Group #28303582 — subgroups

Subgroups — #28303582

Order 13568.

462 conjugacy classes of subgroups; only the 212 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 212 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	#20290647	C₂	1	2	yes	maximal
2	C₂×#16811487	C₂	1	4	yes	maximal
3	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#88324	—	2	4	no	—
5	C₂×C₄⋊#88324	C₂²	1	4	yes	—
6	#16811487	—	2	8	no	—
7	C₂×C₄×#88322	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₅₃	C₄	1	2	yes	—
9	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
10	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
12	C₂²×#88324	—	2	4	no	—
13	C₄⋊#88324	—	2	8	no	—
14	C₄⋊#88324	—	4	8	no	—
15	C₂²×#88322	C₂×C₄	1	2	yes	—
16	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×Dic₅₃	D₄	1	2	yes	—
18	C₂×C₄×Dic₅₃	Q₈	1	2	yes	—
19	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
20	C₄×#88322	C₈	1	4	yes	complemented
21	C₈×Dic₅₃	—	2	4	no	—
22	C₈×Dic₅₃	D₄	1	4	yes	—
23	C₈×Dic₅₃	Q₈	1	4	yes	—
24	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₅₃	—	2	2	no	—
27	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
28	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
29	C₂²×Dic₅₃	C₄⋊C₄	1	2	yes	—
30	C₂²×Dic₅₃	C₂×D₄	1	1	yes	characteristic
31	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
32	C₂×#88324	—	4	4	no	—
33	C₂×#88324	—	4	8	no	—
34	C₂×C₈×C₅₃	—	2	2	no	—
35	C₂×C₈×C₅₃	—	2	2	no	—
36	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
37	C₄²×C₅₃	C₂×C₈	1	2	yes	—
38	C₄²×C₅₃	M₁₆	1	2	yes	—
39	C₄×Dic₅₃	—	2	4	no	—
40	C₄×Dic₅₃	—	2	2	no	—
41	C₄×Dic₅₃	D₈	1	4	yes	—
42	C₄×Dic₅₃	Q₁₆	1	4	yes	—
43	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
44	C₄×Dic₅₃	C₂×Q₈	1	2	yes	—
45	C₂×#88322	C₄⋊C₄	1	4	yes	—
46	C₂×#88322	C₂×C₈	1	4	yes	—
47	C₂×#88322	M₁₆	1	4	yes	—
48	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
50	C₂×C₄×C₅₃	#107	1	4	yes	—
51	C₂×C₄×C₅₃	#108	1	4	yes	—
52	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
53	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
56	C₂×Dic₅₃	—	2	2	no	—
57	C₂×Dic₅₃	—	2	2	no	—
58	C₂×Dic₅₃	—	2	2	no	—
59	C₂×Dic₅₃	#107	1	4	yes	—
60	C₂×Dic₅₃	#127	1	1	yes	characteristic
61	C₂×Dic₅₃	C₄⋊Q₈	1	2	yes	—
62	C₂×Dic₅₃	C₄⋊Q₈	1	1	yes	characteristic
63	C₂×Dic₅₃	C₂×D₈	1	2	yes	—
64	C₂×Dic₅₃	C₂×Q₁₆	1	2	yes	—
65	#88324	—	8	4	no	—
66	C₈×C₅₃	—	2	4	no	—
67	#88322	C₄⋊C₈	1	8	yes	complemented
68	#8806	—	53	1	no	maximal, Sylow
69	C₂²×C₅₃	#384	1	1	yes	characteristic
70	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
71	C₂²×C₅₃	#423	1	1	yes	characteristic
72	C₂²×C₅₃	C₂×#107	1	2	yes	—
73	C₂²×C₅₃	C₂×#108	1	2	yes	—
74	C₄×C₅₃	#335	1	8	yes	complemented
75	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
76	C₄×C₅₃	#423	1	2	yes	—
77	Dic₅₃	—	2	4	no	—
78	Dic₅₃	—	2	2	no	—
79	Dic₅₃	#493	1	1	yes	characteristic
80	Dic₅₃	#494	1	1	yes	characteristic
81	Dic₅₃	#500	1	2	yes	—
82	#1842	—	53	2	no	—
83	C₄²×C₈	—	53	1	no	—
84	C₂×#335	—	53	4	no	—
85	C₂×C₅₃	C₂×#335	1	2	yes	—
86	C₂×C₅₃	#1567	1	2	yes	—
87	C₂×C₅₃	#1842	1	1	yes	characteristic
88	C₂×C₅₃	#1845	1	1	yes	characteristic
89	C₂×C₅₃	#1848	1	1	yes	characteristic
90	C₂×C₄⋊C₈	—	53	4	no	—
91	C₂×C₄⋊C₈	—	106	4	no	—
92	C₂×C₄×C₈	—	53	2	no	—
93	C₂×C₄×C₈	—	53	1	no	—
94	C₂×C₄×C₈	—	53	2	no	—
95	C₂×C₄×C₈	#6157	1	1	yes	characteristic
96	C₄³	—	53	1	no	—
97	#335	—	106	8	no	—
98	C₅₃	#8806	1	1	yes	characteristic, complemented, Sylow
99	C₂²×C₈	—	53	2	no	—
100	C₂²×C₈	—	106	4	no	—
101	C₂²×C₈	C₂×#6157	1	2	yes	—
102	C₂×C₄²	—	53	2	no	—
103	C₂×C₄²	—	53	1	no	—
104	C₂×C₄²	—	53	2	no	—
105	C₂×C₄²	—	53	1	no	—
106	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
107	C₄⋊C₈	—	106	8	no	—
108	C₄⋊C₈	—	212	8	no	—
109	C₄×C₈	—	2	2	no	—
110	C₄×C₈	—	53	4	no	—
111	C₄×C₈	—	53	4	no	—
112	C₄×C₈	—	53	4	no	—
113	C₄×C₈	—	106	4	no	—
114	C₂²×C₄	—	53	1	no	—
115	C₂²×C₄	—	53	2	no	—
116	C₂²×C₄	—	53	1	no	—
117	C₂²×C₄	C₄⋊#6157	1	2	yes	—
118	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	53	4	no	—
122	C₂×C₈	—	53	4	no	—
123	C₂×C₈	—	53	4	no	—
124	C₂×C₈	—	212	4	no	—
125	C₂×C₈	—	212	8	no	—
126	C₂×C₈	C₄⋊#6157	1	4	yes	—
127	C₄²	—	53	4	no	—
128	C₄²	—	53	4	no	—
129	C₄²	—	53	2	no	—
130	C₄²	—	53	2	no	—
131	C₄²	—	106	4	no	—
132	C₄²	—	106	2	no	—
133	C₄²	D₅₃⋊C₈	1	2	yes	—
134	C₄²	#11719940	1	2	yes	—
135	C₂³	C₂×C₄⋊#6157	1	1	yes	characteristic
136	C₂×C₄	—	53	1	no	—
137	C₂×C₄	—	53	1	no	—
138	C₂×C₄	—	53	2	no	—
139	C₂×C₄	—	53	2	no	—
140	C₂×C₄	—	53	2	no	—
141	C₂×C₄	—	53	4	no	—
142	C₂×C₄	—	106	2	no	—
143	C₂×C₄	—	106	2	no	—
144	C₂×C₄	—	106	2	no	—
145	C₂×C₄	#13133284	1	4	yes	—
146	C₂×C₄	#13133286	1	4	yes	—
147	C₂×C₄	#13133287	1	4	yes	—
148	C₂×C₄	C₂×C₄⋊#6157	1	2	yes	—
149	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
150	C₂×C₄	C₂×#11719940	1	1	yes	characteristic
151	C₂×C₄	D₅₃⋊M₁₆	1	2	yes	—
152	C₈	—	2	4	no	—
153	C₈	—	53	8	no	—
154	C₈	—	424	4	no	—
155	C₂²	#16811569	1	1	yes	characteristic
156	C₂²	C₂×#13133284	1	1	yes	characteristic
157	C₂²	#16811628	1	1	yes	characteristic
158	C₂²	C₂×#13133286	1	2	yes	—
159	C₂²	C₂×#13133287	1	2	yes	—
160	C₄	—	53	1	no	—
161	C₄	—	53	1	no	—
162	C₄	—	53	2	no	—
163	C₄	—	106	2	no	—
164	C₄	—	106	4	no	—
165	C₄	#16811487	1	8	yes	complemented
166	C₄	D₅₃⋊(C₄⋊C₈)	1	2	yes	—
167	C₄	#16811626	1	2	yes	—
168	C₂	C₂×#16811487	1	2	yes	—
169	C₂	#20290274	1	2	yes	—
170	C₂	#20290648	1	1	yes	characteristic
171	C₂	#20290651	1	1	yes	characteristic
172	C₂	#20290654	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20290647

C₂

yes

maximal

C₂×#16811487

C₂

yes

maximal

C₄×C₈×Dic₅₃

C₂

yes

maximal, characteristic

C₂×C₄⋊#88324

—

C₂×C₄⋊#88324

C₂²

yes

—

#16811487

—

C₂×C₄×#88322

C₄

yes

characteristic

C₂×C₈×Dic₅₃

C₄

yes

—

C₂×C₈×Dic₅₃

C₂²

yes

—

C₄²×Dic₅₃

C₂²

yes

characteristic

C₂×C₄×C₈×C₅₃

C₄

yes

characteristic

C₂²×#88324

—

C₄⋊#88324

—

C₄⋊#88324

—

C₂²×#88322

C₂×C₄

yes

—

C₂×C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₂×C₄×Dic₅₃

D₄

yes

—

C₂×C₄×Dic₅₃

Q₈

yes

—

C₂×C₄×Dic₅₃

C₂³

yes

characteristic

C₄×#88322

C₈

yes

complemented

C₈×Dic₅₃

—

C₈×Dic₅₃

D₄

yes

—

C₈×Dic₅₃

Q₈

yes

—

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

—

C₂²×C₄×C₅₃

C₄⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×Dic₅₃

C₄⋊C₄

yes

—

C₂²×Dic₅₃

C₂×D₄

yes

characteristic

C₂²×Dic₅₃

C₂×Q₈

yes

characteristic

C₂×#88324

—

C₂×#88324

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₄²×C₅₃

C₂×C₈

yes

—

C₄²×C₅₃

M₁₆

yes

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

D₈

yes

—

C₄×Dic₅₃

Q₁₆

yes

—

C₄×Dic₅₃

C₂×D₄

yes

—

C₄×Dic₅₃

C₂×Q₈

yes

—

C₂×#88322

C₄⋊C₄

yes

—

C₂×#88322

C₂×C₈

yes

—

C₂×#88322

M₁₆

yes

—

C₂³×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₈

yes

—

C₂×C₄×C₅₃

#107

yes

—

C₂×C₄×C₅₃

#108

yes

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₅₃

C₂×M₁₆

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

#107

yes

—

C₂×Dic₅₃

#127

yes

characteristic

C₂×Dic₅₃

C₄⋊Q₈

yes

—

C₂×Dic₅₃

C₄⋊Q₈

yes

characteristic

C₂×Dic₅₃

C₂×D₈

yes

—

C₂×Dic₅₃

C₂×Q₁₆

yes

—

#88324

—

C₈×C₅₃

—

#88322

C₄⋊C₈

yes

—

C₂²×C₅₃

yes

C₂²×C₅₃

C₂×C₄⋊C₈

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

C₂×#108

yes

—

C₄×C₅₃

#335

yes

complemented

C₄×C₅₃

C₂×C₄⋊C₈

yes

—

C₄×C₅₃

#423

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

yes

Dic₅₃

yes

Dic₅₃

yes

—

#1842

—

C₄²×C₈

—

C₂×#335

—

C₂×C₅₃

C₂×#335

yes

—

C₂×C₅₃

#1567

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

106

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

#6157

yes

characteristic

C₄³

—

#335

—

106

—

C₅₃

#8806

yes

characteristic, complemented, Sylow

C₂²×C₈

—

100

C₂²×C₈

—

106

—

101

C₂²×C₈

C₂×#6157

yes

—

102

C₂×C₄²

—

103

C₂×C₄²

—

104

C₂×C₄²

—

105

C₂×C₄²

—

106

C₂×C₄²

C₂×#6157

yes

characteristic

107

C₄⋊C₈

—

106

—

108

C₄⋊C₈

—

212

—

109

C₄×C₈

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

—

106

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

C₄⋊#6157

yes

—

118

C₂²×C₄

C₂²×#6157

yes

characteristic

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

212

—

125

C₂×C₈

—

212

—

126

C₂×C₈

C₄⋊#6157

yes

—

127

C₄²

—

128

C₄²

—

129

C₄²

—

130

C₄²

—

131

C₄²

—

106

—

132

C₄²

—

106

—

133

C₄²

D₅₃⋊C₈

yes

—

134

C₄²

#11719940

yes

—

135

C₂³

C₂×C₄⋊#6157

yes

characteristic

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

106

—

143

C₂×C₄

—

106

—

144

C₂×C₄

—

106

—

145

C₂×C₄

#13133284

yes

—

146

C₂×C₄

#13133286

yes

—

147

C₂×C₄

#13133287

yes

—

148

C₂×C₄

C₂×C₄⋊#6157

yes

—

149

C₂×C₄

C₂×D₅₃⋊C₈

yes

characteristic

150

C₂×C₄

C₂×#11719940

yes

characteristic

151

C₂×C₄

D₅₃⋊M₁₆

yes

—

152

C₈

—

153

C₈

—

154

C₈

—

424

—

155

C₂²

#16811569

yes

characteristic

156

C₂²

C₂×#13133284

yes

characteristic

157

C₂²

#16811628

yes

characteristic

158

C₂²

C₂×#13133286

yes

—

159

C₂²

C₂×#13133287

yes

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

—

106

—

164

C₄

—

106

—

165

C₄

#16811487

yes

complemented

166

C₄

D₅₃⋊(C₄⋊C₈)

yes

—

167

C₄

#16811626

yes

—

168

C₂

C₂×#16811487

yes

—

169

C₂

#20290274

yes

—

170

C₂

#20290648

yes

characteristic

171

C₂

#20290651

yes

characteristic

172

C₂

#20290654

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#6157
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₂×C₄×C₈×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
cosocle	C₂×C₄×Dic₅₃	C₂³
hypercenter	C₂×C₄×C₈	#6157
hypocenter	C₅₃	#8806
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×C₄⋊#6157

commutator

C₂×C₄×C₅₃

C₂²×C₈

Frattini

C₂²×C₄

C₂²×#6157

Fitting

C₂×C₄×C₈×C₅₃

C₄

socle, prime radical

C₂³×C₅₃

C₂×C₄⋊C₄

cosocle

C₂×C₄×Dic₅₃

C₂³

C₂×C₄×C₈

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
20. C₄×#88322	C₈	semidirect
67. #88322	C₄⋊C₈	semidirect
74. C₄×C₅₃	#335	semidirect
98. C₅₃	#8806	semidirect
165. C₄	#16811487	semidirect

H ≅ G/N

type

20. C₄×#88322

C₈

semidirect

67. #88322

C₄⋊C₈

semidirect

74. C₄×C₅₃

#335

semidirect

98. C₅₃

#8806

semidirect

165. C₄

#16811487

semidirect

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

subgroup	index
#20290647	2
C₂×#16811487	2
C₄×C₈×Dic₅₃	2
#8806	53

subgroup

index

#20290647

C₂×#16811487

C₄×C₈×Dic₅₃

#8806

Subgroups — #28303582

Conjugacy classes of subgroups

Complements

Maximal subgroups