Group #28328577 — subgroups

Subgroups — #28328577

Order 13568.

532 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 232 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	Q₈×#11719910	C₂	1	2	yes	maximal, complemented
2	#20291101	C₂	1	6	yes	maximal, complemented
3	#20291074	C₂	1	3	yes	maximal, complemented
4	#20291048	C₂	1	3	yes	maximal, complemented
5	C₈×#11719949	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133292	C₂²	1	6	yes	—
7	#16811544	C₂²	1	1	yes	characteristic
8	#16811544	C₂²	1	3	yes	—
9	C₄⋊#11719910	C₂²	1	6	yes	—
10	#16811518	C₂²	1	6	yes	—
11	C₄×#11719910	C₂²	1	6	yes	—
12	C₄×#11719949	C₄	1	1	yes	characteristic
13	Q₈×#88322	C₂²	1	1	yes	characteristic
14	C₈×D₂₁₂	C₄	1	3	yes	—
15	#16810688	C₂²	1	3	yes	—
16	#16810675	C₄	1	3	yes	—
17	C₄×C₈×D₅₃	C₂²	1	3	yes	—
18	C₈×Q₈×C₅₃	C₄	1	1	yes	characteristic
19	#13133292	—	4	6	no	—
20	C₂×#11719910	C₂³	1	6	yes	—
21	C₂×#11719910	C₂³	1	2	yes	—
22	C₂×#11719949	C₈	1	1	yes	characteristic
23	C₂×C₈×D₅₃	C₂×C₄	1	3	yes	—
24	C₂×C₈×D₅₃	C₂³	1	3	yes	—
25	Q₈×Dic₅₃	C₈	1	1	yes	characteristic
26	C₄×D₂₁₂	C₂×C₄	1	3	yes	—
27	#13133179	C₈	1	3	yes	—
28	#13133175	C₈	1	3	yes	—
29	C₄²×D₅₃	C₂×C₄	1	3	yes	—
30	C₄⋊#88322	C₂×C₄	1	3	yes	—
31	#13133129	C₂×C₄	1	6	yes	—
32	C₄×#88322	C₂³	1	3	yes	—
33	C₈×Dic₅₃	C₂×C₄	1	3	yes	—
34	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
35	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₄⋊C₈×C₅₃	C₂×C₄	1	3	yes	—
37	C₄×C₈×C₅₃	C₂×C₄	1	3	yes	—
38	C₂×Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
39	#11719949	—	2	4	no	—
40	C₂×D₂₁₂	C₂×C₈	1	3	yes	—
41	C₂×C₄×D₅₃	C₂×C₈	1	3	yes	—
42	C₂×C₄×D₅₃	C₂×C₈	1	3	yes	—
43	C₂×C₄×D₅₃	C₂²×C₄	1	3	yes	—
44	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
46	C₄⋊C₄×C₅₃	C₂×C₈	1	3	yes	—
47	C₄²×C₅₃	C₂²×C₄	1	3	yes	—
48	#11719921	C₂×C₈	1	6	yes	—
49	C₄⋊Dic₅₃	C₂×C₈	1	3	yes	—
50	C₄×Dic₅₃	C₂×C₈	1	3	yes	—
51	C₄×Dic₅₃	C₂×C₈	1	3	yes	—
52	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×#88322	C₂²×C₄	1	3	yes	—
54	C₂×#88322	C₂⁴	1	1	yes	characteristic
55	C₈×D₅₃	—	2	3	no	—
56	C₈×D₅₃	—	2	3	no	—
57	C₈×D₅₃	—	4	3	no	—
58	#11719910	—	2	6	no	—
59	#11719910	—	2	2	no	—
60	C₂²×D₅₃	C₂²×C₈	1	3	yes	—
61	Q₈×C₅₃	M₃₂	1	4	yes	complemented
62	C₂×C₄×C₅₃	C₂²×C₈	1	3	yes	—
63	C₂×C₄×C₅₃	C₂²×C₈	1	3	yes	—
64	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
65	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
66	C₂×Dic₅₃	C₂²×C₈	1	3	yes	—
67	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
68	D₂₁₂	—	2	6	no	—
69	C₄×D₅₃	—	2	3	no	—
70	C₄×D₅₃	—	2	3	no	—
71	C₄×D₅₃	—	2	6	no	—
72	C₄×D₅₃	—	2	6	no	—
73	C₈×C₅₃	—	2	3	no	—
74	C₈×C₅₃	—	2	1	no	—
75	#88322	—	2	3	no	—
76	#88322	H(5,𝔽₂)	1	1	yes	characteristic
77	#88322	2₋¹⁺⁴	1	1	yes	characteristic
78	#33505	—	53	1	no	maximal, Sylow
79	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
80	C₂×D₅₃	—	2	3	no	—
81	C₂×D₅₃	—	2	3	no	—
82	C₂×D₅₃	—	2	3	no	—
83	C₄×C₅₃	—	2	3	no	—
84	C₄×C₅₃	C₂×M₃₂	1	6	yes	—
85	C₄×C₅₃	#520	1	1	yes	characteristic
86	C₄×C₅₃	#568	1	1	yes	characteristic
87	Dic₅₃	—	2	1	no	—
88	Dic₅₃	—	2	3	no	—
89	Dic₅₃	#504	1	2	yes	—
90	C₈×#41	—	53	1	no	—
91	Q₈×C₁₆	—	53	2	no	—
92	#2169	—	53	6	no	—
93	#2162	—	53	3	no	—
94	#2150	—	53	3	no	—
95	C₂×C₅₃	#2988	1	1	yes	characteristic
96	C₂×C₅₃	C₂²×M₃₂	1	1	yes	characteristic
97	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
98	D₅₃	—	4	3	no	—
99	C₄×#41	—	53	1	no	—
100	C₂×M₃₂	—	53	6	no	—
101	C₈×Q₈	—	53	1	no	—
102	C₈×Q₈	#6157	1	1	yes	characteristic
103	C₈×D₄	—	53	3	no	—
104	C₈×D₄	—	53	3	no	—
105	#431	—	53	3	no	—
106	C₂×C₄×C₈	—	53	3	no	—
107	C₄⋊C₁₆	—	53	6	no	—
108	C₄⋊C₁₆	—	53	3	no	—
109	C₄⋊C₁₆	—	53	1	no	—
110	C₂²⋊C₁₆	—	53	6	no	—
111	C₄×C₁₆	—	53	6	no	—
112	C₅₃	#33505	1	1	yes	characteristic, complemented, Sylow
113	C₂×#41	—	53	1	no	—
114	C₂²×C₈	—	53	3	no	—
115	C₂²×C₈	—	53	3	no	—
116	C₄×Q₈	—	53	1	no	—
117	C₄×Q₈	C₂×#6157	1	1	yes	characteristic
118	C₄×D₄	—	53	3	no	—
119	C₄×D₄	—	53	3	no	—
120	#117	—	53	3	no	—
121	C₂×C₄²	—	53	3	no	—
122	M₃₂	—	212	6	no	—
123	C₂×C₁₆	—	53	2	no	—
124	C₂×C₁₆	—	53	6	no	—
125	C₄⋊C₈	—	53	3	no	—
126	C₄⋊C₈	C₂×#6157	1	3	yes	—
127	C₂²⋊C₈	—	53	6	no	—
128	C₄×C₈	—	53	3	no	—
129	C₄×C₈	—	53	1	no	—
130	C₄×C₈	—	53	3	no	—
131	C₄×C₈	C₂×#6157	1	3	yes	—
132	#41	—	106	4	no	—
133	C₂×Q₈	D₅₃⋊C₈	1	1	yes	characteristic
134	C₂×D₄	—	53	3	no	—
135	C₂²×C₄	—	53	3	no	—
136	C₂²×C₄	—	53	3	no	—
137	C₂²×C₄	—	53	3	no	—
138	C₂×C₈	—	53	1	no	—
139	C₂×C₈	—	53	3	no	—
140	C₂×C₈	—	106	3	no	—
141	C₂×C₈	—	106	3	no	—
142	C₂×C₈	—	212	3	no	—
143	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
144	C₂×C₈	C₂²×#6157	1	3	yes	—
145	C₄⋊C₄	—	53	3	no	—
146	C₄⋊C₄	D₅₃⋊C₈	1	3	yes	—
147	C₂²⋊C₄	—	53	6	no	—
148	C₄²	—	53	3	no	—
149	C₄²	—	53	3	no	—
150	C₄²	—	53	1	no	—
151	C₄²	C₂²×#6157	1	3	yes	—
152	C₁₆	—	106	6	no	—
153	C₁₆	—	106	2	no	—
154	C₂³	—	53	3	no	—
155	Q₈	#13133292	1	4	yes	complemented
156	D₄	—	106	6	no	—
157	C₂×C₄	—	53	1	no	—
158	C₂×C₄	—	53	1	no	—
159	C₂×C₄	—	53	3	no	—
160	C₂×C₄	—	106	6	no	—
161	C₂×C₄	—	106	6	no	—
162	C₂×C₄	—	106	3	no	—
163	C₂×C₄	—	106	3	no	—
164	C₂×C₄	C₂×D₅₃⋊C₈	1	3	yes	—
165	C₂×C₄	C₂×D₅₃⋊C₈	1	3	yes	—
166	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
167	C₈	—	2	3	no	—
168	C₈	—	2	1	no	—
169	C₈	—	53	1	no	—
170	C₈	—	53	1	no	—
171	C₈	—	106	3	no	—
172	C₂²	—	106	3	no	—
173	C₂²	—	106	3	no	—
174	C₂²	—	106	3	no	—
175	C₂²	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
176	C₄	—	2	3	no	—
177	C₄	—	53	2	no	—
178	C₄	—	106	3	no	—
179	C₄	—	106	1	no	—
180	C₄	C₂×#13133292	1	6	yes	—
181	C₄	#16811692	1	1	yes	characteristic
182	C₄	#16811703	1	1	yes	characteristic
183	C₂	—	212	3	no	—
184	C₂	#20291633	1	1	yes	characteristic
185	C₂	C₂²×#13133292	1	1	yes	characteristic
186	C₂	D₅₃⋊#504	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×#11719910

C₂

yes

maximal, complemented

#20291101

C₂

yes

maximal, complemented

#20291074

C₂

yes

maximal, complemented

#20291048

C₂

yes

maximal, complemented

C₈×#11719949

C₂

yes

maximal, characteristic

C₂×#13133292

C₂²

yes

—

#16811544

C₂²

yes

characteristic

#16811544

C₂²

yes

—

C₄⋊#11719910

C₂²

yes

—

#16811518

C₂²

yes

—

C₄×#11719910

C₂²

yes

—

C₄×#11719949

C₄

yes

characteristic

Q₈×#88322

C₂²

yes

characteristic

C₈×D₂₁₂

C₄

yes

—

#16810688

C₂²

yes

—

#16810675

C₄

yes

—

C₄×C₈×D₅₃

C₂²

yes

—

C₈×Q₈×C₅₃

C₄

yes

characteristic

#13133292

—

C₂×#11719910

C₂³

yes

—

C₂×#11719910

C₂³

yes

—

C₂×#11719949

C₈

yes

characteristic

C₂×C₈×D₅₃

C₂×C₄

yes

—

C₂×C₈×D₅₃

C₂³

yes

—

Q₈×Dic₅₃

C₈

yes

characteristic

C₄×D₂₁₂

C₂×C₄

yes

—

#13133179

C₈

yes

—

#13133175

C₈

yes

—

C₄²×D₅₃

C₂×C₄

yes

—

C₄⋊#88322

C₂×C₄

yes

—

#13133129

C₂×C₄

yes

—

C₄×#88322

C₂³

yes

—

C₈×Dic₅₃

C₂×C₄

yes

—

C₈×Dic₅₃

C₂³

yes

characteristic

C₄×Q₈×C₅₃

C₂×C₄

yes

characteristic

C₄⋊C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₂×Q₈×C₅₃

C₂×C₈

yes

characteristic

#11719949

—

C₂×D₂₁₂

C₂×C₈

yes

—

C₂×C₄×D₅₃

C₂×C₈

yes

—

C₂×C₄×D₅₃

C₂×C₈

yes

—

C₂×C₄×D₅₃

C₂²×C₄

yes

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂²×C₄

yes

—

C₄⋊C₄×C₅₃

C₂×C₈

yes

—

C₄²×C₅₃

C₂²×C₄

yes

—

#11719921

C₂×C₈

yes

—

C₄⋊Dic₅₃

C₂×C₈

yes

—

C₄×Dic₅₃

C₂×C₈

yes

—

C₄×Dic₅₃

C₂×C₈

yes

—

C₄×Dic₅₃

C₂²×C₄

yes

characteristic

C₂×#88322

C₂²×C₄

yes

—

C₂×#88322

C₂⁴

yes

characteristic

C₈×D₅₃

—

C₈×D₅₃

—

C₈×D₅₃

—

#11719910

—

#11719910

—

C₂²×D₅₃

C₂²×C₈

yes

—

Q₈×C₅₃

M₃₂

yes

complemented

C₂×C₄×C₅₃

C₂²×C₈

yes

—

C₂×C₄×C₅₃

C₂²×C₈

yes

—

C₂×C₄×C₅₃

C₂³×C₄

yes

characteristic

C₂×Dic₅₃

C₂²×C₈

yes

characteristic

C₂×Dic₅₃

C₂²×C₈

yes

—

C₂×Dic₅₃

C₂²×C₈

yes

characteristic

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

—

yes

2₋¹⁺⁴

yes

characteristic

#33505

—

maximal, Sylow

C₂²×C₅₃

C₂³×C₈

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×M₃₂

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

#504

yes

—

C₈×#41

—

Q₈×C₁₆

—

#2169

—

#2162

—

#2150

—

C₂×C₅₃

#2988

yes

characteristic

C₂×C₅₃

C₂²×M₃₂

yes

characteristic

C₂×C₅₃

C₂×#504

yes

characteristic

D₅₃

—

C₄×#41

—

100

C₂×M₃₂

—

101

C₈×Q₈

—

102

C₈×Q₈

#6157

yes

characteristic

103

C₈×D₄

—

104

C₈×D₄

—

105

#431

—

106

C₂×C₄×C₈

—

107

C₄⋊C₁₆

—

108

C₄⋊C₁₆

—

109

C₄⋊C₁₆

—

110

C₂²⋊C₁₆

—

111

C₄×C₁₆

—

112

C₅₃

#33505

yes

characteristic, complemented, Sylow

113

C₂×#41

—

114

C₂²×C₈

—

115

C₂²×C₈

—

116

C₄×Q₈

—

117

C₄×Q₈

C₂×#6157

yes

characteristic

118

C₄×D₄

—

119

C₄×D₄

—

120

#117

—

121

C₂×C₄²

—

122

M₃₂

—

212

—

123

C₂×C₁₆

—

124

C₂×C₁₆

—

125

C₄⋊C₈

—

126

C₄⋊C₈

C₂×#6157

yes

—

127

C₂²⋊C₈

—

128

C₄×C₈

—

129

C₄×C₈

—

130

C₄×C₈

—

131

C₄×C₈

C₂×#6157

yes

—

132

#41

—

106

—

133

C₂×Q₈

D₅₃⋊C₈

yes

characteristic

134

C₂×D₄

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

106

—

141

C₂×C₈

—

106

—

142

C₂×C₈

—

212

—

143

C₂×C₈

C₂²×#6157

yes

characteristic

144

C₂×C₈

C₂²×#6157

yes

—

145

C₄⋊C₄

—

146

C₄⋊C₄

D₅₃⋊C₈

yes

—

147

C₂²⋊C₄

—

148

C₄²

—

149

C₄²

—

150

C₄²

—

151

C₄²

C₂²×#6157

yes

—

152

C₁₆

—

106

—

153

C₁₆

—

106

—

154

C₂³

—

155

Q₈

#13133292

yes

complemented

156

D₄

—

106

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

106

—

161

C₂×C₄

—

106

—

162

C₂×C₄

—

106

—

163

C₂×C₄

—

106

—

164

C₂×C₄

C₂×D₅₃⋊C₈

yes

—

165

C₂×C₄

C₂×D₅₃⋊C₈

yes

—

166

C₂×C₄

C₂³×#6157

yes

characteristic

167

C₈

—

168

C₈

—

169

C₈

—

170

C₈

—

171

C₈

—

106

—

172

C₂²

—

106

—

173

C₂²

—

106

—

174

C₂²

—

106

—

175

C₂²

C₂²×D₅₃⋊C₈

yes

characteristic

176

C₄

—

177

C₄

—

178

C₄

—

106

—

179

C₄

—

106

—

180

C₄

C₂×#13133292

yes

—

181

C₄

#16811692

yes

characteristic

182

C₄

#16811703

yes

characteristic

183

C₂

—

212

—

184

C₂

#20291633

yes

characteristic

185

C₂

C₂²×#13133292

yes

characteristic

186

C₂

D₅₃⋊#504

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6157
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₈×Q₈×C₅₃	C₄
prime radical	C₂×#11719949	C₈
cosocle	C₂×#88322	C₂⁴
hypercenter	C₈×Q₈	#6157
hypocenter	C₅₃	#33505
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₈×Q₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×#6157

commutator, socle

C₂²×C₅₃

C₂³×C₈

Fitting

C₈×Q₈×C₅₃

C₄

C₈

C₂⁴

C₈×Q₈

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄ ◁ C₈×Q₈

N	H ≅ G/N	type
1. Q₈×#11719910	C₂	semidirect
2. #20291101	C₂	semidirect
3. #20291074	C₂	semidirect
4. #20291048	C₂	semidirect
61. Q₈×C₅₃	M₃₂	semidirect
112. C₅₃	#33505	semidirect
155. Q₈	#13133292	semidirect

H ≅ G/N

type

1. Q₈×#11719910

C₂

semidirect

2. #20291101

C₂

semidirect

3. #20291074

C₂

semidirect

4. #20291048

C₂

semidirect

61. Q₈×C₅₃

M₃₂

semidirect

112. C₅₃

#33505

semidirect

155. Q₈

#13133292

semidirect

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

subgroup	index
Q₈×#11719910	2
#20291101	2
#20291074	2
#20291048	2
C₈×#11719949	2
#33505	53

subgroup

index

Subgroups — #28328577

Conjugacy classes of subgroups

Complements

Maximal subgroups