Group #36473036 — subgroups

Subgroups — #36473036

Order 22784.

374 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 159 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×#19152000	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151374	C₂	1	1	yes	maximal, characteristic
3	#26018343	C₂	1	4	yes	maximal
4	C₈₉×#1845	C₂	1	1	yes	maximal, characteristic
5	C₂×#15668004	—	2	1	no	—
6	C₂²×#13083549	C₂²	1	2	yes	—
7	#19152000	—	2	2	no	—
8	#19152000	C₄	1	4	yes	complemented
9	C₂×#15667946	C₂²	1	2	yes	—
10	C₂×#15667944	—	2	1	no	—
11	#19151374	—	2	2	no	—
12	C₂×#107×C₈₉	—	2	1	no	—
13	C₂×#107×C₈₉	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
15	#384×C₈₉	—	2	1	no	—
16	#15668004	—	2	4	no	—
17	#15668004	—	4	2	no	—
18	C₂×C₄⋊Dic₈₉	—	2	2	no	—
19	C₂×#13083553	—	4	2	no	—
20	C₂×#13083549	—	2	2	no	—
21	C₂×#13083549	—	2	2	no	—
22	C₂×#13083549	Q₈	1	4	yes	—
23	#15667946	—	2	2	no	—
24	#15667946	C₂×C₄	1	4	yes	—
25	#15667944	—	2	4	no	—
26	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
29	C₂×C₄⋊C₄×C₈₉	—	4	2	no	—
30	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
31	#107×C₈₉	—	2	4	no	—
32	#107×C₈₉	—	4	2	no	—
33	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₈₉	D₄	1	2	yes	—
35	C₂²×Dic₈₉	—	4	2	no	—
36	C₄⋊Dic₈₉	—	2	4	no	—
37	C₄⋊Dic₈₉	—	4	2	no	—
38	#13083553	—	4	8	no	—
39	#13083549	—	2	4	no	—
40	C₂²×C₄×C₈₉	—	4	2	no	—
41	C₂²×C₄×C₈₉	D₈	1	2	yes	—
42	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
44	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
45	C₂×C₈×C₈₉	Q₁₆	1	2	yes	—
46	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
49	C₂×C₈×C₈₉	#41	1	2	yes	—
50	C₄⋊C₄×C₈₉	—	4	2	no	—
51	C₄⋊C₄×C₈₉	—	4	2	no	—
52	C₄⋊C₄×C₈₉	—	8	4	no	—
53	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
54	C₄²×C₈₉	SD₁₆	1	2	yes	—
55	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
56	C₂×C₄×C₈₉	—	2	2	no	—
57	C₂×C₄×C₈₉	—	2	2	no	—
58	C₂×C₄×C₈₉	—	8	2	no	—
59	C₂×C₄×C₈₉	—	8	4	no	—
60	C₂×C₄×C₈₉	#102	1	4	yes	—
61	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
63	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
65	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
66	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
67	C₂×Dic₈₉	—	4	4	no	—
68	C₂×Dic₈₉	—	8	2	no	—
69	C₂×Dic₈₉	—	8	2	no	—
70	C₈×C₈₉	#103	1	4	yes	—
71	C₈×C₈₉	C₄×Q₈	1	2	yes	—
72	C₈×C₈₉	C₂²⋊Q₈	1	2	yes	—
73	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
74	C₂²×C₈₉	#486	1	1	yes	characteristic
75	C₂²×C₈₉	#500	1	1	yes	characteristic
76	C₂²×C₈₉	#509	1	2	yes	—
77	C₂²×C₈₉	#510	1	2	yes	—
78	C₄×C₈₉	—	2	4	no	—
79	C₄×C₈₉	—	16	2	no	—
80	C₄×C₈₉	#391	1	1	yes	characteristic
81	C₄×C₈₉	C₂×#103	1	1	yes	characteristic
82	C₄×C₈₉	#487	1	1	yes	characteristic
83	C₄×C₈₉	#498	1	1	yes	characteristic
84	Dic₈₉	—	8	4	no	—
85	#13284	—	89	1	no	maximal, Sylow
86	C₂×C₈₉	#1990	1	1	yes	characteristic
87	C₂×C₈₉	#2139	1	2	yes	—
88	C₂×C₈₉	#2244	1	1	yes	characteristic
89	C₂×C₈₉	#2245	1	1	yes	characteristic
90	C₂×C₈₉	#2255	1	2	yes	—
91	C₂×#499	—	89	1	no	—
92	C₂×#346	—	89	1	no	—
93	#1845	D₈₉	1	1	yes	characteristic
94	#1380	—	89	4	no	—
95	C₈₉	#13284	1	1	yes	characteristic, complemented, Sylow
96	C₂×#125	—	178	1	no	—
97	C₂²×C₁₆	—	89	2	no	—
98	#499	—	89	4	no	—
99	#499	—	178	2	no	—
100	C₂×#107	—	2	1	no	—
101	C₂×#107	—	89	2	no	—
102	C₂×#107	C₂×D₈₉	1	2	yes	—
103	C₂×#106	—	178	1	no	—
104	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
105	#384	—	2	1	no	—
106	#346	—	178	2	no	—
107	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
108	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
109	#125	—	356	2	no	—
110	#125	—	178	4	no	—
111	C₂×C₄⋊C₄	—	2	2	no	—
112	C₂×C₄⋊C₄	—	4	2	no	—
113	C₂×C₄⋊C₄	—	356	2	no	—
114	C₂×C₄⋊C₄	—	178	2	no	—
115	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
116	C₂×C₁₆	—	89	4	no	—
117	C₂×C₁₆	—	178	2	no	—
118	C₂×C₁₆	—	178	2	no	—
119	#107	—	2	4	no	—
120	#107	—	4	2	no	—
121	#107	—	89	4	no	—
122	#107	—	178	2	no	—
123	#106	—	178	4	no	—
124	C₄×C₈	C₄×D₈₉	1	2	yes	—
125	C₄×C₈	D₃₅₆	1	2	yes	—
126	C₂²×C₄	—	4	2	no	—
127	C₂²×C₄	—	356	2	no	—
128	C₂²×C₄	#13083559	1	2	yes	—
129	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
130	C₂×C₈	#13083551	1	2	yes	—
131	C₂×C₈	#13083562	1	2	yes	—
132	C₂×C₈	#13083564	1	2	yes	—
133	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
134	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
135	C₂×C₈	Q₈×D₈₉	1	2	yes	—
136	C₂×C₈	#13083573	1	2	yes	—
137	C₄⋊C₄	—	4	2	no	—
138	C₄⋊C₄	—	4	2	no	—
139	C₄⋊C₄	—	8	4	no	—
140	C₄⋊C₄	—	356	2	no	—
141	C₄⋊C₄	—	356	8	no	—
142	C₄⋊C₄	—	178	4	no	—
143	C₄²	#13083551	1	2	yes	—
144	C₄²	#13083561	1	2	yes	—
145	C₁₆	—	178	4	no	—
146	C₂³	C₂×#13083559	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	8	2	no	—
150	C₂×C₄	—	8	4	no	—
151	C₂×C₄	—	356	4	no	—
152	C₂×C₄	—	712	2	no	—
153	C₂×C₄	—	712	2	no	—
154	C₂×C₄	#15667932	1	4	yes	—
155	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
156	C₂×C₄	#15667999	1	1	yes	characteristic
157	C₂×C₄	#15668010	1	1	yes	characteristic
158	C₂×C₄	C₂×#13083561	1	1	yes	characteristic
159	C₂×C₄	C₂×#13083562	1	1	yes	characteristic
160	C₂×C₄	C₂×#13083564	1	1	yes	characteristic
161	C₈	#15667935	1	4	yes	—
162	C₈	#15667977	1	2	yes	—
163	C₈	#15667992	1	2	yes	—
164	C₂²	C₂×#15667932	1	1	yes	characteristic
165	C₂²	#19151944	1	1	yes	characteristic
166	C₂²	#19152001	1	1	yes	characteristic
167	C₂²	#19152034	1	2	yes	—
168	C₂²	#19152039	1	2	yes	—
169	C₄	—	2	4	no	—
170	C₄	—	16	2	no	—
171	C₄	—	712	4	no	—
172	C₄	#19151503	1	1	yes	characteristic
173	C₄	C₂×#15667935	1	1	yes	characteristic
174	C₄	#19151948	1	1	yes	characteristic
175	C₄	#19151994	1	1	yes	characteristic
176	C₂	#26021212	1	1	yes	characteristic
177	C₂	#26021962	1	2	yes	—
178	C₂	#26022493	1	1	yes	characteristic
179	C₂	#26022497	1	1	yes	characteristic
180	C₂	#26022535	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19152000

C₂

yes

maximal, characteristic

C₂×#19151374

C₂

yes

maximal, characteristic

#26018343

C₂

yes

maximal

C₈₉×#1845

C₂

yes

maximal, characteristic

C₂×#15668004

—

C₂²×#13083549

C₂²

yes

—

#19152000

—

#19152000

C₄

yes

complemented

C₂×#15667946

C₂²

yes

—

C₂×#15667944

—

#19151374

—

C₂×#107×C₈₉

—

C₂×#107×C₈₉

C₂²

yes

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

#384×C₈₉

—

#15668004

—

#15668004

—

C₂×C₄⋊Dic₈₉

—

—

—

—

Q₈

yes

—

#15667946

—

#15667946

C₂×C₄

yes

—

#15667944

—

C₂²×C₈×C₈₉

D₄

yes

characteristic

C₂²×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄²×C₈₉

D₄

yes

characteristic

#107×C₈₉

—

#107×C₈₉

—

C₄×C₈×C₈₉

C₂×C₄

yes

—

C₄×C₈×C₈₉

D₄

yes

—

C₂²×Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083553

—

#13083549

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

D₈

yes

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

C₂²⋊C₄

yes

—

C₂×C₈×C₈₉

SD₁₆

yes

—

C₂×C₈×C₈₉

Q₁₆

yes

—

C₂×C₈×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

C₂×Q₈

yes

—

C₂×C₈×C₈₉

#41

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂²⋊C₄

yes

—

C₄²×C₈₉

SD₁₆

yes

—

C₂³×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#102

yes

—

C₂×C₄×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₉

#124

yes

characteristic

C₂×C₄×C₈₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₈₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₈₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₁₆

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₈×C₈₉

#103

yes

—

C₈×C₈₉

C₄×Q₈

yes

—

C₈×C₈₉

C₂²⋊Q₈

yes

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

—

C₂²×C₈₉

#510

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₂×#499

—

C₂×#346

—

#1845

D₈₉

yes

characteristic

#1380

—

C₈₉

#13284

yes

characteristic, complemented, Sylow

C₂×#125

—

178

—

C₂²×C₁₆

—

#499

—

#499

—

178

—

100

C₂×#107

—

101

C₂×#107

—

102

C₂×#107

C₂×D₈₉

yes

—

103

C₂×#106

—

178

—

104

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

105

#384

—

106

#346

—

178

—

107

C₂²×C₈

C₂²⋊D₈₉

yes

characteristic

108

C₂²×C₈

C₂²×D₈₉

yes

characteristic

109

#125

—

356

—

110

#125

—

178

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄⋊C₄

—

356

—

114

C₂×C₄⋊C₄

—

178

—

115

C₂×C₄²

C₂²⋊D₈₉

yes

characteristic

116

C₂×C₁₆

—

117

C₂×C₁₆

—

178

—

118

C₂×C₁₆

—

178

—

119

#107

—

120

#107

—

121

#107

—

122

#107

—

178

—

123

#106

—

178

—

124

C₄×C₈

C₄×D₈₉

yes

—

125

C₄×C₈

D₃₅₆

yes

—

126

C₂²×C₄

—

127

C₂²×C₄

—

356

—

128

C₂²×C₄

#13083559

yes

—

129

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

130

C₂×C₈

#13083551

yes

—

131

C₂×C₈

#13083562

yes

—

132

C₂×C₈

#13083564

yes

—

133

C₂×C₈

C₂×C₄×D₈₉

yes

characteristic

134

C₂×C₈

C₂×D₃₅₆

yes

characteristic

135

C₂×C₈

Q₈×D₈₉

yes

—

136

C₂×C₈

#13083573

yes

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

356

—

141

C₄⋊C₄

—

356

—

142

C₄⋊C₄

—

178

—

143

C₄²

#13083551

yes

—

144

C₄²

#13083561

yes

—

145

C₁₆

—

178

—

146

C₂³

C₂×#13083559

yes

characteristic

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

356

—

152

C₂×C₄

—

712

—

153

C₂×C₄

—

712

—

154

C₂×C₄

#15667932

yes

—

155

C₂×C₄

C₂×#13083551

yes

characteristic

156

C₂×C₄

#15667999

yes

characteristic

157

C₂×C₄

#15668010

yes

characteristic

158

C₂×C₄

C₂×#13083561

yes

characteristic

159

C₂×C₄

C₂×#13083562

yes

characteristic

160

C₂×C₄

C₂×#13083564

yes

characteristic

161

C₈

#15667935

yes

—

162

C₈

#15667977

yes

—

163

C₈

#15667992

yes

—

164

C₂²

C₂×#15667932

yes

characteristic

165

C₂²

#19151944

yes

characteristic

166

C₂²

#19152001

yes

characteristic

167

C₂²

#19152034

yes

—

168

C₂²

#19152039

yes

—

169

C₄

—

170

C₄

—

171

C₄

—

712

—

172

C₄

#19151503

yes

characteristic

173

C₄

C₂×#15667935

yes

characteristic

174

C₄

#19151948

yes

characteristic

175

C₄

#19151994

yes

characteristic

176

C₂

#26021212

yes

characteristic

177

C₂

#26021962

yes

—

178

C₂

#26022493

yes

characteristic

179

C₂

#26022497

yes

characteristic

180

C₂

#26022535

yes

—

	subgroup	G/H
center	C₂³	C₂×#13083559
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#1845	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₈
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#1845	D₈₉
hypocenter	C₈₉	#13284
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1845

subgroup

G/H

center

C₂³

C₂×#13083559

commutator

C₂×C₈×C₈₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₈₉

Fitting

C₈₉×#1845

C₂

socle, prime radical

C₂³×C₈₉

C₂×D₈

cosocle

C₂²×C₈×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1845

N	H ≅ G/N	type
8. #19152000	C₄	semidirect
95. C₈₉	#13284	semidirect

H ≅ G/N

type

8. #19152000

C₄

semidirect

95. C₈₉

#13284

semidirect

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

subgroup	index
C₂×#19152000	2
C₂×#19151374	2
#26018343	2
C₈₉×#1845	2
#13284	89

subgroup

index

C₈₉×#1845

Subgroups — #36473036

Conjugacy classes of subgroups

Complements

Maximal subgroups