Group #36472963 — subgroups

Subgroups — #36472963

Order 22784.

334 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 139 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	#26029186	C₂	1	1	yes	maximal, characteristic
2	#26022008	C₂	1	1	yes	maximal, characteristic
3	#26018340	C₂	1	4	yes	maximal
4	C₂×#346×C₈₉	C₂	1	1	yes	maximal, characteristic
5	#19152155	C₄	1	1	yes	characteristic
6	C₂×#15667952	C₂²	1	2	yes	—
7	#19151728	—	2	2	no	—
8	C₈×Dic₁₇₈	—	2	2	no	—
9	#19151624	C₄	1	2	yes	—
10	#19151622	C₄	1	1	yes	characteristic
11	#19151553	C₂²	1	2	yes	—
12	C₄⋊#13083549	—	2	2	no	—
13	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
15	#346×C₈₉	—	2	2	no	—
16	C₂×#13083558	C₂×C₄	1	2	yes	—
17	#15668012	C₈	1	2	yes	—
18	#15668004	C₈	1	2	yes	—
19	#15667991	—	2	2	no	—
20	C₄×Dic₁₇₈	—	2	2	no	—
21	#15667971	C₂×C₄	1	2	yes	—
22	#15667952	—	2	4	no	—
23	C₂×#13083549	—	2	2	no	—
24	C₄⋊#10623494	—	2	2	no	—
25	#15667941	—	2	4	no	—
26	#15667922	—	2	2	no	—
27	C₈×Dic₈₉	—	2	2	no	—
28	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
30	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₆×C₈₉	—	2	2	no	—
32	C₂×C₁₆×C₈₉	—	2	2	no	—
33	C₂×C₁₆×C₈₉	Q₈	1	4	yes	complemented
34	C₄×C₈×C₈₉	D₄	1	2	yes	—
35	C₄×C₈×C₈₉	D₄	1	2	yes	—
36	C₂×Dic₁₇₈	—	2	2	no	—
37	#13083558	—	2	4	no	—
38	C₂×#10623494	—	2	2	no	—
39	C₄⋊Dic₈₉	—	2	2	no	—
40	C₄⋊Dic₈₉	C₂×C₈	1	4	yes	—
41	#13083553	—	2	4	no	—
42	C₂²⋊Dic₈₉	—	2	2	no	—
43	C₄×Dic₈₉	—	2	2	no	—
44	#13083549	—	4	2	no	—
45	C₂²×C₄×C₈₉	M₁₆	1	2	yes	—
46	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₈₉	—	2	2	no	—
48	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
49	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
52	C₂×C₈×C₈₉	#41	1	2	yes	—
53	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
54	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
55	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
56	C₁₆×C₈₉	—	2	4	no	—
57	C₂³×C₈₉	C₂×M₁₆	1	1	yes	characteristic
58	C₂×C₄×C₈₉	—	4	2	no	—
59	C₂×C₄×C₈₉	C₂²⋊C₈	1	2	yes	—
60	C₂×C₄×C₈₉	C₂²⋊C₈	1	2	yes	—
61	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
63	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
64	C₂×C₄×C₈₉	#131	1	2	yes	—
65	C₂×Dic₈₉	—	2	4	no	—
66	Dic₁₇₈	—	4	4	no	—
67	C₈×C₈₉	—	2	2	no	—
68	C₈×C₈₉	C₂²⋊Q₈	1	2	yes	—
69	C₈×C₈₉	#124	1	1	yes	characteristic
70	C₈×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
71	#10623494	—	4	2	no	—
72	C₂²×C₈₉	—	2	2	no	—
73	C₂²×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
74	C₂²×C₈₉	#446	1	2	yes	—
75	C₄×C₈₉	—	2	2	no	—
76	C₄×C₈₉	—	2	2	no	—
77	C₄×C₈₉	#391	1	1	yes	characteristic
78	C₄×C₈₉	#408	1	1	yes	characteristic
79	C₄×C₈₉	C₈×Q₈	1	2	yes	—
80	Dic₈₉	—	4	4	no	—
81	#13267	—	89	1	no	maximal, Sylow
82	C₂×C₈₉	—	2	2	no	—
83	C₂×C₈₉	#1984	1	1	yes	characteristic
84	C₂×C₈₉	#2117	1	2	yes	—
85	#2960	—	89	1	no	—
86	#2150	—	89	1	no	—
87	C₂×#346	D₈₉	1	1	yes	characteristic
88	#1379	—	89	4	no	—
89	C₈₉	#13267	1	1	yes	characteristic, complemented, Sylow
90	#533	—	89	1	no	—
91	C₂×M₃₂	—	89	2	no	—
92	C₂²×C₁₆	C₂×D₈₉	1	2	yes	—
93	#446	—	178	2	no	—
94	C₈×Q₈	—	178	2	no	—
95	#424	—	89	2	no	—
96	#423	—	89	1	no	—
97	#405	—	89	2	no	—
98	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
99	C₄⋊C₁₆	—	178	2	no	—
100	#346	—	2	2	no	—
101	C₂×M₁₆	—	89	2	no	—
102	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
103	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
104	C₄⋊Q₈	—	89	2	no	—
105	#125	—	89	2	no	—
106	C₂²⋊Q₈	—	178	2	no	—
107	C₄×Q₈	—	178	2	no	—
108	#117	—	89	2	no	—
109	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
110	M₃₂	—	178	4	no	—
111	C₂×C₁₆	—	2	2	no	—
112	C₂×C₁₆	—	2	2	no	—
113	C₂×C₁₆	—	178	2	no	—
114	C₂×C₁₆	Dic₁₇₈	1	4	yes	complemented
115	C₄⋊C₈	—	178	4	no	—
116	C₄⋊C₈	—	178	2	no	—
117	#97	—	178	2	no	—
118	C₄×C₈	—	178	2	no	—
119	C₄×C₈	D₃₅₆	1	2	yes	—
120	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
121	C₂×Q₈	—	178	2	no	—
122	C₂²×C₄	#13083557	1	2	yes	—
123	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
124	M₁₆	—	178	4	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	178	2	no	—
127	C₂×C₈	#13083551	1	2	yes	—
128	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
129	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
130	C₂×C₈	C₂×Dic₁₇₈	1	2	yes	—
131	C₂×C₈	#13083574	1	2	yes	—
132	C₄⋊C₄	—	89	4	no	—
133	C₄⋊C₄	—	178	4	no	—
134	C₄⋊C₄	—	178	2	no	—
135	C₂²⋊C₄	—	178	2	no	—
136	C₄²	—	178	2	no	—
137	C₄²	#13083551	1	2	yes	—
138	C₄²	C₈×D₈₉	1	1	yes	characteristic
139	C₄²	C₈×D₈₉	1	1	yes	characteristic
140	C₁₆	—	2	4	no	—
141	C₁₆	—	356	2	no	—
142	C₂³	C₂×#13083557	1	1	yes	characteristic
143	Q₈	—	356	4	no	—
144	C₂×C₄	—	4	2	no	—
145	C₂×C₄	—	178	4	no	—
146	C₂×C₄	#15667924	1	2	yes	—
147	C₂×C₄	#15667924	1	2	yes	—
148	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
149	C₂×C₄	C₄×Dic₁₇₈	1	2	yes	—
150	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
151	C₂×C₄	#15668018	1	2	yes	—
152	C₈	—	2	2	no	—
153	C₈	—	356	2	no	—
154	C₈	#15667991	1	2	yes	—
155	C₈	#15668000	1	1	yes	characteristic
156	C₈	#15668012	1	1	yes	characteristic
157	C₂²	—	2	2	no	—
158	C₂²	C₂×#15667924	1	1	yes	characteristic
159	C₂²	#19151728	1	2	yes	—
160	C₄	—	2	2	no	—
161	C₄	—	2	2	no	—
162	C₄	—	356	4	no	—
163	C₄	#19151504	1	1	yes	characteristic
164	C₄	#19151564	1	1	yes	characteristic
165	C₄	C₈×Dic₁₇₈	1	2	yes	—
166	C₂	—	2	2	no	—
167	C₂	#26021188	1	1	yes	characteristic
168	C₂	#26021879	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26029186

C₂

yes

maximal, characteristic

#26022008

C₂

yes

maximal, characteristic

#26018340

C₂

yes

maximal

C₂×#346×C₈₉

C₂

yes

maximal, characteristic

#19152155

C₄

yes

characteristic

C₂×#15667952

C₂²

yes

—

#19151728

—

C₈×Dic₁₇₈

—

#19151624

C₄

yes

—

#19151622

C₄

yes

characteristic

#19151553

C₂²

yes

—

C₄⋊#13083549

—

C₂²×C₁₆×C₈₉

C₂²

yes

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

#346×C₈₉

—

C₂×#13083558

C₂×C₄

yes

—

#15668012

C₈

yes

—

#15668004

C₈

yes

—

#15667991

—

C₄×Dic₁₇₈

—

#15667971

C₂×C₄

yes

—

—

—

—

—

—

C₈×Dic₈₉

—

C₂²×C₈×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₄²×C₈₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₈₉

—

C₂×C₁₆×C₈₉

—

C₂×C₁₆×C₈₉

Q₈

yes

complemented

C₄×C₈×C₈₉

D₄

yes

—

C₄×C₈×C₈₉

D₄

yes

—

C₂×Dic₁₇₈

—

#13083558

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

C₂×C₈

yes

—

#13083553

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

#13083549

—

C₂²×C₄×C₈₉

M₁₆

yes

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂²⋊C₄

yes

—

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

C₂×Q₈

yes

—

C₂×C₈×C₈₉

#41

yes

—

C₄²×C₈₉

C₂²⋊C₄

yes

—

C₄²×C₈₉

C₂×C₈

yes

characteristic

C₄²×C₈₉

C₂×C₈

yes

characteristic

C₁₆×C₈₉

—

C₂³×C₈₉

C₂×M₁₆

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²⋊C₈

yes

—

C₂×C₄×C₈₉

C₂²⋊C₈

yes

—

C₂×C₄×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₄×Q₈

yes

—

C₂×C₄×C₈₉

C₂²×C₈

yes

characteristic

C₂×C₄×C₈₉

#131

yes

—

C₂×Dic₈₉

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

C₂²⋊Q₈

yes

—

C₈×C₈₉

#124

yes

characteristic

C₈×C₈₉

C₄⋊Q₈

yes

characteristic

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₈₉

#446

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

C₈×Q₈

yes

—

Dic₈₉

—

#13267

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

#1984

yes

characteristic

C₂×C₈₉

#2117

yes

—

#2960

—

#2150

—

C₂×#346

D₈₉

yes

characteristic

#1379

—

C₈₉

#13267

yes

characteristic, complemented, Sylow

#533

—

C₂×M₃₂

—

C₂²×C₁₆

C₂×D₈₉

yes

—

#446

—

178

—

C₈×Q₈

—

178

—

#424

—

#423

—

#405

—

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

C₄⋊C₁₆

—

178

—

100

#346

—

101

C₂×M₁₆

—

102

C₂²×C₈

C₄×D₈₉

yes

characteristic

103

C₂²×C₈

C₂²×D₈₉

yes

characteristic

104

C₄⋊Q₈

—

105

#125

—

106

C₂²⋊Q₈

—

178

—

107

C₄×Q₈

—

178

—

108

#117

—

109

C₂×C₄²

C₄×D₈₉

yes

characteristic

110

M₃₂

—

178

—

111

C₂×C₁₆

—

112

C₂×C₁₆

—

113

C₂×C₁₆

—

178

—

114

C₂×C₁₆

Dic₁₇₈

yes

complemented

115

C₄⋊C₈

—

178

—

116

C₄⋊C₈

—

178

—

117

#97

—

178

—

118

C₄×C₈

—

178

—

119

C₄×C₈

D₃₅₆

yes

—

120

C₄×C₈

C₂²⋊D₈₉

yes

—

121

C₂×Q₈

—

178

—

122

C₂²×C₄

#13083557

yes

—

123

C₂²×C₄

C₂×C₄×D₈₉

yes

characteristic

124

M₁₆

—

178

—

125

C₂×C₈

—

126

C₂×C₈

—

178

—

127

C₂×C₈

#13083551

yes

—

128

C₂×C₈

C₂×C₂²⋊D₈₉

yes

characteristic

129

C₂×C₈

C₂×D₃₅₆

yes

characteristic

130

C₂×C₈

C₂×Dic₁₇₈

yes

—

131

C₂×C₈

#13083574

yes

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

178

—

134

C₄⋊C₄

—

178

—

135

C₂²⋊C₄

—

178

—

136

C₄²

—

178

—

137

C₄²

#13083551

yes

—

138

C₄²

C₈×D₈₉

yes

characteristic

139

C₄²

C₈×D₈₉

yes

characteristic

140

C₁₆

—

141

C₁₆

—

356

—

142

C₂³

C₂×#13083557

yes

characteristic

143

Q₈

—

356

—

144

C₂×C₄

—

145

C₂×C₄

—

178

—

146

C₂×C₄

#15667924

yes

—

147

C₂×C₄

#15667924

yes

—

148

C₂×C₄

C₂×#13083551

yes

characteristic

149

C₂×C₄

C₄×Dic₁₇₈

yes

—

150

C₂×C₄

C₂×C₈×D₈₉

yes

characteristic

151

C₂×C₄

#15668018

yes

—

152

C₈

—

153

C₈

—

356

—

154

C₈

#15667991

yes

—

155

C₈

#15668000

yes

characteristic

156

C₈

#15668012

yes

characteristic

157

C₂²

—

158

C₂²

C₂×#15667924

yes

characteristic

159

C₂²

#19151728

yes

—

160

C₄

—

161

C₄

—

162

C₄

—

356

—

163

C₄

#19151504

yes

characteristic

164

C₄

#19151564

yes

characteristic

165

C₄

C₈×Dic₁₇₈

yes

—

166

C₂

—

167

C₂

#26021188

yes

characteristic

168

C₂

#26021879

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂×#346×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₂²⋊C₈
prime radical	C₂³×C₈₉	C₂×M₁₆
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×#346	D₈₉
hypocenter	C₈₉	#13267
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂×C₈

C₂×D₃₅₆

commutator

C₂×C₄×C₈₉

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₈₉

Fitting

C₂×#346×C₈₉

C₂

socle

C₂²×C₈₉

C₂×C₂²⋊C₈

prime radical

C₂³×C₈₉

C₂×M₁₆

cosocle

C₂²×C₈×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
33. C₂×C₁₆×C₈₉	Q₈	semidirect
89. C₈₉	#13267	semidirect
114. C₂×C₁₆	Dic₁₇₈	semidirect

H ≅ G/N

type

33. C₂×C₁₆×C₈₉

Q₈

semidirect

89. C₈₉

#13267

semidirect

114. C₂×C₁₆

Dic₁₇₈

semidirect

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

subgroup	index
#26029186	2
#26022008	2
#26018340	2
C₂×#346×C₈₉	2
#13267	89

subgroup

index

C₂×#346×C₈₉

Subgroups — #36472963

Conjugacy classes of subgroups

Complements

Maximal subgroups