Group #26019115 — subgroups

Subgroups — #26019115

Order 11392.

182 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 61 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	C₄×D₇₁₂	C₂	1	1	yes	maximal, characteristic
2	#19151675	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151663	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151325	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151325	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151320	C₂	1	1	yes	maximal, characteristic, complemented
7	#334×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₇₁₂	C₄	1	1	yes	characteristic
9	C₂×#13083557	—	2	1	no	—
10	C₂×C₈×D₈₉	—	2	1	no	—
11	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
12	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
13	#15667946	C₄	1	1	yes	characteristic
14	#15667934	C₄	1	1	yes	characteristic
15	#15667934	C₄	1	1	yes	characteristic
16	#15667924	—	2	1	no	—
17	#15667924	—	2	1	no	—
18	C₄×#10623494	C₂²	1	1	yes	characteristic
19	C₄×#10623494	C₂²	1	1	yes	characteristic
20	C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×D₃₅₆	C₂×C₄	1	1	yes	characteristic
24	C₂×D₃₅₆	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×D₈₉	—	2	1	no	—
26	C₂×C₄×D₈₉	—	2	1	no	—
27	D₇₁₂	—	2	2	no	—
28	#13083557	—	4	2	no	—
29	C₂×#10623494	—	2	1	no	—
30	C₂×#10623494	D₄	1	1	yes	characteristic
31	C₂×#10623494	D₄	1	1	yes	characteristic
32	C₈×D₈₉	—	4	2	no	—
33	C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
34	C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
35	#13083551	—	2	1	no	—
36	#13083551	—	2	1	no	—
37	C₂×C₈×C₈₉	—	2	1	no	—
38	C₂×C₈×C₈₉	—	2	1	no	—
39	C₂×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
41	C₄²×C₈₉	C₂³	1	1	yes	characteristic
42	C₂²×D₈₉	—	2	1	no	—
43	C₂²×D₈₉	—	2	1	no	—
44	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
47	C₂×Dic₈₉	—	2	1	no	—
48	C₂×Dic₈₉	—	2	1	no	—
49	D₃₅₆	—	2	1	no	—
50	D₃₅₆	—	2	1	no	—
51	D₃₅₆	—	2	1	no	—
52	D₃₅₆	M₁₆	1	2	yes	complemented
53	C₄×D₈₉	—	4	1	no	—
54	C₄×D₈₉	—	4	1	no	—
55	C₄×D₈₉	—	4	1	no	—
56	C₄×D₈₉	—	4	1	no	—
57	C₈×C₈₉	—	2	1	no	—
58	C₈×C₈₉	—	2	1	no	—
59	C₈×C₈₉	—	4	1	no	—
60	C₈×C₈₉	—	4	1	no	—
61	#10623494	—	2	2	no	—
62	#10623494	—	2	1	no	—
63	#10623494	—	2	1	no	—
64	C₂²×C₈₉	C₄×D₄	1	1	yes	characteristic
65	C₂×D₈₉	—	2	2	no	—
66	C₂×D₈₉	—	4	1	no	—
67	C₂×D₈₉	—	4	1	no	—
68	C₂×D₈₉	—	4	1	no	—
69	C₂×D₈₉	—	4	1	no	—
70	C₂×D₈₉	—	4	1	no	—
71	C₄×C₈₉	—	2	1	no	—
72	C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
73	C₄×C₈₉	#131	1	1	yes	characteristic
74	C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
75	C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
76	Dic₈₉	—	4	1	no	—
77	Dic₈₉	—	4	1	no	—
78	C₂×C₈₉	#435	1	1	yes	characteristic
79	C₂×C₈₉	#442	1	1	yes	characteristic
80	C₂×C₈₉	#443	1	1	yes	characteristic
81	D₈₉	—	4	2	no	—
82	D₈₉	—	8	1	no	—
83	#1586	—	89	1	no	maximal, Sylow
84	C₈₉	#1586	1	1	yes	characteristic, complemented, Sylow
85	C₄×D₈	—	89	1	no	—
86	#436	—	89	1	no	—
87	C₈×D₄	—	89	1	no	—
88	#334	D₈₉	1	1	yes	characteristic, complemented
89	D₄⋊C₈	—	89	1	no	—
90	D₄⋊C₈	—	89	1	no	—
91	#322	—	89	1	no	—
92	C₂×D₈	—	89	1	no	—
93	C₂×M₁₆	—	178	1	no	—
94	C₂²×C₈	—	178	1	no	—
95	C₄×D₄	—	89	1	no	—
96	C₄×D₄	—	89	1	no	—
97	#107	—	89	1	no	—
98	C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
99	C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
100	#102	—	89	1	no	—
101	#102	—	89	1	no	—
102	C₂²⋊C₈	—	178	1	no	—
103	C₂²⋊C₈	—	178	1	no	—
104	C₄×C₈	—	89	1	no	—
105	C₄×C₈	—	89	1	no	—
106	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
107	C₂×D₄	—	89	1	no	—
108	C₂×D₄	—	89	1	no	—
109	C₂²×C₄	—	178	1	no	—
110	C₂²×C₄	—	178	1	no	—
111	D₈	—	178	2	no	—
112	M₁₆	—	356	2	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	89	1	no	—
116	C₂×C₈	—	89	1	no	—
117	C₂×C₈	—	356	2	no	—
118	C₂×C₈	—	178	1	no	—
119	C₂×C₈	C₄×D₈₉	1	1	yes	characteristic
120	C₂×C₈	C₄×D₈₉	1	1	yes	characteristic
121	C₄⋊C₄	—	89	1	no	—
122	C₄⋊C₄	—	89	1	no	—
123	C₂²⋊C₄	—	178	1	no	—
124	C₂²⋊C₄	—	178	1	no	—
125	C₄²	C₂²×D₈₉	1	1	yes	characteristic
126	C₂³	—	178	1	no	—
127	C₂³	—	178	1	no	—
128	D₄	—	89	2	no	—
129	D₄	—	178	1	no	—
130	D₄	—	178	1	no	—
131	D₄	—	178	1	no	—
132	C₂×C₄	—	356	1	no	—
133	C₂×C₄	—	356	1	no	—
134	C₂×C₄	—	356	1	no	—
135	C₂×C₄	—	356	1	no	—
136	C₂×C₄	—	178	1	no	—
137	C₂×C₄	—	178	1	no	—
138	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
139	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
140	C₂×C₄	#13083575	1	1	yes	characteristic
141	C₈	—	2	1	no	—
142	C₈	—	2	1	no	—
143	C₈	—	4	1	no	—
144	C₈	—	4	1	no	—
145	C₈	—	178	2	no	—
146	C₈	—	178	1	no	—
147	C₈	—	178	1	no	—
148	C₂²	—	356	1	no	—
149	C₂²	—	356	1	no	—
150	C₂²	—	356	1	no	—
151	C₂²	—	356	1	no	—
152	C₂²	—	356	1	no	—
153	C₂²	—	178	2	no	—
154	C₂²	#15667974	1	1	yes	characteristic
155	C₄	—	2	1	no	—
156	C₄	—	356	1	no	—
157	C₄	—	356	1	no	—
158	C₄	M₁₆×D₈₉	1	1	yes	characteristic
159	C₄	#15668018	1	1	yes	characteristic
160	C₄	#15668039	1	1	yes	characteristic
161	C₄	#15668039	1	1	yes	characteristic
162	C₂	—	356	2	no	—
163	C₂	—	712	1	no	—
164	C₂	#19151668	1	1	yes	characteristic
165	C₂	#19151709	1	1	yes	characteristic
166	C₂	#19151714	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×D₇₁₂

C₂

yes

maximal, characteristic

#19151675

C₂

yes

maximal, characteristic, complemented

#19151663

C₂

yes

maximal, characteristic, complemented

#19151325

C₂

yes

maximal, characteristic, complemented

#19151325

C₂

yes

maximal, characteristic, complemented

#19151320

C₂

yes

maximal, characteristic, complemented

#334×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×D₇₁₂

C₄

yes

characteristic

C₂×#13083557

—

C₂×C₈×D₈₉

—

C₄×D₃₅₆

C₂²

yes

characteristic

C₄×D₃₅₆

C₂²

yes

characteristic

#15667946

C₄

yes

characteristic

#15667934

C₄

yes

characteristic

#15667934

C₄

yes

—

—

C₂²

yes

characteristic

C₄×#10623494

C₂²

yes

characteristic

C₄⋊C₈×C₈₉

C₂²

yes

characteristic

C₄⋊C₈×C₈₉

C₂²

yes

characteristic

C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂×D₃₅₆

C₂×C₄

yes

characteristic

C₂×D₃₅₆

C₂×C₄

yes

characteristic

C₂×C₄×D₈₉

—

C₂×C₄×D₈₉

—

D₇₁₂

—

#13083557

—

C₂×#10623494

—

C₂×#10623494

D₄

yes

characteristic

C₂×#10623494

D₄

yes

characteristic

C₈×D₈₉

—

C₄⋊Dic₈₉

C₂×C₄

yes

characteristic

C₄⋊Dic₈₉

C₂×C₄

yes

characteristic

#13083551

—

#13083551

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂×C₄

yes

characteristic

C₂×C₈×C₈₉

C₂×C₄

yes

characteristic

C₄²×C₈₉

C₂³

yes

characteristic

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

#41

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

M₁₆

yes

complemented

C₄×D₈₉

—

C₄×D₈₉

—

C₄×D₈₉

—

C₄×D₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

#10623494

—

#10623494

—

#10623494

—

C₂²×C₈₉

C₄×D₄

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂×M₁₆

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

D₈₉

—

D₈₉

—

#1586

—

maximal, Sylow

C₈₉

#1586

yes

characteristic, complemented, Sylow

C₄×D₈

—

#436

—

C₈×D₄

—

#334

D₈₉

yes

characteristic, complemented

D₄⋊C₈

—

D₄⋊C₈

—

#322

—

C₂×D₈

—

C₂×M₁₆

—

178

—

C₂²×C₈

—

178

—

C₄×D₄

—

C₄×D₄

—

#107

—

C₄⋊C₈

C₂×D₈₉

yes

characteristic

C₄⋊C₈

C₂×D₈₉

yes

characteristic

100

#102

—

101

#102

—

102

C₂²⋊C₈

—

178

—

103

C₂²⋊C₈

—

178

—

104

C₄×C₈

—

105

C₄×C₈

—

106

C₄×C₈

C₂×D₈₉

yes

characteristic

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂²×C₄

—

178

—

110

C₂²×C₄

—

178

—

111

D₈

—

178

—

112

M₁₆

—

356

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

356

—

118

C₂×C₈

—

178

—

119

C₂×C₈

C₄×D₈₉

yes

characteristic

120

C₂×C₈

C₄×D₈₉

yes

characteristic

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₂²⋊C₄

—

178

—

124

C₂²⋊C₄

—

178

—

125

C₄²

C₂²×D₈₉

yes

characteristic

126

C₂³

—

178

—

127

C₂³

—

178

—

128

D₄

—

129

D₄

—

178

—

130

D₄

—

178

—

131

D₄

—

178

—

132

C₂×C₄

—

356

—

133

C₂×C₄

—

356

—

134

C₂×C₄

—

356

—

135

C₂×C₄

—

356

—

136

C₂×C₄

—

178

—

137

C₂×C₄

—

178

—

138

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

139

C₂×C₄

D₄×D₈₉

yes

characteristic

140

C₂×C₄

#13083575

yes

characteristic

141

C₈

—

142

C₈

—

143

C₈

—

144

C₈

—

145

C₈

—

178

—

146

C₈

—

178

—

147

C₈

—

178

—

148

C₂²

—

356

—

149

C₂²

—

356

—

150

C₂²

—

356

—

151

C₂²

—

356

—

152

C₂²

—

356

—

153

C₂²

—

178

—

154

C₂²

#15667974

yes

characteristic

155

C₄

—

156

C₄

—

356

—

157

C₄

—

356

—

158

C₄

M₁₆×D₈₉

yes

characteristic

159

C₄

#15668018

yes

characteristic

160

C₄

#15668039

yes

characteristic

161

C₄

#15668039

yes

characteristic

162

C₂

—

356

—

163

C₂

—

712

—

164

C₂

#19151668

yes

characteristic

165

C₂

#19151709

yes

characteristic

166

C₂

#19151714

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂²×C₄
Frattini	C₄²	C₂²×D₈₉
Fitting	#334×C₈₉	C₂
socle	C₂²×C₈₉	C₄×D₄
prime radical	C₂×D₇₁₂	C₄
cosocle	C₄²×C₈₉	C₂³
hypercenter	#334	D₈₉
hypocenter	C₈₉	#1586
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

subgroup

G/H

center

C₂×C₄

D₄×D₈₉

commutator

C₂×C₄×C₈₉

C₂²×C₄

Frattini

C₄²

C₂²×D₈₉

Fitting

#334×C₈₉

C₂

socle

C₂²×C₈₉

C₄×D₄

prime radical

C₂×D₇₁₂

C₄

cosocle

C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

N	H ≅ G/N	type
2. #19151675	C₂	semidirect
3. #19151663	C₂	semidirect
4. #19151325	C₂	semidirect
5. #19151325	C₂	semidirect
6. #19151320	C₂	semidirect
7. #334×C₈₉	C₂	semidirect
52. D₃₅₆	M₁₆	semidirect
84. C₈₉	#1586	semidirect
88. #334	D₈₉	semidirect

H ≅ G/N

type

2. #19151675

C₂

semidirect

3. #19151663

C₂

semidirect

4. #19151325

C₂

semidirect

5. #19151325

C₂

semidirect

6. #19151320

C₂

semidirect

7. #334×C₈₉

C₂

semidirect

52. D₃₅₆

M₁₆

semidirect

84. C₈₉

#1586

semidirect

88. #334

D₈₉

semidirect

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

subgroup	index
C₄×D₇₁₂	2
#19151675	2
#19151663	2
#19151325	2
#19151320	2
#334×C₈₉	2
#1586	89

subgroup

index

C₄×D₇₁₂

#334×C₈₉

Subgroups — #26019115

Conjugacy classes of subgroups

Complements

Maximal subgroups