Group #26020108 — subgroups

Subgroups — #26020108

Order 11392.

260 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 121 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×#15667941	C₂	1	1	yes	maximal, characteristic
2	C₂×#15667941	C₂	1	1	yes	maximal, characteristic
3	C₂×#15667922	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×Dic₈₉	C₂	1	1	yes	maximal, characteristic
5	#19151444	C₂	1	1	yes	maximal, characteristic
6	#19151353	C₂	1	1	yes	maximal, characteristic
7	#336×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×#10623494	C₂²	1	1	yes	characteristic
9	C₂²×#10623494	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
12	#15667941	C₄	1	4	yes	complemented
13	#15667941	C₄	1	4	yes	complemented
14	#15667922	—	2	2	no	—
15	C₈×Dic₈₉	—	2	2	no	—
16	#15667919	C₄	1	2	yes	—
17	#15667919	C₄	1	2	yes	—
18	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
21	C₂²×Dic₈₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×Dic₈₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×Dic₈₉	C₂×C₄	1	2	yes	—
24	C₂×#10623494	—	2	1	no	—
25	C₂×#10623494	—	2	1	no	—
26	C₂×#10623494	—	2	1	no	—
27	C₂×#10623494	—	2	1	no	—
28	C₂×#10623494	C₂×C₄	1	2	yes	—
29	C₂×#10623494	C₂×C₄	1	2	yes	—
30	C₄×Dic₈₉	C₂×C₄	1	2	yes	—
31	C₄×Dic₈₉	C₂×C₄	1	2	yes	—
32	C₄×Dic₈₉	D₄	1	2	yes	—
33	C₄×Dic₈₉	Q₈	1	2	yes	—
34	C₂²×C₄×C₈₉	C₂×C₄	1	2	yes	—
35	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
36	C₂×C₈×C₈₉	—	2	1	no	—
37	C₂×C₈×C₈₉	—	2	1	no	—
38	C₂×C₈×C₈₉	—	2	1	no	—
39	C₂×C₈×C₈₉	—	2	1	no	—
40	C₂×C₈×C₈₉	C₂×C₄	1	2	yes	—
41	C₂×C₈×C₈₉	C₂×C₄	1	2	yes	—
42	C₄²×C₈₉	—	2	1	no	—
43	C₄²×C₈₉	—	2	1	no	—
44	C₂³×C₈₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₈₉	—	2	2	no	—
46	C₂×C₄×C₈₉	—	2	2	no	—
47	C₂×C₄×C₈₉	—	2	2	no	—
48	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
53	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
54	C₂×Dic₈₉	—	2	2	no	—
55	C₂×Dic₈₉	—	2	2	no	—
56	C₂×Dic₈₉	—	2	2	no	—
57	C₂×Dic₈₉	—	2	2	no	—
58	C₂×Dic₈₉	C₄²	1	2	yes	—
59	C₂×Dic₈₉	C₄²	1	2	yes	—
60	C₂×Dic₈₉	C₄⋊C₄	1	4	yes	—
61	C₈×C₈₉	—	2	2	no	—
62	C₈×C₈₉	—	2	2	no	—
63	#10623494	—	2	2	no	—
64	#10623494	—	2	2	no	—
65	C₂²×C₈₉	C₂×C₄²	1	1	yes	characteristic
66	C₂²×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₈₉	#117	1	1	yes	characteristic
68	C₂²×C₈₉	#131	1	2	yes	—
69	C₂²×C₈₉	#131	1	2	yes	—
70	C₄×C₈₉	—	4	2	no	—
71	C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
72	C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
73	C₄×C₈₉	C₄×Q₈	1	1	yes	characteristic
74	C₄×C₈₉	C₄×Q₈	1	1	yes	characteristic
75	Dic₈₉	—	2	2	no	—
76	Dic₈₉	—	2	2	no	—
77	Dic₈₉	—	2	4	no	—
78	C₂×C₈₉	C₄×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₈₉	#405	1	1	yes	characteristic
80	C₂×C₈₉	#405	1	1	yes	characteristic
81	C₂×C₈₉	#424	1	2	yes	—
82	C₂×C₈₉	#433	1	2	yes	—
83	#1768	—	89	1	no	maximal, Sylow
84	C₈₉	#1768	1	1	yes	characteristic, complemented, Sylow
85	C₂×C₄⋊C₈	—	89	1	no	—
86	C₂×C₄⋊C₈	—	89	1	no	—
87	C₂×#97	—	89	1	no	—
88	C₂×C₄×C₈	—	89	1	no	—
89	#376	—	89	1	no	—
90	#336	—	89	1	no	—
91	#336	D₈₉	1	1	yes	characteristic
92	C₂²×C₈	—	89	1	no	—
93	C₂²×C₈	—	89	1	no	—
94	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
95	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
96	C₂×C₄²	—	89	1	no	—
97	C₂×C₄²	—	89	1	no	—
98	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
99	C₄⋊C₈	—	89	4	no	—
100	C₄⋊C₈	—	89	4	no	—
101	#97	—	178	2	no	—
102	C₄×C₈	—	178	2	no	—
103	#95	—	89	2	no	—
104	#95	—	89	2	no	—
105	C₂²×C₄	—	89	1	no	—
106	C₂²×C₄	—	89	1	no	—
107	C₂²×C₄	—	89	2	no	—
108	C₂²×C₄	C₄×D₈₉	1	2	yes	—
109	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	89	2	no	—
115	C₂×C₈	—	89	2	no	—
116	C₂×C₈	—	178	1	no	—
117	C₂×C₈	—	178	1	no	—
118	C₂×C₈	—	178	1	no	—
119	C₂×C₈	—	178	1	no	—
120	C₂×C₈	C₄×D₈₉	1	2	yes	—
121	C₂×C₈	C₄×D₈₉	1	2	yes	—
122	C₄²	—	2	1	no	—
123	C₄²	—	2	1	no	—
124	C₄²	—	89	2	no	—
125	C₄²	—	89	2	no	—
126	C₄²	—	89	2	no	—
127	C₄²	—	89	2	no	—
128	C₂³	C₂×C₄×D₈₉	1	1	yes	characteristic
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	89	2	no	—
133	C₂×C₄	—	89	2	no	—
134	C₂×C₄	—	89	4	no	—
135	C₂×C₄	—	178	2	no	—
136	C₂×C₄	—	178	2	no	—
137	C₂×C₄	—	178	2	no	—
138	C₂×C₄	—	178	2	no	—
139	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
140	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
141	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
142	C₂×C₄	Q₈×D₈₉	1	1	yes	characteristic
143	C₂×C₄	#13083573	1	1	yes	characteristic
144	C₂×C₄	#13083573	1	1	yes	characteristic
145	C₈	—	2	2	no	—
146	C₈	—	2	2	no	—
147	C₈	—	178	2	no	—
148	C₈	—	178	2	no	—
149	C₂²	C₄²×D₈₉	1	1	yes	characteristic
150	C₂²	C₄⋊C₄×D₈₉	1	1	yes	characteristic
151	C₂²	#15667968	1	1	yes	characteristic
152	C₂²	#15668018	1	2	yes	—
153	C₂²	#15668020	1	2	yes	—
154	C₄	—	4	2	no	—
155	C₄	—	178	4	no	—
156	C₄	—	178	2	no	—
157	C₄	—	178	2	no	—
158	C₄	#15667972	1	1	yes	characteristic
159	C₄	#15667972	1	1	yes	characteristic
160	C₄	#15667977	1	1	yes	characteristic
161	C₄	#15667977	1	1	yes	characteristic
162	C₂	#19151449	1	1	yes	characteristic
163	C₂	#19151552	1	1	yes	characteristic
164	C₂	#19151555	1	1	yes	characteristic
165	C₂	#19151623	1	2	yes	—
166	C₂	#19151656	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15667941

C₂

yes

maximal, characteristic

C₂×#15667941

C₂

yes

maximal, characteristic

C₂×#15667922

C₂

yes

maximal, characteristic

C₂×C₈×Dic₈₉

C₂

yes

maximal, characteristic

#19151444

C₂

yes

maximal, characteristic

#19151353

C₂

yes

maximal, characteristic

#336×C₈₉

C₂

yes

maximal, characteristic

C₂²×#10623494

C₂²

yes

characteristic

C₂²×#10623494

C₂²

yes

characteristic

C₂×C₄×Dic₈₉

C₂²

yes

characteristic

C₂×C₄×Dic₈₉

C₂²

yes

characteristic

#15667941

C₄

yes

complemented

#15667941

C₄

yes

complemented

#15667922

—

C₈×Dic₈₉

—

#15667919

C₄

yes

—

#15667919

C₄

yes

—

C₂²×C₈×C₈₉

C₂²

yes

characteristic

C₂²×C₈×C₈₉

C₂²

yes

characteristic

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₂²×Dic₈₉

C₂×C₄

yes

characteristic

C₂²×Dic₈₉

C₂×C₄

yes

characteristic

C₂²×Dic₈₉

C₂×C₄

yes

—

—

—

—

—

C₂×C₄

yes

—

C₂×#10623494

C₂×C₄

yes

—

C₄×Dic₈₉

C₂×C₄

yes

—

C₄×Dic₈₉

C₂×C₄

yes

—

C₄×Dic₈₉

D₄

yes

—

C₄×Dic₈₉

Q₈

yes

—

C₂²×C₄×C₈₉

C₂×C₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂×C₄

yes

—

C₂×C₈×C₈₉

C₂×C₄

yes

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₂³×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₈

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

C₄²

yes

—

C₂×Dic₈₉

C₄²

yes

—

C₂×Dic₈₉

C₄⋊C₄

yes

—

C₈×C₈₉

—

C₈×C₈₉

—

#10623494

—

#10623494

—

C₂²×C₈₉

C₂×C₄²

yes

characteristic

C₂²×C₈₉

C₂×C₄⋊C₄

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

—

C₂²×C₈₉

#131

yes

—

C₄×C₈₉

—

C₄×C₈₉

C₄×D₄

yes

characteristic

C₄×C₈₉

C₄×D₄

yes

characteristic

C₄×C₈₉

C₄×Q₈

yes

characteristic

C₄×C₈₉

C₄×Q₈

yes

characteristic

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

C₄×C₄⋊C₄

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₂×C₈₉

#433

yes

—

#1768

—

maximal, Sylow

C₈₉

#1768

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×#97

—

C₂×C₄×C₈

—

#376

—

#336

—

#336

D₈₉

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₈₉

yes

characteristic

C₂²×C₈

C₂×D₈₉

yes

characteristic

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

C₄⋊C₈

—

100

C₄⋊C₈

—

101

#97

—

178

—

102

C₄×C₈

—

178

—

103

#95

—

104

#95

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

C₄×D₈₉

yes

—

109

C₂²×C₄

C₂²×D₈₉

yes

characteristic

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

178

—

117

C₂×C₈

—

178

—

118

C₂×C₈

—

178

—

119

C₂×C₈

—

178

—

120

C₂×C₈

C₄×D₈₉

yes

—

121

C₂×C₈

C₄×D₈₉

yes

—

122

C₄²

—

123

C₄²

—

124

C₄²

—

125

C₄²

—

126

C₄²

—

127

C₄²

—

128

C₂³

C₂×C₄×D₈₉

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

178

—

136

C₂×C₄

—

178

—

137

C₂×C₄

—

178

—

138

C₂×C₄

—

178

—

139

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

140

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

141

C₂×C₄

D₄×D₈₉

yes

characteristic

142

C₂×C₄

Q₈×D₈₉

yes

characteristic

143

C₂×C₄

#13083573

yes

characteristic

144

C₂×C₄

#13083573

yes

characteristic

145

C₈

—

146

C₈

—

147

C₈

—

178

—

148

C₈

—

178

—

149

C₂²

C₄²×D₈₉

yes

characteristic

150

C₂²

C₄⋊C₄×D₈₉

yes

characteristic

151

C₂²

#15667968

yes

characteristic

152

C₂²

#15668018

yes

—

153

C₂²

#15668020

yes

—

154

C₄

—

155

C₄

—

178

—

156

C₄

—

178

—

157

C₄

—

178

—

158

C₄

#15667972

yes

characteristic

159

C₄

#15667972

yes

characteristic

160

C₄

#15667977

yes

characteristic

161

C₄

#15667977

yes

characteristic

162

C₂

#19151449

yes

characteristic

163

C₂

#19151552

yes

characteristic

164

C₂

#19151555

yes

characteristic

165

C₂

#19151623

yes

—

166

C₂

#19151656

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₈₉
commutator	C₂²×C₈₉	C₂×C₄²
Fitting	#336×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂²×C₄
cosocle	C₂²×C₄×C₈₉	C₂³
hypercenter	#336	D₈₉
hypocenter	C₈₉	#1768
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₈₉

commutator

C₂²×C₈₉

C₂×C₄²

Fitting

#336×C₈₉

C₂

socle, prime radical

C₂³×C₈₉

C₂²×C₄

cosocle

C₂²×C₄×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂²×C₈₉ ▷ C₈₉

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
12. #15667941	C₄	semidirect
13. #15667941	C₄	semidirect
84. C₈₉	#1768	semidirect

H ≅ G/N

type

12. #15667941

C₄

semidirect

13. #15667941

C₄

semidirect

84. C₈₉

#1768

semidirect

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

subgroup	index
C₂×#15667941	2
C₂×#15667922	2
C₂×C₈×Dic₈₉	2
#19151444	2
#19151353	2
#336×C₈₉	2
#1768	89

subgroup

index

C₂×C₈×Dic₈₉

#336×C₈₉

Subgroups — #26020108

Conjugacy classes of subgroups

Complements

Maximal subgroups