212	1	no	—
139	Q₈	Dic₂₁₂	1	2	yes	complemented
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	106	1	no	—
143	C₂×C₄	—	106	1	no	—
144	C₂×C₄	—	106	2	no	—
145	C₂×C₄	—	212	1	no	—
146	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
147	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
148	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
149	C₈	—	4	1	no	—
150	C₈	—	4	1	no	—
151	C₈	—	212	1	no	—
152	C₂²	#13133187	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	—	2	1	no	—
155	C₄	—	4	1	no	—
156	C₄	—	106	2	no	—
157	C₄	—	424	1	no	—
158	C₄	—	212	1	no	—
159	C₄	—	212	2	no	—
160	C₄	SD₁₆×D₅₃	1	1	yes	characteristic
161	C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
162	C₄	#13133249	1	1	yes	characteristic
163	C₄	#13133250	1	1	yes	characteristic
164	C₂	#16810763	1	1	yes	characteristic
165	C₂	#16810781	1	1	yes	characteristic
166	C₂	#16810802	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₁₀₆

C₂

yes

maximal, characteristic

#16810974

C₂

yes

maximal, characteristic

#16810908

C₂

yes

maximal, characteristic

#16810834

C₂

yes

maximal, characteristic

#16810371

C₂

yes

maximal, characteristic

#16810355

C₂

yes

maximal, characteristic

C₅₃×Q₈⋊C₈

C₂

yes

maximal, characteristic

C₂×Dic₂₁₂

—

#13133217

—

#13133217

C₂²

yes

characteristic

#13133217

C₂²

yes

characteristic

#13133216

—

#13133216

—

Q₈×Dic₅₃

—

C₄×Dic₁₀₆

—

C₄×Dic₁₀₆

C₂²

yes

characteristic

#13133148

—

C₄⋊#88322

C₂²

yes

—

—

—

C₄×Q₈×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

D₄

yes

characteristic

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

D₄

yes

characteristic

C₂×Dic₁₀₆

D₄

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

—

C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

D₄

yes

characteristic

#11719919

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₂×#88322

—

Dic₂₁₂

—

Q₈×C₅₃

—

Q₈×C₅₃

Q₁₆

yes

complemented

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

SD₁₆

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

#120

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×SD₁₆

yes

characteristic

C₄×C₅₃

C₂×Q₁₆

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

Q₈²

—

—

—

—

—

Q₈⋊C₈

—

Q₈⋊C₈

D₅₃

yes

characteristic

C₅₃

#1645

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

106

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

106

—

C₄⋊Q₈

—

106

—

C₄⋊Q₈

—

106

—

C₄×Q₈

—

C₄×Q₈

—

106

—

100

C₄×Q₈

—

106

—

101

C₄×Q₈

C₂×D₅₃

yes

characteristic

102

#106

—

106

—

103

C₄⋊C₈

—

104

C₄⋊C₈

C₂×D₅₃

yes

characteristic

105

#103

—

106

—

106

#103

—

106

—

107

#103

—

106

—

108

C₄×C₈

C₂×D₅₃

yes

characteristic

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×Q₈

—

106

—

112

C₂×Q₈

—

106

—

113

C₂×Q₈

D₂₁₂

yes

characteristic

114

Q₁₆

—

212

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

106

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

106

—

121

C₄⋊C₄

—

106

—

122

C₄⋊C₄

—

106

—

123

C₄⋊C₄

—

106

—

124

C₄⋊C₄

—

106

—

125

C₄⋊C₄

—

106

—

126

C₄⋊C₄

—

106

—

127

C₄⋊C₄

D₂₁₂

yes

characteristic

128

C₄²

—

129

C₄²

—

106

—

130

C₄²

—

106

—

131

C₄²

C₂²×D₅₃

yes

characteristic

132

Q₈

—

133

Q₈

—

134

Q₈

—

106

—

135

Q₈

—

424

—

136

Q₈

—

212

—

137

Q₈

—

212

—

138

Q₈

—

212

—

139

Q₈

Dic₂₁₂

yes

complemented

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

106

—

143

C₂×C₄

—

106

—

144

C₂×C₄

—

106

—

145

C₂×C₄

—

212

—

146

C₂×C₄

C₂×D₂₁₂

yes

characteristic

147

C₂×C₄

D₄×D₅₃

yes

characteristic

148

C₂×C₄

D₄×D₅₃

yes

characteristic

149

C₈

—

150

C₈

—

151

C₈

—

212

—

152

C₂²

#13133187

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

106

—

157

C₄

—

424

—

158

C₄

—

212

—

159

C₄

—

212

—

160

C₄

SD₁₆×D₅₃

yes

characteristic

161

C₄

C₂×Dic₂₁₂

yes

characteristic

162

C₄

#13133249

yes

characteristic

163

C₄

#13133250

yes

characteristic

164

C₂

#16810763

yes

characteristic

165

C₂

#16810781

yes

characteristic

166

C₂

#16810802

yes

characteristic

	subgroup	G/H
center	C₂²	#13133187
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×Q₈⋊C₈	C₂
socle	C₂²×C₅₃	#120
hypercenter	Q₈⋊C₈	D₅₃
hypocenter	C₅₃	#1645
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ Q₈⋊C₈

subgroup

G/H

center

C₂²

#13133187

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×Q₈⋊C₈

C₂

socle

C₂²×C₅₃

#120

hypercenter

Q₈⋊C₈

D₅₃

hypocenter

C₅₃

#1645

LCS

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ Q₈⋊C₈

N	H ≅ G/N	type
49. Q₈×C₅₃	Q₁₆	semidirect
91. C₅₃	#1645	semidirect
139. Q₈	Dic₂₁₂	semidirect

H ≅ G/N

type

49. Q₈×C₅₃

Q₁₆

semidirect

91. C₅₃

#1645

semidirect

139. Q₈

Dic₂₁₂

semidirect

Subgroups — #20273939

Conjugacy classes of subgroups

Complements