Group #36470809 — subgroups

Subgroups — #36470809

Order 22784.

310 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 131 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×C₁₆×Dic₈₉	C₂	1	1	yes	maximal, characteristic
2	#26020415	C₂	1	2	yes	maximal
3	#26018344	C₂	1	1	yes	maximal, characteristic
4	#26018343	C₂	1	2	yes	maximal
5	C₈₉×#1380	C₂	1	1	yes	maximal, characteristic
6	C₂²×#13083549	C₂²	1	1	yes	characteristic
7	C₂×#15667946	C₂²	1	2	yes	—
8	C₂×#15667943	—	2	2	no	—
9	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
10	#19151463	—	2	2	no	—
11	C₁₆×Dic₈₉	C₄	1	4	yes	complemented
12	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
13	C₂×#107×C₈₉	C₂²	1	2	yes	—
14	C₂²×#10623494	D₄	1	1	yes	characteristic
15	C₂×C₄⋊Dic₈₉	—	2	2	no	—
16	C₂×C₄×Dic₈₉	D₄	1	1	yes	characteristic
17	C₂×#13083549	—	2	2	no	—
18	C₂×#13083549	C₂×C₄	1	2	yes	—
19	#15667946	—	2	2	no	—
20	#15667946	—	2	2	no	—
21	#15667943	—	4	4	no	—
22	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
23	C₈×Dic₈₉	Q₈	1	2	yes	—
24	#15667919	—	4	2	no	—
25	#15667919	—	4	2	no	—
26	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
27	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
28	C₂×C₁₆×C₈₉	—	2	2	no	—
29	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
30	#107×C₈₉	—	2	2	no	—
31	#107×C₈₉	—	2	2	no	—
32	C₂²×Dic₈₉	—	4	2	no	—
33	C₂²×Dic₈₉	D₈	1	1	yes	characteristic
34	C₂²×Dic₈₉	D₈	1	1	yes	characteristic
35	C₂×#10623494	C₄⋊C₄	1	2	yes	—
36	C₂×#10623494	SD₁₆	1	4	yes	—
37	C₄⋊Dic₈₉	—	4	2	no	—
38	C₄⋊Dic₈₉	—	4	2	no	—
39	C₄×Dic₈₉	C₄⋊C₄	1	2	yes	—
40	C₄×Dic₈₉	Q₁₆	1	2	yes	—
41	#13083549	—	2	2	no	—
42	C₂²×C₄×C₈₉	—	4	2	no	—
43	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₈×C₈₉	#41	1	2	yes	—
47	C₂×C₈×C₈₉	#41	1	2	yes	—
48	C₄⋊C₄×C₈₉	—	4	2	no	—
49	C₄⋊C₄×C₈₉	—	4	2	no	—
50	C₁₆×C₈₉	—	2	2	no	—
51	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
52	C₂×C₄×C₈₉	—	8	2	no	—
53	C₂×C₄×C₈₉	—	8	2	no	—
54	C₂×C₄×C₈₉	—	8	2	no	—
55	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₈₉	#124	1	2	yes	—
57	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
58	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
59	C₂×Dic₈₉	—	2	1	no	—
60	C₂×Dic₈₉	—	2	2	no	—
61	C₂×Dic₈₉	—	2	1	no	—
62	C₂×Dic₈₉	—	8	2	no	—
63	C₂×Dic₈₉	—	8	2	no	—
64	C₂×Dic₈₉	—	8	2	no	—
65	C₂×Dic₈₉	#107	1	2	yes	—
66	C₂×Dic₈₉	#107	1	2	yes	—
67	C₈×C₈₉	#117	1	1	yes	characteristic
68	C₈×C₈₉	#117	1	1	yes	characteristic
69	C₈×C₈₉	#125	1	2	yes	—
70	#10623494	#106	1	4	yes	—
71	C₂²×C₈₉	C₂×#107	1	1	yes	characteristic
72	C₂²×C₈₉	#486	1	2	yes	—
73	C₂²×C₈₉	#508	1	1	yes	characteristic
74	C₂²×C₈₉	#508	1	2	yes	—
75	C₂²×C₈₉	#508	1	1	yes	characteristic
76	C₄×C₈₉	—	16	2	no	—
77	C₄×C₈₉	#382	1	1	yes	characteristic
78	C₄×C₈₉	C₂×#106	1	1	yes	characteristic
79	C₄×C₈₉	#487	1	2	yes	—
80	Dic₈₉	—	2	2	no	—
81	Dic₈₉	—	2	2	no	—
82	Dic₈₉	—	16	2	no	—
83	#12889	—	89	1	no	maximal, Sylow
84	C₂×C₈₉	#1834	1	1	yes	characteristic
85	C₂×C₈₉	#2158	1	1	yes	characteristic
86	C₂×C₈₉	#2158	1	1	yes	characteristic
87	C₂×C₈₉	#2242	1	2	yes	—
88	C₂×C₈₉	#2242	1	2	yes	—
89	C₂×C₄×C₁₆	—	89	1	no	—
90	#1846	—	89	2	no	—
91	#1380	—	89	1	no	—
92	#1380	—	89	2	no	—
93	#1380	D₈₉	1	1	yes	characteristic
94	C₈₉	#12889	1	1	yes	characteristic, complemented, Sylow
95	C₂²×C₁₆	—	89	1	no	—
96	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
97	C₂×#107	—	89	2	no	—
98	C₂×#107	C₂×D₈₉	1	2	yes	—
99	C₂×#106	—	178	2	no	—
100	C₂×C₄×C₈	—	89	1	no	—
101	#382	—	178	2	no	—
102	C₄×C₁₆	—	89	4	no	—
103	C₂²×C₈	—	89	1	no	—
104	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
105	C₂×C₄⋊C₄	—	2	2	no	—
106	C₂×C₄⋊C₄	—	178	2	no	—
107	C₂×C₄²	—	89	1	no	—
108	C₂×C₁₆	—	2	2	no	—
109	C₂×C₁₆	—	89	2	no	—
110	C₂×C₁₆	—	178	2	no	—
111	C₂×C₁₆	C₄×D₈₉	1	2	yes	—
112	#107	—	2	2	no	—
113	#107	—	2	2	no	—
114	#107	—	178	2	no	—
115	#107	—	178	2	no	—
116	#106	—	356	4	no	—
117	C₄×C₈	—	89	2	no	—
118	C₄×C₈	—	89	2	no	—
119	#95	—	356	2	no	—
120	#95	—	356	2	no	—
121	C₂²×C₄	—	4	2	no	—
122	C₂²×C₄	—	89	1	no	—
123	C₂²×C₄	—	89	1	no	—
124	C₂²×C₄	—	356	2	no	—
125	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
126	C₂×C₈	—	89	4	no	—
127	C₂×C₈	—	89	2	no	—
128	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
129	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
130	C₂×C₈	#13083573	1	2	yes	—
131	C₂×C₈	#13083574	1	2	yes	—
132	C₄⋊C₄	—	4	2	no	—
133	C₄⋊C₄	—	4	2	no	—
134	C₄⋊C₄	—	356	2	no	—
135	C₄⋊C₄	—	356	2	no	—
136	C₄²	—	89	2	no	—
137	C₄²	—	89	2	no	—
138	C₁₆	—	2	2	no	—
139	C₁₆	—	178	2	no	—
140	C₂³	D₈×D₈₉	1	1	yes	characteristic
141	C₂×C₄	—	8	2	no	—
142	C₂×C₄	—	8	2	no	—
143	C₂×C₄	—	8	2	no	—
144	C₂×C₄	—	89	2	no	—
145	C₂×C₄	—	89	2	no	—
146	C₂×C₄	—	178	1	no	—
147	C₂×C₄	—	178	2	no	—
148	C₂×C₄	—	178	1	no	—
149	C₂×C₄	—	712	2	no	—
150	C₂×C₄	—	712	2	no	—
151	C₂×C₄	—	712	2	no	—
152	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
153	C₂×C₄	#15667998	1	2	yes	—
154	C₂×C₄	SD₁₆×D₈₉	1	2	yes	—
155	C₂×C₄	Q₁₆×D₈₉	1	1	yes	characteristic
156	C₈	—	89	4	no	—
157	C₈	#15667968	1	1	yes	characteristic
158	C₈	#15667969	1	1	yes	characteristic
159	C₈	#15668002	1	2	yes	—
160	C₂²	#107×D₈₉	1	1	yes	characteristic
161	C₂²	#19151943	1	2	yes	—
162	C₂²	#19152027	1	1	yes	characteristic
163	C₂²	#19152029	1	2	yes	—
164	C₂²	#19152031	1	1	yes	characteristic
165	C₄	—	16	2	no	—
166	C₄	—	1424	2	no	—
167	C₄	—	178	2	no	—
168	C₄	—	178	2	no	—
169	C₄	#19151462	1	1	yes	characteristic
170	C₄	#106×D₈₉	1	1	yes	characteristic
171	C₄	#19151947	1	2	yes	—
172	C₂	#26020378	1	1	yes	characteristic
173	C₂	#26022033	1	1	yes	characteristic
174	C₂	#26022035	1	1	yes	characteristic
175	C₂	#26022482	1	2	yes	—
176	C₂	#26022485	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₆×Dic₈₉

C₂

yes

maximal, characteristic

#26020415

C₂

yes

maximal

#26018344

C₂

yes

maximal, characteristic

#26018343

C₂

yes

maximal

C₈₉×#1380

C₂

yes

maximal, characteristic

C₂²×#13083549

C₂²

yes

characteristic

C₂×#15667946

C₂²

yes

—

C₂×#15667943

—

C₂×C₈×Dic₈₉

C₂²

yes

characteristic

#19151463

—

C₁₆×Dic₈₉

C₄

yes

complemented

C₂²×C₁₆×C₈₉

C₂²

yes

characteristic

C₂×#107×C₈₉

C₂²

yes

—

C₂²×#10623494

D₄

yes

characteristic

C₂×C₄⋊Dic₈₉

—

C₂×C₄×Dic₈₉

D₄

yes

characteristic

C₂×#13083549

—

C₂×#13083549

C₂×C₄

yes

—

#15667946

—

#15667946

—

#15667943

—

C₈×Dic₈₉

C₂×C₄

yes

—

C₈×Dic₈₉

Q₈

yes

—

#15667919

—

#15667919

—

C₂²×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₈₉

—

C₂×C₁₆×C₈₉

—

C₂×C₁₆×C₈₉

C₂×C₄

yes

—

#107×C₈₉

—

#107×C₈₉

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

D₈

yes

characteristic

C₂²×Dic₈₉

D₈

yes

characteristic

C₂×#10623494

C₄⋊C₄

yes

—

C₂×#10623494

SD₁₆

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄×Dic₈₉

C₄⋊C₄

yes

—

C₄×Dic₈₉

Q₁₆

yes

—

#13083549

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₈₉

#41

yes

—

C₂×C₈×C₈₉

#41

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₁₆×C₈₉

—

C₂³×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

#124

yes

—

C₂×C₄×C₈₉

C₂×SD₁₆

yes

—

C₂×C₄×C₈₉

C₂×Q₁₆

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

#107

yes

—

C₂×Dic₈₉

#107

yes

—

C₈×C₈₉

yes

C₈×C₈₉

yes

C₈×C₈₉

yes

—

#10623494

#106

yes

—

C₂²×C₈₉

C₂×#107

yes

characteristic

C₂²×C₈₉

#486

yes

—

C₂²×C₈₉

#508

yes

characteristic

C₂²×C₈₉

#508

yes

—

C₂²×C₈₉

#508

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₂×C₈₉

#2242

yes

—

C₂×C₄×C₁₆

—

—

—

—

D₈₉

yes

characteristic

C₈₉

#12889

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₈₉

yes

characteristic

C₂×#107

—

C₂×#107

C₂×D₈₉

yes

—

C₂×#106

—

178

—

100

C₂×C₄×C₈

—

101

#382

—

178

—

102

C₄×C₁₆

—

103

C₂²×C₈

—

104

C₂²×C₈

C₂²×D₈₉

yes

characteristic

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

178

—

107

C₂×C₄²

—

108

C₂×C₁₆

—

109

C₂×C₁₆

—

110

C₂×C₁₆

—

178

—

111

C₂×C₁₆

C₄×D₈₉

yes

—

112

#107

—

113

#107

—

114

#107

—

178

—

115

#107

—

178

—

116

#106

—

356

—

117

C₄×C₈

—

118

C₄×C₈

—

119

#95

—

356

—

120

#95

—

356

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

356

—

125

C₂²×C₄

D₄×D₈₉

yes

characteristic

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

C₂×C₄×D₈₉

yes

characteristic

129

C₂×C₈

Q₈×D₈₉

yes

characteristic

130

C₂×C₈

#13083573

yes

—

131

C₂×C₈

#13083574

yes

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

356

—

135

C₄⋊C₄

—

356

—

136

C₄²

—

137

C₄²

—

138

C₁₆

—

139

C₁₆

—

178

—

140

C₂³

D₈×D₈₉

yes

characteristic

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

178

—

147

C₂×C₄

—

178

—

148

C₂×C₄

—

178

—

149

C₂×C₄

—

712

—

150

C₂×C₄

—

712

—

151

C₂×C₄

—

712

—

152

C₂×C₄

C₄⋊C₄×D₈₉

yes

characteristic

153

C₂×C₄

#15667998

yes

—

154

C₂×C₄

SD₁₆×D₈₉

yes

—

155

C₂×C₄

Q₁₆×D₈₉

yes

characteristic

156

C₈

—

157

C₈

#15667968

yes

characteristic

158

C₈

#15667969

yes

characteristic

159

C₈

#15668002

yes

—

160

C₂²

#107×D₈₉

yes

characteristic

161

C₂²

#19151943

yes

—

162

C₂²

#19152027

yes

characteristic

163

C₂²

#19152029

yes

—

164

C₂²

#19152031

yes

characteristic

165

C₄

—

166

C₄

—

1424

—

167

C₄

—

178

—

168

C₄

—

178

—

169

C₄

#19151462

yes

characteristic

170

C₄

#106×D₈₉

yes

characteristic

171

C₄

#19151947

yes

—

172

C₂

#26020378

yes

characteristic

173

C₂

#26022033

yes

characteristic

174

C₂

#26022035

yes

characteristic

175

C₂

#26022482

yes

—

176

C₂

#26022485

yes

—

	subgroup	G/H
center	C₂³	D₈×D₈₉
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#1380	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₈
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#1380	D₈₉
hypocenter	C₈₉	#12889
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1380

subgroup

G/H

center

C₂³

D₈×D₈₉

commutator

C₂×C₈×C₈₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₈₉

Fitting

C₈₉×#1380

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 ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1380

N	H ≅ G/N	type
11. C₁₆×Dic₈₉	C₄	semidirect
94. C₈₉	#12889	semidirect

H ≅ G/N

type

11. C₁₆×Dic₈₉

C₄

semidirect

94. C₈₉

#12889

semidirect

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

subgroup	index
C₂×C₁₆×Dic₈₉	2
#26020415	2
#26018344	2
#26018343	2
C₈₉×#1380	2
#12889	89

subgroup

index

C₂×C₁₆×Dic₈₉

C₈₉×#1380

Subgroups — #36470809

Conjugacy classes of subgroups

Complements

Maximal subgroups