Group #26021339 — subgroups

Subgroups — #26021339

Order 11392.

474 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 97 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×#13083561	C₂	1	2	yes	maximal, complemented
2	C₂×#15668009	C₂	1	1	yes	maximal, characteristic
3	C₂×#15667932	C₂	1	2	yes	maximal, complemented
4	#19151353	C₂	1	1	yes	maximal, characteristic, complemented
5	#391×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₃₅₆	—	2	2	no	—
7	C₂²×D₃₅₆	C₂²	1	2	yes	—
8	C₂×#13083561	—	2	4	no	—
9	C₂×#13083561	—	2	4	no	—
10	C₂×#13083561	—	2	4	no	—
11	C₂²×#10623494	C₂²	1	2	yes	—
12	#15668009	—	2	4	no	—
13	#15667932	—	2	4	no	—
14	C₂²×Q₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
17	#95×C₈₉	—	2	2	no	—
18	C₂³×D₈₉	—	2	2	no	—
19	C₂×D₃₅₆	—	2	2	no	—
20	C₂×D₃₅₆	—	2	2	no	—
21	C₂×D₃₅₆	—	2	2	no	—
22	C₂×D₃₅₆	—	2	2	no	—
23	C₂×D₃₅₆	—	2	8	no	—
24	C₂×D₃₅₆	—	2	2	no	—
25	C₂×D₃₅₆	—	4	4	no	—
26	C₂×D₃₅₆	—	4	4	no	—
27	C₂×D₃₅₆	D₄	1	4	yes	—
28	#13083561	—	2	16	no	—
29	C₂×#10623494	—	2	2	no	—
30	C₂×#10623494	—	2	2	no	—
31	C₂×#10623494	D₄	1	4	yes	complemented
32	C₂×Q₈×C₈₉	—	2	1	no	—
33	C₂×Q₈×C₈₉	—	2	2	no	—
34	C₂×Q₈×C₈₉	—	2	2	no	—
35	C₂×Q₈×C₈₉	D₄	1	2	yes	complemented
36	C₂²×C₄×C₈₉	—	2	1	no	—
37	C₂²×C₄×C₈₉	—	2	1	no	—
38	C₂²×C₄×C₈₉	D₄	1	2	yes	—
39	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
40	C₄⋊C₄×C₈₉	—	2	2	no	—
41	C₄²×C₈₉	—	2	2	no	—
42	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
43	C₂²×D₈₉	—	2	4	no	—
44	C₂²×D₈₉	—	4	2	no	—
45	C₂²×D₈₉	—	4	2	no	—
46	C₂²×D₈₉	—	4	4	no	—
47	C₂²×D₈₉	—	4	4	no	—
48	Q₈×C₈₉	—	2	4	no	—
49	Q₈×C₈₉	—	4	2	no	—
50	C₂×C₄×C₈₉	—	2	2	no	—
51	C₂×C₄×C₈₉	—	2	2	no	—
52	C₂×C₄×C₈₉	—	2	2	no	—
53	C₂×C₄×C₈₉	—	4	1	no	—
54	C₂×C₄×C₈₉	—	4	1	no	—
55	C₂×C₄×C₈₉	—	4	1	no	—
56	C₂×C₄×C₈₉	—	4	2	no	—
57	C₂×C₄×C₈₉	SD₁₆	1	4	yes	complemented
58	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
62	D₃₅₆	—	2	4	no	—
63	D₃₅₆	—	2	4	no	—
64	D₃₅₆	—	4	16	no	—
65	D₃₅₆	—	4	2	no	—
66	D₃₅₆	—	4	2	no	—
67	#10623494	—	2	4	no	—
68	C₂²×C₈₉	#120	1	1	yes	characteristic
69	C₂²×C₈₉	#121	1	1	yes	characteristic
70	C₂²×C₈₉	#127	1	1	yes	characteristic
71	C₂²×C₈₉	C₂×SD₁₆	1	2	yes	—
72	C₂²×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
73	C₂×D₈₉	—	4	4	no	—
74	C₂×D₈₉	—	4	8	no	—
75	C₂×D₈₉	—	8	4	no	—
76	C₂×D₈₉	—	8	2	no	—
77	C₂×D₈₉	—	8	2	no	—
78	C₄×C₈₉	—	2	4	no	—
79	C₄×C₈₉	—	4	2	no	—
80	C₄×C₈₉	—	8	1	no	—
81	C₄×C₈₉	#120	1	2	yes	—
82	C₄×C₈₉	#121	1	2	yes	—
83	C₂×C₈₉	#392	1	1	yes	characteristic
84	C₂×C₈₉	#450	1	2	yes	—
85	C₂×C₈₉	#460	1	2	yes	—
86	C₂×C₈₉	#492	1	1	yes	characteristic
87	C₂×C₈₉	#496	1	1	yes	characteristic
88	D₈₉	—	8	4	no	—
89	#2014	—	89	1	no	maximal, Sylow
90	C₈₉	#2014	1	1	yes	characteristic, complemented, Sylow
91	C₂²×SD₁₆	—	89	2	no	—
92	C₂×#127	—	89	1	no	—
93	C₂×#102	—	89	2	no	—
94	#391	D₈₉	1	1	yes	characteristic, complemented
95	#336	—	89	1	no	—
96	C₂²×Q₈	C₂×D₈₉	1	1	yes	characteristic
97	C₂²×D₄	—	89	2	no	—
98	C₂²×D₄	—	178	2	no	—
99	C₂×SD₁₆	—	178	4	no	—
100	C₂×SD₁₆	—	178	4	no	—
101	C₂×SD₁₆	—	178	4	no	—
102	C₂²×C₈	—	89	2	no	—
103	#127	—	178	4	no	—
104	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
105	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
106	#102	—	178	4	no	—
107	#95	—	2	2	no	—
108	C₂⁴	—	178	2	no	—
109	C₂×Q₈	—	2	1	no	—
110	C₂×Q₈	—	2	2	no	—
111	C₂×Q₈	—	2	2	no	—
112	C₂×Q₈	D₃₅₆	1	2	yes	complemented
113	C₂×D₄	—	89	4	no	—
114	C₂×D₄	—	356	4	no	—
115	C₂×D₄	—	356	4	no	—
116	C₂×D₄	—	178	8	no	—
117	C₂×D₄	—	178	2	no	—
118	C₂×D₄	—	178	2	no	—
119	C₂×D₄	—	178	2	no	—
120	C₂×D₄	—	178	2	no	—
121	C₂×D₄	—	178	2	no	—
122	C₂²×C₄	—	2	1	no	—
123	C₂²×C₄	—	2	1	no	—
124	C₂²×C₄	C₂²⋊D₈₉	1	2	yes	—
125	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
126	SD₁₆	—	178	16	no	—
127	C₂×C₈	—	89	4	no	—
128	C₂×C₈	—	178	2	no	—
129	C₂×C₈	—	178	2	no	—
130	C₄⋊C₄	—	2	2	no	—
131	C₄²	—	2	2	no	—
132	C₂³	—	356	2	no	—
133	C₂³	—	356	2	no	—
134	C₂³	—	356	4	no	—
135	C₂³	—	356	4	no	—
136	C₂³	—	178	4	no	—
137	C₂³	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
138	Q₈	—	2	4	no	—
139	Q₈	—	4	2	no	—
140	D₄	—	356	16	no	—
141	D₄	—	356	2	no	—
142	D₄	—	356	2	no	—
143	D₄	—	178	4	no	—
144	D₄	—	178	4	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	#13083561	1	4	yes	complemented
153	C₂×C₄	D₄×D₈₉	1	2	yes	—
154	C₂×C₄	D₄×D₈₉	1	2	yes	—
155	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
156	C₂×C₄	#13083574	1	1	yes	characteristic
157	C₈	—	178	4	no	—
158	C₂²	—	356	8	no	—
159	C₂²	—	356	4	no	—
160	C₂²	—	712	2	no	—
161	C₂²	—	712	2	no	—
162	C₂²	—	712	4	no	—
163	C₂²	#15667980	1	1	yes	characteristic
164	C₂²	#15667986	1	1	yes	characteristic
165	C₂²	#15668008	1	1	yes	characteristic
166	C₂²	C₂×#13083561	1	2	yes	—
167	C₂²	#15668037	1	2	yes	—
168	C₄	—	2	4	no	—
169	C₄	—	4	2	no	—
170	C₄	—	8	1	no	—
171	C₄	#15667982	1	2	yes	—
172	C₄	#15667985	1	2	yes	—
173	C₂	—	712	4	no	—
174	C₂	#19151507	1	1	yes	characteristic
175	C₂	#19151748	1	2	yes	—
176	C₂	#19151798	1	2	yes	—
177	C₂	#19151972	1	1	yes	characteristic
178	C₂	#19151988	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#13083561

C₂

yes

maximal, complemented

C₂×#15668009

C₂

yes

maximal, characteristic

C₂×#15667932

C₂

yes

maximal, complemented

#19151353

C₂

yes

maximal, characteristic, complemented

#391×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₃₅₆

—

C₂²×D₃₅₆

C₂²

yes

—

—

—

—

C₂²

yes

—

#15668009

—

#15667932

—

C₂²×Q₈×C₈₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₈₉

C₂²

yes

characteristic

C₂×C₄²×C₈₉

C₂²

yes

characteristic

#95×C₈₉

—

C₂³×D₈₉

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

D₄

yes

—

—

—

—

D₄

yes

complemented

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

D₄

yes

complemented

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

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

complemented

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

#41

yes

characteristic

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

#10623494

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

C₂×SD₁₆

yes

—

C₂²×C₈₉

AGL(1,ℤ/8)

yes

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#120

yes

—

C₄×C₈₉

#121

yes

—

C₂×C₈₉

#392

yes

characteristic

C₂×C₈₉

#450

yes

—

C₂×C₈₉

#460

yes

—

C₂×C₈₉

yes

C₂×C₈₉

yes

D₈₉

—

#2014

—

maximal, Sylow

C₈₉

#2014

yes

characteristic, complemented, Sylow

C₂²×SD₁₆

—

C₂×#127

—

C₂×#102

—

#391

D₈₉

yes

characteristic, complemented

#336

—

C₂²×Q₈

C₂×D₈₉

yes

characteristic

C₂²×D₄

—

C₂²×D₄

—

178

—

C₂×SD₁₆

—

178

—

100

C₂×SD₁₆

—

178

—

101

C₂×SD₁₆

—

178

—

102

C₂²×C₈

—

103

#127

—

178

—

104

C₂×C₄⋊C₄

C₂×D₈₉

yes

characteristic

105

C₂×C₄²

C₂×D₈₉

yes

characteristic

106

#102

—

178

—

107

#95

—

108

C₂⁴

—

178

—

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂×Q₈

D₃₅₆

yes

complemented

113

C₂×D₄

—

114

C₂×D₄

—

356

—

115

C₂×D₄

—

356

—

116

C₂×D₄

—

178

—

117

C₂×D₄

—

178

—

118

C₂×D₄

—

178

—

119

C₂×D₄

—

178

—

120

C₂×D₄

—

178

—

121

C₂×D₄

—

178

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

C₂²⋊D₈₉

yes

—

125

C₂²×C₄

C₂²×D₈₉

yes

characteristic

126

SD₁₆

—

178

—

127

C₂×C₈

—

128

C₂×C₈

—

178

—

129

C₂×C₈

—

178

—

130

C₄⋊C₄

—

131

C₄²

—

132

C₂³

—

356

—

133

C₂³

—

356

—

134

C₂³

—

356

—

135

C₂³

—

356

—

136

C₂³

—

178

—

137

C₂³

C₂×C₂²⋊D₈₉

yes

characteristic

138

Q₈

—

139

Q₈

—

140

D₄

—

356

—

141

D₄

—

356

—

142

D₄

—

356

—

143

D₄

—

178

—

144

D₄

—

178

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

#13083561

yes

complemented

153

C₂×C₄

D₄×D₈₉

yes

—

154

C₂×C₄

D₄×D₈₉

yes

—

155

C₂×C₄

C₂×D₃₅₆

yes

characteristic

156

C₂×C₄

#13083574

yes

characteristic

157

C₈

—

178

—

158

C₂²

—

356

—

159

C₂²

—

356

—

160

C₂²

—

712

—

161

C₂²

—

712

—

162

C₂²

—

712

—

163

C₂²

#15667980

yes

characteristic

164

C₂²

#15667986

yes

characteristic

165

C₂²

#15668008

yes

characteristic

166

C₂²

C₂×#13083561

yes

—

167

C₂²

#15668037

yes

—

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

#15667982

yes

—

172

C₄

#15667985

yes

—

173

C₂

—

712

—

174

C₂

#19151507

yes

characteristic

175

C₂

#19151748

yes

—

176

C₂

#19151798

yes

—

177

C₂

#19151972

yes

characteristic

178

C₂

#19151988

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₈₉
commutator, cosocle	C₂²×C₄×C₈₉	C₂³
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	#391×C₈₉	C₂
socle	C₂³×C₈₉	C₂×D₄
prime radical	C₂×#15668009	C₂
hypercenter	#391	D₈₉
hypocenter	C₈₉	#2014
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₈₉

commutator, cosocle

C₂²×C₄×C₈₉

C₂³

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

#391×C₈₉

C₂

socle

C₂³×C₈₉

C₂×D₄

C₂

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

N	H ≅ G/N	type
1. C₂²×#13083561	C₂	semidirect
3. C₂×#15667932	C₂	semidirect
4. #19151353	C₂	semidirect
5. #391×C₈₉	C₂	semidirect
31. C₂×#10623494	D₄	semidirect
35. C₂×Q₈×C₈₉	D₄	semidirect
57. C₂×C₄×C₈₉	SD₁₆	semidirect
90. C₈₉	#2014	semidirect
94. #391	D₈₉	semidirect
112. C₂×Q₈	D₃₅₆	semidirect
152. C₂×C₄	#13083561	semidirect

H ≅ G/N

type

1. C₂²×#13083561

C₂

semidirect

3. C₂×#15667932

C₂

semidirect

4. #19151353

C₂

semidirect

5. #391×C₈₉

C₂

semidirect

31. C₂×#10623494

D₄

semidirect

35. C₂×Q₈×C₈₉

D₄

semidirect

57. C₂×C₄×C₈₉

SD₁₆

semidirect

90. C₈₉

#2014

semidirect

94. #391

D₈₉

semidirect

112. C₂×Q₈

D₃₅₆

semidirect

152. C₂×C₄

#13083561

semidirect

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

subgroup	index
C₂²×#13083561	2
C₂×#15668009	2
C₂×#15667932	2
#19151353	2
#391×C₈₉	2
#2014	89

subgroup

index

#391×C₈₉

Subgroups — #26021339

Conjugacy classes of subgroups

Complements

Maximal subgroups