Group #26020002 — subgroups

Subgroups — #26020002

Order 11392.

240 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 113 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×#15667922	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₈₉	C₂	1	1	yes	maximal, characteristic
3	#19151444	C₂	1	1	yes	maximal, characteristic
4	#19151353	C₂	1	1	yes	maximal, characteristic
5	#19151352	C₂	1	1	yes	maximal, characteristic
6	#19151352	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	#15667922	C₄	1	4	yes	complemented
13	C₈×Dic₈₉	C₄	1	4	yes	complemented
14	#15667919	C₄	1	2	yes	—
15	#15667919	C₄	1	2	yes	—
16	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
19	C₂²×Dic₈₉	C₂×C₄	1	2	yes	—
20	C₂²×Dic₈₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×Dic₈₉	C₂×C₄	1	1	yes	characteristic
22	C₂×#10623494	—	2	1	no	—
23	C₂×#10623494	—	2	1	no	—
24	C₂×#10623494	—	2	1	no	—
25	C₂×#10623494	—	2	1	no	—
26	C₂×#10623494	C₂×C₄	1	2	yes	—
27	C₂×#10623494	C₂×C₄	1	2	yes	—
28	C₄×Dic₈₉	—	2	1	no	—
29	C₄×Dic₈₉	—	2	1	no	—
30	C₄×Dic₈₉	C₂×C₄	1	2	yes	—
31	C₄×Dic₈₉	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₈₉	C₂×C₄	1	2	yes	—
33	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₈₉	—	2	1	no	—
35	C₂×C₈×C₈₉	—	2	1	no	—
36	C₂×C₈×C₈₉	—	2	1	no	—
37	C₂×C₈×C₈₉	—	2	1	no	—
38	C₂×C₈×C₈₉	C₂×C₄	1	2	yes	—
39	C₂×C₈×C₈₉	C₂×C₄	1	2	yes	—
40	C₄²×C₈₉	—	2	1	no	—
41	C₄²×C₈₉	—	2	1	no	—
42	C₂³×C₈₉	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₈₉	—	2	2	no	—
44	C₂×C₄×C₈₉	—	2	2	no	—
45	C₂×C₄×C₈₉	—	2	2	no	—
46	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
49	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
50	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
51	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
52	C₂×Dic₈₉	—	2	2	no	—
53	C₂×Dic₈₉	—	2	2	no	—
54	C₂×Dic₈₉	—	2	2	no	—
55	C₂×Dic₈₉	—	2	2	no	—
56	C₂×Dic₈₉	—	2	2	no	—
57	C₂×Dic₈₉	C₄²	1	2	yes	—
58	C₂×Dic₈₉	C₄²	1	2	yes	—
59	C₈×C₈₉	—	2	2	no	—
60	C₈×C₈₉	—	2	2	no	—
61	#10623494	—	2	2	no	—
62	#10623494	—	2	2	no	—
63	C₂²×C₈₉	C₂×C₄²	1	1	yes	characteristic
64	C₂²×C₈₉	#117	1	1	yes	characteristic
65	C₂²×C₈₉	#117	1	1	yes	characteristic
66	C₂²×C₈₉	#131	1	2	yes	—
67	C₂²×C₈₉	#131	1	2	yes	—
68	C₄×C₈₉	—	4	2	no	—
69	C₄×C₈₉	#117	1	1	yes	characteristic
70	C₄×C₈₉	#117	1	1	yes	characteristic
71	C₄×C₈₉	#117	1	1	yes	characteristic
72	C₄×C₈₉	#117	1	1	yes	characteristic
73	Dic₈₉	—	2	2	no	—
74	Dic₈₉	—	2	2	no	—
75	Dic₈₉	—	4	2	no	—
76	C₂×C₈₉	#376	1	1	yes	characteristic
77	C₂×C₈₉	#405	1	1	yes	characteristic
78	C₂×C₈₉	#405	1	1	yes	characteristic
79	C₂×C₈₉	#433	1	2	yes	—
80	C₂×C₈₉	#433	1	2	yes	—
81	#1746	—	89	1	no	maximal, Sylow
82	C₈₉	#1746	1	1	yes	characteristic, complemented, Sylow
83	C₂×#97	—	89	1	no	—
84	C₂×C₄×C₈	—	89	1	no	—
85	#376	—	89	1	no	—
86	#336	—	89	1	no	—
87	#336	—	89	1	no	—
88	#336	—	89	1	no	—
89	#336	D₈₉	1	1	yes	characteristic
90	C₂²×C₈	—	89	1	no	—
91	C₂²×C₈	—	89	1	no	—
92	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
93	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
94	C₂×C₄²	—	89	1	no	—
95	C₂×C₄²	—	89	1	no	—
96	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
97	#97	—	89	4	no	—
98	C₄×C₈	—	89	4	no	—
99	#95	—	89	2	no	—
100	#95	—	89	2	no	—
101	C₂²×C₄	—	89	1	no	—
102	C₂²×C₄	—	89	1	no	—
103	C₂²×C₄	—	89	2	no	—
104	C₂²×C₄	C₄×D₈₉	1	2	yes	—
105	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	89	2	no	—
111	C₂×C₈	—	89	2	no	—
112	C₂×C₈	—	178	1	no	—
113	C₂×C₈	—	178	1	no	—
114	C₂×C₈	—	178	1	no	—
115	C₂×C₈	—	178	1	no	—
116	C₂×C₈	C₄×D₈₉	1	2	yes	—
117	C₂×C₈	C₄×D₈₉	1	2	yes	—
118	C₄²	—	2	1	no	—
119	C₄²	—	2	1	no	—
120	C₄²	—	89	2	no	—
121	C₄²	—	89	2	no	—
122	C₄²	—	178	1	no	—
123	C₄²	—	178	1	no	—
124	C₂³	C₂×C₄×D₈₉	1	1	yes	characteristic
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	89	2	no	—
129	C₂×C₄	—	89	2	no	—
130	C₂×C₄	—	178	2	no	—
131	C₂×C₄	—	178	2	no	—
132	C₂×C₄	—	178	2	no	—
133	C₂×C₄	—	178	2	no	—
134	C₂×C₄	—	178	2	no	—
135	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
136	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
137	C₂×C₄	#13083573	1	1	yes	characteristic
138	C₂×C₄	#13083573	1	1	yes	characteristic
139	C₂×C₄	#13083573	1	1	yes	characteristic
140	C₂×C₄	#13083575	1	1	yes	characteristic
141	C₈	—	2	2	no	—
142	C₈	—	2	2	no	—
143	C₈	—	178	2	no	—
144	C₈	—	178	2	no	—
145	C₂²	C₄²×D₈₉	1	1	yes	characteristic
146	C₂²	#15667968	1	1	yes	characteristic
147	C₂²	#15667970	1	1	yes	characteristic
148	C₂²	#15668018	1	2	yes	—
149	C₂²	#15668020	1	2	yes	—
150	C₄	—	4	2	no	—
151	C₄	—	356	2	no	—
152	C₄	—	178	2	no	—
153	C₄	—	178	2	no	—
154	C₄	#15667968	1	1	yes	characteristic
155	C₄	#15667968	1	1	yes	characteristic
156	C₄	#15667970	1	1	yes	characteristic
157	C₄	#15667970	1	1	yes	characteristic
158	C₂	#19151443	1	1	yes	characteristic
159	C₂	#19151552	1	1	yes	characteristic
160	C₂	#19151555	1	1	yes	characteristic
161	C₂	#19151656	1	2	yes	—
162	C₂	#19151658	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15667922

C₂

yes

maximal, characteristic

C₂×C₈×Dic₈₉

C₂

yes

maximal, characteristic

#19151444

C₂

yes

maximal, characteristic

#19151353

C₂

yes

maximal, characteristic

#19151352

C₂

yes

maximal, characteristic

#19151352

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

#15667922

C₄

yes

complemented

C₈×Dic₈₉

C₄

yes

complemented

#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

—

C₂²×Dic₈₉

C₂×C₄

yes

characteristic

C₂²×Dic₈₉

C₂×C₄

yes

—

—

—

—

C₂×C₄

yes

—

C₂×#10623494

C₂×C₄

yes

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

C₄×Dic₈₉

C₂×C₄

yes

—

C₄×Dic₈₉

C₂×C₄

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

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

C₄²

yes

—

C₂×Dic₈₉

C₄²

yes

—

C₈×C₈₉

—

C₈×C₈₉

—

#10623494

—

#10623494

—

C₂²×C₈₉

C₂×C₄²

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

—

C₂²×C₈₉

#131

yes

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₂×C₈₉

#433

yes

—

#1746

—

maximal, Sylow

C₈₉

#1746

yes

characteristic, complemented, Sylow

C₂×#97

—

C₂×C₄×C₈

—

—

—

—

—

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

#97

—

C₄×C₈

—

#95

—

100

#95

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

C₄×D₈₉

yes

—

105

C₂²×C₄

C₂²×D₈₉

yes

characteristic

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

178

—

113

C₂×C₈

—

178

—

114

C₂×C₈

—

178

—

115

C₂×C₈

—

178

—

116

C₂×C₈

C₄×D₈₉

yes

—

117

C₂×C₈

C₄×D₈₉

yes

—

118

C₄²

—

119

C₄²

—

120

C₄²

—

121

C₄²

—

122

C₄²

—

178

—

123

C₄²

—

178

—

124

C₂³

C₂×C₄×D₈₉

yes

characteristic

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

178

—

131

C₂×C₄

—

178

—

132

C₂×C₄

—

178

—

133

C₂×C₄

—

178

—

134

C₂×C₄

—

178

—

135

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

136

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

137

C₂×C₄

#13083573

yes

characteristic

138

C₂×C₄

#13083573

yes

characteristic

139

C₂×C₄

#13083573

yes

characteristic

140

C₂×C₄

#13083575

yes

characteristic

141

C₈

—

142

C₈

—

143

C₈

—

178

—

144

C₈

—

178

—

145

C₂²

C₄²×D₈₉

yes

characteristic

146

C₂²

#15667968

yes

characteristic

147

C₂²

#15667970

yes

characteristic

148

C₂²

#15668018

yes

—

149

C₂²

#15668020

yes

—

150

C₄

—

151

C₄

—

356

—

152

C₄

—

178

—

153

C₄

—

178

—

154

C₄

#15667968

yes

characteristic

155

C₄

#15667968

yes

characteristic

156

C₄

#15667970

yes

characteristic

157

C₄

#15667970

yes

characteristic

158

C₂

#19151443

yes

characteristic

159

C₂

#19151552

yes

characteristic

160

C₂

#19151555

yes

characteristic

161

C₂

#19151656

yes

—

162

C₂

#19151658

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	#1746
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. #15667922	C₄	semidirect
13. C₈×Dic₈₉	C₄	semidirect
82. C₈₉	#1746	semidirect

H ≅ G/N

type

12. #15667922

C₄

semidirect

13. C₈×Dic₈₉

C₄

semidirect

82. C₈₉

#1746

semidirect

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

subgroup	index
C₂×#15667922	2
C₂×C₈×Dic₈₉	2
#19151444	2
#19151353	2
#19151352	2
#336×C₈₉	2
#1746	89

subgroup

index

C₂×C₈×Dic₈₉

#336×C₈₉

Subgroups — #26020002

Conjugacy classes of subgroups

Complements

Maximal subgroups