Group #36441321 — subgroups

Subgroups — #36441321

Order 22784.

212 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 95 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	C₄×#15667952	C₂	1	1	yes	maximal, characteristic
2	#26018541	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#2163	C₂	1	1	yes	maximal, characteristic
4	C₂×#15667952	—	2	1	no	—
5	#19151651	C₄	1	1	yes	characteristic
6	C₄×#13083558	C₄	1	1	yes	characteristic
7	#19151374	—	2	1	no	—
8	C₄×#13083549	C₄	1	2	yes	—
9	#19151320	—	2	1	no	—
10	#19151319	—	2	1	no	—
11	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
12	C₄⋊C₁₆×C₈₉	C₄	1	2	yes	—
13	C₂²⋊C₁₆×C₈₉	—	2	1	no	—
14	#346×C₈₉	—	2	1	no	—
15	C₂×#13083558	—	2	1	no	—
16	#15667952	—	2	4	no	—
17	C₂×#13083549	—	2	2	no	—
18	C₄⋊#10623494	—	2	1	no	—
19	C₂²⋊#10623494	C₈	1	2	yes	—
20	#15667923	C₈	1	2	yes	—
21	C₄×#10623494	—	2	1	no	—
22	C₄×#10623494	—	2	1	no	—
23	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₈₉	Q₈	1	1	yes	characteristic
25	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
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	—	4	2	no	—
32	C₂×#10623494	—	2	2	no	—
33	C₂×#10623494	—	2	2	no	—
34	#13083549	—	2	4	no	—
35	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₄×C₈₉	M₁₆	1	1	yes	characteristic
38	C₂×C₈×C₈₉	—	2	2	no	—
39	C₂×C₈×C₈₉	—	2	2	no	—
40	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
44	C₄²×C₈₉	C₄²	1	1	yes	characteristic
45	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
46	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
47	C₄²×C₈₉	M₁₆	1	1	yes	characteristic
48	C₁₆×C₈₉	—	4	2	no	—
49	C₂³×C₈₉	C₄⋊C₈	1	1	yes	characteristic
50	C₂×C₄×C₈₉	—	2	1	no	—
51	C₂×C₄×C₈₉	—	2	1	no	—
52	C₂×C₄×C₈₉	—	2	2	no	—
53	C₂×C₄×C₈₉	—	4	1	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	C₈×C₈₉	—	2	1	no	—
61	C₈×C₈₉	—	2	1	no	—
62	C₈×C₈₉	#104	1	2	yes	—
63	C₈×C₈₉	#108	1	2	yes	—
64	#10623494	—	4	2	no	—
65	#10623494	—	4	2	no	—
66	C₂²×C₈₉	—	2	1	no	—
67	C₂²×C₈₉	—	2	1	no	—
68	C₂²×C₈₉	—	2	1	no	—
69	C₂²×C₈₉	#336	1	1	yes	characteristic
70	C₄×C₈₉	—	2	1	no	—
71	C₄×C₈₉	—	2	1	no	—
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	#347	1	2	yes	—
77	#7164	—	89	1	no	maximal, Sylow
78	C₂×C₈₉	—	4	1	no	—
79	C₂×C₈₉	#1278	1	1	yes	characteristic
80	C₂×C₈₉	#1375	1	1	yes	characteristic
81	C₂×C₈₉	#1378	1	1	yes	characteristic
82	#2163	D₈₉	1	1	yes	characteristic
83	C₄×M₃₂	—	89	1	no	—
84	#1451	—	89	1	no	—
85	C₈₉	#7164	1	1	yes	characteristic, complemented, Sylow
86	C₂×M₃₂	—	178	1	no	—
87	#431	—	89	1	no	—
88	C₄×M₁₆	—	89	1	no	—
89	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
90	C₄⋊C₁₆	Dic₈₉	1	2	yes	—
91	C₂²⋊C₁₆	—	2	1	no	—
92	#346	—	2	1	no	—
93	#346	—	178	1	no	—
94	C₄×C₁₆	—	89	2	no	—
95	#322	—	178	1	no	—
96	#322	—	178	1	no	—
97	C₂×M₁₆	—	178	1	no	—
98	C₂²×C₈	Dic₁₇₈	1	1	yes	characteristic
99	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
100	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
101	M₃₂	—	178	4	no	—
102	C₂×C₁₆	—	2	2	no	—
103	C₂×C₁₆	—	178	2	no	—
104	C₄⋊C₈	—	178	1	no	—
105	C₂²⋊C₈	—	89	2	no	—
106	#97	—	89	2	no	—
107	C₄×C₈	—	178	1	no	—
108	C₄×C₈	—	178	1	no	—
109	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
110	C₄×C₈	D₃₅₆	1	1	yes	characteristic
111	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
112	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
113	C₂²×C₄	#13083553	1	1	yes	characteristic
114	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
115	C₂²×C₄	#13083557	1	1	yes	characteristic
116	M₁₆	—	356	2	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	178	2	no	—
120	C₂×C₈	—	178	2	no	—
121	C₂×C₈	#13083551	1	1	yes	characteristic
122	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
123	C₂×C₈	#13083553	1	1	yes	characteristic
124	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
125	C₄²	C₄×Dic₈₉	1	1	yes	characteristic
126	C₄²	#13083551	1	1	yes	characteristic
127	C₄²	C₈×D₈₉	1	1	yes	characteristic
128	C₄²	#13083557	1	1	yes	characteristic
129	C₁₆	—	4	2	no	—
130	C₁₆	—	178	4	no	—
131	C₂³	#15667941	1	1	yes	characteristic
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	#15667919	1	1	yes	characteristic
137	C₂×C₄	C₈×Dic₈₉	1	1	yes	characteristic
138	C₂×C₄	#15667922	1	1	yes	characteristic
139	C₂×C₄	#15667924	1	1	yes	characteristic
140	C₂×C₄	#15667924	1	1	yes	characteristic
141	C₂×C₄	#15667941	1	1	yes	characteristic
142	C₈	—	2	1	no	—
143	C₈	—	2	1	no	—
144	C₈	—	356	2	no	—
145	C₈	—	356	2	no	—
146	C₈	#15667939	1	2	yes	—
147	C₈	#15667947	1	2	yes	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	—	2	1	no	—
151	C₂²	#19151352	1	1	yes	characteristic
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	#19151357	1	1	yes	characteristic
157	C₄	#19151362	1	1	yes	characteristic
158	C₄	#19151375	1	2	yes	—
159	C₂	—	4	1	no	—
160	C₂	#26018088	1	1	yes	characteristic
161	C₂	#26018331	1	1	yes	characteristic
162	C₂	#26018339	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#15667952

C₂

yes

maximal, characteristic

#26018541

C₂

yes

maximal, characteristic

C₈₉×#2163

C₂

yes

maximal, characteristic

C₂×#15667952

—

#19151651

C₄

yes

characteristic

C₄×#13083558

C₄

yes

characteristic

#19151374

—

C₄×#13083549

C₄

yes

—

#19151320

—

#19151319

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

C₄⋊C₁₆×C₈₉

C₄

yes

—

C₂²⋊C₁₆×C₈₉

—

#346×C₈₉

—

—

—

—

—

C₈

yes

—

#15667923

C₈

yes

—

C₄×#10623494

—

C₄×#10623494

—

C₂²×C₈×C₈₉

D₄

yes

characteristic

C₂²×C₈×C₈₉

Q₈

yes

characteristic

C₂×C₄²×C₈₉

C₂×C₄

yes

characteristic

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

—

—

—

—

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

—

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

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

#104

yes

—

C₈×C₈₉

#108

yes

—

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#336

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

#7164

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

D₈₉

yes

characteristic

C₄×M₃₂

—

#1451

—

C₈₉

#7164

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

178

—

#431

—

C₄×M₁₆

—

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

C₄⋊C₁₆

Dic₈₉

yes

—

C₂²⋊C₁₆

—

#346

—

#346

—

178

—

C₄×C₁₆

—

#322

—

178

—

#322

—

178

—

C₂×M₁₆

—

178

—

C₂²×C₈

Dic₁₇₈

yes

characteristic

C₂²×C₈

C₂²⋊D₈₉

yes

characteristic

100

C₂×C₄²

C₄×D₈₉

yes

characteristic

101

M₃₂

—

178

—

102

C₂×C₁₆

—

103

C₂×C₁₆

—

178

—

104

C₄⋊C₈

—

178

—

105

C₂²⋊C₈

—

106

#97

—

107

C₄×C₈

—

178

—

108

C₄×C₈

—

178

—

109

C₄×C₈

C₄×D₈₉

yes

characteristic

110

C₄×C₈

D₃₅₆

yes

characteristic

111

C₄×C₈

C₂×Dic₈₉

yes

characteristic

112

C₄×C₈

C₂²⋊D₈₉

yes

characteristic

113

C₂²×C₄

#13083553

yes

characteristic

114

C₂²×C₄

C₈×D₈₉

yes

characteristic

115

C₂²×C₄

#13083557

yes

characteristic

116

M₁₆

—

356

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

178

—

120

C₂×C₈

—

178

—

121

C₂×C₈

#13083551

yes

characteristic

122

C₂×C₈

C₂²⋊Dic₈₉

yes

characteristic

123

C₂×C₈

#13083553

yes

characteristic

124

C₂×C₈

C₄⋊Dic₈₉

yes

characteristic

125

C₄²

C₄×Dic₈₉

yes

characteristic

126

C₄²

#13083551

yes

characteristic

127

C₄²

C₈×D₈₉

yes

characteristic

128

C₄²

#13083557

yes

characteristic

129

C₁₆

—

130

C₁₆

—

178

—

131

C₂³

#15667941

yes

characteristic

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

#15667919

yes

characteristic

137

C₂×C₄

C₈×Dic₈₉

yes

characteristic

138

C₂×C₄

#15667922

yes

characteristic

139

C₂×C₄

#15667924

yes

characteristic

140

C₂×C₄

#15667924

yes

characteristic

141

C₂×C₄

#15667941

yes

characteristic

142

C₈

—

143

C₈

—

144

C₈

—

356

—

145

C₈

—

356

—

146

C₈

#15667939

yes

—

147

C₈

#15667947

yes

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

#19151352

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

#19151357

yes

characteristic

157

C₄

#19151362

yes

characteristic

158

C₄

#19151375

yes

—

159

C₂

—

160

C₂

#26018088

yes

characteristic

161

C₂

#26018331

yes

characteristic

162

C₂

#26018339

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#15667919
commutator	C₂×C₄×C₈₉	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₈₉
Fitting	C₈₉×#2163	C₂
socle	C₂²×C₈₉	#336
prime radical	C₂³×C₈₉	C₄⋊C₈
cosocle	C₂×C₄×C₈×C₈₉	C₂²
hypercenter	#2163	D₈₉
hypocenter	C₈₉	#7164
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2163

subgroup

G/H

center

C₂×C₄

#15667919

commutator

C₂×C₄×C₈₉

C₄×C₈

Frattini

C₂×C₄×C₈

C₂×D₈₉

Fitting

C₈₉×#2163

C₂

socle

C₂²×C₈₉

#336

prime radical

C₂³×C₈₉

C₄⋊C₈

cosocle

C₂×C₄×C₈×C₈₉

C₂²

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2163

N	H ≅ G/N	type
85. C₈₉	#7164	semidirect

H ≅ G/N

type

85. C₈₉

#7164

semidirect

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

subgroup	index
C₄×#15667952	2
#26018541	2
C₈₉×#2163	2
#7164	89

subgroup

index

C₄×#15667952

#26018541

C₈₉×#2163

#7164

Subgroups — #36441321

Conjugacy classes of subgroups

Complements

Maximal subgroups