Group #36441319 — subgroups

Subgroups — #36441319

Order 22784.

238 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 119 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	#26022051	C₂	1	1	yes	maximal, characteristic
2	C₈×#13083558	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
4	#19151651	C₄	1	1	yes	characteristic
5	C₄×#13083558	C₄	1	1	yes	characteristic
6	C₄⋊#13083549	C₄	1	2	yes	—
7	C₂²⋊#13083549	—	2	1	no	—
8	C₄×#13083549	—	2	1	no	—
9	#19151320	—	2	1	no	—
10	C₈×#10623494	—	2	1	no	—
11	C₂²×C₁₆×C₈₉	—	2	1	no	—
12	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
13	C₄×C₁₆×C₈₉	—	2	1	no	—
14	C₄×C₁₆×C₈₉	C₄	1	2	yes	—
15	C₂×#13083558	C₈	1	2	yes	—
16	C₂×#13083549	—	2	2	no	—
17	C₄⋊#10623494	C₈	1	2	yes	—
18	C₂²⋊#10623494	—	2	1	no	—
19	#15667923	—	2	1	no	—
20	C₄×#10623494	—	2	1	no	—
21	C₄×#10623494	—	2	1	no	—
22	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₈₉	Q₈	1	1	yes	characteristic
24	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₈₉	—	2	4	no	—
26	C₂×C₁₆×C₈₉	—	2	2	no	—
27	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
30	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
31	#13083558	—	2	2	no	—
32	#13083558	C₁₆	1	4	yes	complemented
33	C₂×#10623494	—	2	2	no	—
34	C₂×#10623494	—	2	2	no	—
35	#13083549	—	4	2	no	—
36	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₄×C₈₉	M₁₆	1	1	yes	characteristic
39	C₂×C₈×C₈₉	—	2	2	no	—
40	C₂×C₈×C₈₉	—	2	2	no	—
41	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
45	C₄²×C₈₉	C₄²	1	1	yes	characteristic
46	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
47	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
48	C₄²×C₈₉	M₁₆	1	1	yes	characteristic
49	C₁₆×C₈₉	—	2	4	no	—
50	C₂³×C₈₉	C₄⋊C₈	1	1	yes	characteristic
51	C₂×C₄×C₈₉	—	2	1	no	—
52	C₂×C₄×C₈₉	—	2	1	no	—
53	C₂×C₄×C₈₉	—	2	2	no	—
54	C₂×C₄×C₈₉	#95	1	1	yes	characteristic
55	C₂×C₄×C₈₉	C₄×C₈	1	1	yes	characteristic
56	C₂×C₄×C₈₉	#97	1	1	yes	characteristic
57	C₂×C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
58	C₂×C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
59	C₂×C₄×C₈₉	C₄⋊C₈	1	1	yes	characteristic
60	C₂×C₄×C₈₉	C₂×C₁₆	1	2	yes	—
61	C₂×C₄×C₈₉	M₃₂	1	2	yes	—
62	C₈×C₈₉	—	2	1	no	—
63	C₈×C₈₉	—	2	1	no	—
64	C₈×C₈₉	#104	1	2	yes	—
65	C₈×C₈₉	#108	1	2	yes	—
66	#10623494	—	2	4	no	—
67	#10623494	—	4	2	no	—
68	C₂²×C₈₉	—	2	1	no	—
69	C₂²×C₈₉	—	2	1	no	—
70	C₂²×C₈₉	#336	1	1	yes	characteristic
71	C₂²×C₈₉	C₄⋊C₁₆	1	2	yes	—
72	C₄×C₈₉	—	2	1	no	—
73	C₄×C₈₉	—	2	1	no	—
74	C₄×C₈₉	#339	1	1	yes	characteristic
75	C₄×C₈₉	#341	1	1	yes	characteristic
76	C₄×C₈₉	C₄×C₁₆	1	1	yes	characteristic
77	C₄×C₈₉	#346	1	1	yes	characteristic
78	C₄×C₈₉	C₂²⋊C₁₆	1	2	yes	—
79	C₄×C₈₉	#350	1	2	yes	—
80	#7163	—	89	1	no	maximal, Sylow
81	C₂×C₈₉	—	2	2	no	—
82	C₂×C₈₉	#1278	1	1	yes	characteristic
83	C₂×C₈₉	#1374	1	1	yes	characteristic
84	C₂×C₈₉	#1379	1	1	yes	characteristic
85	#2162	—	89	1	no	—
86	C₂×C₄×C₁₆	D₈₉	1	1	yes	characteristic
87	C₈×M₁₆	—	89	1	no	—
88	C₈₉	#7163	1	1	yes	characteristic, complemented, Sylow
89	C₂²×C₁₆	—	2	1	no	—
90	#431	—	89	1	no	—
91	C₄×M₁₆	—	89	1	no	—
92	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
93	C₄⋊C₁₆	—	89	2	no	—
94	C₂²⋊C₁₆	—	178	1	no	—
95	C₄×C₁₆	—	2	1	no	—
96	C₄×C₁₆	—	178	1	no	—
97	C₄×C₁₆	Dic₈₉	1	2	yes	—
98	#322	—	178	1	no	—
99	C₈²	—	178	1	no	—
100	C₂×M₁₆	—	89	2	no	—
101	C₂²×C₈	Dic₁₇₈	1	1	yes	characteristic
102	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
103	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
104	C₂×C₁₆	—	2	4	no	—
105	C₂×C₁₆	—	2	2	no	—
106	C₂×C₁₆	—	178	2	no	—
107	C₄⋊C₈	—	89	2	no	—
108	C₂²⋊C₈	—	178	1	no	—
109	#97	—	178	1	no	—
110	C₄×C₈	—	178	1	no	—
111	C₄×C₈	—	178	1	no	—
112	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
113	C₄×C₈	D₃₅₆	1	1	yes	characteristic
114	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
115	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
116	C₂²×C₄	#13083553	1	1	yes	characteristic
117	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
118	C₂²×C₄	#13083557	1	1	yes	characteristic
119	M₁₆	—	89	4	no	—
120	M₁₆	—	178	2	no	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	178	2	no	—
124	C₂×C₈	—	178	2	no	—
125	C₂×C₈	#13083551	1	1	yes	characteristic
126	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
127	C₂×C₈	#13083553	1	1	yes	characteristic
128	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
129	C₄²	C₄×Dic₈₉	1	1	yes	characteristic
130	C₄²	#13083551	1	1	yes	characteristic
131	C₄²	C₈×D₈₉	1	1	yes	characteristic
132	C₄²	#13083557	1	1	yes	characteristic
133	C₁₆	—	2	4	no	—
134	C₁₆	—	356	2	no	—
135	C₂³	#15667941	1	1	yes	characteristic
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	#15667919	1	1	yes	characteristic
140	C₂×C₄	C₈×Dic₈₉	1	1	yes	characteristic
141	C₂×C₄	#15667922	1	1	yes	characteristic
142	C₂×C₄	#15667924	1	1	yes	characteristic
143	C₂×C₄	#15667924	1	1	yes	characteristic
144	C₂×C₄	#15667941	1	1	yes	characteristic
145	C₂×C₄	C₁₆×D₈₉	1	2	yes	—
146	C₂×C₄	#15667951	1	2	yes	—
147	C₈	—	2	1	no	—
148	C₈	—	2	1	no	—
149	C₈	—	356	2	no	—
150	C₈	—	178	4	no	—
151	C₈	#15667940	1	2	yes	—
152	C₈	#15667948	1	2	yes	—
153	C₂²	—	2	1	no	—
154	C₂²	—	2	1	no	—
155	C₂²	#19151352	1	1	yes	characteristic
156	C₂²	#19151416	1	2	yes	—
157	C₄	—	2	1	no	—
158	C₄	—	2	1	no	—
159	C₄	#19151358	1	1	yes	characteristic
160	C₄	#19151363	1	1	yes	characteristic
161	C₄	C₁₆×Dic₈₉	1	1	yes	characteristic
162	C₄	#19151373	1	1	yes	characteristic
163	C₄	#19151377	1	2	yes	—
164	C₄	#19151381	1	2	yes	—
165	C₂	—	2	2	no	—
166	C₂	#26018089	1	1	yes	characteristic
167	C₂	#26018329	1	1	yes	characteristic
168	C₂	#26018340	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26022051

C₂

yes

maximal, characteristic

C₈×#13083558

C₂

yes

maximal, characteristic

C₂×C₄×C₁₆×C₈₉

C₂

yes

maximal, characteristic

#19151651

C₄

yes

characteristic

C₄×#13083558

C₄

yes

characteristic

C₄⋊#13083549

C₄

yes

—

—

—

—

—

C₂²×C₁₆×C₈₉

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

C₄×C₁₆×C₈₉

—

C₄×C₁₆×C₈₉

C₄

yes

—

C₂×#13083558

C₈

yes

—

C₂×#13083549

—

C₄⋊#10623494

C₈

yes

—

—

—

—

—

C₂²×C₈×C₈₉

D₄

yes

characteristic

C₂²×C₈×C₈₉

Q₈

yes

characteristic

C₂×C₄²×C₈₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₈₉

—

C₂×C₁₆×C₈₉

—

C₄×C₈×C₈₉

C₂×C₄

yes

characteristic

C₄×C₈×C₈₉

C₂×C₄

yes

characteristic

C₄×C₈×C₈₉

D₄

yes

characteristic

C₄×C₈×C₈₉

D₄

yes

characteristic

#13083558

—

#13083558

C₁₆

yes

—

—

—

C₂²×C₄×C₈₉

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×C₈

yes

characteristic

C₂²×C₄×C₈₉

M₁₆

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₈₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₈₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₈₉

C₄⋊C₄

yes

characteristic

C₄²×C₈₉

C₄²

yes

characteristic

C₄²×C₈₉

C₂²⋊C₄

yes

characteristic

C₄²×C₈₉

C₂×C₈

yes

characteristic

C₄²×C₈₉

M₁₆

yes

characteristic

C₁₆×C₈₉

—

C₂³×C₈₉

C₄⋊C₈

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#95

yes

characteristic

C₂×C₄×C₈₉

C₄×C₈

yes

characteristic

C₂×C₄×C₈₉

#97

yes

characteristic

C₂×C₄×C₈₉

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₈₉

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₈₉

C₄⋊C₈

yes

characteristic

C₂×C₄×C₈₉

C₂×C₁₆

yes

—

C₂×C₄×C₈₉

M₃₂

yes

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

#104

yes

—

C₈×C₈₉

#108

yes

—

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#336

yes

characteristic

C₂²×C₈₉

C₄⋊C₁₆

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

C₄×C₁₆

yes

characteristic

C₄×C₈₉

#346

yes

characteristic

C₄×C₈₉

C₂²⋊C₁₆

yes

—

C₄×C₈₉

#350

yes

—

#7163

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₂×C₄×C₁₆

D₈₉

yes

characteristic

C₈×M₁₆

—

C₈₉

#7163

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

#431

—

C₄×M₁₆

—

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

178

—

C₄×C₁₆

—

C₄×C₁₆

—

178

—

C₄×C₁₆

Dic₈₉

yes

—

#322

—

178

—

C₈²

—

178

—

100

C₂×M₁₆

—

101

C₂²×C₈

Dic₁₇₈

yes

characteristic

102

C₂²×C₈

C₂²⋊D₈₉

yes

characteristic

103

C₂×C₄²

C₄×D₈₉

yes

characteristic

104

C₂×C₁₆

—

105

C₂×C₁₆

—

106

C₂×C₁₆

—

178

—

107

C₄⋊C₈

—

108

C₂²⋊C₈

—

178

—

109

#97

—

178

—

110

C₄×C₈

—

178

—

111

C₄×C₈

—

178

—

112

C₄×C₈

C₄×D₈₉

yes

characteristic

113

C₄×C₈

D₃₅₆

yes

characteristic

114

C₄×C₈

C₂×Dic₈₉

yes

characteristic

115

C₄×C₈

C₂²⋊D₈₉

yes

characteristic

116

C₂²×C₄

#13083553

yes

characteristic

117

C₂²×C₄

C₈×D₈₉

yes

characteristic

118

C₂²×C₄

#13083557

yes

characteristic

119

M₁₆

—

120

M₁₆

—

178

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

178

—

124

C₂×C₈

—

178

—

125

C₂×C₈

#13083551

yes

characteristic

126

C₂×C₈

C₂²⋊Dic₈₉

yes

characteristic

127

C₂×C₈

#13083553

yes

characteristic

128

C₂×C₈

C₄⋊Dic₈₉

yes

characteristic

129

C₄²

C₄×Dic₈₉

yes

characteristic

130

C₄²

#13083551

yes

characteristic

131

C₄²

C₈×D₈₉

yes

characteristic

132

C₄²

#13083557

yes

characteristic

133

C₁₆

—

134

C₁₆

—

356

—

135

C₂³

#15667941

yes

characteristic

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

#15667919

yes

characteristic

140

C₂×C₄

C₈×Dic₈₉

yes

characteristic

141

C₂×C₄

#15667922

yes

characteristic

142

C₂×C₄

#15667924

yes

characteristic

143

C₂×C₄

#15667924

yes

characteristic

144

C₂×C₄

#15667941

yes

characteristic

145

C₂×C₄

C₁₆×D₈₉

yes

—

146

C₂×C₄

#15667951

yes

—

147

C₈

—

148

C₈

—

149

C₈

—

356

—

150

C₈

—

178

—

151

C₈

#15667940

yes

—

152

C₈

#15667948

yes

—

153

C₂²

—

154

C₂²

—

155

C₂²

#19151352

yes

characteristic

156

C₂²

#19151416

yes

—

157

C₄

—

158

C₄

—

159

C₄

#19151358

yes

characteristic

160

C₄

#19151363

yes

characteristic

161

C₄

C₁₆×Dic₈₉

yes

characteristic

162

C₄

#19151373

yes

characteristic

163

C₄

#19151377

yes

—

164

C₄

#19151381

yes

—

165

C₂

—

166

C₂

#26018089

yes

characteristic

167

C₂

#26018329

yes

characteristic

168

C₂

#26018340

yes

characteristic

	subgroup	G/H
center	C₄×C₈	D₃₅₆
commutator	C₄×C₈₉	C₄×C₁₆
Frattini	C₂×C₄×C₈	C₂×D₈₉
Fitting	C₂×C₄×C₁₆×C₈₉	C₂
socle	C₂²×C₈₉	#336
prime radical	C₂³×C₈₉	C₄⋊C₈
cosocle	C₂×C₄×C₈×C₈₉	C₂²
hypercenter	C₂×C₄×C₁₆	D₈₉
hypocenter	C₈₉	#7163
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

subgroup

G/H

center

C₄×C₈

D₃₅₆

commutator

C₄×C₈₉

C₄×C₁₆

Frattini

C₂×C₄×C₈

C₂×D₈₉

Fitting

C₂×C₄×C₁₆×C₈₉

C₂

socle

C₂²×C₈₉

#336

prime radical

C₂³×C₈₉

C₄⋊C₈

cosocle

C₂×C₄×C₈×C₈₉

C₂²

hypercenter

C₂×C₄×C₁₆

D₈₉

hypocenter

C₈₉

#7163

LCS

G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

N	H ≅ G/N	type
32. #13083558	C₁₆	semidirect
88. C₈₉	#7163	semidirect

H ≅ G/N

type

32. #13083558

C₁₆

semidirect

88. C₈₉

#7163

semidirect

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

subgroup	index
#26022051	2
C₈×#13083558	2
C₂×C₄×C₁₆×C₈₉	2
#7163	89

subgroup

index

#26022051

C₈×#13083558

C₂×C₄×C₁₆×C₈₉

#7163

Subgroups — #36441319

Conjugacy classes of subgroups

Complements

Maximal subgroups