Group #26026082 — subgroups

Subgroups — #26026082

Order 11392.

472 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 165 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×#15668004	C₂	1	1	yes	maximal, characteristic, complemented
2	#19151531	C₂	1	4	yes	maximal, complemented
3	#19151521	C₂	1	4	yes	maximal
4	#19151496	C₂	1	2	yes	maximal, complemented
5	#19151468	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151458	C₂	1	2	yes	maximal
7	C₂×#117×C₈₉	C₂	1	1	yes	maximal, characteristic
8	#15668004	—	2	4	no	—
9	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₈₉	C₂²	1	4	yes	—
12	C₂×#13083553	C₂²	1	4	yes	—
13	C₂×#13083553	C₂²	1	4	yes	—
14	C₂×C₂²⋊Dic₈₉	C₂²	1	4	yes	—
15	#15667919	C₂²	1	4	yes	—
16	#15667919	C₂²	1	4	yes	—
17	C₂³×C₄×C₈₉	C₂²	1	1	yes	characteristic
18	#117×C₈₉	—	2	4	no	—
19	C₂×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₄×C₈₉	C₂²	1	2	yes	—
21	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
22	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
23	C₂²×Dic₈₉	C₂³	1	4	yes	—
24	C₂²×Dic₈₉	C₂³	1	4	yes	—
25	C₄⋊Dic₈₉	—	2	4	no	—
26	C₄⋊Dic₈₉	—	2	4	no	—
27	C₄⋊Dic₈₉	—	4	4	no	—
28	#13083553	—	2	8	no	—
29	#13083553	—	4	4	no	—
30	C₂²⋊Dic₈₉	—	4	4	no	—
31	C₂⁴×C₈₉	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₈₉	—	2	2	no	—
33	C₂²×C₄×C₈₉	—	2	2	no	—
34	C₂²×C₄×C₈₉	Q₈	1	4	yes	—
35	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
36	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
37	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
39	C₄⋊C₄×C₈₉	—	2	4	no	—
40	C₂²⋊C₄×C₈₉	—	2	4	no	—
41	C₄²×C₈₉	—	2	2	no	—
42	C₄²×C₈₉	D₄	1	4	yes	—
43	C₂³×C₈₉	—	2	1	no	—
44	C₂³×C₈₉	—	2	1	no	—
45	C₂³×C₈₉	—	2	2	no	—
46	C₂³×C₈₉	—	2	2	no	—
47	C₂³×C₈₉	C₂×Q₈	1	2	yes	—
48	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
49	C₂×C₄×C₈₉	—	2	1	no	—
50	C₂×C₄×C₈₉	—	2	1	no	—
51	C₂×C₄×C₈₉	—	2	2	no	—
52	C₂×C₄×C₈₉	—	2	2	no	—
53	C₂×C₄×C₈₉	—	2	8	no	—
54	C₂×C₄×C₈₉	—	2	2	no	—
55	C₂×C₄×C₈₉	—	2	2	no	—
56	C₂×C₄×C₈₉	—	2	2	no	—
57	C₂×C₄×C₈₉	C₂×D₄	1	4	yes	—
58	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₈₉	C₂×Q₈	1	2	yes	—
60	C₂×C₄×C₈₉	C₂×Q₈	1	2	yes	—
61	C₂×C₄×C₈₉	#41	1	2	yes	—
62	C₂×Dic₈₉	—	2	4	no	—
63	C₂×Dic₈₉	—	2	4	no	—
64	C₂×Dic₈₉	—	2	4	no	—
65	C₂×Dic₈₉	—	2	4	no	—
66	C₂×Dic₈₉	—	2	4	no	—
67	C₂×Dic₈₉	—	2	4	no	—
68	C₂²×C₈₉	—	2	2	no	—
69	C₂²×C₈₉	—	2	4	no	—
70	C₂²×C₈₉	—	4	2	no	—
71	C₂²×C₈₉	—	4	1	no	—
72	C₂²×C₈₉	—	4	1	no	—
73	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
74	C₂²×C₈₉	C₂²×Q₈	1	1	yes	characteristic
75	C₂²×C₈₉	C₂×#41	1	1	yes	characteristic
76	C₂²×C₈₉	H(5,𝔽₂)	1	2	yes	—
77	C₂²×C₈₉	2₋¹⁺⁴	1	2	yes	—
78	C₄×C₈₉	—	2	2	no	—
79	C₄×C₈₉	—	2	4	no	—
80	C₄×C₈₉	—	4	2	no	—
81	C₄×C₈₉	C₂²⋊Q₈	1	4	yes	—
82	Dic₈₉	—	4	4	no	—
83	Dic₈₉	—	4	4	no	—
84	C₂×C₈₉	—	4	2	no	—
85	C₂×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
86	C₂×C₈₉	#537	1	2	yes	—
87	C₂×C₈₉	#540	1	1	yes	characteristic
88	C₂×C₈₉	#541	1	1	yes	characteristic
89	C₂×C₈₉	#544	1	2	yes	—
90	#2629	—	89	1	no	maximal, Sylow
91	C₈₉	#2629	1	1	yes	characteristic, complemented, Sylow
92	C₂×#125	—	89	1	no	—
93	C₂×#117	D₈₉	1	1	yes	characteristic
94	#398	—	89	4	no	—
95	#396	—	89	4	no	—
96	#389	—	89	2	no	—
97	#384	—	89	1	no	—
98	#380	—	89	2	no	—
99	C₂³×C₄	C₂×D₈₉	1	1	yes	characteristic
100	#125	—	178	4	no	—
101	#117	—	2	4	no	—
102	C₂×C₄⋊C₄	—	89	4	no	—
103	C₂×C₄⋊C₄	—	89	4	no	—
104	C₂×C₄⋊C₄	—	89	2	no	—
105	C₂×C₄⋊C₄	—	89	4	no	—
106	C₂×C₄⋊C₄	—	89	2	no	—
107	C₂×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
108	C₂×C₂²⋊C₄	—	89	4	no	—
109	C₂×C₂²⋊C₄	C₂×D₈₉	1	2	yes	—
110	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
111	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
112	#95	—	89	4	no	—
113	#95	—	89	4	no	—
114	C₂⁴	C₂²×D₈₉	1	1	yes	characteristic
115	C₂²×C₄	—	2	2	no	—
116	C₂²×C₄	—	2	2	no	—
117	C₂²×C₄	—	89	4	no	—
118	C₂²×C₄	—	89	4	no	—
119	C₂²×C₄	Dic₁₇₈	1	4	yes	—
120	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
121	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
122	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
123	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
124	C₄⋊C₄	—	2	4	no	—
125	C₄⋊C₄	—	356	4	no	—
126	C₄⋊C₄	—	356	4	no	—
127	C₄⋊C₄	—	178	8	no	—
128	C₄⋊C₄	—	178	4	no	—
129	C₄⋊C₄	—	178	4	no	—
130	C₂²⋊C₄	—	2	4	no	—
131	C₂²⋊C₄	—	356	4	no	—
132	C₄²	—	2	2	no	—
133	C₄²	C₂²⋊D₈₉	1	4	yes	—
134	C₂³	—	2	1	no	—
135	C₂³	—	2	1	no	—
136	C₂³	—	2	2	no	—
137	C₂³	—	2	2	no	—
138	C₂³	C₂×Dic₁₇₈	1	2	yes	—
139	C₂³	C₂³×D₈₉	1	1	yes	characteristic
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	8	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	178	4	no	—
149	C₂×C₄	—	178	4	no	—
150	C₂×C₄	—	178	4	no	—
151	C₂×C₄	—	178	4	no	—
152	C₂×C₄	—	178	4	no	—
153	C₂×C₄	—	178	4	no	—
154	C₂×C₄	C₂×C₂²⋊D₈₉	1	4	yes	—
155	C₂×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
156	C₂×C₄	C₂×Dic₁₇₈	1	2	yes	—
157	C₂×C₄	C₂×Dic₁₇₈	1	2	yes	—
158	C₂×C₄	#13083574	1	2	yes	—
159	C₂²	—	2	4	no	—
160	C₂²	—	2	2	no	—
161	C₂²	—	4	2	no	—
162	C₂²	—	4	1	no	—
163	C₂²	—	4	1	no	—
164	C₂²	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
165	C₂²	C₂²×Dic₁₇₈	1	1	yes	characteristic
166	C₂²	C₂×#13083574	1	1	yes	characteristic
167	C₂²	#15668058	1	2	yes	—
168	C₂²	#15668059	1	2	yes	—
169	C₄	—	2	4	no	—
170	C₄	—	2	2	no	—
171	C₄	—	4	2	no	—
172	C₄	—	356	4	no	—
173	C₄	—	356	4	no	—
174	C₄	#15667991	1	4	yes	—
175	C₂	—	4	2	no	—
176	C₂	C₂×#15667991	1	1	yes	characteristic
177	C₂	#19152176	1	2	yes	—
178	C₂	#19152203	1	1	yes	characteristic
179	C₂	#19152210	1	1	yes	characteristic
180	C₂	#19152226	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15668004

C₂

yes

maximal, characteristic, complemented

#19151531

C₂

yes

maximal, complemented

#19151521

C₂

yes

maximal

#19151496

C₂

yes

maximal, complemented

#19151468

C₂

yes

maximal, characteristic, complemented

#19151458

C₂

yes

maximal

C₂×#117×C₈₉

C₂

yes

maximal, characteristic

#15668004

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×#13083553

C₂²

yes

—

C₂×#13083553

C₂²

yes

—

C₂×C₂²⋊Dic₈₉

C₂²

yes

—

#15667919

C₂²

yes

—

#15667919

C₂²

yes

—

C₂³×C₄×C₈₉

C₂²

yes

characteristic

#117×C₈₉

—

C₂×C₄⋊C₄×C₈₉

C₂²

yes

—

C₂×C₂²⋊C₄×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₂²×Dic₈₉

C₂³

yes

—

C₂²×Dic₈₉

C₂³

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083553

—

#13083553

—

C₂²⋊Dic₈₉

—

C₂⁴×C₈₉

C₂³

yes

characteristic

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

Q₈

yes

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

D₄

yes

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

C₂×Q₈

yes

—

C₂³×C₈₉

C₂⁴

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

#41

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₂²×C₈₉

C₂²×Q₈

yes

characteristic

C₂²×C₈₉

C₂×#41

yes

characteristic

C₂²×C₈₉

H(5,𝔽₂)

yes

—

C₂²×C₈₉

2₋¹⁺⁴

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂²⋊Q₈

yes

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

C₂×C₂²⋊Q₈

yes

characteristic

C₂×C₈₉

#537

yes

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

#2629

—

maximal, Sylow

C₈₉

#2629

yes

characteristic, complemented, Sylow

C₂×#125

—

C₂×#117

D₈₉

yes

—

—

—

—

—

C₂³×C₄

C₂×D₈₉

yes

characteristic

100

#125

—

178

—

101

#117

—

102

C₂×C₄⋊C₄

—

103

C₂×C₄⋊C₄

—

104

C₂×C₄⋊C₄

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

C₂×D₈₉

yes

—

108

C₂×C₂²⋊C₄

—

109

C₂×C₂²⋊C₄

C₂×D₈₉

yes

—

110

C₂×C₄²

C₂×D₈₉

yes

characteristic

111

C₂×C₄²

C₂×D₈₉

yes

characteristic

112

#95

—

113

#95

—

114

C₂⁴

C₂²×D₈₉

yes

characteristic

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

Dic₁₇₈

yes

—

120

C₂²×C₄

C₂²×D₈₉

yes

—

121

C₂²×C₄

C₂²×D₈₉

yes

—

122

C₂²×C₄

C₂²×D₈₉

yes

characteristic

123

C₂²×C₄

C₂²×D₈₉

yes

characteristic

124

C₄⋊C₄

—

125

C₄⋊C₄

—

356

—

126

C₄⋊C₄

—

356

—

127

C₄⋊C₄

—

178

—

128

C₄⋊C₄

—

178

—

129

C₄⋊C₄

—

178

—

130

C₂²⋊C₄

—

131

C₂²⋊C₄

—

356

—

132

C₄²

—

133

C₄²

C₂²⋊D₈₉

yes

—

134

C₂³

—

135

C₂³

—

136

C₂³

—

137

C₂³

—

138

C₂³

C₂×Dic₁₇₈

yes

—

139

C₂³

C₂³×D₈₉

yes

characteristic

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

178

—

149

C₂×C₄

—

178

—

150

C₂×C₄

—

178

—

151

C₂×C₄

—

178

—

152

C₂×C₄

—

178

—

153

C₂×C₄

—

178

—

154

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

155

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

156

C₂×C₄

C₂×Dic₁₇₈

yes

—

157

C₂×C₄

C₂×Dic₁₇₈

yes

—

158

C₂×C₄

#13083574

yes

—

159

C₂²

—

160

C₂²

—

161

C₂²

—

162

C₂²

—

163

C₂²

—

164

C₂²

C₂²×C₂²⋊D₈₉

yes

characteristic

165

C₂²

C₂²×Dic₁₇₈

yes

characteristic

166

C₂²

C₂×#13083574

yes

characteristic

167

C₂²

#15668058

yes

—

168

C₂²

#15668059

yes

—

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

—

356

—

173

C₄

—

356

—

174

C₄

#15667991

yes

—

175

C₂

—

176

C₂

C₂×#15667991

yes

characteristic

177

C₂

#19152176

yes

—

178

C₂

#19152203

yes

characteristic

179

C₂

#19152210

yes

characteristic

180

C₂

#19152226

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator, socle, cosocle	C₂³×C₈₉	C₂⁴
Fitting	C₂×#117×C₈₉	C₂
prime radical	C₂⁴×C₈₉	C₂³
hypercenter	C₂×#117	D₈₉
hypocenter	C₈₉	#2629
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×#117

subgroup

G/H

center, Frattini

C₂³

C₂³×D₈₉

commutator, socle, cosocle

C₂³×C₈₉

C₂⁴

Fitting

C₂×#117×C₈₉

C₂

prime radical

C₂⁴×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₈₉

C₁ ◁ C₂³ ◁ C₂×#117

N	H ≅ G/N	type
1. C₂×#15668004	C₂	semidirect
2. #19151531	C₂	semidirect
4. #19151496	C₂	semidirect
5. #19151468	C₂	semidirect
91. C₈₉	#2629	semidirect

H ≅ G/N

type

1. C₂×#15668004

C₂

semidirect

2. #19151531

C₂

semidirect

4. #19151496

C₂

semidirect

5. #19151468

C₂

semidirect

91. C₈₉

#2629

semidirect

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

subgroup	index
C₂×#15668004	2
#19151531	2
#19151521	2
#19151496	2
#19151468	2
#19151458	2
C₂×#117×C₈₉	2
#2629	89

subgroup

index

C₂×#117×C₈₉

Subgroups — #26026082

Conjugacy classes of subgroups

Complements

Maximal subgroups