Group #26029903 — subgroups

Subgroups — #26029903

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	C₂×#15668018	C₂	1	1	yes	maximal, characteristic
2	#19152172	C₂	1	2	yes	maximal
3	#19151853	C₂	1	4	yes	maximal, complemented
4	#19151836	C₂	1	4	yes	maximal, complemented
5	#19151636	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151607	C₂	1	2	yes	maximal, complemented
7	C₂²×Q₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13083574	C₂²	1	1	yes	characteristic
9	#15668018	—	2	4	no	—
10	C₂×#13083558	C₂²	1	1	yes	characteristic
11	C₂×#13083557	C₂²	1	2	yes	—
12	C₂×C₈×D₈₉	C₂²	1	2	yes	—
13	#15668010	—	2	2	no	—
14	#15668001	—	2	2	no	—
15	#15667995	—	2	4	no	—
16	#15667988	C₂²	1	8	yes	—
17	#15667971	C₂²	1	2	yes	—
18	#15667946	C₂²	1	2	yes	—
19	#15667944	C₂²	1	2	yes	—
20	#15667936	C₂²	1	8	yes	—
21	C₂²×Q₈×C₈₉	C₂²	1	2	yes	—
22	C₂×Q₁₆×C₈₉	—	2	4	no	—
23	C₂×Q₁₆×C₈₉	C₂²	1	2	yes	—
24	C₂×Q₁₆×C₈₉	C₂²	1	2	yes	—
25	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
26	#13083574	—	2	2	no	—
27	#13083574	—	2	2	no	—
28	C₂×Dic₁₇₈	D₄	1	1	yes	characteristic
29	C₂×D₃₅₆	D₄	1	1	yes	characteristic
30	C₂×C₂²⋊D₈₉	D₄	1	2	yes	—
31	C₂×C₄×D₈₉	C₂³	1	2	yes	—
32	#13083558	—	2	2	no	—
33	#13083557	—	2	4	no	—
34	C₂×#10623494	C₂³	1	2	yes	—
35	C₈×D₈₉	—	2	4	no	—
36	C₄⋊Dic₈₉	C₂³	1	4	yes	—
37	#13083553	—	2	8	no	—
38	C₂²⋊Dic₈₉	—	2	2	no	—
39	#13083551	—	2	8	no	—
40	C₄×Dic₈₉	—	2	2	no	—
41	C₂×Q₈×C₈₉	—	2	2	no	—
42	C₂×Q₈×C₈₉	—	2	4	no	—
43	C₂×Q₈×C₈₉	—	2	4	no	—
44	C₂×Q₈×C₈₉	C₂³	1	4	yes	—
45	C₂²×C₄×C₈₉	—	2	2	no	—
46	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
47	Q₁₆×C₈₉	—	2	4	no	—
48	Q₁₆×C₈₉	—	2	4	no	—
49	C₂×C₈×C₈₉	D₄	1	4	yes	—
50	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
51	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
52	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
53	C₂²×D₈₉	C₂×D₄	1	2	yes	—
54	Q₈×C₈₉	—	2	4	no	—
55	Q₈×C₈₉	—	2	4	no	—
56	Q₈×C₈₉	—	4	4	no	—
57	C₂×C₄×C₈₉	—	2	4	no	—
58	C₂×C₄×C₈₉	—	4	4	no	—
59	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₈₉	#41	1	2	yes	—
62	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
63	C₂²⋊D₈₉	—	2	4	no	—
64	C₂×Dic₈₉	—	2	4	no	—
65	C₂×Dic₈₉	C₂×D₄	1	2	yes	—
66	D₃₅₆	—	2	2	no	—
67	C₄×D₈₉	—	2	2	no	—
68	C₄×D₈₉	—	2	2	no	—
69	Dic₁₇₈	—	2	2	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	D₈₉	1	1	yes	characteristic, complemented
91	C₂×#131	—	89	1	no	—
92	#536	—	89	2	no	—
93	#470	—	89	4	no	—
94	#467	—	89	4	no	—
95	#427	—	89	1	no	—
96	#419	—	89	2	no	—
97	C₂×#41	—	89	1	no	—
98	C₂²×Q₈	C₂×D₈₉	1	2	yes	—
99	C₂×Q₁₆	—	2	4	no	—
100	C₂×Q₁₆	C₂×D₈₉	1	2	yes	—
101	C₂×Q₁₆	C₂×D₈₉	1	2	yes	—
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	—	89	2	no	—
112	#107	—	89	2	no	—
113	#106	—	89	2	no	—
114	#103	—	89	8	no	—
115	#41	—	178	2	no	—
116	#41	—	178	2	no	—
117	C₂×Q₈	—	2	2	no	—
118	C₂×Q₈	—	2	4	no	—
119	C₂×Q₈	—	2	4	no	—
120	C₂×Q₈	—	89	1	no	—
121	C₂×Q₈	C₂²×D₈₉	1	4	yes	—
122	C₂×D₄	—	89	2	no	—
123	C₂×D₄	—	89	1	no	—
124	C₂²×C₄	—	2	2	no	—
125	C₂²×C₄	—	89	2	no	—
126	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
127	Q₁₆	—	2	4	no	—
128	Q₁₆	—	2	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	C₂²⋊D₈₉	1	4	yes	—
134	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
135	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
136	C₄⋊C₄	—	89	4	no	—
137	C₄⋊C₄	—	178	8	no	—
138	C₂²⋊C₄	—	178	8	no	—
139	C₂²⋊C₄	—	178	2	no	—
140	C₄²	—	178	2	no	—
141	C₂³	—	89	2	no	—
142	C₂³	D₄×D₈₉	1	1	yes	characteristic
143	Q₈	—	2	4	no	—
144	Q₈	—	2	4	no	—
145	Q₈	—	4	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	—	4	4	no	—
151	C₂×C₄	—	89	2	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×C₂²⋊D₈₉	1	2	yes	—
157	C₂×C₄	#13083573	1	2	yes	—
158	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
159	C₈	—	178	2	no	—
160	C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
161	C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
162	C₂²	—	2	2	no	—
163	C₂²	—	178	4	no	—
164	C₂²	—	178	2	no	—
165	C₂²	#15667983	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	#15667983	1	2	yes	—
171	C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
172	C₄	C₂×#13083573	1	1	yes	characteristic
173	C₂	—	2	2	no	—
174	C₂	—	356	2	no	—
175	C₂	C₂×#15667983	1	1	yes	characteristic
176	C₂	#19152411	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15668018

C₂

yes

maximal, characteristic

#19152172

C₂

yes

maximal

#19151853

C₂

yes

maximal, complemented

#19151836

C₂

yes

maximal, complemented

#19151636

C₂

yes

maximal, characteristic, complemented

#19151607

C₂

yes

maximal, complemented

C₂²×Q₁₆×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×#13083574

C₂²

yes

characteristic

#15668018

—

C₂×#13083558

C₂²

yes

characteristic

C₂×#13083557

C₂²

yes

—

C₂×C₈×D₈₉

C₂²

yes

—

—

—

—

C₂²

yes

—

#15667971

C₂²

yes

—

#15667946

C₂²

yes

—

#15667944

C₂²

yes

—

#15667936

C₂²

yes

—

C₂²×Q₈×C₈₉

C₂²

yes

—

C₂×Q₁₆×C₈₉

—

C₂×Q₁₆×C₈₉

C₂²

yes

—

C₂×Q₁₆×C₈₉

C₂²

yes

—

C₂²×C₈×C₈₉

C₂²

yes

characteristic

#13083574

—

#13083574

—

C₂×Dic₁₇₈

D₄

yes

characteristic

C₂×D₃₅₆

D₄

yes

characteristic

C₂×C₂²⋊D₈₉

D₄

yes

—

C₂×C₄×D₈₉

C₂³

yes

—

#13083558

—

#13083557

—

C₂×#10623494

C₂³

yes

—

C₈×D₈₉

—

C₄⋊Dic₈₉

C₂³

yes

—

#13083553

—

C₂²⋊Dic₈₉

—

#13083551

—

C₄×Dic₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

Q₁₆×C₈₉

—

Q₁₆×C₈₉

—

C₂×C₈×C₈₉

D₄

yes

—

C₂×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₈×C₈₉

C₂³

yes

characteristic

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂²×D₈₉

C₂×D₄

yes

—

Q₈×C₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

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×D₄

yes

—

D₃₅₆

—

C₄×D₈₉

—

C₄×D₈₉

—

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

—

—

—

—

—

—

C₂×#41

—

C₂²×Q₈

C₂×D₈₉

yes

—

C₂×Q₁₆

—

100

C₂×Q₁₆

C₂×D₈₉

yes

—

101

C₂×Q₁₆

C₂×D₈₉

yes

—

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

—

112

#107

—

113

#106

—

114

#103

—

115

#41

—

178

—

116

#41

—

178

—

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂×Q₈

—

120

C₂×Q₈

—

121

C₂×Q₈

C₂²×D₈₉

yes

—

122

C₂×D₄

—

123

C₂×D₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

C₂²×D₈₉

yes

characteristic

127

Q₁₆

—

128

Q₁₆

—

129

M₁₆

—

178

—

130

M₁₆

—

178

—

131

C₂×C₈

—

132

C₂×C₈

—

178

—

133

C₂×C₈

C₂²⋊D₈₉

yes

—

134

C₂×C₈

C₂²×D₈₉

yes

characteristic

135

C₂×C₈

C₂²×D₈₉

yes

characteristic

136

C₄⋊C₄

—

137

C₄⋊C₄

—

178

—

138

C₂²⋊C₄

—

178

—

139

C₂²⋊C₄

—

178

—

140

C₄²

—

178

—

141

C₂³

—

142

C₂³

D₄×D₈₉

yes

characteristic

143

Q₈

—

144

Q₈

—

145

Q₈

—

146

Q₈

—

178

—

147

D₄

—

178

—

148

D₄

—

178

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

178

—

153

C₂×C₄

—

178

—

154

C₂×C₄

—

178

—

155

C₂×C₄

D₄×D₈₉

yes

characteristic

156

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

157

C₂×C₄

#13083573

yes

—

158

C₂×C₄

C₂³×D₈₉

yes

characteristic

159

C₈

—

178

—

160

C₈

C₂×C₂²⋊D₈₉

yes

—

161

C₈

C₂×C₂²⋊D₈₉

yes

—

162

C₂²

—

163

C₂²

—

178

—

164

C₂²

—

178

—

165

C₂²

#15667983

yes

—

166

C₂²

C₂×D₄×D₈₉

yes

characteristic

167

C₄

—

168

C₄

—

356

—

169

C₄

—

178

—

170

C₄

#15667983

yes

—

171

C₄

C₂²×C₂²⋊D₈₉

yes

characteristic

172

C₄

C₂×#13083573

yes

characteristic

173

C₂

—

174

C₂

—

356

—

175

C₂

C₂×#15667983

yes

characteristic

176

C₂

#19152411

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	C₂²×Q₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂×#13083574	C₂²
hypercenter	C₂²×Q₁₆	D₈₉
hypocenter	C₈₉	#3059
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

subgroup

G/H

center

C₂²

C₂×D₄×D₈₉

commutator, cosocle

C₂×C₄×C₈₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₉

Fitting

C₂²×Q₁₆×C₈₉

C₂

socle

C₂²×C₈₉

C₂²×D₄

prime radical

C₂×#13083574

C₂²

hypercenter

C₂²×Q₁₆

D₈₉

hypocenter

C₈₉

#3059

LCS

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
3. #19151853	C₂	semidirect
4. #19151836	C₂	semidirect
5. #19151636	C₂	semidirect
6. #19151607	C₂	semidirect
7. C₂²×Q₁₆×C₈₉	C₂	semidirect
89. C₈₉	#3059	semidirect
90. C₂²×Q₁₆	D₈₉	semidirect

H ≅ G/N

type

3. #19151853

C₂

semidirect

4. #19151836

C₂

semidirect

5. #19151636

C₂

semidirect

6. #19151607

C₂

semidirect

7. C₂²×Q₁₆×C₈₉

C₂

semidirect

89. C₈₉

#3059

semidirect

90. C₂²×Q₁₆

D₈₉

semidirect

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

subgroup	index
C₂×#15668018	2
#19152172	2
#19151853	2
#19151836	2
#19151636	2
#19151607	2
C₂²×Q₁₆×C₈₉	2
#3059	89

subgroup

index

C₂²×Q₁₆×C₈₉

Subgroups — #26029903

Conjugacy classes of subgroups

Complements

Maximal subgroups