Group #26020434 — subgroups

Subgroups — #26020434

Order 11392.

324 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 109 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	C₂²×#13083558	C₂	1	1	yes	maximal, characteristic
2	C₂×#15667941	C₂	1	2	yes	maximal, complemented
3	C₄×C₂²⋊Dic₈₉	C₂	1	1	yes	maximal, characteristic
4	#19151352	C₂	1	2	yes	maximal, complemented
5	C₂×C₂²⋊C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13083558	—	2	4	no	—
7	C₂×#13083558	—	2	2	no	—
8	C₂²×#10623494	C₂²	1	2	yes	—
9	C₂×C₂²⋊Dic₈₉	C₄	1	2	yes	—
10	C₂×C₄×Dic₈₉	C₂²	1	2	yes	—
11	#15667941	—	2	4	no	—
12	#15667919	C₄	1	1	yes	characteristic
13	#15667919	C₄	1	1	yes	characteristic
14	C₂³×C₄×C₈₉	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	C₂²⋊C₈×C₈₉	—	2	2	no	—
18	C₂²×Dic₈₉	C₂×C₄	1	2	yes	—
19	C₂²×Dic₈₉	C₂×C₄	1	2	yes	—
20	#13083558	—	2	8	no	—
21	C₂×#10623494	—	2	2	no	—
22	C₂×#10623494	—	2	2	no	—
23	C₂×#10623494	D₄	1	4	yes	—
24	C₂²⋊Dic₈₉	—	2	4	no	—
25	C₄×Dic₈₉	—	2	2	no	—
26	C₄×Dic₈₉	—	2	2	no	—
27	C₂⁴×C₈₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₈₉	—	2	2	no	—
29	C₂²×C₄×C₈₉	—	2	2	no	—
30	C₂²×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₈₉	D₄	1	2	yes	—
32	C₂²×C₄×C₈₉	Q₈	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	2	no	—
37	C₂×C₈×C₈₉	—	2	2	no	—
38	C₂³×C₈₉	—	2	2	no	—
39	C₂³×C₈₉	—	2	2	no	—
40	C₂³×C₈₉	—	2	1	no	—
41	C₂³×C₈₉	—	2	1	no	—
42	C₂³×C₈₉	C₄⋊C₄	1	2	yes	—
43	C₂³×C₈₉	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₈₉	—	2	2	no	—
45	C₂×C₄×C₈₉	—	2	2	no	—
46	C₂×C₄×C₈₉	—	2	4	no	—
47	C₂×C₄×C₈₉	—	2	2	no	—
48	C₂×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
49	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
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	2	yes	—
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	—	2	2	no	—
58	C₂×Dic₈₉	M₁₆	1	4	yes	complemented
59	C₈×C₈₉	—	4	1	no	—
60	C₈×C₈₉	—	4	1	no	—
61	#10623494	—	2	4	no	—
62	C₂²×C₈₉	—	2	2	no	—
63	C₂²×C₈₉	—	2	2	no	—
64	C₂²×C₈₉	—	2	2	no	—
65	C₂²×C₈₉	—	4	2	no	—
66	C₂²×C₈₉	—	4	2	no	—
67	C₂²×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂²×C₈₉	C₄×D₄	1	2	yes	—
69	C₂²×C₈₉	C₂×M₁₆	1	2	yes	—
70	C₂²×C₈₉	#131	1	1	yes	characteristic
71	C₂²×C₈₉	#131	1	1	yes	characteristic
72	C₄×C₈₉	—	2	2	no	—
73	C₄×C₈₉	#120	1	1	yes	characteristic
74	C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
75	C₄×C₈₉	#123	1	1	yes	characteristic
76	Dic₈₉	—	2	4	no	—
77	Dic₈₉	—	4	2	no	—
78	C₂×C₈₉	—	4	2	no	—
79	C₂×C₈₉	#385	1	1	yes	characteristic
80	C₂×C₈₉	#423	1	1	yes	characteristic
81	C₂×C₈₉	#424	1	1	yes	characteristic
82	C₂×C₈₉	#435	1	2	yes	—
83	C₂×C₈₉	#436	1	2	yes	—
84	#1851	—	89	1	no	maximal, Sylow
85	C₈₉	#1851	1	1	yes	characteristic, complemented, Sylow
86	C₂²×M₁₆	—	89	1	no	—
87	C₂×C₄⋊C₈	—	89	2	no	—
88	C₂×C₂²⋊C₈	D₈₉	1	1	yes	characteristic
89	C₄×C₂²⋊C₄	—	89	1	no	—
90	#336	—	89	2	no	—
91	C₂³×C₄	C₂×D₈₉	1	1	yes	characteristic
92	C₂×M₁₆	—	178	4	no	—
93	C₂×M₁₆	—	178	2	no	—
94	C₂²×C₈	—	89	2	no	—
95	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
96	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
97	C₂×C₂²⋊C₄	—	89	2	no	—
98	C₂×C₄²	—	89	2	no	—
99	C₄⋊C₈	—	178	4	no	—
100	C₂²⋊C₈	—	2	2	no	—
101	#95	—	89	1	no	—
102	#95	—	89	1	no	—
103	C₂⁴	C₄×D₈₉	1	1	yes	characteristic
104	C₂²×C₄	—	2	2	no	—
105	C₂²×C₄	—	2	2	no	—
106	C₂²×C₄	—	89	2	no	—
107	C₂²×C₄	—	89	2	no	—
108	C₂²×C₄	Dic₁₇₈	1	2	yes	—
109	C₂²×C₄	C₄×D₈₉	1	1	yes	characteristic
110	C₂²×C₄	C₂²⋊D₈₉	1	2	yes	—
111	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
112	M₁₆	—	178	8	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	89	4	no	—
118	C₂×C₈	—	178	2	no	—
119	C₂×C₈	—	178	2	no	—
120	C₂²⋊C₄	—	178	4	no	—
121	C₄²	—	178	2	no	—
122	C₄²	—	178	2	no	—
123	C₂³	—	2	2	no	—
124	C₂³	—	2	2	no	—
125	C₂³	—	2	1	no	—
126	C₂³	—	2	1	no	—
127	C₂³	#13083553	1	2	yes	—
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	4	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	89	4	no	—
134	C₂×C₄	—	178	2	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	#13083553	1	2	yes	—
140	C₂×C₄	D₄×D₈₉	1	2	yes	—
141	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
142	C₂×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
143	C₂×C₄	#13083573	1	2	yes	—
144	C₈	—	4	1	no	—
145	C₈	—	4	1	no	—
146	C₈	—	178	4	no	—
147	C₂²	—	2	2	no	—
148	C₂²	—	2	2	no	—
149	C₂²	—	2	2	no	—
150	C₂²	—	4	2	no	—
151	C₂²	—	4	2	no	—
152	C₂²	C₂×#13083553	1	1	yes	characteristic
153	C₂²	#15667972	1	2	yes	—
154	C₂²	M₁₆×D₈₉	1	2	yes	—
155	C₂²	#15668018	1	1	yes	characteristic
156	C₂²	#15668018	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	—	356	2	no	—
159	C₄	—	178	4	no	—
160	C₄	#15667980	1	1	yes	characteristic
161	C₄	#15667989	1	2	yes	—
162	C₄	#15667993	1	1	yes	characteristic
163	C₂	—	4	2	no	—
164	C₂	#19151470	1	1	yes	characteristic
165	C₂	#19151621	1	1	yes	characteristic
166	C₂	#19151624	1	1	yes	characteristic
167	C₂	#19151672	1	2	yes	—
168	C₂	#19151673	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#13083558

C₂

yes

maximal, characteristic

C₂×#15667941

C₂

yes

maximal, complemented

C₄×C₂²⋊Dic₈₉

C₂

yes

maximal, characteristic

#19151352

C₂

yes

maximal, complemented

C₂×C₂²⋊C₈×C₈₉

C₂

yes

maximal, characteristic

C₂×#13083558

—

C₂×#13083558

—

C₂²×#10623494

C₂²

yes

—

C₂×C₂²⋊Dic₈₉

C₄

yes

—

C₂×C₄×Dic₈₉

C₂²

yes

—

#15667941

—

#15667919

C₄

yes

characteristic

#15667919

C₄

yes

characteristic

C₂³×C₄×C₈₉

C₂²

yes

characteristic

C₂²×C₈×C₈₉

C₂²

yes

characteristic

C₂²×C₈×C₈₉

C₂²

yes

characteristic

C₂²⋊C₈×C₈₉

—

C₂²×Dic₈₉

C₂×C₄

yes

—

C₂²×Dic₈₉

C₂×C₄

yes

—

—

—

—

D₄

yes

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

C₂⁴×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

Q₈

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

C₄⋊C₄

yes

—

C₂³×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₄⋊C₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₈₉

#41

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

M₁₆

yes

complemented

C₈×C₈₉

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₈₉

C₄×D₄

yes

—

C₂²×C₈₉

C₂×M₁₆

yes

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₄×C₈₉

—

C₄×C₈₉

#120

yes

characteristic

C₄×C₈₉

C₂²⋊Q₈

yes

—

C₄×C₈₉

#123

yes

characteristic

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₂×C₈₉

#436

yes

—

#1851

—

maximal, Sylow

C₈₉

#1851

yes

characteristic, complemented, Sylow

C₂²×M₁₆

—

C₂×C₄⋊C₈

—

C₂×C₂²⋊C₈

D₈₉

yes

characteristic

C₄×C₂²⋊C₄

—

#336

—

C₂³×C₄

C₂×D₈₉

yes

characteristic

C₂×M₁₆

—

178

—

C₂×M₁₆

—

178

—

C₂²×C₈

—

C₂²×C₈

C₂×D₈₉

yes

characteristic

C₂²×C₈

C₂×D₈₉

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₄⋊C₈

—

178

—

100

C₂²⋊C₈

—

101

#95

—

102

#95

—

103

C₂⁴

C₄×D₈₉

yes

characteristic

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

Dic₁₇₈

yes

—

109

C₂²×C₄

C₄×D₈₉

yes

characteristic

110

C₂²×C₄

C₂²⋊D₈₉

yes

—

111

C₂²×C₄

C₂²×D₈₉

yes

characteristic

112

M₁₆

—

178

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

178

—

119

C₂×C₈

—

178

—

120

C₂²⋊C₄

—

178

—

121

C₄²

—

178

—

122

C₄²

—

178

—

123

C₂³

—

124

C₂³

—

125

C₂³

—

126

C₂³

—

127

C₂³

#13083553

yes

—

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

—

178

—

135

C₂×C₄

—

178

—

136

C₂×C₄

—

178

—

137

C₂×C₄

—

178

—

138

C₂×C₄

—

178

—

139

C₂×C₄

#13083553

yes

—

140

C₂×C₄

D₄×D₈₉

yes

—

141

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

142

C₂×C₄

C₂×Dic₁₇₈

yes

characteristic

143

C₂×C₄

#13083573

yes

—

144

C₈

—

145

C₈

—

146

C₈

—

178

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

C₂×#13083553

yes

characteristic

153

C₂²

#15667972

yes

—

154

C₂²

M₁₆×D₈₉

yes

—

155

C₂²

#15668018

yes

characteristic

156

C₂²

#15668018

yes

characteristic

157

C₄

—

158

C₄

—

356

—

159

C₄

—

178

—

160

C₄

#15667980

yes

characteristic

161

C₄

#15667989

yes

—

162

C₄

#15667993

yes

characteristic

163

C₂

—

164

C₂

#19151470

yes

characteristic

165

C₂

#19151621

yes

characteristic

166

C₂

#19151624

yes

characteristic

167

C₂

#19151672

yes

—

168

C₂

#19151673

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₈₉
commutator, socle	C₂³×C₈₉	C₂²×C₄
Fitting	C₂×C₂²⋊C₈×C₈₉	C₂
prime radical	C₂⁴×C₈₉	C₂×C₄
cosocle	C₂²×C₄×C₈₉	C₂³
hypercenter	C₂×C₂²⋊C₈	D₈₉
hypocenter	C₈₉	#1851
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₈₉

commutator, socle

C₂³×C₈₉

C₂²×C₄

Fitting

C₂×C₂²⋊C₈×C₈₉

C₂

prime radical

C₂⁴×C₈₉

C₂×C₄

cosocle

C₂²×C₄×C₈₉

C₂³

hypercenter

C₂×C₂²⋊C₈

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₈₉

C₁ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈

N	H ≅ G/N	type
2. C₂×#15667941	C₂	semidirect
4. #19151352	C₂	semidirect
58. C₂×Dic₈₉	M₁₆	semidirect
85. C₈₉	#1851	semidirect

H ≅ G/N

type

2. C₂×#15667941

C₂

semidirect

4. #19151352

C₂

semidirect

58. C₂×Dic₈₉

M₁₆

semidirect

85. C₈₉

#1851

semidirect

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

subgroup	index
C₂²×#13083558	2
C₂×#15667941	2
C₄×C₂²⋊Dic₈₉	2
#19151352	2
C₂×C₂²⋊C₈×C₈₉	2
#1851	89

subgroup

index

C₄×C₂²⋊Dic₈₉

C₂×C₂²⋊C₈×C₈₉

Subgroups — #26020434

Conjugacy classes of subgroups

Complements

Maximal subgroups