Group #27300285 — subgroups

Subgroups — #27300285

Order 13568.

354 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 143 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×#16810453	C₂	1	2	yes	maximal, complemented
2	#20274765	C₂	1	2	yes	maximal
3	#20272819	C₂	1	2	yes	maximal, complemented
4	C₂×C₅₃×#504	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133153	C₂²	1	4	yes	—
6	#16810607	—	2	2	no	—
7	#16810453	—	2	4	no	—
8	#16810381	—	2	2	no	—
9	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
10	C₅₃×#504	—	2	4	no	—
11	C₂×M₃₂×C₅₃	C₄	1	2	yes	—
12	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₅₃	C₄	1	2	yes	—
14	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#11719917	—	2	4	no	—
16	#13133153	—	2	4	no	—
17	#13133153	—	2	4	no	—
18	#13133136	—	4	2	no	—
19	#13133134	—	4	2	no	—
20	C₂×#41×C₅₃	D₄	1	1	yes	characteristic
21	#131×C₅₃	—	2	2	no	—
22	#131×C₅₃	—	2	2	no	—
23	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
24	C₂×M₁₆×C₅₃	Q₈	1	2	yes	—
25	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
27	M₃₂×C₅₃	—	2	2	no	—
28	M₃₂×C₅₃	—	2	4	no	—
29	C₂×C₁₆×C₅₃	—	2	4	no	—
30	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
31	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₁₆×C₅₃	D₄	1	4	yes	—
34	#41×C₅₃	—	2	2	no	—
35	#41×C₅₃	—	2	2	no	—
36	C₂×Q₈×C₅₃	D₈	1	1	yes	characteristic
37	C₂×D₄×C₅₃	D₈	1	1	yes	characteristic
38	C₂×D₄×C₅₃	Q₁₆	1	2	yes	—
39	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
40	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
41	M₁₆×C₅₃	—	2	2	no	—
42	M₁₆×C₅₃	—	2	4	no	—
43	C₂×C₈×C₅₃	—	2	2	no	—
44	C₂×C₈×C₅₃	—	2	2	no	—
45	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
46	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
48	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₅₃	#41	1	2	yes	—
50	#11719917	—	4	4	no	—
51	#11719917	—	4	4	no	—
52	C₂×#88322	—	4	4	no	—
53	C₁₆×C₅₃	—	2	2	no	—
54	C₁₆×C₅₃	C₂²⋊C₄	1	2	yes	—
55	C₁₆×C₅₃	C₂²⋊C₄	1	2	yes	—
56	C₂³×C₅₃	#107	1	2	yes	—
57	C₂³×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
58	Q₈×C₅₃	—	2	2	no	—
59	D₄×C₅₃	—	2	4	no	—
60	D₄×C₅₃	—	2	2	no	—
61	C₂×C₄×C₅₃	—	2	2	no	—
62	C₂×C₄×C₅₃	—	2	2	no	—
63	C₂×C₄×C₅₃	#107	1	2	yes	—
64	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₅₃	#121	1	2	yes	—
66	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₅₃	#135	1	2	yes	—
68	C₈×C₅₃	—	2	2	no	—
69	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₈×C₅₃	#117	1	1	yes	characteristic
71	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
72	#88322	—	8	4	no	—
73	#12840	—	53	1	no	maximal, Sylow
74	C₂²×C₅₃	—	2	2	no	—
75	C₂²×C₅₃	—	4	1	no	—
76	C₂²×C₅₃	—	4	2	no	—
77	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
78	C₂²×C₅₃	#467	1	2	yes	—
79	C₄×C₅₃	—	2	2	no	—
80	C₄×C₅₃	#380	1	1	yes	characteristic
81	C₄×C₅₃	#427	1	1	yes	characteristic
82	C₄×C₅₃	#466	1	2	yes	—
83	C₂×#504	D₅₃	1	1	yes	characteristic
84	C₂×#368	—	53	2	no	—
85	#1813	—	53	2	no	—
86	#1385	—	53	2	no	—
87	C₂×C₅₃	—	2	1	no	—
88	C₂×C₅₃	—	2	2	no	—
89	C₂×C₅₃	—	4	2	no	—
90	C₂×C₅₃	#1808	1	1	yes	characteristic
91	C₂×#131	C₂×D₅₃	1	1	yes	characteristic
92	#504	—	2	4	no	—
93	C₂×M₃₂	Dic₅₃	1	2	yes	—
94	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
95	C₂²×C₁₆	Dic₅₃	1	2	yes	—
96	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
97	C₂×#108	—	53	4	no	—
98	#413	—	106	2	no	—
99	#368	—	106	4	no	—
100	#338	—	106	2	no	—
101	C₅₃	#12840	1	1	yes	characteristic, complemented, Sylow
102	C₂×#41	D₂₁₂	1	1	yes	characteristic
103	#131	—	2	2	no	—
104	#131	—	2	2	no	—
105	C₂×M₁₆	—	106	4	no	—
106	C₂×M₁₆	Dic₁₀₆	1	2	yes	—
107	C₂×M₁₆	D₂₁₂	1	1	yes	characteristic
108	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
109	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
110	M₃₂	—	2	2	no	—
111	M₃₂	—	2	4	no	—
112	C₂×C₁₆	—	2	4	no	—
113	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
114	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
115	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
116	C₂×C₁₆	C₂²⋊D₅₃	1	4	yes	—
117	#108	—	106	4	no	—
118	#108	—	106	4	no	—
119	#101	—	212	2	no	—
120	#100	—	212	2	no	—
121	#41	—	2	2	no	—
122	#41	—	2	2	no	—
123	C₂×Q₈	D₄₂₄	1	1	yes	characteristic
124	C₂×D₄	D₄₂₄	1	1	yes	characteristic
125	C₂×D₄	Dic₂₁₂	1	2	yes	—
126	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
127	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
128	M₁₆	—	2	2	no	—
129	M₁₆	—	2	4	no	—
130	M₁₆	—	212	4	no	—
131	M₁₆	—	212	4	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	212	4	no	—
135	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
136	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
137	C₂×C₈	#11719945	1	2	yes	—
138	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
139	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
140	C₁₆	—	2	2	no	—
141	C₁₆	C₂²⋊Dic₅₃	1	2	yes	—
142	C₁₆	C₂²⋊Dic₅₃	1	2	yes	—
143	C₂³	#13133151	1	2	yes	—
144	C₂³	C₂×Dic₂₁₂	1	1	yes	characteristic
145	Q₈	—	2	2	no	—
146	D₄	—	2	4	no	—
147	D₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	#13133151	1	2	yes	—
151	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
152	C₂×C₄	#13133191	1	2	yes	—
153	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
154	C₂×C₄	#13133239	1	2	yes	—
155	C₈	—	2	2	no	—
156	C₈	—	424	4	no	—
157	C₈	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
158	C₈	#13133176	1	1	yes	characteristic
159	C₈	#13133196	1	2	yes	—
160	C₂²	—	2	2	no	—
161	C₂²	—	4	1	no	—
162	C₂²	—	4	2	no	—
163	C₂²	C₂×#13133151	1	1	yes	characteristic
164	C₂²	#16810864	1	2	yes	—
165	C₄	—	2	2	no	—
166	C₄	#16810483	1	1	yes	characteristic
167	C₄	#16810661	1	1	yes	characteristic
168	C₄	#16810859	1	2	yes	—
169	C₂	—	2	1	no	—
170	C₂	—	2	2	no	—
171	C₂	—	4	2	no	—
172	C₂	#20274744	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16810453

C₂

yes

maximal, complemented

#20274765

C₂

yes

maximal

#20272819

C₂

yes

maximal, complemented

C₂×C₅₃×#504

C₂

yes

maximal, characteristic

C₂×#13133153

C₂²

yes

—

#16810607

—

#16810453

—

#16810381

—

C₂×#131×C₅₃

C₂²

yes

characteristic

C₅₃×#504

—

C₂×M₃₂×C₅₃

C₄

yes

—

C₂×M₃₂×C₅₃

C₂²

yes

characteristic

C₂²×C₁₆×C₅₃

C₄

yes

—

C₂²×C₁₆×C₅₃

C₂²

yes

—

—

—

—

—

C₂×#41×C₅₃

D₄

yes

characteristic

#131×C₅₃

—

#131×C₅₃

—

C₂×M₁₆×C₅₃

D₄

yes

characteristic

C₂×M₁₆×C₅₃

Q₈

yes

—

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂²×C₈×C₅₃

C₂³

yes

characteristic

M₃₂×C₅₃

—

M₃₂×C₅₃

—

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

C₂×C₄

yes

—

C₂×C₁₆×C₅₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₃

D₄

yes

—

#41×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

D₈

yes

characteristic

C₂×D₄×C₅₃

D₈

yes

characteristic

C₂×D₄×C₅₃

Q₁₆

yes

—

C₂²×C₄×C₅₃

C₄⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₂×C₈×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₃

#41

yes

—

#11719917

—

#11719917

—

C₂×#88322

—

C₁₆×C₅₃

—

C₁₆×C₅₃

C₂²⋊C₄

yes

—

C₁₆×C₅₃

C₂²⋊C₄

yes

—

C₂³×C₅₃

#107

yes

—

C₂³×C₅₃

C₂×Q₁₆

yes

characteristic

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#107

yes

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

#121

yes

—

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

#135

yes

—

C₈×C₅₃

—

C₈×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₅₃

#117

yes

characteristic

C₈×C₅₃

C₂²⋊Q₈

yes

—

#88322

—

#12840

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×#107

yes

characteristic

C₂²×C₅₃

#467

yes

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₂×#504

D₅₃

yes

—

—

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#1808

yes

characteristic

C₂×#131

C₂×D₅₃

yes

characteristic

#504

—

C₂×M₃₂

Dic₅₃

yes

—

C₂×M₃₂

C₂×D₅₃

yes

characteristic

C₂²×C₁₆

Dic₅₃

yes

—

C₂²×C₁₆

C₂×D₅₃

yes

characteristic

C₂×#108

—

#413

—

106

—

#368

—

106

—

100

#338

—

106

—

101

C₅₃

#12840

yes

characteristic, complemented, Sylow

102

C₂×#41

D₂₁₂

yes

characteristic

103

#131

—

104

#131

—

105

C₂×M₁₆

—

106

—

106

C₂×M₁₆

Dic₁₀₆

yes

—

107

C₂×M₁₆

D₂₁₂

yes

characteristic

108

C₂²×C₈

C₂×Dic₅₃

yes

—

109

C₂²×C₈

C₂²×D₅₃

yes

characteristic

110

M₃₂

—

111

M₃₂

—

112

C₂×C₁₆

—

113

C₂×C₁₆

C₂×Dic₅₃

yes

—

114

C₂×C₁₆

C₂×Dic₅₃

yes

characteristic

115

C₂×C₁₆

C₂×Dic₅₃

yes

characteristic

116

C₂×C₁₆

C₂²⋊D₅₃

yes

—

117

#108

—

106

—

118

#108

—

106

—

119

#101

—

212

—

120

#100

—

212

—

121

#41

—

122

#41

—

123

C₂×Q₈

D₄₂₄

yes

characteristic

124

C₂×D₄

D₄₂₄

yes

characteristic

125

C₂×D₄

Dic₂₁₂

yes

—

126

C₂²×C₄

C₄⋊Dic₅₃

yes

—

127

C₂²×C₄

C₂×D₂₁₂

yes

characteristic

128

M₁₆

—

129

M₁₆

—

130

M₁₆

—

212

—

131

M₁₆

—

212

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

212

—

135

C₂×C₈

C₄⋊Dic₅₃

yes

—

136

C₂×C₈

C₂×Dic₁₀₆

yes

characteristic

137

C₂×C₈

#11719945

yes

—

138

C₂×C₈

C₂²×Dic₅₃

yes

characteristic

139

C₂×C₈

C₂×C₂²⋊D₅₃

yes

—

140

C₁₆

—

141

C₁₆

C₂²⋊Dic₅₃

yes

—

142

C₁₆

C₂²⋊Dic₅₃

yes

—

143

C₂³

#13133151

yes

—

144

C₂³

C₂×Dic₂₁₂

yes

characteristic

145

Q₈

—

146

D₄

—

147

D₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

#13133151

yes

—

151

C₂×C₄

C₂×C₄⋊Dic₅₃

yes

characteristic

152

C₂×C₄

#13133191

yes

—

153

C₂×C₄

C₂×D₄₂₄

yes

characteristic

154

C₂×C₄

#13133239

yes

—

155

C₈

—

156

C₈

—

424

—

157

C₈

C₂×C₂²⋊Dic₅₃

yes

characteristic

158

C₈

#13133176

yes

characteristic

159

C₈

#13133196

yes

—

160

C₂²

—

161

C₂²

—

162

C₂²

—

163

C₂²

C₂×#13133151

yes

characteristic

164

C₂²

#16810864

yes

—

165

C₄

—

166

C₄

#16810483

yes

characteristic

167

C₄

#16810661

yes

characteristic

168

C₄

#16810859

yes

—

169

C₂

—

170

C₂

—

171

C₂

—

172

C₂

#20274744

yes

characteristic

	subgroup	G/H
center	C₄	#16810661
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#504	C₂
socle	C₂×C₅₃	#1808
prime radical	C₂×#41×C₅₃	D₄
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#504	D₅₃
hypocenter	C₅₃	#12840
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131 ◁ C₂×#504

subgroup

G/H

center

C₄

#16810661

commutator

C₂×C₈×C₅₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₂×C₅₃×#504

C₂

socle

C₂×C₅₃

#1808

prime radical

C₂×#41×C₅₃

D₄

cosocle

C₂²×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131 ◁ C₂×#504

N	H ≅ G/N	type
1. C₂×#16810453	C₂	semidirect
3. #20272819	C₂	semidirect
101. C₅₃	#12840	semidirect

H ≅ G/N

type

1. C₂×#16810453

C₂

semidirect

3. #20272819

C₂

semidirect

101. C₅₃

#12840

semidirect

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

subgroup	index
C₂×#16810453	2
#20274765	2
#20272819	2
C₂×C₅₃×#504	2
#12840	53

subgroup

index

C₂×C₅₃×#504

Subgroups — #27300285

Conjugacy classes of subgroups

Complements

Maximal subgroups