Group #20273114 — subgroups

Subgroups — #20273114

Order 6784.

260 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 97 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	#16810676	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄⋊#88322	C₂	1	4	yes	maximal, complemented
4	C₅₃×#532	C₂	1	1	yes	maximal, characteristic
5	C₂²×#88322	—	2	1	no	—
6	C₄⋊#88322	—	2	1	no	—
7	C₄⋊#88322	—	2	1	no	—
8	C₄⋊#88322	C₂²	1	2	yes	—
9	C₂²⋊#88322	—	2	1	no	—
10	C₄×#88322	C₂²	1	2	yes	—
11	C₂×#41×C₅₃	—	2	1	no	—
12	C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
13	#127×C₅₃	C₄	1	1	yes	characteristic
14	#124×C₅₃	—	2	1	no	—
15	#121×C₅₃	—	2	1	no	—
16	#121×C₅₃	C₄	1	2	yes	—
17	C₄×D₄×C₅₃	—	2	1	no	—
18	C₄×D₄×C₅₃	C₂²	1	2	yes	—
19	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
20	#41×C₅₃	—	4	2	no	—
21	#41×C₅₃	—	4	2	no	—
22	C₂×Q₈×C₅₃	—	2	1	no	—
23	C₂×D₄×C₅₃	—	2	2	no	—
24	C₂×D₄×C₅₃	—	2	1	no	—
25	C₂×D₄×C₅₃	—	2	1	no	—
26	C₂×D₄×C₅₃	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₅₃	—	2	2	no	—
28	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
31	C₄⋊C₄×C₅₃	—	2	1	no	—
32	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
33	C₂²⋊C₄×C₅₃	—	2	2	no	—
34	C₂²⋊C₄×C₅₃	—	2	2	no	—
35	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₄²×C₅₃	D₄	1	1	yes	characteristic
37	C₄²×C₅₃	D₄	1	1	yes	characteristic
38	C₄²×C₅₃	C₂³	1	1	yes	characteristic
39	C₂×#88322	—	2	2	no	—
40	C₂×#88322	—	2	2	no	—
41	C₂×#88322	—	4	2	no	—
42	C₂³×C₅₃	—	2	2	no	—
43	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
44	Q₈×C₅₃	—	4	2	no	—
45	D₄×C₅₃	—	2	4	no	—
46	D₄×C₅₃	—	4	2	no	—
47	D₄×C₅₃	—	4	4	no	—
48	D₄×C₅₃	—	4	2	no	—
49	C₂×C₄×C₅₃	—	2	2	no	—
50	C₂×C₄×C₅₃	—	2	2	no	—
51	C₂×C₄×C₅₃	—	4	1	no	—
52	C₂×C₄×C₅₃	—	4	2	no	—
53	C₂×C₄×C₅₃	—	4	2	no	—
54	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₅₃	D₈	1	2	yes	—
58	C₂×C₄×C₅₃	SD₁₆	1	2	yes	—
59	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
62	#88322	—	4	2	no	—
63	#88322	—	4	2	no	—
64	C₂²×C₅₃	—	2	1	no	—
65	C₂²×C₅₃	—	2	1	no	—
66	C₂²×C₅₃	—	4	2	no	—
67	C₂²×C₅₃	—	4	2	no	—
68	C₂²×C₅₃	—	4	2	no	—
69	C₂²×C₅₃	#102	1	2	yes	—
70	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₄×C₅₃	—	2	1	no	—
72	C₄×C₅₃	—	2	1	no	—
73	C₄×C₅₃	—	2	1	no	—
74	C₄×C₅₃	—	4	2	no	—
75	C₄×C₅₃	#102	1	2	yes	—
76	C₄×C₅₃	#131	1	2	yes	—
77	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
78	C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
79	#1480	—	53	1	no	maximal, Sylow
80	C₂×C₅₃	—	2	2	no	—
81	C₂×C₅₃	—	8	2	no	—
82	C₂×C₅₃	#408	1	1	yes	characteristic
83	C₂×C₅₃	C₂×#102	1	1	yes	characteristic
84	C₂×C₅₃	#421	1	1	yes	characteristic
85	#532	D₅₃	1	1	yes	characteristic
86	#431	—	53	1	no	—
87	C₂×C₄⋊C₈	—	53	1	no	—
88	D₄⋊C₈	—	53	4	no	—
89	C₅₃	#1480	1	1	yes	characteristic, complemented, Sylow
90	C₂×#41	—	2	1	no	—
91	C₂²×C₈	—	106	1	no	—
92	C₄⋊Q₈	Dic₅₃	1	1	yes	characteristic
93	#127	Dic₅₃	1	1	yes	characteristic
94	#124	—	2	1	no	—
95	#121	—	2	1	no	—
96	#121	Dic₅₃	1	2	yes	—
97	C₄×D₄	—	2	1	no	—
98	C₄×D₄	C₂×D₅₃	1	2	yes	—
99	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
100	C₄⋊C₈	—	53	2	no	—
101	C₄⋊C₈	—	106	1	no	—
102	C₄⋊C₈	—	106	1	no	—
103	C₂²⋊C₈	—	106	1	no	—
104	C₄×C₈	—	53	2	no	—
105	#41	—	4	2	no	—
106	#41	—	4	2	no	—
107	C₂×Q₈	—	2	1	no	—
108	C₂×D₄	—	2	2	no	—
109	C₂×D₄	—	2	1	no	—
110	C₂×D₄	—	2	1	no	—
111	C₂×D₄	C₂×Dic₅₃	1	2	yes	—
112	C₂²×C₄	—	2	2	no	—
113	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
114	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
115	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
116	C₂×C₈	—	106	2	no	—
117	C₂×C₈	—	106	2	no	—
118	C₂×C₈	—	212	2	no	—
119	C₄⋊C₄	—	2	1	no	—
120	C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
121	C₂²⋊C₄	—	2	2	no	—
122	C₂²⋊C₄	—	2	2	no	—
123	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
124	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
125	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
126	C₄²	C₂²×D₅₃	1	1	yes	characteristic
127	C₂³	—	2	2	no	—
128	C₂³	C₂²⋊Dic₅₃	1	1	yes	characteristic
129	Q₈	—	4	2	no	—
130	D₄	—	2	4	no	—
131	D₄	—	4	2	no	—
132	D₄	—	4	4	no	—
133	D₄	—	4	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	4	2	no	—
138	C₂×C₄	—	4	2	no	—
139	C₂×C₄	#11719922	1	2	yes	—
140	C₂×C₄	#11719923	1	2	yes	—
141	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
142	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
143	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
144	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
145	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
146	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
147	C₈	—	212	2	no	—
148	C₈	—	212	2	no	—
149	C₂²	—	2	1	no	—
150	C₂²	—	2	1	no	—
151	C₂²	—	4	2	no	—
152	C₂²	—	4	2	no	—
153	C₂²	—	4	2	no	—
154	C₂²	#13133138	1	2	yes	—
155	C₂²	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	—	2	1	no	—
159	C₄	—	4	2	no	—
160	C₄	#13133138	1	2	yes	—
161	C₄	#13133224	1	2	yes	—
162	C₄	C₂×#11719922	1	1	yes	characteristic
163	C₄	C₂×#11719923	1	1	yes	characteristic
164	C₂	—	2	2	no	—
165	C₂	—	8	2	no	—
166	C₂	#16810590	1	1	yes	characteristic
167	C₂	C₂×#13133138	1	1	yes	characteristic
168	C₂	#16810640	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810676

C₂

yes

maximal, characteristic, complemented

C₂×C₄⋊#88322

C₂

yes

maximal, characteristic, complemented

D₄⋊#88322

C₂

yes

maximal, complemented

C₅₃×#532

C₂

yes

maximal, characteristic

C₂²×#88322

—

C₄⋊#88322

—

C₄⋊#88322

—

C₄⋊#88322

C₂²

yes

—

C₂²⋊#88322

—

C₄×#88322

C₂²

yes

—

C₂×#41×C₅₃

—

C₄⋊Q₈×C₅₃

C₄

yes

characteristic

#127×C₅₃

C₄

yes

characteristic

#124×C₅₃

—

#121×C₅₃

—

#121×C₅₃

C₄

yes

—

C₄×D₄×C₅₃

—

C₄×D₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

#41×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

—

—

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂²⋊C₄

yes

characteristic

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

D₈

yes

—

C₂×C₄×C₅₃

SD₁₆

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#102

yes

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#102

yes

—

C₄×C₅₃

#131

yes

—

C₄×C₅₃

C₂×D₈

yes

characteristic

C₄×C₅₃

C₂×SD₁₆

yes

characteristic

#1480

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

yes

characteristic

#431

—

C₂×C₄⋊C₈

—

D₄⋊C₈

—

C₅₃

#1480

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂²×C₈

—

106

—

C₄⋊Q₈

Dic₅₃

yes

characteristic

#127

Dic₅₃

yes

—

—

Dic₅₃

yes

—

C₄×D₄

—

C₄×D₄

C₂×D₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

100

C₄⋊C₈

—

101

C₄⋊C₈

—

106

—

102

C₄⋊C₈

—

106

—

103

C₂²⋊C₈

—

106

—

104

C₄×C₈

—

105

#41

—

106

#41

—

107

C₂×Q₈

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂×D₄

C₂×Dic₅₃

yes

—

112

C₂²×C₄

—

113

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

114

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

115

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

116

C₂×C₈

—

106

—

117

C₂×C₈

—

106

—

118

C₂×C₈

—

212

—

119

C₄⋊C₄

—

120

C₄⋊C₄

C₂×Dic₅₃

yes

—

121

C₂²⋊C₄

—

122

C₂²⋊C₄

—

123

C₄²

C₂×Dic₅₃

yes

characteristic

124

C₄²

C₂²⋊D₅₃

yes

characteristic

125

C₄²

C₂²⋊D₅₃

yes

characteristic

126

C₄²

C₂²×D₅₃

yes

characteristic

127

C₂³

—

128

C₂³

C₂²⋊Dic₅₃

yes

characteristic

129

Q₈

—

130

D₄

—

131

D₄

—

132

D₄

—

133

D₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

#11719922

yes

—

140

C₂×C₄

#11719923

yes

—

141

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

142

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

143

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

144

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

145

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

146

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

147

C₈

—

212

—

148

C₈

—

212

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

154

C₂²

#13133138

yes

—

155

C₂²

C₂×C₂²⋊Dic₅₃

yes

characteristic

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

#13133138

yes

—

161

C₄

#13133224

yes

—

162

C₄

C₂×#11719922

yes

characteristic

163

C₄

C₂×#11719923

yes

characteristic

164

C₂

—

165

C₂

—

166

C₂

#16810590

yes

characteristic

167

C₂

C₂×#13133138

yes

characteristic

168

C₂

#16810640

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#532	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	#532	D₅₃
hypocenter	C₅₃	#1480
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#532

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
1. #16810676	C₂	semidirect
2. C₂×C₄⋊#88322	C₂	semidirect
3. D₄⋊#88322	C₂	semidirect
89. C₅₃	#1480	semidirect

H ≅ G/N

type

1. #16810676

C₂

semidirect

2. C₂×C₄⋊#88322

C₂

semidirect

3. D₄⋊#88322

C₂

semidirect

89. C₅₃

#1480

semidirect

Subgroups — #20273114

Conjugacy classes of subgroups

Complements