Group #36472491 — subgroups

Subgroups — #36472491

Order 22784.

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

normal subgroups only

This group has 274 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	C₂×#19151428	C₂	1	1	yes	maximal, characteristic
2	#26021848	C₂	1	1	yes	maximal, characteristic
3	#26021035	C₂	1	2	yes	maximal
4	#26018360	C₂	1	2	yes	maximal
5	C₂×#367×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#15667952	C₂²	1	1	yes	characteristic
7	C₂×#15667948	C₂²	1	2	yes	—
8	#19151638	—	2	2	no	—
9	#19151553	C₂²	1	1	yes	characteristic
10	#19151428	C₄	1	4	yes	complemented
11	#19151373	—	2	1	no	—
12	C₁₆×Dic₈₉	—	2	1	no	—
13	#19151367	—	2	2	no	—
14	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×#107×C₈₉	C₂²	1	2	yes	—
16	#367×C₈₉	—	2	1	no	—
17	#367×C₈₉	—	2	1	no	—
18	C₂×#13083558	—	2	2	no	—
19	C₂×#13083558	D₄	1	1	yes	characteristic
20	#15667971	D₄	1	1	yes	characteristic
21	#15667952	—	2	1	no	—
22	#15667952	Q₈	1	2	yes	—
23	C₂×#13083549	—	2	1	no	—
24	#15667948	—	2	2	no	—
25	#15667948	C₂×C₄	1	4	yes	—
26	#15667945	—	4	2	no	—
27	#15667943	—	4	2	no	—
28	#15667922	—	2	1	no	—
29	C₈×Dic₈₉	—	2	1	no	—
30	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
32	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
33	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
34	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
35	C₂×C₁₆×C₈₉	Q₈	1	2	yes	—
36	#107×C₈₉	—	2	2	no	—
37	#107×C₈₉	—	2	2	no	—
38	#13083558	—	2	1	no	—
39	#13083558	—	2	4	no	—
40	#13083558	—	4	2	no	—
41	#13083558	Q₁₆	1	2	yes	—
42	C₂×#10623494	—	2	1	no	—
43	C₂×#10623494	—	4	2	no	—
44	C₄⋊Dic₈₉	D₈	1	2	yes	—
45	C₂²⋊Dic₈₉	—	2	1	no	—
46	C₄×Dic₈₉	—	2	1	no	—
47	#13083549	—	2	2	no	—
48	C₂²×C₄×C₈₉	—	4	2	no	—
49	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
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	C₄⋊C₄×C₈₉	—	4	2	no	—
57	C₄⋊C₄×C₈₉	—	4	2	no	—
58	C₁₆×C₈₉	C₄⋊C₄	1	2	yes	—
59	C₁₆×C₈₉	C₄⋊C₄	1	2	yes	—
60	C₂³×C₈₉	#135	1	1	yes	characteristic
61	C₂×C₄×C₈₉	—	8	2	no	—
62	C₂×C₄×C₈₉	—	16	2	no	—
63	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
64	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
65	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
68	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
69	C₂×Dic₈₉	—	2	2	no	—
70	C₈×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₈×C₈₉	C₄×Q₈	1	1	yes	characteristic
72	C₈×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
73	C₈×C₈₉	#125	1	1	yes	characteristic
74	#10623494	—	2	2	no	—
75	#10623494	—	4	4	no	—
76	C₂²×C₈₉	—	2	2	no	—
77	C₂²×C₈₉	C₄×Q₁₆	1	1	yes	characteristic
78	C₂²×C₈₉	#466	1	1	yes	characteristic
79	C₂²×C₈₉	#499	1	1	yes	characteristic
80	C₄×C₈₉	—	16	2	no	—
81	C₄×C₈₉	#389	1	1	yes	characteristic
82	C₄×C₈₉	C₄×D₈	1	1	yes	characteristic
83	C₄×C₈₉	#467	1	1	yes	characteristic
84	C₄×C₈₉	#500	1	1	yes	characteristic
85	Dic₈₉	—	4	2	no	—
86	#13172	—	89	1	no	maximal, Sylow
87	C₂×C₈₉	—	2	1	no	—
88	C₂×C₈₉	—	2	1	no	—
89	C₂×C₈₉	#1942	1	1	yes	characteristic
90	C₂×C₈₉	#2223	1	2	yes	—
91	C₂×#368	—	89	1	no	—
92	C₂×#367	D₈₉	1	1	yes	characteristic
93	#2108	—	89	1	no	—
94	#1952	—	89	2	no	—
95	#1387	—	89	2	no	—
96	C₈₉	#13172	1	1	yes	characteristic, complemented, Sylow
97	C₂×M₃₂	—	89	1	no	—
98	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
99	C₂×#108	—	89	2	no	—
100	#428	—	178	2	no	—
101	C₂×#107	C₂×D₈₉	1	2	yes	—
102	#405	—	89	1	no	—
103	#368	—	89	4	no	—
104	#367	—	2	1	no	—
105	#367	—	2	1	no	—
106	#346	—	178	1	no	—
107	C₄×C₁₆	—	178	1	no	—
108	#343	—	178	2	no	—
109	C₂×M₁₆	—	89	1	no	—
110	C₂×M₁₆	—	178	2	no	—
111	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
112	#117	—	89	1	no	—
113	C₂×C₄⋊C₄	—	2	2	no	—
114	M₃₂	—	89	2	no	—
115	M₃₂	—	178	1	no	—
116	C₂×C₁₆	—	178	1	no	—
117	C₂×C₁₆	Dic₁₇₈	1	2	yes	—
118	C₂×C₁₆	C₄×D₈₉	1	2	yes	—
119	C₂×C₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
120	C₂×C₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
121	#108	—	89	4	no	—
122	#108	—	178	2	no	—
123	#107	—	2	2	no	—
124	#107	—	2	2	no	—
125	#107	—	356	2	no	—
126	#106	—	356	2	no	—
127	#97	—	178	1	no	—
128	C₄×C₈	—	178	1	no	—
129	C₂²×C₄	—	4	2	no	—
130	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
131	M₁₆	—	89	2	no	—
132	M₁₆	—	356	2	no	—
133	M₁₆	—	178	4	no	—
134	M₁₆	—	178	1	no	—
135	C₂×C₈	—	356	2	no	—
136	C₂×C₈	—	178	1	no	—
137	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
138	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
139	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
140	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
141	C₂×C₈	#13083573	1	1	yes	characteristic
142	C₂×C₈	#13083575	1	1	yes	characteristic
143	C₄⋊C₄	—	4	2	no	—
144	C₄⋊C₄	—	4	2	no	—
145	C₄⋊C₄	—	89	2	no	—
146	C₂²⋊C₄	—	178	1	no	—
147	C₄²	—	178	1	no	—
148	C₁₆	—	178	2	no	—
149	C₁₆	#13083553	1	2	yes	—
150	C₁₆	#13083553	1	2	yes	—
151	C₂³	#15668031	1	1	yes	characteristic
152	C₂×C₄	—	8	2	no	—
153	C₂×C₄	—	16	2	no	—
154	C₂×C₄	—	178	2	no	—
155	C₂×C₄	#15667974	1	1	yes	characteristic
156	C₂×C₄	#15667983	1	1	yes	characteristic
157	C₂×C₄	#15668011	1	1	yes	characteristic
158	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
159	C₂×C₄	Q₁₆×D₈₉	1	1	yes	characteristic
160	C₂×C₄	#15668035	1	1	yes	characteristic
161	C₈	—	356	4	no	—
162	C₈	—	178	2	no	—
163	C₈	C₂×#13083553	1	1	yes	characteristic
164	C₈	#15667977	1	1	yes	characteristic
165	C₈	#15667988	1	1	yes	characteristic
166	C₈	#15668003	1	1	yes	characteristic
167	C₂²	—	2	2	no	—
168	C₂²	#19151692	1	1	yes	characteristic
169	C₂²	#19151831	1	1	yes	characteristic
170	C₂²	#19151999	1	1	yes	characteristic
171	C₄	—	16	2	no	—
172	C₄	—	356	2	no	—
173	C₄	#19151494	1	1	yes	characteristic
174	C₄	#19151680	1	1	yes	characteristic
175	C₄	#19151836	1	1	yes	characteristic
176	C₄	#19152002	1	1	yes	characteristic
177	C₂	—	2	1	no	—
178	C₂	—	2	1	no	—
179	C₂	#26020983	1	1	yes	characteristic
180	C₂	#26022379	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19151428

C₂

yes

maximal, characteristic

#26021848

C₂

yes

maximal, characteristic

#26021035

C₂

yes

maximal

#26018360

C₂

yes

maximal

C₂×#367×C₈₉

C₂

yes

maximal, characteristic

C₂×#15667952

C₂²

yes

characteristic

C₂×#15667948

C₂²

yes

—

#19151638

—

#19151553

C₂²

yes

characteristic

#19151428

C₄

yes

complemented

#19151373

—

C₁₆×Dic₈₉

—

#19151367

—

C₂²×C₁₆×C₈₉

C₂²

yes

characteristic

C₂×#107×C₈₉

C₂²

yes

—

#367×C₈₉

—

#367×C₈₉

—

C₂×#13083558

—

C₂×#13083558

D₄

yes

characteristic

#15667971

D₄

yes

characteristic

#15667952

—

#15667952

Q₈

yes

—

C₂×#13083549

—

#15667948

—

#15667948

C₂×C₄

yes

—

#15667945

—

#15667943

—

#15667922

—

C₈×Dic₈₉

—

C₂²×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₈₉

—

C₂×C₁₆×C₈₉

C₂×C₄

yes

—

C₂×C₁₆×C₈₉

D₄

yes

characteristic

C₂×C₁₆×C₈₉

D₄

yes

characteristic

C₂×C₁₆×C₈₉

Q₈

yes

—

#107×C₈₉

—

#107×C₈₉

—

—

—

—

Q₁₆

yes

—

C₂×#10623494

—

C₂×#10623494

—

C₄⋊Dic₈₉

D₈

yes

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

#13083549

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

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

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₁₆×C₈₉

C₄⋊C₄

yes

—

C₁₆×C₈₉

C₄⋊C₄

yes

—

C₂³×C₈₉

#135

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

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

—

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×Q₁₆

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₄×C₈₉

—

C₄×C₈₉

#389

yes

characteristic

C₄×C₈₉

C₄×D₈

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

#13172

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#1942

yes

characteristic

C₂×C₈₉

#2223

yes

—

C₂×#368

—

C₂×#367

D₈₉

yes

—

—

—

C₈₉

#13172

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂²×C₁₆

C₂×D₈₉

yes

characteristic

C₂×#108

—

100

#428

—

178

—

101

C₂×#107

C₂×D₈₉

yes

—

102

#405

—

103

#368

—

104

#367

—

105

#367

—

106

#346

—

178

—

107

C₄×C₁₆

—

178

—

108

#343

—

178

—

109

C₂×M₁₆

—

110

C₂×M₁₆

—

178

—

111

C₂²×C₈

C₂²×D₈₉

yes

characteristic

112

#117

—

113

C₂×C₄⋊C₄

—

114

M₃₂

—

115

M₃₂

—

178

—

116

C₂×C₁₆

—

178

—

117

C₂×C₁₆

Dic₁₇₈

yes

—

118

C₂×C₁₆

C₄×D₈₉

yes

—

119

C₂×C₁₆

C₂²⋊D₈₉

yes

characteristic

120

C₂×C₁₆

C₂²⋊D₈₉

yes

characteristic

121

#108

—

122

#108

—

178

—

123

#107

—

124

#107

—

125

#107

—

356

—

126

#106

—

356

—

127

#97

—

178

—

128

C₄×C₈

—

178

—

129

C₂²×C₄

—

130

C₂²×C₄

D₄×D₈₉

yes

characteristic

131

M₁₆

—

132

M₁₆

—

356

—

133

M₁₆

—

178

—

134

M₁₆

—

178

—

135

C₂×C₈

—

356

—

136

C₂×C₈

—

178

—

137

C₂×C₈

C₂×C₄×D₈₉

yes

characteristic

138

C₂×C₈

C₂×C₂²⋊D₈₉

yes

characteristic

139

C₂×C₈

Q₈×D₈₉

yes

characteristic

140

C₂×C₈

C₂×Dic₁₇₈

yes

characteristic

141

C₂×C₈

#13083573

yes

characteristic

142

C₂×C₈

#13083575

yes

characteristic

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₂²⋊C₄

—

178

—

147

C₄²

—

178

—

148

C₁₆

—

178

—

149

C₁₆

#13083553

yes

—

150

C₁₆

#13083553

yes

—

151

C₂³

#15668031

yes

characteristic

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

178

—

155

C₂×C₄

#15667974

yes

characteristic

156

C₂×C₄

#15667983

yes

characteristic

157

C₂×C₄

#15668011

yes

characteristic

158

C₂×C₄

D₈×D₈₉

yes

characteristic

159

C₂×C₄

Q₁₆×D₈₉

yes

characteristic

160

C₂×C₄

#15668035

yes

characteristic

161

C₈

—

356

—

162

C₈

—

178

—

163

C₈

C₂×#13083553

yes

characteristic

164

C₈

#15667977

yes

characteristic

165

C₈

#15667988

yes

characteristic

166

C₈

#15668003

yes

characteristic

167

C₂²

—

168

C₂²

#19151692

yes

characteristic

169

C₂²

#19151831

yes

characteristic

170

C₂²

#19151999

yes

characteristic

171

C₄

—

172

C₄

—

356

—

173

C₄

#19151494

yes

characteristic

174

C₄

#19151680

yes

characteristic

175

C₄

#19151836

yes

characteristic

176

C₄

#19152002

yes

characteristic

177

C₂

—

178

C₂

—

179

C₂

#26020983

yes

characteristic

180

C₂

#26022379

yes

—

	subgroup	G/H
center	C₂²	#19151831
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂×#367×C₈₉	C₂
socle	C₂²×C₈₉	#466
prime radical	C₂³×C₈₉	#135
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×#367	D₈₉
hypocenter	C₈₉	#13172
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#367

subgroup

G/H

center

C₂²

#19151831

commutator

C₂×C₈×C₈₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₈₉

Fitting

C₂×#367×C₈₉

C₂

C₂²×C₈₉

C₂³×C₈₉

C₂²×C₈×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#367

N	H ≅ G/N	type
10. #19151428	C₄	semidirect
96. C₈₉	#13172	semidirect

H ≅ G/N

type

10. #19151428

C₄

semidirect

96. C₈₉

#13172

semidirect

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

subgroup	index
C₂×#19151428	2
#26021848	2
#26021035	2
#26018360	2
C₂×#367×C₈₉	2
#13172	89

subgroup

index

C₂×#367×C₈₉

Subgroups — #36472491

Conjugacy classes of subgroups

Complements

Maximal subgroups