Group #36472476 — subgroups

Subgroups — #36472476

Order 22784.

286 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 115 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	#26022036	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×Dic₈₉	C₂	1	1	yes	maximal, characteristic
3	#26020998	C₂	1	2	yes	maximal
4	#26018349	C₂	1	2	yes	maximal
5	C₂×#368×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#13083549	C₂²	1	1	yes	characteristic
7	#19151636	C₂²	1	2	yes	—
8	#19151626	—	2	2	no	—
9	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
10	#19151426	C₄	1	2	yes	complemented
11	#19151424	C₄	1	2	yes	complemented
12	C₁₆×Dic₈₉	—	2	2	no	—
13	#19151359	—	2	2	no	—
14	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×#108×C₈₉	C₂²	1	2	yes	—
16	#368×C₈₉	—	2	2	no	—
17	C₂²×#10623494	D₄	1	1	yes	characteristic
18	#15667971	—	2	2	no	—
19	C₂×C₄×Dic₈₉	D₄	1	1	yes	characteristic
20	C₂×#13083549	—	2	1	no	—
21	C₂×#13083549	—	2	1	no	—
22	C₂×#13083549	Q₈	1	2	yes	—
23	#15667946	C₂×C₄	1	4	yes	—
24	#15667944	—	2	2	no	—
25	#15667941	—	4	4	no	—
26	C₈×Dic₈₉	—	2	1	no	—
27	C₈×Dic₈₉	—	2	1	no	—
28	C₂×M₁₆×C₈₉	—	2	2	no	—
29	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
30	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₆×C₈₉	D₄	1	2	yes	—
33	C₂×C₁₆×C₈₉	Q₈	1	2	yes	—
34	#108×C₈₉	—	2	2	no	—
35	#108×C₈₉	—	2	2	no	—
36	C₂²×Dic₈₉	—	2	1	no	—
37	C₂×#10623494	—	2	1	no	—
38	C₂×#10623494	—	2	1	no	—
39	C₂×#10623494	D₈	1	2	yes	—
40	C₄⋊Dic₈₉	—	2	4	no	—
41	C₂²⋊Dic₈₉	—	4	2	no	—
42	C₄×Dic₈₉	—	2	1	no	—
43	C₄×Dic₈₉	—	4	2	no	—
44	C₄×Dic₈₉	Q₁₆	1	2	yes	—
45	#13083549	—	2	2	no	—
46	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
47	M₁₆×C₈₉	—	4	2	no	—
48	M₁₆×C₈₉	—	4	2	no	—
49	C₂×C₈×C₈₉	—	4	2	no	—
50	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
55	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
56	C₁₆×C₈₉	C₄⋊C₄	1	2	yes	—
57	C₁₆×C₈₉	C₄⋊C₄	1	2	yes	—
58	C₂³×C₈₉	#135	1	1	yes	characteristic
59	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
60	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
61	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
62	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
63	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
64	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
65	C₂×Dic₈₉	—	2	4	no	—
66	C₂×Dic₈₉	—	2	2	no	—
67	C₂×Dic₈₉	—	4	4	no	—
68	C₈×C₈₉	—	8	2	no	—
69	C₈×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₈×C₈₉	C₄×Q₈	1	1	yes	characteristic
71	C₈×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
72	C₈×C₈₉	#125	1	1	yes	characteristic
73	#10623494	—	2	2	no	—
74	C₂²×C₈₉	—	2	2	no	—
75	C₂²×C₈₉	C₄×D₈	1	1	yes	characteristic
76	C₂²×C₈₉	#467	1	1	yes	characteristic
77	C₂²×C₈₉	#499	1	1	yes	characteristic
78	C₄×C₈₉	#389	1	1	yes	characteristic
79	C₄×C₈₉	C₄×Q₁₆	1	1	yes	characteristic
80	C₄×C₈₉	#466	1	1	yes	characteristic
81	C₄×C₈₉	#500	1	1	yes	characteristic
82	Dic₈₉	—	2	4	no	—
83	Dic₈₉	—	8	4	no	—
84	#13169	—	89	1	no	maximal, Sylow
85	C₂×C₈₉	—	2	1	no	—
86	C₂×C₈₉	—	2	1	no	—
87	C₂×C₈₉	#1942	1	1	yes	characteristic
88	C₂×C₈₉	#2178	1	2	yes	—
89	C₂×#368	D₈₉	1	1	yes	characteristic
90	#2158	—	89	1	no	—
91	C₂×C₄×C₁₆	—	89	1	no	—
92	#1945	—	89	2	no	—
93	#1382	—	89	2	no	—
94	C₈₉	#13169	1	1	yes	characteristic, complemented, Sylow
95	C₂²×C₁₆	—	89	1	no	—
96	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
97	C₂×#108	C₂×D₈₉	1	2	yes	—
98	#427	—	89	2	no	—
99	#424	—	178	2	no	—
100	C₂×C₄×C₈	—	89	1	no	—
101	#368	—	2	2	no	—
102	#367	—	89	2	no	—
103	#366	—	89	2	no	—
104	C₄×C₁₆	—	178	2	no	—
105	#339	—	178	2	no	—
106	C₂×M₁₆	—	2	2	no	—
107	C₂²×C₈	—	89	1	no	—
108	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
109	#117	—	178	2	no	—
110	C₂×C₄²	—	89	1	no	—
111	C₂×C₁₆	—	89	2	no	—
112	C₂×C₁₆	—	178	1	no	—
113	C₂×C₁₆	—	178	1	no	—
114	C₂×C₁₆	Dic₁₇₈	1	2	yes	—
115	C₂×C₁₆	C₄×D₈₉	1	1	yes	characteristic
116	C₂×C₁₆	C₄×D₈₉	1	1	yes	characteristic
117	C₂×C₁₆	C₂²⋊D₈₉	1	2	yes	—
118	#108	—	2	2	no	—
119	#108	—	2	2	no	—
120	#107	—	89	4	no	—
121	#106	—	178	2	no	—
122	C₄⋊C₈	—	356	4	no	—
123	C₄×C₈	—	178	1	no	—
124	C₄×C₈	—	178	1	no	—
125	C₂²×C₄	—	178	1	no	—
126	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
127	M₁₆	—	4	2	no	—
128	M₁₆	—	4	2	no	—
129	C₂×C₈	—	4	2	no	—
130	C₂×C₈	—	89	2	no	—
131	C₂×C₈	—	178	1	no	—
132	C₂×C₈	—	178	1	no	—
133	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
134	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
135	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
136	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
137	C₂×C₈	#13083573	1	1	yes	characteristic
138	C₂×C₈	#13083575	1	1	yes	characteristic
139	C₄⋊C₄	—	178	4	no	—
140	C₂²⋊C₄	—	356	2	no	—
141	C₄²	—	89	2	no	—
142	C₄²	—	356	2	no	—
143	C₄²	—	178	1	no	—
144	C₁₆	—	178	2	no	—
145	C₁₆	#13083553	1	2	yes	—
146	C₁₆	#13083553	1	2	yes	—
147	C₂³	#15668035	1	1	yes	characteristic
148	C₂×C₄	—	356	4	no	—
149	C₂×C₄	—	178	4	no	—
150	C₂×C₄	—	178	2	no	—
151	C₂×C₄	#15667974	1	1	yes	characteristic
152	C₂×C₄	#15667983	1	1	yes	characteristic
153	C₂×C₄	#15668011	1	1	yes	characteristic
154	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
155	C₂×C₄	Q₁₆×D₈₉	1	1	yes	characteristic
156	C₂×C₄	#15668031	1	1	yes	characteristic
157	C₈	—	8	2	no	—
158	C₈	—	178	2	no	—
159	C₈	C₂×#13083553	1	1	yes	characteristic
160	C₈	#15667977	1	1	yes	characteristic
161	C₈	#15667988	1	1	yes	characteristic
162	C₈	#15668003	1	1	yes	characteristic
163	C₂²	—	2	2	no	—
164	C₂²	#19151680	1	1	yes	characteristic
165	C₂²	#19151836	1	1	yes	characteristic
166	C₂²	#19151999	1	1	yes	characteristic
167	C₄	—	178	4	no	—
168	C₄	—	712	4	no	—
169	C₄	#19151494	1	1	yes	characteristic
170	C₄	#19151692	1	1	yes	characteristic
171	C₄	#19151831	1	1	yes	characteristic
172	C₄	#19152002	1	1	yes	characteristic
173	C₂	—	2	1	no	—
174	C₂	—	2	1	no	—
175	C₂	#26020983	1	1	yes	characteristic
176	C₂	#26022132	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26022036

C₂

yes

maximal, characteristic

C₂×C₁₆×Dic₈₉

C₂

yes

maximal, characteristic

#26020998

C₂

yes

maximal

#26018349

C₂

yes

maximal

C₂×#368×C₈₉

C₂

yes

maximal, characteristic

C₂²×#13083549

C₂²

yes

characteristic

#19151636

C₂²

yes

—

#19151626

—

C₂×C₈×Dic₈₉

C₂²

yes

characteristic

#19151426

C₄

yes

complemented

#19151424

C₄

yes

complemented

C₁₆×Dic₈₉

—

#19151359

—

C₂²×C₁₆×C₈₉

C₂²

yes

characteristic

C₂×#108×C₈₉

C₂²

yes

—

#368×C₈₉

—

C₂²×#10623494

D₄

yes

characteristic

#15667971

—

C₂×C₄×Dic₈₉

D₄

yes

—

—

Q₈

yes

—

#15667946

C₂×C₄

yes

—

#15667944

—

#15667941

—

C₈×Dic₈₉

—

C₈×Dic₈₉

—

C₂×M₁₆×C₈₉

—

C₂²×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₁₆×C₈₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₈₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₈₉

D₄

yes

—

C₂×C₁₆×C₈₉

Q₈

yes

—

#108×C₈₉

—

#108×C₈₉

—

C₂²×Dic₈₉

—

C₂×#10623494

—

C₂×#10623494

—

C₂×#10623494

D₈

yes

—

C₄⋊Dic₈₉

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

C₄×Dic₈₉

Q₁₆

yes

—

#13083549

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

M₁₆×C₈₉

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₈₉

C₂×Q₈

yes

C₂×C₈×C₈₉

yes

C₂×C₈×C₈₉

yes

C₁₆×C₈₉

C₄⋊C₄

yes

—

C₁₆×C₈₉

C₄⋊C₄

yes

—

C₂³×C₈₉

#135

yes

characteristic

C₂×C₄×C₈₉

C₄×D₄

yes

characteristic

C₂×C₄×C₈₉

#121

yes

characteristic

C₂×C₄×C₈₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₈₉

#135

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₈×C₈₉

—

C₈×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₈×C₈₉

C₄×Q₈

yes

characteristic

C₈×C₈₉

C₂²⋊Q₈

yes

C₈×C₈₉

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

C₄×D₈

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

C₄×Q₁₆

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

Dic₈₉

—

#13169

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#1942

yes

characteristic

C₂×C₈₉

#2178

yes

—

C₂×#368

D₈₉

yes

characteristic

#2158

—

C₂×C₄×C₁₆

—

#1945

—

#1382

—

C₈₉

#13169

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₈₉

yes

characteristic

C₂×#108

C₂×D₈₉

yes

—

#427

—

#424

—

178

—

100

C₂×C₄×C₈

—

101

#368

—

102

#367

—

103

#366

—

104

C₄×C₁₆

—

178

—

105

#339

—

178

—

106

C₂×M₁₆

—

107

C₂²×C₈

—

108

C₂²×C₈

C₂²×D₈₉

yes

characteristic

109

#117

—

178

—

110

C₂×C₄²

—

111

C₂×C₁₆

—

112

C₂×C₁₆

—

178

—

113

C₂×C₁₆

—

178

—

114

C₂×C₁₆

Dic₁₇₈

yes

—

115

C₂×C₁₆

C₄×D₈₉

yes

characteristic

116

C₂×C₁₆

C₄×D₈₉

yes

characteristic

117

C₂×C₁₆

C₂²⋊D₈₉

yes

—

118

#108

—

119

#108

—

120

#107

—

121

#106

—

178

—

122

C₄⋊C₈

—

356

—

123

C₄×C₈

—

178

—

124

C₄×C₈

—

178

—

125

C₂²×C₄

—

178

—

126

C₂²×C₄

D₄×D₈₉

yes

characteristic

127

M₁₆

—

128

M₁₆

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

178

—

132

C₂×C₈

—

178

—

133

C₂×C₈

C₂×C₄×D₈₉

yes

characteristic

134

C₂×C₈

C₂×C₂²⋊D₈₉

yes

characteristic

135

C₂×C₈

Q₈×D₈₉

yes

characteristic

136

C₂×C₈

C₂×Dic₁₇₈

yes

characteristic

137

C₂×C₈

#13083573

yes

characteristic

138

C₂×C₈

#13083575

yes

characteristic

139

C₄⋊C₄

—

178

—

140

C₂²⋊C₄

—

356

—

141

C₄²

—

142

C₄²

—

356

—

143

C₄²

—

178

—

144

C₁₆

—

178

—

145

C₁₆

#13083553

yes

—

146

C₁₆

#13083553

yes

—

147

C₂³

#15668035

yes

characteristic

148

C₂×C₄

—

356

—

149

C₂×C₄

—

178

—

150

C₂×C₄

—

178

—

151

C₂×C₄

#15667974

yes

characteristic

152

C₂×C₄

#15667983

yes

characteristic

153

C₂×C₄

#15668011

yes

characteristic

154

C₂×C₄

D₈×D₈₉

yes

characteristic

155

C₂×C₄

Q₁₆×D₈₉

yes

characteristic

156

C₂×C₄

#15668031

yes

characteristic

157

C₈

—

158

C₈

—

178

—

159

C₈

C₂×#13083553

yes

characteristic

160

C₈

#15667977

yes

characteristic

161

C₈

#15667988

yes

characteristic

162

C₈

#15668003

yes

characteristic

163

C₂²

—

164

C₂²

#19151680

yes

characteristic

165

C₂²

#19151836

yes

characteristic

166

C₂²

#19151999

yes

characteristic

167

C₄

—

178

—

168

C₄

—

712

—

169

C₄

#19151494

yes

characteristic

170

C₄

#19151692

yes

characteristic

171

C₄

#19151831

yes

characteristic

172

C₄

#19152002

yes

characteristic

173

C₂

—

174

C₂

—

175

C₂

#26020983

yes

characteristic

176

C₂

#26022132

yes

—

	subgroup	G/H
center	C₂×C₄	D₈×D₈₉
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂×#368×C₈₉	C₂
socle	C₂²×C₈₉	C₄×D₈
prime radical	C₂³×C₈₉	#135
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×#368	D₈₉
hypocenter	C₈₉	#13169
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

subgroup

G/H

center

C₂×C₄

D₈×D₈₉

commutator

C₂×C₈×C₈₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₈₉

Fitting

C₂×#368×C₈₉

C₂

socle

C₂²×C₈₉

C₄×D₈

prime radical

C₂³×C₈₉

#135

cosocle

C₂²×C₈×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

N	H ≅ G/N	type
10. #19151426	C₄	semidirect
11. #19151424	C₄	semidirect
94. C₈₉	#13169	semidirect

H ≅ G/N

type

10. #19151426

C₄

semidirect

11. #19151424

C₄

semidirect

94. C₈₉

#13169

semidirect

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

subgroup	index
#26022036	2
C₂×C₁₆×Dic₈₉	2
#26020998	2
#26018349	2
C₂×#368×C₈₉	2
#13169	89

subgroup

index

C₂×C₁₆×Dic₈₉

C₂×#368×C₈₉

Subgroups — #36472476

Conjugacy classes of subgroups

Complements

Maximal subgroups