Group #36469826 — subgroups

Subgroups — #36469826

Order 22784.

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

normal subgroups only

This group has 366 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×#19151423	C₂	1	2	yes	maximal
2	C₂×C₁₆×Dic₈₉	C₂	1	1	yes	maximal, characteristic
3	#107×Dic₈₉	C₂	1	2	yes	maximal
4	#26018344	C₂	1	1	yes	maximal, characteristic
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×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
9	C₄⋊C₄×Dic₈₉	—	2	2	no	—
10	#19151423	C₄	1	8	yes	complemented
11	C₁₆×Dic₈₉	C₄	1	4	yes	complemented
12	#19151360	—	2	2	no	—
13	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×#107×C₈₉	C₂²	1	2	yes	—
15	C₂²×#10623494	D₄	1	1	yes	characteristic
16	C₂×C₄⋊Dic₈₉	—	2	2	no	—
17	C₂×C₄×Dic₈₉	—	4	2	no	—
18	C₂×C₄×Dic₈₉	D₄	1	1	yes	characteristic
19	C₂×#13083549	C₂×C₄	1	2	yes	—
20	C₂×#13083549	C₂×C₄	1	4	yes	—
21	#15667946	—	2	2	no	—
22	#15667946	C₂×C₄	1	4	yes	—
23	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
24	C₈×Dic₈₉	Q₈	1	2	yes	—
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	C₂×C₄	1	4	yes	—
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	—	2	2	no	—
36	C₂×#10623494	C₄⋊C₄	1	2	yes	—
37	C₄⋊Dic₈₉	—	2	4	no	—
38	C₄⋊Dic₈₉	—	4	2	no	—
39	C₄×Dic₈₉	—	8	4	no	—
40	C₄×Dic₈₉	C₄⋊C₄	1	2	yes	—
41	C₄×Dic₈₉	Q₁₆	1	2	yes	—
42	#13083549	C₄²	1	4	yes	complemented
43	C₂²×C₄×C₈₉	—	4	2	no	—
44	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
47	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₈×C₈₉	#41	1	2	yes	—
49	C₄⋊C₄×C₈₉	—	2	4	no	—
50	C₄⋊C₄×C₈₉	—	4	2	no	—
51	C₁₆×C₈₉	—	2	2	no	—
52	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
53	C₂×C₄×C₈₉	—	4	4	no	—
54	C₂×C₄×C₈₉	—	8	2	no	—
55	C₂×C₄×C₈₉	—	8	2	no	—
56	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
58	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
59	C₂×C₄×C₈₉	#135	1	2	yes	—
60	C₂×Dic₈₉	—	2	2	no	—
61	C₂×Dic₈₉	—	4	4	no	—
62	C₂×Dic₈₉	—	8	2	no	—
63	C₂×Dic₈₉	—	8	2	no	—
64	C₂×Dic₈₉	#107	1	2	yes	—
65	C₂×Dic₈₉	#107	1	2	yes	—
66	C₂×Dic₈₉	D₁₆	1	2	yes	—
67	C₂×Dic₈₉	Q₃₂	1	2	yes	—
68	C₈×C₈₉	C₂×C₄²	1	1	yes	characteristic
69	C₈×C₈₉	#117	1	1	yes	characteristic
70	C₈×C₈₉	C₄×Q₈	1	2	yes	—
71	#10623494	—	2	2	no	—
72	C₂²×C₈₉	C₂×#107	1	1	yes	characteristic
73	C₂²×C₈₉	C₄×D₈	1	2	yes	—
74	C₂²×C₈₉	C₂×D₁₆	1	1	yes	characteristic
75	C₂²×C₈₉	C₂×Q₃₂	1	1	yes	characteristic
76	C₂²×C₈₉	#508	1	2	yes	—
77	C₄×C₈₉	—	8	4	no	—
78	C₄×C₈₉	C₄×C₄⋊C₄	1	1	yes	characteristic
79	C₄×C₈₉	#427	1	1	yes	characteristic
80	C₄×C₈₉	C₄×Q₁₆	1	2	yes	—
81	Dic₈₉	—	2	2	no	—
82	Dic₈₉	—	8	4	no	—
83	Dic₈₉	#366	1	4	yes	complemented
84	C₄×#366	—	89	1	no	maximal, Sylow
85	C₂×C₈₉	C₄×#107	1	1	yes	characteristic
86	C₂×C₈₉	C₂×#366	1	1	yes	characteristic
87	C₂×C₈₉	#2158	1	1	yes	characteristic
88	C₂×C₈₉	C₄×D₁₆	1	2	yes	—
89	C₂×C₈₉	C₄×Q₃₂	1	2	yes	—
90	C₂×#366	—	89	2	no	—
91	C₂×C₄×C₁₆	—	89	1	no	—
92	C₄×#107	—	89	2	no	—
93	#1380	—	89	1	no	—
94	#1380	D₈₉	1	1	yes	characteristic
95	C₈₉	C₄×#366	1	1	yes	characteristic, complemented, Sylow
96	C₂²×C₁₆	—	89	1	no	—
97	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
98	C₂×#107	—	89	2	no	—
99	C₂×#107	C₂×D₈₉	1	2	yes	—
100	C₂×C₄×C₈	—	89	1	no	—
101	C₄×C₄⋊C₄	—	178	2	no	—
102	#366	—	89	8	no	—
103	C₄×C₁₆	—	89	4	no	—
104	#340	—	178	2	no	—
105	C₂²×C₈	—	89	1	no	—
106	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
107	C₂×C₄⋊C₄	—	2	2	no	—
108	C₂×C₄⋊C₄	—	178	2	no	—
109	C₂×C₄²	—	89	1	no	—
110	C₂×C₄²	—	356	2	no	—
111	C₂×C₁₆	—	2	2	no	—
112	C₂×C₁₆	—	89	2	no	—
113	C₂×C₁₆	—	89	4	no	—
114	C₂×C₁₆	C₄×D₈₉	1	2	yes	—
115	#107	—	2	2	no	—
116	#107	—	89	4	no	—
117	#107	—	178	2	no	—
118	#107	C₄×D₈₉	1	4	yes	—
119	C₄×C₈	—	89	2	no	—
120	C₄×C₈	—	89	2	no	—
121	#95	—	356	2	no	—
122	C₂²×C₄	—	4	2	no	—
123	C₂²×C₄	—	89	1	no	—
124	C₂²×C₄	—	89	1	no	—
125	C₂²×C₄	—	356	2	no	—
126	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
127	C₂×C₈	—	89	2	no	—
128	C₂×C₈	—	178	2	no	—
129	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
130	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
131	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
132	C₂×C₈	#13083573	1	2	yes	—
133	C₄⋊C₄	—	2	4	no	—
134	C₄⋊C₄	—	4	2	no	—
135	C₄⋊C₄	—	356	2	no	—
136	C₄⋊C₄	—	178	4	no	—
137	C₄²	—	89	2	no	—
138	C₄²	—	89	2	no	—
139	C₄²	—	712	4	no	—
140	C₁₆	—	2	2	no	—
141	C₁₆	—	89	4	no	—
142	C₂³	D₈×D₈₉	1	1	yes	characteristic
143	C₂×C₄	—	4	4	no	—
144	C₂×C₄	—	8	2	no	—
145	C₂×C₄	—	8	2	no	—
146	C₂×C₄	—	89	2	no	—
147	C₂×C₄	—	89	2	no	—
148	C₂×C₄	—	89	2	no	—
149	C₂×C₄	—	89	2	no	—
150	C₂×C₄	—	356	4	no	—
151	C₂×C₄	—	178	2	no	—
152	C₂×C₄	—	712	2	no	—
153	C₂×C₄	—	712	2	no	—
154	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
155	C₂×C₄	#15667972	1	2	yes	—
156	C₂×C₄	Q₁₆×D₈₉	1	1	yes	characteristic
157	C₂×C₄	#15668033	1	2	yes	—
158	C₈	—	178	2	no	—
159	C₈	C₄²×D₈₉	1	1	yes	characteristic
160	C₈	#15667968	1	1	yes	characteristic
161	C₈	#15667977	1	2	yes	—
162	C₂²	#107×D₈₉	1	1	yes	characteristic
163	C₂²	#19151678	1	2	yes	—
164	C₂²	D₁₆×D₈₉	1	1	yes	characteristic
165	C₂²	Q₃₂×D₈₉	1	1	yes	characteristic
166	C₂²	#19152029	1	2	yes	—
167	C₄	—	8	4	no	—
168	C₄	—	89	4	no	—
169	C₄	—	178	2	no	—
170	C₄	—	712	4	no	—
171	C₄	#19151449	1	1	yes	characteristic
172	C₄	#19151633	1	1	yes	characteristic
173	C₄	#19151690	1	2	yes	—
174	C₂	#26020144	1	1	yes	characteristic
175	C₂	#366×D₈₉	1	1	yes	characteristic
176	C₂	#26022035	1	1	yes	characteristic
177	C₂	#26022096	1	2	yes	—
178	C₂	#26022108	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19151423

C₂

yes

maximal

C₂×C₁₆×Dic₈₉

C₂

yes

maximal, characteristic

#107×Dic₈₉

C₂

yes

maximal

#26018344

C₂

yes

maximal, characteristic

C₈₉×#1380

C₂

yes

maximal, characteristic

C₂²×#13083549

C₂²

yes

characteristic

C₂×#15667946

C₂²

yes

—

C₂×C₈×Dic₈₉

C₂²

yes

characteristic

C₄⋊C₄×Dic₈₉

—

#19151423

C₄

yes

complemented

C₁₆×Dic₈₉

C₄

yes

complemented

#19151360

—

C₂²×C₁₆×C₈₉

C₂²

yes

characteristic

C₂×#107×C₈₉

C₂²

yes

—

C₂²×#10623494

D₄

yes

characteristic

C₂×C₄⋊Dic₈₉

—

C₂×C₄×Dic₈₉

—

C₂×C₄×Dic₈₉

D₄

yes

characteristic

C₂×#13083549

C₂×C₄

yes

—

C₂×#13083549

C₂×C₄

yes

—

#15667946

—

#15667946

C₂×C₄

yes

—

C₈×Dic₈₉

C₂×C₄

yes

—

C₈×Dic₈₉

Q₈

yes

—

#15667919

—

C₂²×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₈₉

—

C₂×C₁₆×C₈₉

—

C₂×C₁₆×C₈₉

C₂×C₄

yes

—

#107×C₈₉

—

#107×C₈₉

C₂×C₄

yes

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

D₈

yes

characteristic

C₂²×Dic₈₉

D₈

yes

characteristic

C₂×#10623494

—

C₂×#10623494

C₄⋊C₄

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄×Dic₈₉

—

C₄×Dic₈₉

C₄⋊C₄

yes

—

C₄×Dic₈₉

Q₁₆

yes

—

#13083549

C₄²

yes

complemented

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₉

C₂²×C₄

yes

—

C₂×C₈×C₈₉

C₂×Q₈

yes

characteristic

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

C₄×D₄

yes

—

C₂×C₄×C₈₉

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₈₉

#135

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

#107

yes

—

C₂×Dic₈₉

#107

yes

—

C₂×Dic₈₉

D₁₆

yes

—

C₂×Dic₈₉

Q₃₂

yes

—

C₈×C₈₉

C₂×C₄²

yes

characteristic

C₈×C₈₉

#117

yes

characteristic

C₈×C₈₉

C₄×Q₈

yes

—

#10623494

—

C₂²×C₈₉

C₂×#107

yes

characteristic

C₂²×C₈₉

C₄×D₈

yes

—

C₂²×C₈₉

C₂×D₁₆

yes

characteristic

C₂²×C₈₉

C₂×Q₃₂

yes

characteristic

C₂²×C₈₉

#508

yes

—

C₄×C₈₉

—

C₄×C₈₉

C₄×C₄⋊C₄

yes

characteristic

C₄×C₈₉

#427

yes

characteristic

C₄×C₈₉

C₄×Q₁₆

yes

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

yes

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

C₄×D₁₆

yes

—

C₂×C₈₉

C₄×Q₃₂

yes

—

C₂×#366

—

C₂×C₄×C₁₆

—

C₄×#107

—

#1380

—

#1380

D₈₉

yes

characteristic

C₈₉

C₄×#366

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₈₉

yes

characteristic

C₂×#107

—

C₂×#107

C₂×D₈₉

yes

—

100

C₂×C₄×C₈

—

101

C₄×C₄⋊C₄

—

178

—

102

#366

—

103

C₄×C₁₆

—

104

#340

—

178

—

105

C₂²×C₈

—

106

C₂²×C₈

C₂²×D₈₉

yes

characteristic

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

178

—

109

C₂×C₄²

—

110

C₂×C₄²

—

356

—

111

C₂×C₁₆

—

112

C₂×C₁₆

—

113

C₂×C₁₆

—

114

C₂×C₁₆

C₄×D₈₉

yes

—

115

#107

—

116

#107

—

117

#107

—

178

—

118

#107

C₄×D₈₉

yes

—

119

C₄×C₈

—

120

C₄×C₈

—

121

#95

—

356

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

356

—

126

C₂²×C₄

D₄×D₈₉

yes

characteristic

127

C₂×C₈

—

128

C₂×C₈

—

178

—

129

C₂×C₈

C₂×C₄×D₈₉

yes

characteristic

130

C₂×C₈

C₂×C₄×D₈₉

yes

—

131

C₂×C₈

Q₈×D₈₉

yes

characteristic

132

C₂×C₈

#13083573

yes

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

356

—

136

C₄⋊C₄

—

178

—

137

C₄²

—

138

C₄²

—

139

C₄²

—

712

—

140

C₁₆

—

141

C₁₆

—

142

C₂³

D₈×D₈₉

yes

characteristic

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

356

—

151

C₂×C₄

—

178

—

152

C₂×C₄

—

712

—

153

C₂×C₄

—

712

—

154

C₂×C₄

C₄⋊C₄×D₈₉

yes

characteristic

155

C₂×C₄

#15667972

yes

—

156

C₂×C₄

Q₁₆×D₈₉

yes

characteristic

157

C₂×C₄

#15668033

yes

—

158

C₈

—

178

—

159

C₈

C₄²×D₈₉

yes

characteristic

160

C₈

#15667968

yes

characteristic

161

C₈

#15667977

yes

—

162

C₂²

#107×D₈₉

yes

characteristic

163

C₂²

#19151678

yes

—

164

C₂²

D₁₆×D₈₉

yes

characteristic

165

C₂²

Q₃₂×D₈₉

yes

characteristic

166

C₂²

#19152029

yes

—

167

C₄

—

168

C₄

—

169

C₄

—

178

—

170

C₄

—

712

—

171

C₄

#19151449

yes

characteristic

172

C₄

#19151633

yes

characteristic

173

C₄

#19151690

yes

—

174

C₂

#26020144

yes

characteristic

175

C₂

#366×D₈₉

yes

characteristic

176

C₂

#26022035

yes

characteristic

177

C₂

#26022096

yes

—

178

C₂

#26022108

yes

—

	subgroup	G/H
center	C₂³	D₈×D₈₉
commutator	C₈×C₈₉	C₂×C₄²
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	C₄×#366
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1380

subgroup

G/H

center

C₂³

D₈×D₈₉

commutator

C₈×C₈₉

C₂×C₄²

Frattini

C₂²×C₈

C₂²×D₈₉

Fitting

C₈₉×#1380

C₂

socle, prime radical

C₂³×C₈₉

C₂×D₈

cosocle

C₂²×C₈×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1380

N	H ≅ G/N	type
10. #19151423	C₄	semidirect
11. C₁₆×Dic₈₉	C₄	semidirect
42. #13083549	C₄²	semidirect
83. Dic₈₉	#366	semidirect
95. C₈₉	C₄×#366	semidirect

H ≅ G/N

type

10. #19151423

C₄

semidirect

11. C₁₆×Dic₈₉

C₄

semidirect

42. #13083549

C₄²

semidirect

83. Dic₈₉

#366

semidirect

95. C₈₉

C₄×#366

semidirect

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

subgroup	index
C₂×#19151423	2
C₂×C₁₆×Dic₈₉	2
#107×Dic₈₉	2
#26018344	2
C₈₉×#1380	2
C₄×#366	89

subgroup

index

C₂×#19151423

C₂×C₁₆×Dic₈₉

#107×Dic₈₉

#26018344

C₈₉×#1380

C₄×#366

Subgroups — #36469826

Conjugacy classes of subgroups

Complements

Maximal subgroups