Group #36453922 — subgroups

Subgroups — #36453922

Order 22784.

390 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 119 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×#19152003	C₂	1	1	yes	maximal, characteristic
2	#26018938	C₂	1	2	yes	maximal
3	#26018934	C₂	1	2	yes	maximal
4	#26018544	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1837	C₂	1	1	yes	maximal, characteristic
6	C₂×#15668012	C₂²	1	2	yes	—
7	#19152003	—	2	2	no	—
8	#19152003	C₄	1	4	yes	—
9	#19151651	C₂²	1	1	yes	characteristic
10	#19151651	C₂²	1	1	yes	characteristic
11	C₂×#15667946	—	2	2	no	—
12	#19151346	—	2	4	no	—
13	#19151343	—	2	4	no	—
14	#19151320	—	2	2	no	—
15	#423×C₈₉	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	#341×C₈₉	—	2	2	no	—
18	C₂²×Dic₁₇₈	—	2	2	no	—
19	#15668012	—	2	2	no	—
20	#15668012	—	2	2	no	—
21	#15668012	—	2	2	no	—
22	#15668012	C₂×C₄	1	4	yes	—
23	C₂×C₄⋊Dic₈₉	—	2	2	no	—
24	C₂×C₄⋊Dic₈₉	—	2	2	no	—
25	#15667946	—	2	8	no	—
26	C₄⋊#10623494	D₄	1	2	yes	—
27	C₄⋊#10623494	D₄	1	2	yes	—
28	C₂²⋊#10623494	—	2	1	no	—
29	C₂²⋊#10623494	—	2	1	no	—
30	C₄×#10623494	—	2	1	no	—
31	C₄×#10623494	—	2	1	no	—
32	C₂×M₁₆×C₈₉	—	2	2	no	—
33	C₂²×C₈×C₈₉	D₄	1	2	yes	—
34	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₈₉	—	2	2	no	—
36	C₄⋊C₈×C₈₉	—	2	2	no	—
37	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₈₉	D₄	1	2	yes	—
39	C₂×Dic₁₇₈	—	2	4	no	—
40	C₂×Dic₁₇₈	—	4	2	no	—
41	C₂×Dic₁₇₈	—	8	4	no	—
42	C₂²×Dic₈₉	—	4	2	no	—
43	C₂×#10623494	—	2	2	no	—
44	C₂×#10623494	—	2	2	no	—
45	C₄⋊Dic₈₉	—	2	4	no	—
46	C₄⋊Dic₈₉	—	2	4	no	—
47	C₄⋊Dic₈₉	—	4	2	no	—
48	C₄⋊Dic₈₉	—	4	2	no	—
49	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
51	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
52	M₁₆×C₈₉	—	4	4	no	—
53	C₂×C₈×C₈₉	—	2	4	no	—
54	C₂×C₈×C₈₉	—	4	2	no	—
55	C₂×C₈×C₈₉	C₂²⋊C₄	1	4	yes	—
56	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
60	C₂³×C₈₉	#124	1	1	yes	characteristic
61	C₂×C₄×C₈₉	—	2	2	no	—
62	C₂×C₄×C₈₉	—	2	2	no	—
63	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
65	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
66	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
67	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
68	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
69	C₂×C₄×C₈₉	#137	1	4	yes	—
70	C₂×Dic₈₉	—	4	4	no	—
71	C₂×Dic₈₉	—	8	4	no	—
72	Dic₁₇₈	—	8	4	no	—
73	Dic₁₇₈	—	8	4	no	—
74	C₈×C₈₉	—	2	4	no	—
75	C₈×C₈₉	—	8	2	no	—
76	#10623494	—	4	2	no	—
77	#10623494	—	4	2	no	—
78	C₂²×C₈₉	—	2	1	no	—
79	C₂²×C₈₉	—	2	1	no	—
80	C₂²×C₈₉	#390	1	1	yes	characteristic
81	C₂²×C₈₉	#491	1	2	yes	—
82	C₄×C₈₉	—	2	1	no	—
83	C₄×C₈₉	—	2	1	no	—
84	C₄×C₈₉	#419	1	2	yes	—
85	C₄×C₈₉	#472	1	2	yes	—
86	C₄×C₈₉	#473	1	2	yes	—
87	C₄×C₈₉	#497	1	2	yes	—
88	Dic₈₉	—	8	4	no	—
89	#9778	—	89	1	no	maximal, Sylow
90	C₂×C₈₉	—	2	2	no	—
91	C₂×C₈₉	#1960	1	1	yes	characteristic
92	C₂×C₈₉	#1978	1	1	yes	characteristic
93	C₂×C₈₉	#1979	1	1	yes	characteristic
94	C₂×#500	—	89	1	no	—
95	#1837	D₈₉	1	1	yes	characteristic
96	#1549	—	89	2	no	—
97	#1548	—	89	2	no	—
98	#1452	—	89	1	no	—
99	C₈₉	#9778	1	1	yes	characteristic, complemented, Sylow
100	C₂×C₄⋊Q₈	—	89	2	no	—
101	#500	—	89	4	no	—
102	#500	—	178	2	no	—
103	#431	—	89	1	no	—
104	#431	—	89	1	no	—
105	C₂×#107	—	178	2	no	—
106	#423	C₂×D₈₉	1	2	yes	—
107	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
108	#341	—	2	2	no	—
109	#333	—	178	4	no	—
110	#332	—	178	4	no	—
111	#322	—	178	2	no	—
112	C₂²×Q₈	—	178	2	no	—
113	C₂×M₁₆	—	2	2	no	—
114	C₂²×C₈	C₂²⋊D₈₉	1	2	yes	—
115	C₄⋊Q₈	—	89	4	no	—
116	C₄⋊Q₈	—	178	2	no	—
117	C₄⋊Q₈	—	178	2	no	—
118	C₄⋊Q₈	—	178	2	no	—
119	C₂×C₄⋊C₄	—	178	2	no	—
120	C₂×C₄⋊C₄	—	178	2	no	—
121	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
122	#107	—	178	8	no	—
123	C₄⋊C₈	—	2	2	no	—
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	—	89	2	no	—
126	C₄⋊C₈	—	89	2	no	—
127	C₂²⋊C₈	—	178	1	no	—
128	C₂²⋊C₈	—	178	1	no	—
129	C₄×C₈	—	178	1	no	—
130	C₄×C₈	—	178	1	no	—
131	C₄×C₈	C₄×D₈₉	1	2	yes	—
132	C₄×C₈	D₃₅₆	1	2	yes	—
133	C₂×Q₈	—	356	2	no	—
134	C₂×Q₈	—	178	4	no	—
135	C₂×Q₈	—	712	4	no	—
136	C₂²×C₄	—	356	2	no	—
137	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
138	C₂²×C₄	#13083575	1	1	yes	characteristic
139	C₂²×C₄	#13083575	1	1	yes	characteristic
140	M₁₆	—	4	4	no	—
141	C₂×C₈	—	2	4	no	—
142	C₂×C₈	—	4	2	no	—
143	C₂×C₈	—	178	2	no	—
144	C₂×C₈	—	178	2	no	—
145	C₂×C₈	#13083551	1	4	yes	—
146	C₄⋊C₄	—	356	2	no	—
147	C₄⋊C₄	—	356	2	no	—
148	C₄⋊C₄	—	178	4	no	—
149	C₄⋊C₄	—	178	4	no	—
150	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
151	C₄²	D₄×D₈₉	1	1	yes	characteristic
152	C₄²	D₄×D₈₉	1	1	yes	characteristic
153	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
154	C₂³	#15668001	1	1	yes	characteristic
155	Q₈	—	712	4	no	—
156	Q₈	—	712	4	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	356	4	no	—
160	C₂×C₄	—	712	4	no	—
161	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
162	C₂×C₄	#15667974	1	1	yes	characteristic
163	C₂×C₄	#15667974	1	1	yes	characteristic
164	C₂×C₄	#15667987	1	1	yes	characteristic
165	C₂×C₄	#15667987	1	1	yes	characteristic
166	C₂×C₄	#15668008	1	1	yes	characteristic
167	C₂×C₄	#15668042	1	4	yes	—
168	C₈	—	2	4	no	—
169	C₈	—	8	2	no	—
170	C₈	—	356	2	no	—
171	C₈	—	356	2	no	—
172	C₂²	—	2	1	no	—
173	C₂²	—	2	1	no	—
174	C₂²	#19151499	1	1	yes	characteristic
175	C₂²	#19151968	1	2	yes	—
176	C₄	—	2	1	no	—
177	C₄	—	2	1	no	—
178	C₄	—	712	4	no	—
179	C₄	#19151606	1	2	yes	—
180	C₄	#19151869	1	2	yes	—
181	C₄	#19151874	1	2	yes	—
182	C₄	#19151993	1	2	yes	—
183	C₂	—	2	2	no	—
184	C₂	#26021071	1	1	yes	characteristic
185	C₂	#26021160	1	1	yes	characteristic
186	C₂	#26021165	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19152003

C₂

yes

maximal, characteristic

#26018938

C₂

yes

maximal

#26018934

C₂

yes

maximal

#26018544

C₂

yes

maximal, characteristic

C₈₉×#1837

C₂

yes

maximal, characteristic

C₂×#15668012

C₂²

yes

—

#19152003

—

#19152003

C₄

yes

—

#19151651

C₂²

yes

characteristic

#19151651

C₂²

yes

—

—

—

—

#423×C₈₉

C₂²

yes

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

#341×C₈₉

—

C₂²×Dic₁₇₈

—

—

—

—

C₂×C₄

yes

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

#15667946

—

C₄⋊#10623494

D₄

yes

—

C₄⋊#10623494

D₄

yes

—

—

—

—

—

C₂×M₁₆×C₈₉

—

C₂²×C₈×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄⋊C₈×C₈₉

—

C₄⋊C₈×C₈₉

—

C₄×C₈×C₈₉

C₂×C₄

yes

—

C₄×C₈×C₈₉

D₄

yes

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂²×Dic₈₉

—

C₂×#10623494

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

C₂²×C₄×C₈₉

yes

C₂²×C₄×C₈₉

yes

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂²⋊C₄

yes

—

C₄²×C₈₉

C₂²×C₄

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₂³×C₈₉

#124

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₄×D₄

yes

characteristic

C₂×C₄×C₈₉

C₄×D₄

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

—

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#390

yes

characteristic

C₂²×C₈₉

#491

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#419

yes

—

C₄×C₈₉

#472

yes

—

C₄×C₈₉

#473

yes

—

C₄×C₈₉

#497

yes

—

Dic₈₉

—

#9778

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

D₈₉

yes

—

—

—

C₈₉

#9778

yes

characteristic, complemented, Sylow

100

C₂×C₄⋊Q₈

—

101

#500

—

102

#500

—

178

—

103

#431

—

104

#431

—

105

C₂×#107

—

178

—

106

#423

C₂×D₈₉

yes

—

107

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

108

#341

—

109

#333

—

178

—

110

#332

—

178

—

111

#322

—

178

—

112

C₂²×Q₈

—

178

—

113

C₂×M₁₆

—

114

C₂²×C₈

C₂²⋊D₈₉

yes

—

115

C₄⋊Q₈

—

116

C₄⋊Q₈

—

178

—

117

C₄⋊Q₈

—

178

—

118

C₄⋊Q₈

—

178

—

119

C₂×C₄⋊C₄

—

178

—

120

C₂×C₄⋊C₄

—

178

—

121

C₂×C₄²

C₂²×D₈₉

yes

characteristic

122

#107

—

178

—

123

C₄⋊C₈

—

124

C₄⋊C₈

—

125

C₄⋊C₈

—

126

C₄⋊C₈

—

127

C₂²⋊C₈

—

178

—

128

C₂²⋊C₈

—

178

—

129

C₄×C₈

—

178

—

130

C₄×C₈

—

178

—

131

C₄×C₈

C₄×D₈₉

yes

—

132

C₄×C₈

D₃₅₆

yes

—

133

C₂×Q₈

—

356

—

134

C₂×Q₈

—

178

—

135

C₂×Q₈

—

712

—

136

C₂²×C₄

—

356

—

137

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

138

C₂²×C₄

#13083575

yes

characteristic

139

C₂²×C₄

#13083575

yes

characteristic

140

M₁₆

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

178

—

144

C₂×C₈

—

178

—

145

C₂×C₈

#13083551

yes

—

146

C₄⋊C₄

—

356

—

147

C₄⋊C₄

—

356

—

148

C₄⋊C₄

—

178

—

149

C₄⋊C₄

—

178

—

150

C₄²

C₂×C₄×D₈₉

yes

characteristic

151

C₄²

D₄×D₈₉

yes

characteristic

152

C₄²

D₄×D₈₉

yes

characteristic

153

C₄²

C₂×D₃₅₆

yes

characteristic

154

C₂³

#15668001

yes

characteristic

155

Q₈

—

712

—

156

Q₈

—

712

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

356

—

160

C₂×C₄

—

712

—

161

C₂×C₄

C₂×#13083551

yes

characteristic

162

C₂×C₄

#15667974

yes

characteristic

163

C₂×C₄

#15667974

yes

characteristic

164

C₂×C₄

#15667987

yes

characteristic

165

C₂×C₄

#15667987

yes

characteristic

166

C₂×C₄

#15668008

yes

characteristic

167

C₂×C₄

#15668042

yes

—

168

C₈

—

169

C₈

—

170

C₈

—

356

—

171

C₈

—

356

—

172

C₂²

—

173

C₂²

—

174

C₂²

#19151499

yes

characteristic

175

C₂²

#19151968

yes

—

176

C₄

—

177

C₄

—

178

C₄

—

712

—

179

C₄

#19151606

yes

—

180

C₄

#19151869

yes

—

181

C₄

#19151874

yes

—

182

C₄

#19151993

yes

—

183

C₂

—

184

C₂

#26021071

yes

characteristic

185

C₂

#26021160

yes

characteristic

186

C₂

#26021165

yes

characteristic

	subgroup	G/H
center	C₂²	#19151499
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1837	C₂
socle	C₂²×C₈₉	#390
prime radical	C₂³×C₈₉	#124
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1837	D₈₉
hypocenter	C₈₉	#9778
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1837

subgroup

G/H

center

C₂²

#19151499

commutator

C₄²×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1837

C₂

C₂²×C₈₉

C₂³×C₈₉

C₂×C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1837

N	H ≅ G/N	type
99. C₈₉	#9778	semidirect

H ≅ G/N

type

99. C₈₉

#9778

semidirect

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

subgroup	index
C₂×#19152003	2
#26018938	2
#26018934	2
#26018544	2
C₈₉×#1837	2
#9778	89

subgroup

index

C₈₉×#1837

Subgroups — #36453922

Conjugacy classes of subgroups

Complements

Maximal subgroups