Group #26021495 — subgroups

Subgroups — #26021495

Order 11392.

298 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 95 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×#15667936	C₂	1	2	yes	maximal
2	C₂×#15667936	C₂	1	2	yes	maximal
3	#19151496	C₂	1	1	yes	maximal, characteristic
4	#19151353	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10623494	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
9	C₂×#13083553	—	2	1	no	—
10	#15667936	—	2	4	no	—
11	#15667936	—	2	4	no	—
12	#15667919	—	2	1	no	—
13	C₂²×Q₈×C₈₉	C₂²	1	2	yes	—
14	C₄⋊Q₈×C₈₉	—	2	4	no	—
15	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
16	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
17	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
18	C₂²×Dic₈₉	—	2	1	no	—
19	C₂²×Dic₈₉	—	2	1	no	—
20	C₂×#10623494	—	2	2	no	—
21	C₂×#10623494	—	2	2	no	—
22	C₂×#10623494	—	2	2	no	—
23	C₄⋊Dic₈₉	—	2	1	no	—
24	C₄⋊Dic₈₉	—	2	2	no	—
25	C₄⋊Dic₈₉	D₄	1	2	yes	—
26	#13083553	—	4	2	no	—
27	C₂×Q₈×C₈₉	—	2	2	no	—
28	C₂×Q₈×C₈₉	—	2	2	no	—
29	C₂×Q₈×C₈₉	—	2	2	no	—
30	C₂×Q₈×C₈₉	—	2	2	no	—
31	C₂×Q₈×C₈₉	—	2	2	no	—
32	C₂×Q₈×C₈₉	D₄	1	4	yes	—
33	C₂²×C₄×C₈₉	—	2	2	no	—
34	C₂²×C₄×C₈₉	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₈₉	D₄	1	1	yes	characteristic
36	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
37	C₄⋊C₄×C₈₉	—	2	4	no	—
38	C₄⋊C₄×C₈₉	—	2	4	no	—
39	C₄²×C₈₉	—	2	2	no	—
40	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
41	Q₈×C₈₉	—	2	4	no	—
42	Q₈×C₈₉	—	2	4	no	—
43	Q₈×C₈₉	—	4	2	no	—
44	Q₈×C₈₉	—	4	2	no	—
45	C₂×C₄×C₈₉	—	2	1	no	—
46	C₂×C₄×C₈₉	—	2	1	no	—
47	C₂×C₄×C₈₉	—	2	1	no	—
48	C₂×C₄×C₈₉	—	2	1	no	—
49	C₂×C₄×C₈₉	—	2	4	no	—
50	C₂×C₄×C₈₉	—	4	2	no	—
51	C₂×C₄×C₈₉	—	4	2	no	—
52	C₂×C₄×C₈₉	SD₁₆	1	2	yes	—
53	C₂×C₄×C₈₉	Q₁₆	1	2	yes	—
54	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₈₉	#41	1	2	yes	—
57	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
58	C₂×Dic₈₉	—	2	2	no	—
59	C₂×Dic₈₉	—	4	1	no	—
60	C₂×Dic₈₉	—	4	1	no	—
61	C₂×Dic₈₉	—	4	1	no	—
62	C₂×Dic₈₉	—	4	1	no	—
63	C₂×Dic₈₉	—	4	1	no	—
64	#10623494	—	4	2	no	—
65	C₂²×C₈₉	#120	1	1	yes	characteristic
66	C₂²×C₈₉	#121	1	1	yes	characteristic
67	C₂²×C₈₉	#124	1	1	yes	characteristic
68	C₂²×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
69	C₂²×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
70	C₂²×C₈₉	#137	1	1	yes	characteristic
71	C₂²×C₈₉	#137	1	1	yes	characteristic
72	C₄×C₈₉	—	2	2	no	—
73	C₄×C₈₉	—	2	2	no	—
74	C₄×C₈₉	—	4	4	no	—
75	C₄×C₈₉	#121	1	2	yes	—
76	C₄×C₈₉	#123	1	2	yes	—
77	Dic₈₉	—	4	2	no	—
78	Dic₈₉	—	8	1	no	—
79	C₂×C₈₉	#394	1	1	yes	characteristic
80	C₂×C₈₉	#448	1	1	yes	characteristic
81	C₂×C₈₉	#451	1	1	yes	characteristic
82	C₂×C₈₉	#461	1	1	yes	characteristic
83	C₂×C₈₉	#462	1	1	yes	characteristic
84	C₂×C₈₉	#487	1	1	yes	characteristic
85	C₂×C₈₉	#491	1	1	yes	characteristic
86	#2043	—	89	1	no	maximal, Sylow
87	C₈₉	#2043	1	1	yes	characteristic, complemented, Sylow
88	C₂×C₄⋊Q₈	D₈₉	1	1	yes	characteristic
89	C₂×#103	—	89	2	no	—
90	C₂×#103	—	89	2	no	—
91	#389	—	89	1	no	—
92	#336	—	89	1	no	—
93	C₂²×Q₈	C₂×D₈₉	1	2	yes	—
94	C₂²×C₈	—	89	2	no	—
95	C₄⋊Q₈	—	2	4	no	—
96	C₂×C₄⋊C₄	—	2	1	no	—
97	C₂×C₄⋊C₄	—	2	1	no	—
98	C₂×C₄⋊C₄	—	89	1	no	—
99	C₂×C₄⋊C₄	—	89	1	no	—
100	C₂×C₄⋊C₄	—	178	1	no	—
101	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
102	#103	—	178	4	no	—
103	#103	—	178	4	no	—
104	#95	—	178	1	no	—
105	C₂×Q₈	—	2	2	no	—
106	C₂×Q₈	—	2	2	no	—
107	C₂×Q₈	—	2	2	no	—
108	C₂×Q₈	—	2	2	no	—
109	C₂×Q₈	—	2	2	no	—
110	C₂×Q₈	C₂²⋊D₈₉	1	4	yes	—
111	C₂²×C₄	—	2	2	no	—
112	C₂²×C₄	—	178	1	no	—
113	C₂²×C₄	—	178	1	no	—
114	C₂²×C₄	C₂²⋊D₈₉	1	1	yes	characteristic
115	C₂²×C₄	C₂²⋊D₈₉	1	1	yes	characteristic
116	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
117	C₂×C₈	—	178	2	no	—
118	C₂×C₈	—	178	2	no	—
119	C₂×C₈	—	178	2	no	—
120	C₄⋊C₄	—	2	4	no	—
121	C₄⋊C₄	—	2	4	no	—
122	C₄⋊C₄	—	89	2	no	—
123	C₄⋊C₄	—	356	2	no	—
124	C₄⋊C₄	—	178	2	no	—
125	C₄⋊C₄	—	178	1	no	—
126	C₄²	—	2	2	no	—
127	C₂³	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
128	Q₈	—	2	4	no	—
129	Q₈	—	2	4	no	—
130	Q₈	—	4	2	no	—
131	Q₈	—	4	2	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	4	no	—
137	C₂×C₄	—	4	2	no	—
138	C₂×C₄	—	4	2	no	—
139	C₂×C₄	—	356	1	no	—
140	C₂×C₄	—	356	1	no	—
141	C₂×C₄	—	356	1	no	—
142	C₂×C₄	—	356	1	no	—
143	C₂×C₄	—	356	1	no	—
144	C₂×C₄	—	178	2	no	—
145	C₂×C₄	#13083561	1	2	yes	—
146	C₂×C₄	#13083564	1	2	yes	—
147	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
148	C₂×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
149	C₂×C₄	#13083573	1	2	yes	—
150	C₂×C₄	#13083573	1	1	yes	characteristic
151	C₈	—	356	2	no	—
152	C₂²	#15667981	1	1	yes	characteristic
153	C₂²	#15667983	1	1	yes	characteristic
154	C₂²	#15667999	1	1	yes	characteristic
155	C₂²	C₂×#13083561	1	1	yes	characteristic
156	C₂²	C₂×#13083564	1	1	yes	characteristic
157	C₂²	#15668043	1	1	yes	characteristic
158	C₂²	#15668043	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	2	2	no	—
161	C₄	—	4	4	no	—
162	C₄	—	356	2	no	—
163	C₄	—	712	1	no	—
164	C₄	#15667984	1	2	yes	—
165	C₄	#15667993	1	2	yes	—
166	C₂	#19151512	1	1	yes	characteristic
167	C₂	#19151737	1	1	yes	characteristic
168	C₂	#19151754	1	1	yes	characteristic
169	C₂	#19151800	1	1	yes	characteristic
170	C₂	#19151805	1	1	yes	characteristic
171	C₂	#19151948	1	1	yes	characteristic
172	C₂	#19151966	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15667936

C₂

yes

maximal

C₂×#15667936

C₂

yes

maximal

#19151496

C₂

yes

maximal, characteristic

#19151353

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₈₉

C₂

yes

maximal, characteristic

C₂²×#10623494

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₈₉

C₂²

yes

—

—

—

—

C₂²×Q₈×C₈₉

C₂²

yes

—

C₄⋊Q₈×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

C₂×#10623494

—

C₂×#10623494

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

D₄

yes

—

#13083553

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

D₄

yes

characteristic

C₂²×C₄×C₈₉

D₄

yes

characteristic

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

Q₈×C₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

—

Q₈×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

SD₁₆

yes

—

C₂×C₄×C₈₉

Q₁₆

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

#41

yes

—

C₂×C₄×C₈₉

#41

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

#10623494

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

C₂×SD₁₆

yes

characteristic

C₂²×C₈₉

C₂×Q₁₆

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#121

yes

—

C₄×C₈₉

#123

yes

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

D₈₉

yes

—

—

—

—

C₂²×Q₈

C₂×D₈₉

yes

—

C₂²×C₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

100

C₂×C₄⋊C₄

—

178

—

101

C₂×C₄²

C₂×D₈₉

yes

characteristic

102

#103

—

178

—

103

#103

—

178

—

104

#95

—

178

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂×Q₈

C₂²⋊D₈₉

yes

—

111

C₂²×C₄

—

112

C₂²×C₄

—

178

—

113

C₂²×C₄

—

178

—

114

C₂²×C₄

C₂²⋊D₈₉

yes

characteristic

115

C₂²×C₄

C₂²⋊D₈₉

yes

characteristic

116

C₂²×C₄

C₂²×D₈₉

yes

characteristic

117

C₂×C₈

—

178

—

118

C₂×C₈

—

178

—

119

C₂×C₈

—

178

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

356

—

124

C₄⋊C₄

—

178

—

125

C₄⋊C₄

—

178

—

126

C₄²

—

127

C₂³

C₂×C₂²⋊D₈₉

yes

characteristic

128

Q₈

—

129

Q₈

—

130

Q₈

—

131

Q₈

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

356

—

140

C₂×C₄

—

356

—

141

C₂×C₄

—

356

—

142

C₂×C₄

—

356

—

143

C₂×C₄

—

356

—

144

C₂×C₄

—

178

—

145

C₂×C₄

#13083561

yes

—

146

C₂×C₄

#13083564

yes

—

147

C₂×C₄

D₄×D₈₉

yes

characteristic

148

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

149

C₂×C₄

#13083573

yes

—

150

C₂×C₄

#13083573

yes

characteristic

151

C₈

—

356

—

152

C₂²

#15667981

yes

characteristic

153

C₂²

#15667983

yes

characteristic

154

C₂²

#15667999

yes

characteristic

155

C₂²

C₂×#13083561

yes

characteristic

156

C₂²

C₂×#13083564

yes

characteristic

157

C₂²

#15668043

yes

characteristic

158

C₂²

#15668043

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

356

—

163

C₄

—

712

—

164

C₄

#15667984

yes

—

165

C₄

#15667993

yes

—

166

C₂

#19151512

yes

characteristic

167

C₂

#19151737

yes

characteristic

168

C₂

#19151754

yes

characteristic

169

C₂

#19151800

yes

characteristic

170

C₂

#19151805

yes

characteristic

171

C₂

#19151948

yes

characteristic

172

C₂

#19151966

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₈₉
commutator, cosocle	C₂²×C₄×C₈₉	C₂³
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	C₂×C₄⋊Q₈×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₈₉
hypocenter	C₈₉	#2043
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₈₉

commutator, cosocle

C₂²×C₄×C₈₉

C₂³

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

C₂×C₄⋊Q₈×C₈₉

C₂

socle, prime radical

C₂³×C₈₉

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₈₉

hypocenter

C₈₉

#2043

LCS

G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
87. C₈₉	#2043	semidirect

H ≅ G/N

type

87. C₈₉

#2043

semidirect

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

subgroup	index
C₂×#15667936	2
#19151496	2
#19151353	2
C₂×C₄⋊Q₈×C₈₉	2
#2043	89

subgroup

index

C₂×C₄⋊Q₈×C₈₉

Subgroups — #26021495

Conjugacy classes of subgroups

Complements

Maximal subgroups