Group #27275102 — subgroups

Subgroups — #27275102

Order 13568.

384 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 133 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	#20283650	C₂	1	1	yes	maximal, characteristic
2	#20273380	C₂	1	1	yes	maximal, characteristic
3	M₁₆×#88322	C₂	1	2	yes	maximal
4	#20272605	C₂	1	2	yes	maximal
5	C₅₃×#1526	C₂	1	1	yes	maximal, characteristic
6	#16810754	—	2	2	no	—
7	Q₈×#88322	—	2	2	no	—
8	#16810676	C₂²	1	2	yes	—
9	#16810650	C₄	1	2	yes	—
10	#16810647	C₂²	1	1	yes	characteristic
11	#16810579	C₄	1	2	yes	—
12	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
13	#16810369	C₄	1	4	yes	—
14	#16810344	—	2	2	no	—
15	C₈×#88322	—	2	2	no	—
16	C₅₃×#533	C₂²	1	1	yes	characteristic
17	C₄×M₁₆×C₅₃	C₂²	1	2	yes	—
18	C₅₃×#330	C₄	1	4	yes	—
19	C₂×#11719917	C₂×C₄	1	2	yes	—
20	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
21	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
22	C₄⋊#88322	—	2	2	no	—
23	C₄⋊#88322	—	2	4	no	—
24	C₄⋊#88322	C₂×C₄	1	4	yes	—
25	C₂²⋊#88322	—	2	2	no	—
26	#13133128	—	2	2	no	—
27	C₄×#88322	—	2	2	no	—
28	C₄×#88322	—	2	2	no	—
29	C₄×#88322	D₄	1	2	yes	—
30	C₄×#88322	D₄	1	2	yes	—
31	C₂×M₁₆×C₅₃	—	2	2	no	—
32	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
33	#125×C₅₃	C₂×C₄	1	2	yes	—
34	C₂²⋊Q₈×C₅₃	—	2	2	no	—
35	C₄×Q₈×C₅₃	—	2	2	no	—
36	#117×C₅₃	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
38	#97×C₅₃	C₂×C₄	1	4	yes	—
39	C₄×C₈×C₅₃	—	2	2	no	—
40	C₂×Q₈×C₅₃	—	2	2	no	—
41	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	#41	1	2	yes	—
43	M₁₆×C₅₃	—	2	8	no	—
44	C₂×C₈×C₅₃	—	2	4	no	—
45	C₄⋊C₄×C₅₃	—	2	4	no	—
46	C₄⋊C₄×C₅₃	—	2	2	no	—
47	C₄⋊C₄×C₅₃	C₄²	1	4	yes	—
48	C₂²⋊C₄×C₅₃	—	2	2	no	—
49	C₄²×C₅₃	—	2	2	no	—
50	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₅₃	C₂×D₄	1	2	yes	—
53	#11719917	—	4	2	no	—
54	C₂×#88322	—	2	2	no	—
55	C₂×#88322	—	2	2	no	—
56	C₂×#88322	—	2	4	no	—
57	C₂×#88322	—	4	1	no	—
58	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×#88322	C₂²⋊C₄	1	2	yes	—
60	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
61	C₂³×C₅₃	#117	1	1	yes	characteristic
62	Q₈×C₅₃	—	4	4	no	—
63	C₂×C₄×C₅₃	—	2	1	no	—
64	C₂×C₄×C₅₃	—	2	1	no	—
65	C₂×C₄×C₅₃	—	2	4	no	—
66	C₂×C₄×C₅₃	—	2	4	no	—
67	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
70	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
71	C₈×C₅₃	—	2	8	no	—
72	#88322	—	2	1	no	—
73	#88322	—	2	2	no	—
74	#88322	—	4	2	no	—
75	#88322	—	4	4	no	—
76	#88322	#101	1	2	yes	—
77	#8242	—	53	1	no	maximal, Sylow
78	C₂²×C₅₃	—	2	1	no	—
79	C₂²×C₅₃	—	2	2	no	—
80	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
81	C₄×C₅₃	—	2	1	no	—
82	C₄×C₅₃	—	2	2	no	—
83	C₄×C₅₃	—	4	4	no	—
84	C₄×C₅₃	C₂×#101	1	1	yes	characteristic
85	C₄×C₅₃	#413	1	1	yes	characteristic
86	C₄×C₅₃	#443	1	4	yes	—
87	#2960	—	53	1	no	—
88	#1544	—	53	1	no	—
89	#1526	D₅₃	1	1	yes	characteristic
90	C₈×M₁₆	—	53	2	no	—
91	#1299	—	53	2	no	—
92	C₂×C₅₃	—	4	1	no	—
93	C₂×C₅₃	C₄×#101	1	1	yes	characteristic
94	C₂×C₅₃	#1764	1	2	yes	—
95	#533	C₂×D₅₃	1	1	yes	characteristic
96	#446	—	106	2	no	—
97	C₈×Q₈	—	106	2	no	—
98	#431	—	53	2	no	—
99	#424	—	53	2	no	—
100	#423	—	53	1	no	—
101	#405	—	53	2	no	—
102	C₄×M₁₆	C₂×D₅₃	1	2	yes	—
103	C₂×C₄×C₈	—	53	1	no	—
104	#332	—	53	4	no	—
105	#330	Dic₅₃	1	4	yes	—
106	#322	—	106	2	no	—
107	C₈²	—	106	2	no	—
108	C₅₃	#8242	1	1	yes	characteristic, complemented, Sylow
109	C₂×M₁₆	—	2	2	no	—
110	C₂×M₁₆	—	53	2	no	—
111	C₂²×C₈	—	53	1	no	—
112	C₂²×C₈	—	53	1	no	—
113	C₄⋊Q₈	C₄×D₅₃	1	2	yes	—
114	#125	C₂×Dic₅₃	1	2	yes	—
115	C₂²⋊Q₈	—	2	2	no	—
116	C₄×Q₈	—	2	2	no	—
117	#117	C₄×D₅₃	1	2	yes	—
118	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
119	C₄⋊C₈	—	53	4	no	—
120	C₄⋊C₈	—	106	4	no	—
121	C₄⋊C₈	—	106	2	no	—
122	C₂²⋊C₈	—	106	2	no	—
123	#97	—	106	2	no	—
124	#97	C₂×Dic₅₃	1	4	yes	—
125	C₄×C₈	—	2	2	no	—
126	C₄×C₈	—	53	2	no	—
127	C₄×C₈	—	53	2	no	—
128	C₄×C₈	—	106	2	no	—
129	C₄×C₈	—	106	2	no	—
130	C₂×Q₈	—	2	2	no	—
131	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
132	C₂²×C₄	#11719947	1	2	yes	—
133	M₁₆	—	2	8	no	—
134	M₁₆	—	212	2	no	—
135	C₂×C₈	—	2	4	no	—
136	C₂×C₈	—	53	1	no	—
137	C₂×C₈	—	53	1	no	—
138	C₂×C₈	—	53	2	no	—
139	C₂×C₈	—	106	2	no	—
140	C₂×C₈	—	106	2	no	—
141	C₂×C₈	—	106	4	no	—
142	C₂×C₈	—	212	1	no	—
143	C₄⋊C₄	—	2	2	no	—
144	C₄⋊C₄	—	2	4	no	—
145	C₄⋊C₄	C₄×Dic₅₃	1	4	yes	—
146	C₂²⋊C₄	—	2	2	no	—
147	C₄²	—	2	2	no	—
148	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
149	C₄²	D₄×D₅₃	1	2	yes	—
150	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
151	C₂³	#13133173	1	1	yes	characteristic
152	Q₈	—	4	4	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
158	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
159	C₂×C₄	#13133177	1	2	yes	—
160	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
161	C₈	—	2	8	no	—
162	C₈	—	53	2	no	—
163	C₈	—	106	1	no	—
164	C₈	—	106	2	no	—
165	C₈	—	212	2	no	—
166	C₈	—	212	4	no	—
167	C₂²	—	2	1	no	—
168	C₂²	—	2	2	no	—
169	C₂²	C₂²⋊C₄×Dic₅₃	1	1	yes	characteristic
170	C₄	—	2	1	no	—
171	C₄	—	2	2	no	—
172	C₄	—	4	4	no	—
173	C₄	#101×D₅₃	1	1	yes	characteristic
174	C₄	#16810606	1	1	yes	characteristic
175	C₄	#16810737	1	4	yes	—
176	C₂	—	4	1	no	—
177	C₂	#101×Dic₅₃	1	1	yes	characteristic
178	C₂	#20274553	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20283650

C₂

yes

maximal, characteristic

#20273380

C₂

yes

maximal, characteristic

M₁₆×#88322

C₂

yes

maximal

#20272605

C₂

yes

maximal

C₅₃×#1526

C₂

yes

maximal, characteristic

#16810754

—

Q₈×#88322

—

#16810676

C₂²

yes

—

#16810650

C₄

yes

—

#16810647

C₂²

yes

characteristic

#16810579

C₄

yes

—

C₂×C₄×#88322

C₂²

yes

characteristic

#16810369

C₄

yes

—

#16810344

—

C₈×#88322

—

C₅₃×#533

C₂²

yes

characteristic

C₄×M₁₆×C₅₃

C₂²

yes

—

C₅₃×#330

C₄

yes

—

C₂×#11719917

C₂×C₄

yes

—

C₂²×#88322

C₂×C₄

yes

characteristic

C₂²×#88322

C₂×C₄

yes

—

—

C₂×C₄

yes

—

—

—

—

—

D₄

yes

—

C₄×#88322

D₄

yes

—

C₂×M₁₆×C₅₃

—

C₄⋊Q₈×C₅₃

C₂×C₄

yes

—

#125×C₅₃

C₂×C₄

yes

—

C₂²⋊Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

#117×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#97×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₄²

yes

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

—

—

—

—

—

—

C₂²⋊C₄

yes

characteristic

C₂×#88322

C₂²⋊C₄

yes

—

C₂×#88322

C₂²⋊C₄

yes

characteristic

C₂³×C₅₃

#117

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄×D₄

yes

—

C₂×C₄×C₅₃

C₄×D₄

yes

—

C₈×C₅₃

—

—

—

—

—

yes

—

#8242

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₄×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

#2960

—

#1544

—

#1526

D₅₃

yes

characteristic

C₈×M₁₆

—

#1299

—

C₂×C₅₃

—

C₂×C₅₃

C₄×#101

yes

characteristic

C₂×C₅₃

#1764

yes

—

#533

C₂×D₅₃

yes

characteristic

#446

—

106

—

C₈×Q₈

—

106

—

#431

—

#424

—

100

#423

—

101

#405

—

102

C₄×M₁₆

C₂×D₅₃

yes

—

103

C₂×C₄×C₈

—

104

#332

—

105

#330

Dic₅₃

yes

—

106

#322

—

106

—

107

C₈²

—

106

—

108

C₅₃

#8242

yes

characteristic, complemented, Sylow

109

C₂×M₁₆

—

110

C₂×M₁₆

—

111

C₂²×C₈

—

112

C₂²×C₈

—

113

C₄⋊Q₈

C₄×D₅₃

yes

—

114

#125

C₂×Dic₅₃

yes

—

115

C₂²⋊Q₈

—

116

C₄×Q₈

—

117

#117

C₄×D₅₃

yes

—

118

C₂×C₄²

C₂²×D₅₃

yes

characteristic

119

C₄⋊C₈

—

120

C₄⋊C₈

—

106

—

121

C₄⋊C₈

—

106

—

122

C₂²⋊C₈

—

106

—

123

#97

—

106

—

124

#97

C₂×Dic₅₃

yes

—

125

C₄×C₈

—

126

C₄×C₈

—

127

C₄×C₈

—

128

C₄×C₈

—

106

—

129

C₄×C₈

—

106

—

130

C₂×Q₈

—

131

C₂²×C₄

C₂×C₄×D₅₃

yes

characteristic

132

C₂²×C₄

#11719947

yes

—

133

M₁₆

—

134

M₁₆

—

212

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

106

—

140

C₂×C₈

—

106

—

141

C₂×C₈

—

106

—

142

C₂×C₈

—

212

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

C₄×Dic₅₃

yes

—

146

C₂²⋊C₄

—

147

C₄²

—

148

C₄²

C₂×C₄×D₅₃

yes

characteristic

149

C₄²

D₄×D₅₃

yes

—

150

C₄²

C₂²×Dic₅₃

yes

characteristic

151

C₂³

#13133173

yes

characteristic

152

Q₈

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

C₂×C₄×Dic₅₃

yes

characteristic

158

C₂×C₄

C₂²⋊C₄×D₅₃

yes

characteristic

159

C₂×C₄

#13133177

yes

—

160

C₂×C₄

D₄×Dic₅₃

yes

—

161

C₈

—

162

C₈

—

163

C₈

—

106

—

164

C₈

—

106

—

165

C₈

—

212

—

166

C₈

—

212

—

167

C₂²

—

168

C₂²

—

169

C₂²

C₂²⋊C₄×Dic₅₃

yes

characteristic

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

#101×D₅₃

yes

characteristic

174

C₄

#16810606

yes

characteristic

175

C₄

#16810737

yes

—

176

C₂

—

177

C₂

#101×Dic₅₃

yes

characteristic

178

C₂

#20274553

yes

—

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1526	C₂
socle	C₂²×C₅₃	C₄×C₂²⋊C₄
prime radical	C₂³×C₅₃	#117
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1526	D₅₃
hypocenter	C₅₃	#8242
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1526

subgroup

G/H

center

C₂×C₄

C₂²⋊C₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1526

C₂

socle

C₂²×C₅₃

C₄×C₂²⋊C₄

prime radical

C₂³×C₅₃

#117

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1526

N	H ≅ G/N	type
108. C₅₃	#8242	semidirect

H ≅ G/N

type

108. C₅₃

#8242

semidirect

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

subgroup	index
#20283650	2
#20273380	2
M₁₆×#88322	2
#20272605	2
C₅₃×#1526	2
#8242	53

subgroup

index

M₁₆×#88322

C₅₃×#1526

Subgroups — #27275102

Conjugacy classes of subgroups

Complements

Maximal subgroups