Group #20273437 — subgroups

Subgroups — #20273437

Order 6784.

246 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 81 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	#16810680	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810647	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810369	C₂	1	2	yes	maximal, complemented
4	#16810367	C₂	1	2	yes	maximal, complemented
5	C₅₃×#532	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719917	—	2	1	no	—
7	C₄⋊#88322	—	2	1	no	—
8	C₄⋊#88322	C₂²	1	2	yes	—
9	C₄⋊#88322	C₂²	1	2	yes	—
10	C₂²⋊#88322	—	2	1	no	—
11	#13133128	—	2	1	no	—
12	C₂×#41×C₅₃	—	2	1	no	—
13	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
14	#127×C₅₃	C₂²	1	1	yes	characteristic
15	#124×C₅₃	—	2	1	no	—
16	#121×C₅₃	—	2	1	no	—
17	#121×C₅₃	C₄	1	2	yes	—
18	C₄×D₄×C₅₃	—	2	1	no	—
19	C₄×D₄×C₅₃	C₄	1	2	yes	—
20	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
21	#41×C₅₃	—	4	4	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	#11719917	—	4	2	no	—
40	C₂×#88322	—	2	2	no	—
41	C₂×#88322	—	2	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	—	4	2	no	—
46	D₄×C₅₃	—	4	4	no	—
47	D₄×C₅₃	—	4	1	no	—
48	D₄×C₅₃	—	4	2	no	—
49	D₄×C₅₃	—	4	1	no	—
50	C₂×C₄×C₅₃	—	2	2	no	—
51	C₂×C₄×C₅₃	—	2	1	no	—
52	C₂×C₄×C₅₃	—	2	1	no	—
53	C₂×C₄×C₅₃	—	2	2	no	—
54	C₂×C₄×C₅₃	—	4	1	no	—
55	C₂×C₄×C₅₃	—	4	4	no	—
56	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
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	—	2	1	no	—
67	C₂²×C₅₃	—	4	2	no	—
68	C₂²×C₅₃	—	4	2	no	—
69	C₂²×C₅₃	—	4	2	no	—
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	—	2	1	no	—
75	C₄×C₅₃	—	4	2	no	—
76	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
77	C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
78	C₄×C₅₃	#137	1	1	yes	characteristic
79	#1557	—	53	1	no	maximal, Sylow
80	C₂×C₅₃	—	4	1	no	—
81	C₂×C₅₃	—	8	2	no	—
82	C₂×C₅₃	#413	1	1	yes	characteristic
83	C₂×C₅₃	#417	1	1	yes	characteristic
84	C₂×C₅₃	#418	1	1	yes	characteristic
85	#532	D₅₃	1	1	yes	characteristic
86	#432	—	53	1	no	—
87	#423	—	53	1	no	—
88	#332	—	53	2	no	—
89	#331	—	53	2	no	—
90	C₅₃	#1557	1	1	yes	characteristic, complemented, Sylow
91	C₂×#41	—	2	1	no	—
92	C₂×M₁₆	—	106	1	no	—
93	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
94	#127	C₂×D₅₃	1	1	yes	characteristic
95	#124	—	2	1	no	—
96	#121	—	2	1	no	—
97	#121	Dic₅₃	1	2	yes	—
98	C₄×D₄	—	2	1	no	—
99	C₄×D₄	Dic₅₃	1	2	yes	—
100	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
101	C₄⋊C₈	—	53	2	no	—
102	C₄⋊C₈	—	53	2	no	—
103	C₄⋊C₈	—	106	1	no	—
104	C₂²⋊C₈	—	106	1	no	—
105	#97	—	106	1	no	—
106	#41	—	4	4	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	M₁₆	—	212	2	no	—
117	C₂×C₈	—	106	2	no	—
118	C₂×C₈	—	106	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	—	4	2	no	—
131	D₄	—	4	4	no	—
132	D₄	—	4	1	no	—
133	D₄	—	4	2	no	—
134	D₄	—	4	1	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	4	1	no	—
140	C₂×C₄	—	4	4	no	—
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	—	2	1	no	—
152	C₂²	—	4	2	no	—
153	C₂²	—	4	2	no	—
154	C₂²	—	4	2	no	—
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	—	2	1	no	—
160	C₄	—	4	2	no	—
161	C₄	#13133242	1	1	yes	characteristic
162	C₄	#13133243	1	2	yes	—
163	C₄	#13133247	1	1	yes	characteristic
164	C₂	—	4	1	no	—
165	C₂	—	8	2	no	—
166	C₂	#16810607	1	1	yes	characteristic
167	C₂	#16810624	1	1	yes	characteristic
168	C₂	#16810628	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810680

C₂

yes

maximal, characteristic, complemented

#16810647

C₂

yes

maximal, characteristic, complemented

#16810369

C₂

yes

maximal, complemented

#16810367

C₂

yes

maximal, complemented

C₅₃×#532

C₂

yes

maximal, characteristic

C₂×#11719917

—

C₄⋊#88322

—

C₄⋊#88322

C₂²

yes

—

C₄⋊#88322

C₂²

yes

—

C₂²⋊#88322

—

#13133128

—

C₂×#41×C₅₃

—

C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

#127×C₅₃

C₂²

yes

characteristic

#124×C₅₃

—

#121×C₅₃

—

#121×C₅₃

C₄

yes

—

C₄×D₄×C₅₃

—

C₄×D₄×C₅₃

C₄

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

#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

—

D₄×C₅₃

—

C₂×C₄×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

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

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

AGL(1,ℤ/8)

yes

characteristic

C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₄×C₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

yes

—

—

—

—

C₅₃

#1557

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×M₁₆

—

106

—

C₄⋊Q₈

C₂×D₅₃

yes

characteristic

#127

C₂×D₅₃

yes

—

—

Dic₅₃

yes

—

C₄×D₄

—

C₄×D₄

Dic₅₃

yes

—

100

C₂×C₄²

C₂×D₅₃

yes

characteristic

101

C₄⋊C₈

—

102

C₄⋊C₈

—

103

C₄⋊C₈

—

106

—

104

C₂²⋊C₈

—

106

—

105

#97

—

106

—

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

M₁₆

—

212

—

117

C₂×C₈

—

106

—

118

C₂×C₈

—

106

—

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

D₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

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

—

155

C₂²

C₂×C₂²⋊Dic₅₃

yes

characteristic

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

#13133242

yes

characteristic

162

C₄

#13133243

yes

—

163

C₄

#13133247

yes

characteristic

164

C₂

—

165

C₂

—

166

C₂

#16810607

yes

characteristic

167

C₂

#16810624

yes

characteristic

168

C₂

#16810628

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₅₃
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	#1557
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₅₃

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 ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
1. #16810680	C₂	semidirect
2. #16810647	C₂	semidirect
3. #16810369	C₂	semidirect
4. #16810367	C₂	semidirect
90. C₅₃	#1557	semidirect

H ≅ G/N

type

1. #16810680

C₂

semidirect

2. #16810647

C₂

semidirect

3. #16810369

C₂

semidirect

4. #16810367

C₂

semidirect

90. C₅₃

#1557

semidirect

Subgroups — #20273437

Conjugacy classes of subgroups

Complements