Group #20273873 — subgroups

Subgroups — #20273873

Order 6784.

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

normal subgroups only

This group has 188 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	#16810975	C₂	1	1	yes	maximal, characteristic
2	#16810918	C₂	1	1	yes	maximal, characteristic
3	#16810918	C₂	1	1	yes	maximal, characteristic
4	#16810369	C₂	1	1	yes	maximal, characteristic
5	Q₈⋊#88322	C₂	1	1	yes	maximal, characteristic
6	Q₈⋊#88322	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#557	C₂	1	1	yes	maximal, characteristic
8	#13133217	C₂²	1	1	yes	characteristic
9	#13133150	—	2	1	no	—
10	#13133150	—	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	#13133141	—	2	1	no	—
15	#13133140	—	2	2	no	—
16	C₄×#88322	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₅₃	—	2	1	no	—
18	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
19	#125×C₅₃	—	2	2	no	—
20	#125×C₅₃	—	2	1	no	—
21	C₄×Q₈×C₅₃	—	2	1	no	—
22	C₄×Q₈×C₅₃	—	2	1	no	—
23	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×Q₈×C₅₃	—	2	1	no	—
26	C₂×Q₈×C₅₃	D₄	1	1	yes	characteristic
27	C₂×Q₈×C₅₃	D₄	1	1	yes	characteristic
28	C₂×Dic₁₀₆	—	2	1	no	—
29	C₄⋊C₄×C₅₃	—	2	2	no	—
30	C₄⋊C₄×C₅₃	—	2	2	no	—
31	C₄⋊C₄×C₅₃	—	2	2	no	—
32	C₄⋊C₄×C₅₃	—	2	1	no	—
33	C₄⋊C₄×C₅₃	—	2	1	no	—
34	C₄⋊C₄×C₅₃	—	2	1	no	—
35	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
36	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
37	C₄⋊C₄×C₅₃	D₄	1	2	yes	—
38	C₄²×C₅₃	—	2	2	no	—
39	C₄²×C₅₃	—	2	1	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	Q₈×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	Q₈×C₅₃	Q₁₆	1	2	yes	—
53	C₂×C₄×C₅₃	—	2	1	no	—
54	C₂×C₄×C₅₃	—	2	2	no	—
55	C₂×C₄×C₅₃	—	2	1	no	—
56	C₂×C₄×C₅₃	—	2	1	no	—
57	C₂×C₄×C₅₃	—	2	1	no	—
58	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₂×Dic₅₃	—	4	1	no	—
62	Dic₁₀₆	—	4	2	no	—
63	#88322	—	4	1	no	—
64	#88322	—	4	1	no	—
65	#88322	—	4	1	no	—
66	C₂²×C₅₃	#120	1	1	yes	characteristic
67	C₄×C₅₃	—	2	2	no	—
68	C₄×C₅₃	—	2	1	no	—
69	C₄×C₅₃	—	4	1	no	—
70	C₄×C₅₃	—	4	2	no	—
71	C₄×C₅₃	—	4	1	no	—
72	C₄×C₅₃	—	4	1	no	—
73	C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
74	C₄×C₅₃	#135	1	1	yes	characteristic
75	C₄×C₅₃	#137	1	1	yes	characteristic
76	C₄×C₅₃	#137	1	1	yes	characteristic
77	Dic₅₃	—	8	1	no	—
78	#1636	—	53	1	no	maximal, Sylow
79	C₂×C₅₃	#451	1	1	yes	characteristic
80	C₂×C₅₃	#452	1	1	yes	characteristic
81	C₂×C₅₃	#454	1	1	yes	characteristic
82	#557	D₅₃	1	1	yes	characteristic
83	#487	—	53	1	no	—
84	#477	—	53	1	no	—
85	#477	—	53	1	no	—
86	#332	—	53	1	no	—
87	Q₈⋊C₈	—	53	1	no	—
88	Q₈⋊C₈	—	53	1	no	—
89	C₅₃	#1636	1	1	yes	characteristic, complemented, Sylow
90	C₄⋊Q₈	—	2	1	no	—
91	C₄⋊Q₈	—	53	1	no	—
92	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
93	#125	—	2	2	no	—
94	#125	—	2	1	no	—
95	C₄×Q₈	—	2	1	no	—
96	C₄×Q₈	—	2	1	no	—
97	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
98	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
99	#107	—	106	1	no	—
100	#107	—	106	1	no	—
101	C₄⋊C₈	—	53	1	no	—
102	C₄⋊C₈	—	53	1	no	—
103	#103	—	106	1	no	—
104	#103	—	106	1	no	—
105	#103	—	106	2	no	—
106	C₄×C₈	—	53	1	no	—
107	C₂×Q₈	—	2	1	no	—
108	C₂×Q₈	—	106	1	no	—
109	C₂×Q₈	C₂²⋊D₅₃	1	1	yes	characteristic
110	C₂×Q₈	C₂²⋊D₅₃	1	1	yes	characteristic
111	C₂×C₈	—	106	1	no	—
112	C₂×C₈	—	106	1	no	—
113	C₂×C₈	—	106	1	no	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	—	2	2	no	—
116	C₄⋊C₄	—	2	2	no	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	2	1	no	—
119	C₄⋊C₄	—	2	1	no	—
120	C₄⋊C₄	—	106	1	no	—
121	C₄⋊C₄	—	106	1	no	—
122	C₄⋊C₄	C₂²⋊D₅₃	1	2	yes	—
123	C₄⋊C₄	C₂²⋊D₅₃	1	1	yes	characteristic
124	C₄⋊C₄	C₂²⋊D₅₃	1	1	yes	characteristic
125	C₄²	—	2	2	no	—
126	C₄²	—	2	1	no	—
127	C₄²	—	2	1	no	—
128	C₄²	C₂²×D₅₃	1	1	yes	characteristic
129	Q₈	—	2	1	no	—
130	Q₈	—	2	1	no	—
131	Q₈	—	2	1	no	—
132	Q₈	—	4	1	no	—
133	Q₈	—	4	1	no	—
134	Q₈	—	212	2	no	—
135	Q₈	#11719925	1	2	yes	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	212	1	no	—
142	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
143	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
144	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
145	C₈	—	212	1	no	—
146	C₈	—	212	1	no	—
147	C₈	—	212	1	no	—
148	C₂²	#13133186	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	2	1	no	—
151	C₄	—	4	1	no	—
152	C₄	—	4	2	no	—
153	C₄	—	4	1	no	—
154	C₄	—	4	1	no	—
155	C₄	—	424	1	no	—
156	C₄	C₂×#11719925	1	1	yes	characteristic
157	C₄	#13133237	1	1	yes	characteristic
158	C₄	#13133247	1	1	yes	characteristic
159	C₄	#13133248	1	1	yes	characteristic
160	C₂	#16810779	1	1	yes	characteristic
161	C₂	#16810785	1	1	yes	characteristic
162	C₂	#16810796	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810975

C₂

yes

maximal, characteristic

#16810918

C₂

yes

maximal, characteristic

#16810918

C₂

yes

maximal, characteristic

#16810369

C₂

yes

maximal, characteristic

Q₈⋊#88322

C₂

yes

maximal, characteristic

Q₈⋊#88322

C₂

yes

maximal, characteristic

C₅₃×#557

C₂

yes

maximal, characteristic

#13133217

C₂²

yes

—

—

C₂²

yes

characteristic

C₄⋊#88322

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

#125×C₅₃

—

#125×C₅₃

—

C₄×Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

C₄×Q₈×C₅₃

C₂²

yes

characteristic

C₄×Q₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

D₄

yes

characteristic

C₂×Q₈×C₅₃

D₄

yes

characteristic

C₂×Dic₁₀₆

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

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

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

Q₁₆

yes

—

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

#120

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×Q₁₆

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

yes

—

—

—

—

Q₈⋊C₈

—

Q₈⋊C₈

—

C₅₃

#1636

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₅₃

yes

characteristic

#125

—

#125

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂×D₅₃

yes

characteristic

C₄×Q₈

C₂×D₅₃

yes

characteristic

#107

—

106

—

100

#107

—

106

—

101

C₄⋊C₈

—

102

C₄⋊C₈

—

103

#103

—

106

—

104

#103

—

106

—

105

#103

—

106

—

106

C₄×C₈

—

107

C₂×Q₈

—

108

C₂×Q₈

—

106

—

109

C₂×Q₈

C₂²⋊D₅₃

yes

characteristic

110

C₂×Q₈

C₂²⋊D₅₃

yes

characteristic

111

C₂×C₈

—

106

—

112

C₂×C₈

—

106

—

113

C₂×C₈

—

106

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

106

—

121

C₄⋊C₄

—

106

—

122

C₄⋊C₄

C₂²⋊D₅₃

yes

—

123

C₄⋊C₄

C₂²⋊D₅₃

yes

characteristic

124

C₄⋊C₄

C₂²⋊D₅₃

yes

characteristic

125

C₄²

—

126

C₄²

—

127

C₄²

—

128

C₄²

C₂²×D₅₃

yes

characteristic

129

Q₈

—

130

Q₈

—

131

Q₈

—

132

Q₈

—

133

Q₈

—

134

Q₈

—

212

—

135

Q₈

#11719925

yes

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

212

—

142

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

143

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

144

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

145

C₈

—

212

—

146

C₈

—

212

—

147

C₈

—

212

—

148

C₂²

#13133186

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

424

—

156

C₄

C₂×#11719925

yes

characteristic

157

C₄

#13133237

yes

characteristic

158

C₄

#13133247

yes

characteristic

159

C₄

#13133248

yes

characteristic

160

C₂

#16810779

yes

characteristic

161

C₂

#16810785

yes

characteristic

162

C₂

#16810796

yes

characteristic

	subgroup	G/H
center	C₂²	#13133186
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#557	C₂
socle	C₂²×C₅₃	#120
hypercenter	#557	D₅₃
hypocenter	C₅₃	#1636
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #557

subgroup

G/H

center

C₂²

#13133186

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#557

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #557

N	H ≅ G/N	type
89. C₅₃	#1636	semidirect

H ≅ G/N

type

89. C₅₃

#1636

semidirect

Subgroups — #20273873

Conjugacy classes of subgroups

Complements