Group #26029908 — subgroups

Subgroups — #26029908

Order 11392.

460 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 141 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	C₂²×Dic₃₅₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15668018	C₂	1	1	yes	maximal, characteristic
3	#19152174	C₂	1	2	yes	maximal
4	#19151857	C₂	1	4	yes	maximal, complemented
5	#19151840	C₂	1	4	yes	maximal, complemented
6	#19151606	C₂	1	2	yes	maximal, complemented
7	#427×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13083574	C₂²	1	1	yes	characteristic
9	C₂²×Dic₁₇₈	C₂²	1	2	yes	—
10	C₂×Dic₃₅₆	—	2	4	no	—
11	C₂×Dic₃₅₆	C₂²	1	2	yes	—
12	C₂×Dic₃₅₆	C₂²	1	2	yes	—
13	#15668018	—	2	4	no	—
14	C₂×#13083558	C₂²	1	1	yes	characteristic
15	C₂×#13083557	C₂²	1	2	yes	—
16	C₂×C₈×D₈₉	C₂²	1	2	yes	—
17	#15668012	—	2	2	no	—
18	#15668000	—	2	2	no	—
19	#15667996	—	2	4	no	—
20	#15667992	C₂²	1	8	yes	—
21	#15667935	C₂²	1	8	yes	—
22	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
23	#117×C₈₉	C₂²	1	2	yes	—
24	#107×C₈₉	C₂²	1	2	yes	—
25	#106×C₈₉	C₂²	1	2	yes	—
26	#13083574	—	2	2	no	—
27	#13083574	—	2	2	no	—
28	C₂×Dic₁₇₈	—	2	4	no	—
29	C₂×Dic₁₇₈	—	2	4	no	—
30	C₂×Dic₁₇₈	—	2	2	no	—
31	C₂×Dic₁₇₈	D₄	1	1	yes	characteristic
32	C₂×Dic₁₇₈	C₂³	1	4	yes	—
33	C₂×D₃₅₆	D₄	1	1	yes	characteristic
34	C₂×C₂²⋊D₈₉	D₄	1	2	yes	—
35	C₂²×Dic₈₉	—	2	2	no	—
36	C₂×C₄×D₈₉	C₂³	1	2	yes	—
37	Dic₃₅₆	—	2	4	no	—
38	Dic₃₅₆	—	2	4	no	—
39	#13083558	—	2	2	no	—
40	#13083557	—	2	4	no	—
41	C₂×#10623494	C₂³	1	2	yes	—
42	C₈×D₈₉	—	2	4	no	—
43	C₄⋊Dic₈₉	—	2	8	no	—
44	#13083551	—	2	8	no	—
45	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
46	C₂×C₈×C₈₉	D₄	1	4	yes	—
47	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
48	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
49	C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
50	C₂²⋊C₄×C₈₉	—	2	2	no	—
51	C₄²×C₈₉	—	2	2	no	—
52	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
53	C₂²×D₈₉	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₈₉	—	2	4	no	—
55	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₈₉	#41	1	2	yes	—
58	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
59	C₂²⋊D₈₉	—	2	4	no	—
60	C₂×Dic₈₉	—	2	4	no	—
61	C₂×Dic₈₉	—	4	4	no	—
62	C₂×Dic₈₉	C₂×D₄	1	2	yes	—
63	D₃₅₆	—	2	2	no	—
64	C₄×D₈₉	—	2	2	no	—
65	C₄×D₈₉	—	2	2	no	—
66	Dic₁₇₈	—	2	2	no	—
67	Dic₁₇₈	—	2	4	no	—
68	Dic₁₇₈	—	2	4	no	—
69	Dic₁₇₈	—	4	4	no	—
70	C₈×C₈₉	C₂×D₄	1	2	yes	—
71	C₈×C₈₉	C₂×D₄	1	2	yes	—
72	#10623494	—	2	2	no	—
73	C₂²×C₈₉	—	2	2	no	—
74	C₂²×C₈₉	#121	1	2	yes	—
75	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
76	C₂×D₈₉	—	2	2	no	—
77	C₂×D₈₉	—	2	4	no	—
78	C₄×C₈₉	—	4	4	no	—
79	C₄×C₈₉	#121	1	2	yes	—
80	C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
81	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
82	Dic₈₉	—	2	2	no	—
83	Dic₈₉	—	4	4	no	—
84	C₂×C₈₉	—	2	2	no	—
85	C₂×C₈₉	C₂×#121	1	1	yes	characteristic
86	C₂×C₈₉	#578	1	2	yes	—
87	D₈₉	—	4	2	no	—
88	#3059	—	89	1	no	maximal, Sylow
89	C₈₉	#3059	1	1	yes	characteristic, complemented, Sylow
90	C₂²×Q₁₆	—	89	1	no	—
91	C₂×#131	—	89	1	no	—
92	#536	—	89	2	no	—
93	#470	—	89	4	no	—
94	#467	—	89	4	no	—
95	#427	D₈₉	1	1	yes	characteristic, complemented
96	#419	—	89	2	no	—
97	C₂×#41	—	89	1	no	—
98	C₂²×Q₈	—	89	2	no	—
99	C₂×Q₁₆	—	89	2	no	—
100	C₂×Q₁₆	—	89	2	no	—
101	C₂×Q₁₆	—	178	4	no	—
102	#131	—	178	4	no	—
103	C₂×M₁₆	—	89	1	no	—
104	C₂×M₁₆	—	89	2	no	—
105	C₂²×C₈	—	89	2	no	—
106	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
107	C₄⋊Q₈	—	178	2	no	—
108	#124	—	178	2	no	—
109	#123	—	178	4	no	—
110	C₂²⋊Q₈	—	89	8	no	—
111	#117	C₂×D₈₉	1	2	yes	—
112	#107	C₂×D₈₉	1	2	yes	—
113	#106	C₂×D₈₉	1	2	yes	—
114	#103	—	89	8	no	—
115	#41	—	178	2	no	—
116	#41	—	178	2	no	—
117	C₂×Q₈	—	89	1	no	—
118	C₂×Q₈	—	89	4	no	—
119	C₂×Q₈	—	178	4	no	—
120	C₂×Q₈	—	178	4	no	—
121	C₂×Q₈	—	178	2	no	—
122	C₂×D₄	—	89	2	no	—
123	C₂×D₄	—	89	1	no	—
124	C₂²×C₄	—	89	2	no	—
125	C₂²×C₄	—	178	2	no	—
126	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
127	Q₁₆	—	178	4	no	—
128	Q₁₆	—	178	4	no	—
129	M₁₆	—	178	2	no	—
130	M₁₆	—	178	4	no	—
131	C₂×C₈	—	89	2	no	—
132	C₂×C₈	—	178	4	no	—
133	C₂×C₈	D₃₅₆	1	4	yes	—
134	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
135	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
136	C₄⋊C₄	—	178	8	no	—
137	C₄⋊C₄	C₂²×D₈₉	1	4	yes	—
138	C₂²⋊C₄	—	2	2	no	—
139	C₂²⋊C₄	—	178	8	no	—
140	C₄²	—	2	2	no	—
141	C₂³	—	89	2	no	—
142	C₂³	D₄×D₈₉	1	1	yes	characteristic
143	Q₈	—	356	4	no	—
144	Q₈	—	178	4	no	—
145	Q₈	—	178	4	no	—
146	Q₈	—	178	2	no	—
147	D₄	—	178	4	no	—
148	D₄	—	178	2	no	—
149	C₂×C₄	—	2	4	no	—
150	C₂×C₄	—	89	2	no	—
151	C₂×C₄	—	356	4	no	—
152	C₂×C₄	—	178	4	no	—
153	C₂×C₄	—	178	2	no	—
154	C₂×C₄	—	178	2	no	—
155	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
156	C₂×C₄	C₂×D₃₅₆	1	2	yes	—
157	C₂×C₄	#13083575	1	2	yes	—
158	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
159	C₈	—	178	2	no	—
160	C₈	C₂×D₃₅₆	1	2	yes	—
161	C₈	C₂×D₃₅₆	1	2	yes	—
162	C₂²	—	2	2	no	—
163	C₂²	—	178	4	no	—
164	C₂²	—	178	2	no	—
165	C₂²	#15667987	1	2	yes	—
166	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
167	C₄	—	4	4	no	—
168	C₄	—	356	4	no	—
169	C₄	—	178	2	no	—
170	C₄	#15667987	1	2	yes	—
171	C₄	C₂²×D₃₅₆	1	1	yes	characteristic
172	C₄	C₂×#13083575	1	1	yes	characteristic
173	C₂	—	2	2	no	—
174	C₂	—	356	2	no	—
175	C₂	C₂×#15667987	1	1	yes	characteristic
176	C₂	#19152412	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₃₅₆

C₂

yes

maximal, characteristic, complemented

C₂×#15668018

C₂

yes

maximal, characteristic

#19152174

C₂

yes

maximal

#19151857

C₂

yes

maximal, complemented

#19151840

C₂

yes

maximal, complemented

#19151606

C₂

yes

maximal, complemented

#427×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×#13083574

C₂²

yes

characteristic

C₂²×Dic₁₇₈

C₂²

yes

—

C₂×Dic₃₅₆

—

C₂×Dic₃₅₆

C₂²

yes

—

C₂×Dic₃₅₆

C₂²

yes

—

#15668018

—

C₂×#13083558

C₂²

yes

characteristic

C₂×#13083557

C₂²

yes

—

C₂×C₈×D₈₉

C₂²

yes

—

—

—

—

C₂²

yes

—

#15667935

C₂²

yes

—

C₂²×C₈×C₈₉

C₂²

yes

characteristic

#117×C₈₉

C₂²

yes

—

#107×C₈₉

C₂²

yes

—

#106×C₈₉

C₂²

yes

—

#13083574

—

#13083574

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

D₄

yes

characteristic

C₂×Dic₁₇₈

C₂³

yes

—

C₂×D₃₅₆

D₄

yes

characteristic

C₂×C₂²⋊D₈₉

D₄

yes

—

C₂²×Dic₈₉

—

C₂×C₄×D₈₉

C₂³

yes

—

Dic₃₅₆

—

Dic₃₅₆

—

#13083558

—

#13083557

—

C₂×#10623494

C₂³

yes

—

C₈×D₈₉

—

C₄⋊Dic₈₉

—

#13083551

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂×C₈×C₈₉

D₄

yes

—

C₂×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₈×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

C₂³

yes

—

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂²×D₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

#41

yes

—

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

C₂²⋊D₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

C₂×D₄

yes

—

D₃₅₆

—

C₄×D₈₉

—

C₄×D₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

C₂×D₄

yes

—

C₈×C₈₉

C₂×D₄

yes

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

#121

yes

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#121

yes

—

C₄×C₈₉

C₂²×D₄

yes

characteristic

C₄×C₈₉

C₂×#41

yes

characteristic

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

C₂×#121

yes

characteristic

C₂×C₈₉

#578

yes

—

D₈₉

—

#3059

—

maximal, Sylow

C₈₉

#3059

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

—

—

—

—

D₈₉

yes

characteristic, complemented

#419

—

C₂×#41

—

C₂²×Q₈

—

C₂×Q₁₆

—

100

C₂×Q₁₆

—

101

C₂×Q₁₆

—

178

—

102

#131

—

178

—

103

C₂×M₁₆

—

104

C₂×M₁₆

—

105

C₂²×C₈

—

106

C₂²×C₈

C₂×D₈₉

yes

characteristic

107

C₄⋊Q₈

—

178

—

108

#124

—

178

—

109

#123

—

178

—

110

C₂²⋊Q₈

—

111

#117

C₂×D₈₉

yes

—

112

#107

C₂×D₈₉

yes

—

113

#106

C₂×D₈₉

yes

—

114

#103

—

115

#41

—

178

—

116

#41

—

178

—

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂×Q₈

—

178

—

120

C₂×Q₈

—

178

—

121

C₂×Q₈

—

178

—

122

C₂×D₄

—

123

C₂×D₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

178

—

126

C₂²×C₄

C₂²×D₈₉

yes

characteristic

127

Q₁₆

—

178

—

128

Q₁₆

—

178

—

129

M₁₆

—

178

—

130

M₁₆

—

178

—

131

C₂×C₈

—

132

C₂×C₈

—

178

—

133

C₂×C₈

D₃₅₆

yes

—

134

C₂×C₈

C₂²×D₈₉

yes

characteristic

135

C₂×C₈

C₂²×D₈₉

yes

characteristic

136

C₄⋊C₄

—

178

—

137

C₄⋊C₄

C₂²×D₈₉

yes

—

138

C₂²⋊C₄

—

139

C₂²⋊C₄

—

178

—

140

C₄²

—

141

C₂³

—

142

C₂³

D₄×D₈₉

yes

characteristic

143

Q₈

—

356

—

144

Q₈

—

178

—

145

Q₈

—

178

—

146

Q₈

—

178

—

147

D₄

—

178

—

148

D₄

—

178

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

356

—

152

C₂×C₄

—

178

—

153

C₂×C₄

—

178

—

154

C₂×C₄

—

178

—

155

C₂×C₄

D₄×D₈₉

yes

characteristic

156

C₂×C₄

C₂×D₃₅₆

yes

—

157

C₂×C₄

#13083575

yes

—

158

C₂×C₄

C₂³×D₈₉

yes

characteristic

159

C₈

—

178

—

160

C₈

C₂×D₃₅₆

yes

—

161

C₈

C₂×D₃₅₆

yes

—

162

C₂²

—

163

C₂²

—

178

—

164

C₂²

—

178

—

165

C₂²

#15667987

yes

—

166

C₂²

C₂×D₄×D₈₉

yes

characteristic

167

C₄

—

168

C₄

—

356

—

169

C₄

—

178

—

170

C₄

#15667987

yes

—

171

C₄

C₂²×D₃₅₆

yes

characteristic

172

C₄

C₂×#13083575

yes

characteristic

173

C₂

—

174

C₂

—

356

—

175

C₂

C₂×#15667987

yes

characteristic

176

C₂

#19152412

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#427×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂×#13083574	C₂²
hypercenter	#427	D₈₉
hypocenter	C₈₉	#3059
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂²

C₂×D₄×D₈₉

commutator, cosocle

C₂×C₄×C₈₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₉

Fitting

#427×C₈₉

C₂

socle

C₂²×C₈₉

C₂²×D₄

C₂²

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
1. C₂²×Dic₃₅₆	C₂	semidirect
4. #19151857	C₂	semidirect
5. #19151840	C₂	semidirect
6. #19151606	C₂	semidirect
7. #427×C₈₉	C₂	semidirect
89. C₈₉	#3059	semidirect
95. #427	D₈₉	semidirect

H ≅ G/N

type

1. C₂²×Dic₃₅₆

C₂

semidirect

4. #19151857

C₂

semidirect

5. #19151840

C₂

semidirect

6. #19151606

C₂

semidirect

7. #427×C₈₉

C₂

semidirect

89. C₈₉

#3059

semidirect

95. #427

D₈₉

semidirect

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

subgroup	index
C₂²×Dic₃₅₆	2
C₂×#15668018	2
#19152174	2
#19151857	2
#19151840	2
#19151606	2
#427×C₈₉	2
#3059	89

subgroup

index

C₂²×Dic₃₅₆

#427×C₈₉

Subgroups — #26029908

Conjugacy classes of subgroups

Complements

Maximal subgroups