Group #27392479 — subgroups

Subgroups — #27392479

Order 13568.

1230 conjugacy classes of subgroups; only the 367 normal subgroups are shown.

normal subgroups only

This group has 1230 conjugacy classes of subgroups — too many to draw. Only its 367 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	C₂×#16810701	C₂	1	6	yes	maximal, complemented
2	#20275510	C₂	1	6	yes	maximal, complemented
3	#20274867	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄²×#88322	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×#127×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719917	C₂²	1	4	yes	—
7	#16810701	—	2	24	no	—
8	C₂×C₄⋊#88322	C₂²	1	6	yes	—
9	C₂×C₂²⋊#88322	C₂²	1	12	yes	—
10	C₂×C₄×#88322	C₂²	1	3	yes	—
11	C₂×C₄×#88322	C₂²	1	3	yes	—
12	C₂×#127×C₅₃	C₄	1	1	yes	characteristic
13	C₂×C₄×D₄×C₅₃	C₂²	1	6	yes	—
14	C₅₃×#390	C₄	1	6	yes	—
15	C₅₃×#384	C₄	1	1	yes	characteristic
16	C₄³×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#11719917	—	2	24	no	—
18	C₂²×#88322	C₂³	1	4	yes	—
19	C₂²×#88322	C₂³	1	3	yes	—
20	C₂²×#88322	C₂³	1	1	yes	characteristic
21	C₄⋊#88322	—	2	12	no	—
22	C₂²⋊#88322	—	2	24	no	—
23	C₄×#88322	—	2	6	no	—
24	C₄×#88322	D₄	1	12	yes	complemented
25	C₂²×D₄×C₅₃	C₂×C₄	1	6	yes	—
26	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
27	#127×C₅₃	—	2	4	no	—
28	C₄×D₄×C₅₃	—	2	12	no	—
29	C₄×D₄×C₅₃	—	2	12	no	—
30	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	6	yes	—
31	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	12	yes	—
32	C₂×C₄²×C₅₃	C₂×C₄	1	3	yes	—
33	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
35	C₂⁴×C₅₃	C₂²×C₄	1	4	yes	—
36	C₂×D₄×C₅₃	—	2	12	no	—
37	C₂×D₄×C₅₃	—	2	12	no	—
38	C₂×D₄×C₅₃	—	2	12	no	—
39	C₂²×C₄×C₅₃	—	2	12	no	—
40	C₂²×C₄×C₅₃	—	2	12	no	—
41	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
42	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
43	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
44	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
45	C₄⋊C₄×C₅₃	—	2	12	no	—
46	C₂²⋊C₄×C₅₃	—	2	24	no	—
47	C₄²×C₅₃	—	2	6	no	—
48	C₄²×C₅₃	M₁₆	1	4	yes	complemented
49	C₄²×C₅₃	C₂×D₄	1	6	yes	—
50	C₄²×C₅₃	#41	1	6	yes	—
51	#11719917	—	4	16	no	—
52	C₂×#88322	—	2	6	no	—
53	C₂×#88322	—	2	12	no	—
54	C₂×#88322	C₂×D₄	1	6	yes	—
55	C₂×#88322	C₂×D₄	1	6	yes	—
56	C₂³×C₅₃	—	2	12	no	—
57	C₂³×C₅₃	—	2	4	no	—
58	C₂³×C₅₃	—	2	12	no	—
59	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
60	D₄×C₅₃	—	4	24	no	—
61	C₂×C₄×C₅₃	—	2	6	no	—
62	C₂×C₄×C₅₃	—	2	12	no	—
63	C₂×C₄×C₅₃	—	4	4	no	—
64	C₂×C₄×C₅₃	—	4	12	no	—
65	C₂×C₄×C₅₃	C₄×D₄	1	6	yes	—
66	C₂×C₄×C₅₃	C₄×D₄	1	6	yes	—
67	C₂×C₄×C₅₃	C₂×M₁₆	1	6	yes	—
68	C₂×C₄×C₅₃	#131	1	6	yes	—
69	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
70	C₂×C₄×C₅₃	C₂×#41	1	3	yes	—
71	#88322	—	2	6	no	—
72	#88322	—	4	4	no	—
73	#88322	#127	1	4	yes	complemented
74	#21681	—	53	1	no	maximal, Sylow
75	C₂²×C₅₃	—	4	4	no	—
76	C₂²×C₅₃	—	4	12	no	—
77	C₂²×C₅₃	—	4	12	no	—
78	C₂²×C₅₃	C₂×C₄×D₄	1	3	yes	—
79	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
80	C₂²×C₅₃	C₂×#131	1	3	yes	—
81	C₄×C₅₃	—	2	6	no	—
82	C₄×C₅₃	—	4	4	no	—
83	C₄×C₅₃	#436	1	12	yes	complemented
84	C₄×C₅₃	C₂×#127	1	1	yes	characteristic
85	C₄×C₅₃	#532	1	3	yes	—
86	C₂×#436	—	53	6	no	—
87	C₄×#127	D₅₃	1	1	yes	characteristic
88	#1954	—	53	6	no	—
89	#1842	—	53	1	no	—
90	C₄²×C₈	—	53	1	no	—
91	C₂×C₅₃	—	8	4	no	—
92	C₂×C₅₃	C₄×#127	1	1	yes	characteristic
93	C₂×C₅₃	C₂×#436	1	3	yes	—
94	C₂×C₅₃	#2931	1	3	yes	—
95	C₂²×M₁₆	—	53	4	no	—
96	C₂×#127	Dic₅₃	1	1	yes	characteristic
97	C₂×C₄×D₄	C₂×D₅₃	1	6	yes	—
98	#436	—	106	24	no	—
99	C₂×C₄⋊C₈	—	53	6	no	—
100	C₂×C₂²⋊C₈	—	53	12	no	—
101	C₂×C₄×C₈	—	53	3	no	—
102	C₂×C₄×C₈	—	53	3	no	—
103	#390	Dic₅₃	1	6	yes	—
104	#384	Dic₅₃	1	1	yes	characteristic
105	C₄³	C₂×D₅₃	1	1	yes	characteristic
106	C₅₃	#21681	1	1	yes	characteristic, complemented, Sylow
107	C₂²×D₄	C₂×Dic₅₃	1	6	yes	—
108	C₂³×C₄	C₂²×D₅₃	1	4	yes	—
109	C₂×M₁₆	—	106	24	no	—
110	C₂²×C₈	—	53	1	no	—
111	C₂²×C₈	—	53	3	no	—
112	C₂²×C₈	—	53	4	no	—
113	#127	—	2	4	no	—
114	C₄×D₄	—	2	12	no	—
115	C₄×D₄	—	2	12	no	—
116	C₂×C₄⋊C₄	C₂×Dic₅₃	1	6	yes	—
117	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	12	yes	—
118	C₂×C₄²	C₂×Dic₅₃	1	3	yes	—
119	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
120	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
121	C₄⋊C₈	—	106	12	no	—
122	C₂²⋊C₈	—	106	24	no	—
123	C₄×C₈	—	53	12	no	—
124	C₄×C₈	—	106	6	no	—
125	C₂⁴	C₂²×Dic₅₃	1	4	yes	—
126	C₂×D₄	—	2	12	no	—
127	C₂×D₄	—	2	12	no	—
128	C₂×D₄	—	2	12	no	—
129	C₂²×C₄	—	2	12	no	—
130	C₂²×C₄	—	2	12	no	—
131	C₂²×C₄	C₂²×Dic₅₃	1	4	yes	—
132	C₂²×C₄	C₂²×Dic₅₃	1	3	yes	—
133	C₂²×C₄	C₂²×Dic₅₃	1	3	yes	—
134	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
135	M₁₆	—	212	16	no	—
136	C₂×C₈	—	53	6	no	—
137	C₂×C₈	—	53	6	no	—
138	C₂×C₈	—	106	6	no	—
139	C₂×C₈	—	106	12	no	—
140	C₄⋊C₄	—	2	12	no	—
141	C₂²⋊C₄	—	2	24	no	—
142	C₄²	—	2	6	no	—
143	C₄²	#11719917	1	4	yes	complemented
144	C₄²	D₄×D₅₃	1	6	yes	—
145	C₄²	#11719947	1	6	yes	—
146	C₂³	—	2	12	no	—
147	C₂³	—	2	4	no	—
148	C₂³	—	2	12	no	—
149	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
150	D₄	—	4	24	no	—
151	C₂×C₄	—	2	6	no	—
152	C₂×C₄	—	2	12	no	—
153	C₂×C₄	—	4	4	no	—
154	C₂×C₄	—	4	12	no	—
155	C₂×C₄	D₄×Dic₅₃	1	6	yes	—
156	C₂×C₄	D₄×Dic₅₃	1	6	yes	—
157	C₂×C₄	C₂×#11719917	1	6	yes	—
158	C₂×C₄	#13133224	1	6	yes	—
159	C₂×C₄	C₂×D₄×D₅₃	1	3	yes	—
160	C₂×C₄	C₂×#11719947	1	3	yes	—
161	C₈	—	53	4	no	—
162	C₈	—	106	6	no	—
163	C₈	—	212	4	no	—
164	C₂²	—	4	12	no	—
165	C₂²	—	4	4	no	—
166	C₂²	—	4	12	no	—
167	C₂²	C₂×D₄×Dic₅₃	1	3	yes	—
168	C₂²	C₂²×#11719917	1	1	yes	characteristic
169	C₂²	C₂×#13133224	1	3	yes	—
170	C₄	—	2	6	no	—
171	C₄	—	4	4	no	—
172	C₄	#16810701	1	12	yes	complemented
173	C₄	#127×D₅₃	1	1	yes	characteristic
174	C₄	#16811171	1	3	yes	—
175	C₂	—	8	4	no	—
176	C₂	#127×Dic₅₃	1	1	yes	characteristic
177	C₂	C₂×#16810701	1	3	yes	—
178	C₂	#20283427	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16810701

C₂

yes

maximal, complemented

#20275510

C₂

yes

maximal, complemented

#20274867

C₂

yes

maximal, characteristic, complemented

C₄²×#88322

C₂

yes

maximal, characteristic, complemented

C₄×#127×C₅₃

C₂

yes

maximal, characteristic

C₂²×#11719917

C₂²

yes

—

#16810701

—

C₂×C₄⋊#88322

C₂²

yes

—

C₂×C₂²⋊#88322

C₂²

yes

—

C₂×C₄×#88322

C₂²

yes

—

C₂×C₄×#88322

C₂²

yes

—

C₂×#127×C₅₃

C₄

yes

characteristic

C₂×C₄×D₄×C₅₃

C₂²

yes

—

C₅₃×#390

C₄

yes

—

C₅₃×#384

C₄

yes

characteristic

C₄³×C₅₃

C₂²

yes

characteristic

C₂×#11719917

—

C₂²×#88322

C₂³

yes

—

C₂²×#88322

C₂³

yes

—

C₂²×#88322

C₂³

yes

—

—

—

D₄

yes

complemented

C₂²×D₄×C₅₃

C₂×C₄

yes

—

C₂³×C₄×C₅₃

C₂³

yes

—

#127×C₅₃

—

C₄×D₄×C₅₃

—

C₄×D₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₂×C₂²⋊C₄×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₅₃

C₂³

yes

—

C₂⁴×C₅₃

C₂²×C₄

yes

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

—

C₂²×C₄×C₅₃

C₂⁴

yes

characteristic

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

M₁₆

yes

complemented

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

#41

yes

—

—

—

—

C₂×D₄

yes

—

C₂×#88322

C₂×D₄

yes

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂³×C₄

yes

characteristic

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄×D₄

yes

—

C₂×C₄×C₅₃

C₄×D₄

yes

—

C₂×C₄×C₅₃

C₂×M₁₆

yes

—

C₂×C₄×C₅₃

#131

yes

—

C₂×C₄×C₅₃

C₂²×D₄

yes

—

C₂×C₄×C₅₃

C₂×#41

yes

—

—

—

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₄×D₄

yes

—

C₂²×C₅₃

C₂²×M₁₆

yes

characteristic

C₂²×C₅₃

C₂×#131

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₂×#436

—

C₄×#127

D₅₃

yes

characteristic

#1954

—

#1842

—

C₄²×C₈

—

C₂×C₅₃

—

C₂×C₅₃

C₄×#127

yes

characteristic

C₂×C₅₃

C₂×#436

yes

—

C₂×C₅₃

#2931

yes

—

C₂²×M₁₆

—

C₂×#127

Dic₅₃

yes

characteristic

C₂×C₄×D₄

C₂×D₅₃

yes

—

#436

—

106

—

C₂×C₄⋊C₈

—

100

C₂×C₂²⋊C₈

—

101

C₂×C₄×C₈

—

102

C₂×C₄×C₈

—

103

#390

Dic₅₃

yes

—

104

#384

Dic₅₃

yes

characteristic

105

C₄³

C₂×D₅₃

yes

characteristic

106

C₅₃

#21681

yes

characteristic, complemented, Sylow

107

C₂²×D₄

C₂×Dic₅₃

yes

—

108

C₂³×C₄

C₂²×D₅₃

yes

—

109

C₂×M₁₆

—

106

—

110

C₂²×C₈

—

111

C₂²×C₈

—

112

C₂²×C₈

—

113

#127

—

114

C₄×D₄

—

115

C₄×D₄

—

116

C₂×C₄⋊C₄

C₂×Dic₅₃

yes

—

117

C₂×C₂²⋊C₄

C₂×Dic₅₃

yes

—

118

C₂×C₄²

C₂×Dic₅₃

yes

—

119

C₂×C₄²

C₂×Dic₅₃

yes

characteristic

120

C₂×C₄²

C₂²×D₅₃

yes

—

121

C₄⋊C₈

—

106

—

122

C₂²⋊C₈

—

106

—

123

C₄×C₈

—

124

C₄×C₈

—

106

—

125

C₂⁴

C₂²×Dic₅₃

yes

—

126

C₂×D₄

—

127

C₂×D₄

—

128

C₂×D₄

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

C₂²×Dic₅₃

yes

—

132

C₂²×C₄

C₂²×Dic₅₃

yes

—

133

C₂²×C₄

C₂²×Dic₅₃

yes

—

134

C₂²×C₄

C₂³×D₅₃

yes

characteristic

135

M₁₆

—

212

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

106

—

139

C₂×C₈

—

106

—

140

C₄⋊C₄

—

141

C₂²⋊C₄

—

142

C₄²

—

143

C₄²

#11719917

yes

complemented

144

C₄²

D₄×D₅₃

yes

—

145

C₄²

#11719947

yes

—

146

C₂³

—

147

C₂³

—

148

C₂³

—

149

C₂³

C₂³×Dic₅₃

yes

characteristic

150

D₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

D₄×Dic₅₃

yes

—

156

C₂×C₄

D₄×Dic₅₃

yes

—

157

C₂×C₄

C₂×#11719917

yes

—

158

C₂×C₄

#13133224

yes

—

159

C₂×C₄

C₂×D₄×D₅₃

yes

—

160

C₂×C₄

C₂×#11719947

yes

—

161

C₈

—

162

C₈

—

106

—

163

C₈

—

212

—

164

C₂²

—

165

C₂²

—

166

C₂²

—

167

C₂²

C₂×D₄×Dic₅₃

yes

—

168

C₂²

C₂²×#11719917

yes

characteristic

169

C₂²

C₂×#13133224

yes

—

170

C₄

—

171

C₄

—

172

C₄

#16810701

yes

complemented

173

C₄

#127×D₅₃

yes

characteristic

174

C₄

#16811171

yes

—

175

C₂

—

176

C₂

#127×Dic₅₃

yes

characteristic

177

C₂

C₂×#16810701

yes

—

178

C₂

#20283427

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂³×C₄
Fitting	C₄×#127×C₅₃	C₂
prime radical	C₂×#127×C₅₃	C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₄×#127	D₅₃
hypocenter	C₅₃	#21681
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#127

subgroup

G/H

center, Frattini

C₂²×C₄

C₂³×D₅₃

commutator, socle

C₂³×C₅₃

C₂³×C₄

Fitting

C₄×#127×C₅₃

C₂

prime radical

C₂×#127×C₅₃

C₄

cosocle

C₂²×C₄×C₅₃

C₂⁴

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂²×C₄ ◁ C₄×#127

N	H ≅ G/N	type
1. C₂×#16810701	C₂	semidirect
2. #20275510	C₂	semidirect
3. #20274867	C₂	semidirect
4. C₄²×#88322	C₂	semidirect
24. C₄×#88322	D₄	semidirect
48. C₄²×C₅₃	M₁₆	semidirect
73. #88322	#127	semidirect
83. C₄×C₅₃	#436	semidirect
106. C₅₃	#21681	semidirect
143. C₄²	#11719917	semidirect
172. C₄	#16810701	semidirect

H ≅ G/N

type

1. C₂×#16810701

C₂

semidirect

2. #20275510

C₂

semidirect

3. #20274867

C₂

semidirect

4. C₄²×#88322

C₂

semidirect

24. C₄×#88322

D₄

semidirect

48. C₄²×C₅₃

M₁₆

semidirect

73. #88322

#127

semidirect

83. C₄×C₅₃

#436

semidirect

106. C₅₃

#21681

semidirect

143. C₄²

#11719917

semidirect

172. C₄

#16810701

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#16810701	2
#20275510	2
#20274867	2
C₄²×#88322	2
C₄×#127×C₅₃	2
#21681	53

subgroup

index

C₂×#16810701

#20275510

#20274867

C₄²×#88322

C₄×#127×C₅₃

#21681

Subgroups — #27392479

Conjugacy classes of subgroups

Complements

Maximal subgroups