Group #36474078 — subgroups

Subgroups — #36474078

Order 22784.

324 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 105 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×#19151420	C₂	1	2	yes	maximal
2	#26021000	C₂	1	1	yes	maximal, characteristic
3	#26018359	C₂	1	2	yes	maximal
4	#26018334	C₂	1	1	yes	maximal, characteristic
5	C₂×#499×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#15667952	C₂²	1	2	yes	—
7	C₂×#15667948	C₂²	1	1	yes	characteristic
8	#19151636	C₂²	1	1	yes	characteristic
9	#19151624	—	2	1	no	—
10	#19151420	—	2	4	no	—
11	#19151358	—	2	1	no	—
12	C₂×#125×C₈₉	—	2	1	no	—
13	#499×C₈₉	—	2	4	no	—
14	C₂×#107×C₈₉	—	2	1	no	—
15	C₂×#106×C₈₉	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×#13083558	—	2	1	no	—
18	#15667971	—	2	1	no	—
19	#15667952	—	2	2	no	—
20	#15667952	Q₈	1	4	yes	—
21	C₂×#13083549	—	2	2	no	—
22	#15667948	—	2	2	no	—
23	#15667946	Q₈	1	2	yes	—
24	#15667944	—	2	1	no	—
25	#15667941	—	4	2	no	—
26	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
28	#125×C₈₉	—	2	4	no	—
29	#125×C₈₉	—	4	2	no	—
30	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
31	C₂×C₄⋊C₄×C₈₉	—	4	2	no	—
32	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
33	#107×C₈₉	—	2	4	no	—
34	#106×C₈₉	—	2	2	no	—
35	#106×C₈₉	D₄	1	4	yes	—
36	C₄×C₈×C₈₉	—	2	2	no	—
37	#13083558	—	4	1	no	—
38	#13083558	—	4	1	no	—
39	C₂×#10623494	—	4	1	no	—
40	C₄⋊Dic₈₉	—	2	2	no	—
41	C₂²⋊Dic₈₉	—	4	1	no	—
42	C₄×Dic₈₉	—	4	1	no	—
43	#13083549	—	2	4	no	—
44	C₂²×C₄×C₈₉	—	2	1	no	—
45	C₂²×C₄×C₈₉	—	4	2	no	—
46	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₈₉	—	2	1	no	—
48	C₂×C₈×C₈₉	—	2	1	no	—
49	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
50	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
51	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
52	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
54	C₄⋊C₄×C₈₉	—	2	4	no	—
55	C₄⋊C₄×C₈₉	—	4	8	no	—
56	C₄⋊C₄×C₈₉	—	4	2	no	—
57	C₄²×C₈₉	—	2	1	no	—
58	C₄²×C₈₉	SD₁₆	1	2	yes	—
59	C₂³×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
60	C₂×C₄×C₈₉	—	2	2	no	—
61	C₂×C₄×C₈₉	—	4	2	no	—
62	C₂×C₄×C₈₉	—	4	4	no	—
63	C₂×C₄×C₈₉	—	8	4	no	—
64	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
65	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
68	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
69	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
70	C₂×Dic₈₉	—	4	2	no	—
71	C₈×C₈₉	—	2	2	no	—
72	C₈×C₈₉	C₂²⋊Q₈	1	2	yes	—
73	C₈×C₈₉	C₄⋊Q₈	1	2	yes	—
74	#10623494	—	8	1	no	—
75	C₂²×C₈₉	—	2	2	no	—
76	C₂²×C₈₉	#450	1	1	yes	characteristic
77	C₂²×C₈₉	#476	1	1	yes	characteristic
78	C₂²×C₈₉	#501	1	1	yes	characteristic
79	C₄×C₈₉	—	4	2	no	—
80	C₄×C₈₉	—	8	4	no	—
81	C₄×C₈₉	#395	1	1	yes	characteristic
82	C₄×C₈₉	#448	1	1	yes	characteristic
83	C₄×C₈₉	#475	1	1	yes	characteristic
84	C₄×C₈₉	#498	1	1	yes	characteristic
85	Dic₈₉	—	8	2	no	—
86	#13458	—	89	1	no	maximal, Sylow
87	C₂×C₈₉	—	2	1	no	—
88	C₂×C₈₉	—	2	1	no	—
89	C₂×C₈₉	#2058	1	1	yes	characteristic
90	C₂×C₈₉	#2231	1	2	yes	—
91	C₂×#499	D₈₉	1	1	yes	characteristic
92	C₂×#365	—	89	2	no	—
93	#1945	—	89	1	no	—
94	#1386	—	89	2	no	—
95	#1376	—	89	1	no	—
96	C₈₉	#13458	1	1	yes	characteristic, complemented, Sylow
97	C₂×#125	—	2	1	no	—
98	C₂×M₃₂	—	89	2	no	—
99	#499	—	2	4	no	—
100	C₂×#108	—	89	1	no	—
101	#427	—	89	1	no	—
102	C₂×#107	—	2	1	no	—
103	C₂×#106	C₂×D₈₉	1	2	yes	—
104	#424	—	178	1	no	—
105	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
106	#365	—	178	4	no	—
107	#339	—	178	1	no	—
108	C₂×M₁₆	—	178	1	no	—
109	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
110	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
111	#125	—	2	4	no	—
112	#125	—	4	2	no	—
113	#117	—	178	1	no	—
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	—	4	2	no	—
116	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
117	M₃₂	—	89	4	no	—
118	M₃₂	—	178	2	no	—
119	C₂×C₁₆	—	178	2	no	—
120	#108	—	178	2	no	—
121	#107	—	2	4	no	—
122	#107	—	89	2	no	—
123	#106	—	2	2	no	—
124	#106	—	178	1	no	—
125	#106	C₂²⋊D₈₉	1	4	yes	—
126	C₄⋊C₈	—	356	2	no	—
127	C₄×C₈	—	2	2	no	—
128	C₂²×C₄	—	2	1	no	—
129	C₂²×C₄	—	4	2	no	—
130	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
131	M₁₆	—	356	1	no	—
132	M₁₆	—	356	1	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	356	1	no	—
136	C₂×C₈	#13083561	1	2	yes	—
137	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
138	C₂×C₈	Q₈×D₈₉	1	2	yes	—
139	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
140	C₂×C₈	#13083575	1	1	yes	characteristic
141	C₄⋊C₄	—	2	4	no	—
142	C₄⋊C₄	—	4	8	no	—
143	C₄⋊C₄	—	4	2	no	—
144	C₄⋊C₄	—	178	2	no	—
145	C₂²⋊C₄	—	356	1	no	—
146	C₄²	—	2	1	no	—
147	C₄²	—	356	1	no	—
148	C₄²	#13083562	1	2	yes	—
149	C₁₆	—	178	4	no	—
150	C₂³	#15668038	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	4	4	no	—
154	C₂×C₄	—	8	4	no	—
155	C₂×C₄	—	356	2	no	—
156	C₂×C₄	#15667981	1	1	yes	characteristic
157	C₂×C₄	#15667988	1	1	yes	characteristic
158	C₂×C₄	#15668010	1	1	yes	characteristic
159	C₂×C₄	C₂×#13083561	1	1	yes	characteristic
160	C₂×C₄	C₂×#13083562	1	1	yes	characteristic
161	C₂×C₄	#15668043	1	1	yes	characteristic
162	C₈	—	2	2	no	—
163	C₈	—	712	1	no	—
164	C₈	#15667988	1	2	yes	—
165	C₈	#15668010	1	2	yes	—
166	C₂²	—	2	2	no	—
167	C₂²	#19151749	1	1	yes	characteristic
168	C₂²	#19151885	1	1	yes	characteristic
169	C₂²	#19152004	1	1	yes	characteristic
170	C₄	—	4	2	no	—
171	C₄	—	8	4	no	—
172	C₄	—	712	2	no	—
173	C₄	#19151517	1	1	yes	characteristic
174	C₄	#19151737	1	1	yes	characteristic
175	C₄	#19151880	1	1	yes	characteristic
176	C₄	#19151994	1	1	yes	characteristic
177	C₂	—	2	1	no	—
178	C₂	—	2	1	no	—
179	C₂	#26021582	1	1	yes	characteristic
180	C₂	#26022425	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19151420

C₂

yes

maximal

#26021000

C₂

yes

maximal, characteristic

#26018359

C₂

yes

maximal

#26018334

C₂

yes

maximal, characteristic

C₂×#499×C₈₉

C₂

yes

maximal, characteristic

C₂×#15667952

C₂²

yes

—

C₂×#15667948

C₂²

yes

characteristic

#19151636

C₂²

yes

—

—

—

C₂×#125×C₈₉

—

#499×C₈₉

—

C₂×#107×C₈₉

—

C₂×#106×C₈₉

C₂²

yes

—

C₂×C₄×C₈×C₈₉

C₂²

yes

—

—

—

Q₈

yes

—

C₂×#13083549

—

#15667948

—

#15667946

Q₈

yes

—

#15667944

—

#15667941

—

C₂²×C₈×C₈₉

D₄

yes

characteristic

C₂²×C₈×C₈₉

C₂³

yes

characteristic

#125×C₈₉

—

#125×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄²×C₈₉

D₄

yes

characteristic

#107×C₈₉

—

#106×C₈₉

—

#106×C₈₉

D₄

yes

—

C₄×C₈×C₈₉

—

#13083558

—

#13083558

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

#13083549

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

SD₁₆

yes

—

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₂×C₈×C₈₉

C₂×Q₈

yes

—

C₂×C₈×C₈₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₈₉

#41

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

SD₁₆

yes

—

C₂³×C₈₉

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#120

yes

characteristic

C₂×C₄×C₈₉

C₂²⋊Q₈

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

#137

yes

characteristic

C₂×Dic₈₉

—

C₈×C₈₉

—

C₈×C₈₉

C₂²⋊Q₈

yes

—

C₈×C₈₉

C₄⋊Q₈

yes

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

#13458

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#2058

yes

characteristic

C₂×C₈₉

#2231

yes

—

C₂×#499

D₈₉

yes

—

—

—

—

C₈₉

#13458

yes

characteristic, complemented, Sylow

C₂×#125

—

C₂×M₃₂

—

#499

—

100

C₂×#108

—

101

#427

—

102

C₂×#107

—

103

C₂×#106

C₂×D₈₉

yes

—

104

#424

—

178

—

105

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

106

#365

—

178

—

107

#339

—

178

—

108

C₂×M₁₆

—

178

—

109

C₂²×C₈

C₂²⋊D₈₉

yes

characteristic

110

C₂²×C₈

C₂²×D₈₉

yes

characteristic

111

#125

—

112

#125

—

113

#117

—

178

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄²

C₂²⋊D₈₉

yes

characteristic

117

M₃₂

—

118

M₃₂

—

178

—

119

C₂×C₁₆

—

178

—

120

#108

—

178

—

121

#107

—

122

#107

—

123

#106

—

124

#106

—

178

—

125

#106

C₂²⋊D₈₉

yes

—

126

C₄⋊C₈

—

356

—

127

C₄×C₈

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

131

M₁₆

—

356

—

132

M₁₆

—

356

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

356

—

136

C₂×C₈

#13083561

yes

—

137

C₂×C₈

C₂×C₂²⋊D₈₉

yes

—

138

C₂×C₈

Q₈×D₈₉

yes

—

139

C₂×C₈

Q₈×D₈₉

yes

characteristic

140

C₂×C₈

#13083575

yes

characteristic

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

178

—

145

C₂²⋊C₄

—

356

—

146

C₄²

—

147

C₄²

—

356

—

148

C₄²

#13083562

yes

—

149

C₁₆

—

178

—

150

C₂³

#15668038

yes

characteristic

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

356

—

156

C₂×C₄

#15667981

yes

characteristic

157

C₂×C₄

#15667988

yes

characteristic

158

C₂×C₄

#15668010

yes

characteristic

159

C₂×C₄

C₂×#13083561

yes

characteristic

160

C₂×C₄

C₂×#13083562

yes

characteristic

161

C₂×C₄

#15668043

yes

characteristic

162

C₈

—

163

C₈

—

712

—

164

C₈

#15667988

yes

—

165

C₈

#15668010

yes

—

166

C₂²

—

167

C₂²

#19151749

yes

characteristic

168

C₂²

#19151885

yes

characteristic

169

C₂²

#19152004

yes

characteristic

170

C₄

—

171

C₄

—

172

C₄

—

712

—

173

C₄

#19151517

yes

characteristic

174

C₄

#19151737

yes

characteristic

175

C₄

#19151880

yes

characteristic

176

C₄

#19151994

yes

characteristic

177

C₂

—

178

C₂

—

179

C₂

#26021582

yes

characteristic

180

C₂

#26022425

yes

—

	subgroup	G/H
center	C₂²	#19151885
commutator, cosocle	C₂²×C₈×C₈₉	C₂³
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂×#499×C₈₉	C₂
socle	C₂²×C₈₉	#476
prime radical	C₂³×C₈₉	AGL(1,ℤ/8)
hypercenter	C₂×#499	D₈₉
hypocenter	C₈₉	#13458
LCS	G ▷ C₂²×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#499

subgroup

G/H

center

C₂²

#19151885

commutator, cosocle

C₂²×C₈×C₈₉

C₂³

Frattini

C₂²×C₈

C₂²×D₈₉

Fitting

C₂×#499×C₈₉

C₂

C₂²×C₈₉

C₂³×C₈₉

D₈₉

C₈₉

G ▷ C₂²×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#499

N	H ≅ G/N	type
96. C₈₉	#13458	semidirect

H ≅ G/N

type

96. C₈₉

#13458

semidirect

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

subgroup	index
C₂×#19151420	2
#26021000	2
#26018359	2
#26018334	2
C₂×#499×C₈₉	2
#13458	89

subgroup

index

C₂×#499×C₈₉

Subgroups — #36474078

Conjugacy classes of subgroups

Complements

Maximal subgroups