Group #26021871 — subgroups

Subgroups — #26021871

Order 11392.

220 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 75 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×#15668020	C₂	1	1	yes	maximal, characteristic
2	M₃₂×D₈₉	C₂	1	2	yes	maximal, complemented
3	#19151380	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151379	C₂	1	2	yes	maximal, complemented
5	#349×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13083574	C₄	1	1	yes	characteristic
7	#15668020	—	2	2	no	—
8	#15668020	—	2	2	no	—
9	C₂×#13083557	C₄	1	1	yes	characteristic
10	C₂×#13083557	C₂²	1	1	yes	characteristic
11	C₂²×#10623494	C₄	1	1	yes	characteristic
12	C₂×C₈×D₈₉	C₄	1	1	yes	characteristic
13	C₂×C₈×D₈₉	C₂²	1	1	yes	characteristic
14	#15667952	C₂²	1	2	yes	—
15	#15667951	—	2	2	no	—
16	C₁₆×D₈₉	—	2	2	no	—
17	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
18	M₃₂×C₈₉	C₂²	1	2	yes	—
19	#13083574	—	2	2	no	—
20	#13083574	—	2	2	no	—
21	C₂×Dic₁₇₈	C₈	1	1	yes	characteristic
22	C₂×D₃₅₆	C₈	1	1	yes	characteristic
23	C₂×C₂²⋊D₈₉	C₈	1	1	yes	characteristic
24	C₂×C₂²⋊D₈₉	C₈	1	1	yes	characteristic
25	C₂×C₄×D₈₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×D₈₉	C₂×C₄	1	1	yes	characteristic
27	#13083557	—	2	2	no	—
28	#13083557	—	2	2	no	—
29	C₂×#10623494	—	2	1	no	—
30	C₂×#10623494	—	2	1	no	—
31	C₂×#10623494	C₂×C₄	1	1	yes	characteristic
32	C₂×#10623494	C₂×C₄	1	1	yes	characteristic
33	C₈×D₈₉	—	2	1	no	—
34	C₈×D₈₉	—	2	1	no	—
35	C₈×D₈₉	D₄	1	2	yes	—
36	C₈×D₈₉	D₄	1	2	yes	—
37	#13083549	—	2	2	no	—
38	C₂²×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
39	M₁₆×C₈₉	—	2	2	no	—
40	C₂×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
41	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
42	C₁₆×C₈₉	—	2	2	no	—
43	C₂³×C₈₉	C₂×C₈	1	1	yes	characteristic
44	C₂²×D₈₉	C₂×C₈	1	1	yes	characteristic
45	C₂²×D₈₉	C₂×C₈	1	1	yes	characteristic
46	C₂×C₄×C₈₉	—	2	1	no	—
47	C₂×C₄×C₈₉	—	2	1	no	—
48	C₂×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
49	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
50	C₂²⋊D₈₉	—	2	2	no	—
51	C₂²⋊D₈₉	—	2	2	no	—
52	C₂×Dic₈₉	C₂×C₈	1	1	yes	characteristic
53	C₂×Dic₈₉	C₂×C₈	1	1	yes	characteristic
54	D₃₅₆	—	2	2	no	—
55	C₄×D₈₉	—	2	1	no	—
56	C₄×D₈₉	—	2	1	no	—
57	C₄×D₈₉	C₂²⋊C₄	1	2	yes	—
58	C₄×D₈₉	M₁₆	1	2	yes	—
59	Dic₁₇₈	—	2	2	no	—
60	C₈×C₈₉	—	2	1	no	—
61	C₈×C₈₉	C₂×D₄	1	2	yes	—
62	#10623494	—	2	1	no	—
63	#10623494	C₂²⋊C₄	1	1	yes	characteristic
64	#10623494	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₈₉	—	2	1	no	—
66	C₂²×C₈₉	—	4	1	no	—
67	C₂²×C₈₉	C₂²×C₈	1	1	yes	characteristic
68	C₂×D₈₉	—	2	1	no	—
69	C₂×D₈₉	—	4	1	no	—
70	C₂×D₈₉	—	4	1	no	—
71	C₂×D₈₉	C₂²⋊C₈	1	2	yes	—
72	C₄×C₈₉	—	2	1	no	—
73	C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
75	Dic₈₉	—	2	1	no	—
76	Dic₈₉	C₂²⋊C₈	1	2	yes	—
77	C₂×C₈₉	—	2	1	no	—
78	C₂×C₈₉	—	4	1	no	—
79	C₂×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
80	D₈₉	—	2	2	no	—
81	D₈₉	—	4	1	no	—
82	#2115	—	89	1	no	maximal, Sylow
83	C₈₉	#2115	1	1	yes	characteristic, complemented, Sylow
84	C₂×#131	—	89	1	no	—
85	C₂×M₃₂	—	89	2	no	—
86	#349	—	89	1	no	—
87	#349	—	89	2	no	—
88	#349	D₈₉	1	1	yes	characteristic, complemented
89	C₂×#41	—	89	1	no	—
90	#131	—	178	2	no	—
91	#131	—	178	2	no	—
92	C₂×M₁₆	—	89	1	no	—
93	C₂×M₁₆	—	89	1	no	—
94	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
95	C₂²×C₈	—	89	1	no	—
96	C₂²×C₈	—	89	1	no	—
97	C₂²×C₈	—	89	1	no	—
98	M₃₂	—	89	2	no	—
99	M₃₂	—	178	2	no	—
100	M₃₂	C₂×D₈₉	1	2	yes	—
101	C₂×C₁₆	—	178	2	no	—
102	#41	—	178	2	no	—
103	#41	—	178	2	no	—
104	C₂×Q₈	—	89	1	no	—
105	C₂×D₄	—	89	1	no	—
106	C₂×D₄	—	89	1	no	—
107	C₂×D₄	—	89	1	no	—
108	C₂²×C₄	—	89	1	no	—
109	C₂²×C₄	—	89	1	no	—
110	C₂²×C₄	C₄×D₈₉	1	1	yes	characteristic
111	M₁₆	—	2	2	no	—
112	M₁₆	—	178	2	no	—
113	M₁₆	—	178	2	no	—
114	C₂×C₈	—	89	1	no	—
115	C₂×C₈	—	89	2	no	—
116	C₂×C₈	—	89	2	no	—
117	C₂×C₈	—	89	1	no	—
118	C₂×C₈	—	178	1	no	—
119	C₂×C₈	—	178	1	no	—
120	C₂×C₈	—	178	1	no	—
121	C₂×C₈	—	178	1	no	—
122	C₂×C₈	C₄×D₈₉	1	1	yes	characteristic
123	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
124	C₁₆	—	2	2	no	—
125	C₁₆	—	178	2	no	—
126	C₂³	—	89	1	no	—
127	C₂³	—	89	1	no	—
128	C₂³	C₈×D₈₉	1	1	yes	characteristic
129	Q₈	—	178	2	no	—
130	D₄	—	178	2	no	—
131	D₄	—	178	2	no	—
132	D₄	—	178	2	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	89	1	no	—
136	C₂×C₄	—	89	2	no	—
137	C₂×C₄	—	89	2	no	—
138	C₂×C₄	—	89	1	no	—
139	C₂×C₄	—	178	1	no	—
140	C₂×C₄	—	178	1	no	—
141	C₂×C₄	C₈×D₈₉	1	1	yes	characteristic
142	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
143	C₈	—	2	1	no	—
144	C₈	—	89	1	no	—
145	C₈	—	89	1	no	—
146	C₈	—	178	1	no	—
147	C₈	D₄×D₈₉	1	2	yes	—
148	C₂²	—	2	1	no	—
149	C₂²	—	4	1	no	—
150	C₂²	—	89	2	no	—
151	C₂²	—	356	1	no	—
152	C₂²	—	356	1	no	—
153	C₂²	—	178	1	no	—
154	C₂²	C₂×C₈×D₈₉	1	1	yes	characteristic
155	C₄	—	2	1	no	—
156	C₄	—	89	2	no	—
157	C₄	—	178	1	no	—
158	C₄	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
159	C₄	M₁₆×D₈₉	1	1	yes	characteristic
160	C₂	—	2	1	no	—
161	C₂	—	4	1	no	—
162	C₂	—	356	1	no	—
163	C₂	—	178	2	no	—
164	C₂	C₂²⋊C₈×D₈₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15668020

C₂

yes

maximal, characteristic

M₃₂×D₈₉

C₂

yes

maximal, complemented

#19151380

C₂

yes

maximal, characteristic, complemented

#19151379

C₂

yes

maximal, complemented

#349×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×#13083574

C₄

yes

—

—

C₄

yes

characteristic

C₂×#13083557

C₂²

yes

characteristic

C₂²×#10623494

C₄

yes

characteristic

C₂×C₈×D₈₉

C₄

yes

characteristic

C₂×C₈×D₈₉

C₂²

yes

characteristic

#15667952

C₂²

yes

—

#15667951

—

C₁₆×D₈₉

—

C₂×M₁₆×C₈₉

C₂²

yes

characteristic

M₃₂×C₈₉

C₂²

yes

—

#13083574

—

#13083574

—

C₂×Dic₁₇₈

C₈

yes

characteristic

C₂×D₃₅₆

C₈

yes

characteristic

C₂×C₂²⋊D₈₉

C₈

yes

characteristic

C₂×C₂²⋊D₈₉

C₈

yes

characteristic

C₂×C₄×D₈₉

C₂×C₄

yes

characteristic

C₂×C₄×D₈₉

C₂×C₄

yes

—

—

—

—

C₂×C₄

yes

characteristic

C₂×#10623494

C₂×C₄

yes

characteristic

C₈×D₈₉

—

C₈×D₈₉

—

C₈×D₈₉

D₄

yes

—

C₈×D₈₉

D₄

yes

—

#13083549

—

C₂²×C₄×C₈₉

C₂×C₄

yes

characteristic

M₁₆×C₈₉

—

C₂×C₈×C₈₉

C₂×C₄

yes

characteristic

C₂×C₈×C₈₉

C₂³

yes

characteristic

C₁₆×C₈₉

—

C₂³×C₈₉

C₂×C₈

yes

characteristic

C₂²×D₈₉

C₂×C₈

yes

characteristic

C₂²×D₈₉

C₂×C₈

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×C₈

yes

characteristic

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂²⋊D₈₉

—

C₂²⋊D₈₉

—

C₂×Dic₈₉

C₂×C₈

yes

characteristic

C₂×Dic₈₉

C₂×C₈

yes

characteristic

D₃₅₆

—

C₄×D₈₉

—

C₄×D₈₉

—

C₄×D₈₉

C₂²⋊C₄

yes

—

C₄×D₈₉

M₁₆

yes

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

C₂×D₄

yes

—

#10623494

—

#10623494

C₂²⋊C₄

yes

characteristic

#10623494

C₂²⋊C₄

yes

characteristic

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂²×C₈

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

C₂²⋊C₈

yes

—

C₄×C₈₉

—

C₄×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₈₉

C₂×M₁₆

yes

characteristic

Dic₈₉

—

Dic₈₉

C₂²⋊C₈

yes

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

C₂×C₂²⋊C₈

yes

characteristic

D₈₉

—

D₈₉

—

#2115

—

maximal, Sylow

C₈₉

#2115

yes

characteristic, complemented, Sylow

C₂×#131

—

C₂×M₃₂

—

#349

—

#349

—

#349

D₈₉

yes

characteristic, complemented

C₂×#41

—

#131

—

178

—

#131

—

178

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₈₉

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

M₃₂

—

M₃₂

—

178

—

100

M₃₂

C₂×D₈₉

yes

—

101

C₂×C₁₆

—

178

—

102

#41

—

178

—

103

#41

—

178

—

104

C₂×Q₈

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

C₄×D₈₉

yes

characteristic

111

M₁₆

—

112

M₁₆

—

178

—

113

M₁₆

—

178

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

178

—

119

C₂×C₈

—

178

—

120

C₂×C₈

—

178

—

121

C₂×C₈

—

178

—

122

C₂×C₈

C₄×D₈₉

yes

characteristic

123

C₂×C₈

C₂²×D₈₉

yes

characteristic

124

C₁₆

—

125

C₁₆

—

178

—

126

C₂³

—

127

C₂³

—

128

C₂³

C₈×D₈₉

yes

characteristic

129

Q₈

—

178

—

130

D₄

—

178

—

131

D₄

—

178

—

132

D₄

—

178

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

178

—

140

C₂×C₄

—

178

—

141

C₂×C₄

C₈×D₈₉

yes

characteristic

142

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

—

178

—

147

C₈

D₄×D₈₉

yes

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

356

—

152

C₂²

—

356

—

153

C₂²

—

178

—

154

C₂²

C₂×C₈×D₈₉

yes

characteristic

155

C₄

—

156

C₄

—

157

C₄

—

178

—

158

C₄

C₂²⋊C₄×D₈₉

yes

characteristic

159

C₄

M₁₆×D₈₉

yes

characteristic

160

C₂

—

161

C₂

—

162

C₂

—

356

—

163

C₂

—

178

—

164

C₂

C₂²⋊C₈×D₈₉

yes

characteristic

	subgroup	G/H
center	C₄	C₂²⋊C₄×D₈₉
commutator	C₂²×C₈₉	C₂²×C₈
Frattini	C₂×C₈	C₂²×D₈₉
Fitting	#349×C₈₉	C₂
socle	C₂×C₈₉	C₂×C₂²⋊C₈
prime radical	C₂×#13083574	C₄
cosocle	C₂×C₈×C₈₉	C₂³
hypercenter	#349	D₈₉
hypocenter	C₈₉	#2115
LCS	G ▷ C₂²×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #349

subgroup

G/H

center

C₄

C₂²⋊C₄×D₈₉

commutator

C₂²×C₈₉

C₂²×C₈

Frattini

C₂×C₈

C₂²×D₈₉

Fitting

#349×C₈₉

C₂

socle

C₂×C₈₉

C₂×C₂²⋊C₈

prime radical

C₂×#13083574

C₄

cosocle

C₂×C₈×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂²×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ #349

N	H ≅ G/N	type
2. M₃₂×D₈₉	C₂	semidirect
3. #19151380	C₂	semidirect
4. #19151379	C₂	semidirect
5. #349×C₈₉	C₂	semidirect
83. C₈₉	#2115	semidirect
88. #349	D₈₉	semidirect

H ≅ G/N

type

2. M₃₂×D₈₉

C₂

semidirect

3. #19151380

C₂

semidirect

4. #19151379

C₂

semidirect

5. #349×C₈₉

C₂

semidirect

83. C₈₉

#2115

semidirect

88. #349

D₈₉

semidirect

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

subgroup	index
C₂×#15668020	2
M₃₂×D₈₉	2
#19151380	2
#19151379	2
#349×C₈₉	2
#2115	89

subgroup

index

M₃₂×D₈₉

#349×C₈₉

Subgroups — #26021871

Conjugacy classes of subgroups

Complements

Maximal subgroups