Group #20273887 — subgroups

Subgroups — #20273887

Order 6784.

194 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 57 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	#16810975	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810918	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810913	C₂	1	1	yes	maximal, characteristic
4	#16810369	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₈⋊#88322	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄⋊#88322	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#556	C₂	1	1	yes	maximal, characteristic
8	#13133217	C₂²	1	1	yes	characteristic
9	#13133150	—	2	1	no	—
10	#13133148	—	2	1	no	—
11	C₄⋊#88322	C₂²	1	1	yes	characteristic
12	C₄⋊#88322	C₂²	1	1	yes	characteristic
13	#13133141	—	2	1	no	—
14	#13133140	—	2	2	no	—
15	#13133138	—	2	1	no	—
16	C₄×#88322	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
18	#126×C₅₃	—	2	2	no	—
19	#124×C₅₃	—	2	1	no	—
20	C₂²⋊Q₈×C₅₃	—	2	1	no	—
21	C₄×Q₈×C₅₃	—	2	1	no	—
22	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
24	#117×C₅₃	—	2	1	no	—
25	C₂×Q₈×C₅₃	—	2	1	no	—
26	C₂×Q₈×C₅₃	D₄	1	1	yes	characteristic
27	C₂×D₄×C₅₃	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₅₃	—	2	1	no	—
29	C₂×Dic₁₀₆	—	2	1	no	—
30	C₄⋊C₄×C₅₃	—	2	2	no	—
31	C₄⋊C₄×C₅₃	—	2	1	no	—
32	C₄⋊C₄×C₅₃	—	2	1	no	—
33	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
34	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
35	C₄⋊C₄×C₅₃	D₄	1	2	yes	—
36	C₂²⋊C₄×C₅₃	—	2	2	no	—
37	C₂²⋊C₄×C₅₃	—	2	2	no	—
38	C₂²⋊C₄×C₅₃	—	2	1	no	—
39	C₄²×C₅₃	—	2	2	no	—
40	C₄²×C₅₃	—	2	1	no	—
41	C₄²×C₅₃	C₂³	1	1	yes	characteristic
42	C₄⋊Dic₅₃	—	2	1	no	—
43	C₄⋊Dic₅₃	—	2	1	no	—
44	C₂×#88322	—	2	1	no	—
45	C₂×#88322	—	2	1	no	—
46	C₂×#88322	—	2	1	no	—
47	C₂³×C₅₃	—	2	1	no	—
48	Q₈×C₅₃	—	2	1	no	—
49	Q₈×C₅₃	—	2	1	no	—
50	Q₈×C₅₃	—	4	1	no	—
51	Q₈×C₅₃	—	4	1	no	—
52	D₄×C₅₃	—	2	1	no	—
53	D₄×C₅₃	—	2	1	no	—
54	C₂×C₄×C₅₃	—	2	1	no	—
55	C₂×C₄×C₅₃	—	2	2	no	—
56	C₂×C₄×C₅₃	—	2	1	no	—
57	C₂×C₄×C₅₃	—	2	1	no	—
58	C₂×C₄×C₅₃	—	4	1	no	—
59	C₂×C₄×C₅₃	—	4	1	no	—
60	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
63	C₂×Dic₅₃	—	4	1	no	—
64	Dic₁₀₆	—	4	2	no	—
65	#88322	—	4	1	no	—
66	#88322	—	4	1	no	—
67	#88322	—	4	1	no	—
68	C₂²×C₅₃	—	4	1	no	—
69	C₂²×C₅₃	—	4	1	no	—
70	C₂²×C₅₃	—	4	1	no	—
71	C₂²×C₅₃	#120	1	1	yes	characteristic
72	C₄×C₅₃	—	2	1	no	—
73	C₄×C₅₃	—	4	1	no	—
74	C₄×C₅₃	—	4	1	no	—
75	C₄×C₅₃	—	4	2	no	—
76	C₄×C₅₃	—	4	1	no	—
77	C₄×C₅₃	#135	1	1	yes	characteristic
78	C₄×C₅₃	#135	1	1	yes	characteristic
79	C₄×C₅₃	#137	1	1	yes	characteristic
80	C₄×C₅₃	#137	1	1	yes	characteristic
81	Dic₅₃	—	8	1	no	—
82	#1638	—	53	1	no	maximal, Sylow
83	C₂×C₅₃	—	8	1	no	—
84	C₂×C₅₃	#452	1	1	yes	characteristic
85	C₂×C₅₃	#452	1	1	yes	characteristic
86	C₂×C₅₃	#454	1	1	yes	characteristic
87	#556	D₅₃	1	1	yes	characteristic
88	#487	—	53	1	no	—
89	#477	—	53	1	no	—
90	#476	—	53	1	no	—
91	#332	—	53	1	no	—
92	Q₈⋊C₈	—	53	1	no	—
93	D₄⋊C₈	—	53	1	no	—
94	C₅₃	#1638	1	1	yes	characteristic, complemented, Sylow
95	C₄⋊Q₈	—	53	1	no	—
96	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
97	#126	—	2	2	no	—
98	#124	—	2	1	no	—
99	C₂²⋊Q₈	—	2	1	no	—
100	C₄×Q₈	—	2	1	no	—
101	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
102	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
103	#117	—	2	1	no	—
104	#107	—	106	1	no	—
105	#106	—	106	1	no	—
106	C₄⋊C₈	—	53	1	no	—
107	C₄⋊C₈	—	53	1	no	—
108	#103	—	106	1	no	—
109	#103	—	106	2	no	—
110	#102	—	106	1	no	—
111	C₄×C₈	—	53	1	no	—
112	C₂×Q₈	—	2	1	no	—
113	C₂×Q₈	—	106	1	no	—
114	C₂×Q₈	C₂²⋊D₅₃	1	1	yes	characteristic
115	C₂×D₄	C₂²⋊D₅₃	1	1	yes	characteristic
116	C₂²×C₄	—	2	1	no	—
117	C₂×C₈	—	106	1	no	—
118	C₂×C₈	—	106	1	no	—
119	C₂×C₈	—	106	1	no	—
120	C₄⋊C₄	—	2	2	no	—
121	C₄⋊C₄	—	2	1	no	—
122	C₄⋊C₄	—	2	1	no	—
123	C₄⋊C₄	—	106	1	no	—
124	C₄⋊C₄	—	106	1	no	—
125	C₄⋊C₄	C₂²⋊D₅₃	1	2	yes	—
126	C₄⋊C₄	C₂²⋊D₅₃	1	1	yes	characteristic
127	C₄⋊C₄	C₂²⋊D₅₃	1	1	yes	characteristic
128	C₂²⋊C₄	—	2	2	no	—
129	C₂²⋊C₄	—	2	2	no	—
130	C₂²⋊C₄	—	2	1	no	—
131	C₄²	—	2	2	no	—
132	C₄²	—	2	1	no	—
133	C₄²	C₂²×D₅₃	1	1	yes	characteristic
134	C₂³	—	2	1	no	—
135	Q₈	—	2	1	no	—
136	Q₈	—	2	1	no	—
137	Q₈	—	4	1	no	—
138	Q₈	—	4	1	no	—
139	Q₈	—	212	2	no	—
140	D₄	—	2	1	no	—
141	D₄	—	2	1	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	4	1	no	—
147	C₂×C₄	—	4	1	no	—
148	C₂×C₄	—	212	1	no	—
149	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
150	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
151	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
152	C₈	—	212	1	no	—
153	C₈	—	212	1	no	—
154	C₈	—	212	1	no	—
155	C₂²	—	4	1	no	—
156	C₂²	—	4	1	no	—
157	C₂²	—	4	1	no	—
158	C₂²	#13133186	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	4	1	no	—
161	C₄	—	4	1	no	—
162	C₄	—	4	2	no	—
163	C₄	—	4	1	no	—
164	C₄	—	424	1	no	—
165	C₄	#13133237	1	1	yes	characteristic
166	C₄	#13133237	1	1	yes	characteristic
167	C₄	#13133247	1	1	yes	characteristic
168	C₄	#13133247	1	1	yes	characteristic
169	C₂	—	8	1	no	—
170	C₂	#16810785	1	1	yes	characteristic
171	C₂	#16810785	1	1	yes	characteristic
172	C₂	#16810796	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810975

C₂

yes

maximal, characteristic, complemented

#16810918

C₂

yes

maximal, characteristic, complemented

#16810913

C₂

yes

maximal, characteristic

#16810369

C₂

yes

maximal, characteristic, complemented

Q₈⋊#88322

C₂

yes

maximal, characteristic, complemented

D₄⋊#88322

C₂

yes

maximal, characteristic

C₅₃×#556

C₂

yes

maximal, characteristic

#13133217

C₂²

yes

—

—

C₂²

yes

characteristic

C₄⋊#88322

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

#126×C₅₃

—

#124×C₅₃

—

C₂²⋊Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

C₄×Q₈×C₅₃

C₂²

yes

characteristic

C₄×D₄×C₅₃

C₂²

yes

characteristic

#117×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

D₄

yes

characteristic

C₂×D₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

—

C₂×Dic₁₀₆

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

D₄

yes

characteristic

C₄⋊C₄×C₅₃

D₄

yes

characteristic

C₄⋊C₄×C₅₃

D₄

yes

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

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

—

Dic₁₀₆

—

#88322

—

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#120

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

#1638

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

yes

—

—

—

—

Q₈⋊C₈

—

D₄⋊C₈

—

C₅₃

#1638

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₅₃

yes

characteristic

#126

—

#124

—

C₂²⋊Q₈

—

100

C₄×Q₈

—

101

C₄×Q₈

C₂×D₅₃

yes

characteristic

102

C₄×D₄

C₂×D₅₃

yes

characteristic

103

#117

—

104

#107

—

106

—

105

#106

—

106

—

106

C₄⋊C₈

—

107

C₄⋊C₈

—

108

#103

—

106

—

109

#103

—

106

—

110

#102

—

106

—

111

C₄×C₈

—

112

C₂×Q₈

—

113

C₂×Q₈

—

106

—

114

C₂×Q₈

C₂²⋊D₅₃

yes

characteristic

115

C₂×D₄

C₂²⋊D₅₃

yes

characteristic

116

C₂²×C₄

—

117

C₂×C₈

—

106

—

118

C₂×C₈

—

106

—

119

C₂×C₈

—

106

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

106

—

124

C₄⋊C₄

—

106

—

125

C₄⋊C₄

C₂²⋊D₅₃

yes

—

126

C₄⋊C₄

C₂²⋊D₅₃

yes

characteristic

127

C₄⋊C₄

C₂²⋊D₅₃

yes

characteristic

128

C₂²⋊C₄

—

129

C₂²⋊C₄

—

130

C₂²⋊C₄

—

131

C₄²

—

132

C₄²

—

133

C₄²

C₂²×D₅₃

yes

characteristic

134

C₂³

—

135

Q₈

—

136

Q₈

—

137

Q₈

—

138

Q₈

—

139

Q₈

—

212

—

140

D₄

—

141

D₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

212

—

149

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

150

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

151

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

152

C₈

—

212

—

153

C₈

—

212

—

154

C₈

—

212

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₂²

#13133186

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

—

424

—

165

C₄

#13133237

yes

characteristic

166

C₄

#13133237

yes

characteristic

167

C₄

#13133247

yes

characteristic

168

C₄

#13133247

yes

characteristic

169

C₂

—

170

C₂

#16810785

yes

characteristic

171

C₂

#16810785

yes

characteristic

172

C₂

#16810796

yes

characteristic

	subgroup	G/H
center	C₂²	#13133186
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#556	C₂
socle	C₂²×C₅₃	#120
hypercenter	#556	D₅₃
hypocenter	C₅₃	#1638
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #556

subgroup

G/H

center

C₂²

#13133186

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#556

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #556

N	H ≅ G/N	type
1. #16810975	C₂	semidirect
2. #16810918	C₂	semidirect
4. #16810369	C₂	semidirect
5. Q₈⋊#88322	C₂	semidirect
94. C₅₃	#1638	semidirect

H ≅ G/N

type

1. #16810975

C₂

semidirect

2. #16810918

C₂

semidirect

4. #16810369

C₂

semidirect

5. Q₈⋊#88322

C₂

semidirect

94. C₅₃

#1638

semidirect

Subgroups — #20273887

Conjugacy classes of subgroups

Complements