Group #36448180 — subgroups

Subgroups — #36448180

Order 22784.

402 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 187 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	#26020404	C₂	1	1	yes	maximal, characteristic
2	#26019998	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151348	C₂	1	4	yes	maximal
4	C₈₉×#1842	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#10623494	—	2	2	no	—
6	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
7	C₂×C₄×#10623494	C₂²	1	2	yes	—
8	#19151353	C₄	1	2	yes	—
9	#19151353	C₄	1	2	yes	—
10	#19151348	—	2	8	no	—
11	C₂×C₄⋊C₈×C₈₉	—	2	2	no	—
12	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
13	C₄³×C₈₉	C₂²	1	1	yes	characteristic
14	C₂²×#10623494	—	2	2	no	—
15	C₂²×#10623494	C₂×C₄	1	4	yes	—
16	C₄⋊#10623494	—	2	4	no	—
17	C₄⋊#10623494	—	4	4	no	—
18	C₄×#10623494	D₄	1	4	yes	—
19	C₄×#10623494	Q₈	1	4	yes	—
20	C₂²×C₈×C₈₉	—	2	2	no	—
21	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₈₉	D₄	1	2	yes	—
24	C₂×C₄²×C₈₉	Q₈	1	2	yes	—
25	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
26	C₄⋊C₈×C₈₉	—	2	4	no	—
27	C₄⋊C₈×C₈₉	—	4	4	no	—
28	C₂×#10623494	—	2	4	no	—
29	C₂×#10623494	—	2	4	no	—
30	C₂×#10623494	—	4	2	no	—
31	C₂×#10623494	—	4	4	no	—
32	C₂×#10623494	C₄⋊C₄	1	8	yes	—
33	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
34	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
35	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₈×C₈₉	—	4	2	no	—
39	C₂×C₈×C₈₉	—	4	4	no	—
40	C₄²×C₈₉	—	2	2	no	—
41	C₄²×C₈₉	—	2	2	no	—
42	C₄²×C₈₉	—	4	2	no	—
43	C₄²×C₈₉	SD₁₆	1	4	yes	—
44	C₄²×C₈₉	SD₁₆	1	4	yes	—
45	C₄²×C₈₉	C₂×D₄	1	2	yes	—
46	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
47	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₈₉	—	2	2	no	—
49	C₂×C₄×C₈₉	—	2	2	no	—
50	C₂×C₄×C₈₉	—	2	2	no	—
51	C₂×C₄×C₈₉	—	2	2	no	—
52	C₂×C₄×C₈₉	—	2	2	no	—
53	C₂×C₄×C₈₉	#106	1	4	yes	—
54	C₂×C₄×C₈₉	#106	1	4	yes	—
55	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
56	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
57	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
58	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
59	C₂×C₄×C₈₉	#131	1	2	yes	—
60	C₂×C₄×C₈₉	#131	1	2	yes	—
61	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
62	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
63	C₈×C₈₉	—	8	2	no	—
64	#10623494	—	2	8	no	—
65	#10623494	—	8	2	no	—
66	C₂²×C₈₉	#384	1	1	yes	characteristic
67	C₂²×C₈₉	#424	1	1	yes	characteristic
68	C₂²×C₈₉	#424	1	1	yes	characteristic
69	C₂²×C₈₉	C₂×#106	1	2	yes	—
70	C₂²×C₈₉	#430	1	2	yes	—
71	C₄×C₈₉	—	2	4	no	—
72	C₄×C₈₉	—	2	4	no	—
73	C₄×C₈₉	—	2	2	no	—
74	C₄×C₈₉	#424	1	4	yes	—
75	C₄×C₈₉	#492	1	1	yes	characteristic
76	C₄×C₈₉	#492	1	1	yes	characteristic
77	C₄×C₈₉	#498	1	2	yes	—
78	#8809	—	89	1	no	maximal, Sylow
79	C₂×C₈₉	#1570	1	2	yes	—
80	C₂×C₈₉	#1570	1	2	yes	—
81	C₂×C₈₉	#1843	1	1	yes	characteristic
82	C₂×C₈₉	#1844	1	1	yes	characteristic
83	C₂×C₈₉	#1849	1	1	yes	characteristic
84	#1842	—	89	1	no	—
85	#1842	D₈₉	1	1	yes	characteristic
86	#1744	—	89	1	no	—
87	C₂×#334	—	89	4	no	—
88	C₈₉	#8809	1	1	yes	characteristic, complemented, Sylow
89	C₂×C₄⋊C₈	—	2	2	no	—
90	C₂×C₄⋊C₈	—	89	2	no	—
91	C₂×C₄⋊C₈	—	178	2	no	—
92	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
93	C₂×C₄×C₈	—	89	2	no	—
94	C₄³	C₂×D₈₉	1	1	yes	characteristic
95	#336	—	89	2	no	—
96	#336	—	89	2	no	—
97	#334	—	178	8	no	—
98	C₂²×C₈	—	2	2	no	—
99	C₂²×C₈	—	89	4	no	—
100	C₂²×C₈	—	178	2	no	—
101	C₂×C₄²	Dic₁₇₈	1	2	yes	—
102	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
103	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
104	C₂×C₄²	C₂²⋊D₈₉	1	2	yes	—
105	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
106	C₄⋊C₈	—	2	4	no	—
107	C₄⋊C₈	—	4	4	no	—
108	C₄⋊C₈	—	356	4	no	—
109	C₄⋊C₈	—	178	4	no	—
110	C₄×C₈	—	89	4	no	—
111	C₄×C₈	—	89	4	no	—
112	C₂²×C₄	#13083553	1	2	yes	—
113	C₂²×C₄	#13083553	1	2	yes	—
114	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
115	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
116	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
117	C₂×C₈	—	4	2	no	—
118	C₂×C₈	—	4	4	no	—
119	C₂×C₈	—	89	8	no	—
120	C₂×C₈	—	356	2	no	—
121	C₂×C₈	—	356	4	no	—
122	C₂×C₈	—	178	4	no	—
123	C₂×C₈	—	178	4	no	—
124	C₄²	—	2	2	no	—
125	C₄²	—	2	2	no	—
126	C₄²	—	4	2	no	—
127	C₄²	#13083561	1	4	yes	—
128	C₄²	#13083562	1	4	yes	—
129	C₄²	D₄×D₈₉	1	2	yes	—
130	C₄²	Q₈×D₈₉	1	2	yes	—
131	C₂³	C₂×#13083553	1	1	yes	characteristic
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	#15667943	1	4	yes	—
138	C₂×C₄	#15667943	1	4	yes	—
139	C₂×C₄	C₄⋊C₄×D₈₉	1	2	yes	—
140	C₂×C₄	#15668008	1	1	yes	characteristic
141	C₂×C₄	#15668010	1	1	yes	characteristic
142	C₂×C₄	#15668011	1	2	yes	—
143	C₂×C₄	#15668018	1	2	yes	—
144	C₂×C₄	#15668020	1	2	yes	—
145	C₂×C₄	C₂×#13083561	1	2	yes	—
146	C₂×C₄	C₂×#13083562	1	2	yes	—
147	C₈	—	8	2	no	—
148	C₈	—	178	8	no	—
149	C₈	—	712	2	no	—
150	C₂²	#19151467	1	1	yes	characteristic
151	C₂²	#19151624	1	1	yes	characteristic
152	C₂²	#19151626	1	1	yes	characteristic
153	C₂²	C₂×#15667943	1	2	yes	—
154	C₂²	#19151646	1	2	yes	—
155	C₄	—	2	2	no	—
156	C₄	—	2	4	no	—
157	C₄	—	2	4	no	—
158	C₄	#19151623	1	4	yes	—
159	C₄	#19151970	1	1	yes	characteristic
160	C₄	#19151972	1	1	yes	characteristic
161	C₄	#19151994	1	2	yes	—
162	C₂	#26019022	1	2	yes	—
163	C₂	#26019024	1	2	yes	—
164	C₂	#26020405	1	1	yes	characteristic
165	C₂	#26020408	1	1	yes	characteristic
166	C₂	#26020426	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26020404

C₂

yes

maximal, characteristic

#26019998

C₂

yes

maximal, characteristic

C₂×#19151348

C₂

yes

maximal

C₈₉×#1842

C₂

yes

maximal, characteristic

C₂×C₄⋊#10623494

—

C₂×C₄⋊#10623494

C₂²

yes

—

C₂×C₄×#10623494

C₂²

yes

—

#19151353

C₄

yes

—

#19151353

C₄

yes

—

#19151348

—

C₂×C₄⋊C₈×C₈₉

—

C₂×C₄⋊C₈×C₈₉

C₂²

yes

—

C₄³×C₈₉

C₂²

yes

characteristic

C₂²×#10623494

—

C₂²×#10623494

C₂×C₄

yes

—

C₄⋊#10623494

—

C₄⋊#10623494

—

C₄×#10623494

D₄

yes

—

C₄×#10623494

Q₈

yes

—

C₂²×C₈×C₈₉

—

C₂×C₄²×C₈₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₈₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

Q₈

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄⋊C₈×C₈₉

—

C₄⋊C₈×C₈₉

—

—

—

—

—

C₄⋊C₄

yes

—

C₂²×C₄×C₈₉

C₄⋊C₄

yes

—

C₂²×C₄×C₈₉

C₄⋊C₄

yes

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

SD₁₆

yes

—

C₄²×C₈₉

SD₁₆

yes

—

C₄²×C₈₉

C₂×D₄

yes

—

C₄²×C₈₉

C₂×Q₈

yes

—

C₂³×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#106

yes

—

C₂×C₄×C₈₉

#106

yes

—

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₈₉

#127

yes

characteristic

C₂×C₄×C₈₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₈₉

C₄⋊Q₈

yes

—

C₂×C₄×C₈₉

#131

yes

—

C₂×C₄×C₈₉

#131

yes

—

C₂×C₄×C₈₉

C₂×SD₁₆

yes

—

C₂×C₄×C₈₉

C₂×SD₁₆

yes

—

C₈×C₈₉

—

#10623494

—

#10623494

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

—

C₂²×C₈₉

#430

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#424

yes

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

#8809

—

maximal, Sylow

C₂×C₈₉

#1570

yes

—

C₂×C₈₉

#1570

yes

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

D₈₉

yes

characteristic

#1744

—

C₂×#334

—

C₈₉

#8809

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

178

—

C₂×C₄⋊C₈

C₂×D₈₉

yes

—

C₂×C₄×C₈

—

C₄³

C₂×D₈₉

yes

—

—

—

178

—

C₂²×C₈

—

C₂²×C₈

—

100

C₂²×C₈

—

178

—

101

C₂×C₄²

Dic₁₇₈

yes

—

102

C₂×C₄²

C₄×D₈₉

yes

characteristic

103

C₂×C₄²

C₄×D₈₉

yes

characteristic

104

C₂×C₄²

C₂²⋊D₈₉

yes

—

105

C₂×C₄²

C₂²×D₈₉

yes

characteristic

106

C₄⋊C₈

—

107

C₄⋊C₈

—

108

C₄⋊C₈

—

356

—

109

C₄⋊C₈

—

178

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₂²×C₄

#13083553

yes

—

113

C₂²×C₄

#13083553

yes

—

114

C₂²×C₄

C₂×C₄×D₈₉

yes

characteristic

115

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

116

C₂²×C₄

C₂×Dic₁₇₈

yes

characteristic

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

356

—

121

C₂×C₈

—

356

—

122

C₂×C₈

—

178

—

123

C₂×C₈

—

178

—

124

C₄²

—

125

C₄²

—

126

C₄²

—

127

C₄²

#13083561

yes

—

128

C₄²

#13083562

yes

—

129

C₄²

D₄×D₈₉

yes

—

130

C₄²

Q₈×D₈₉

yes

—

131

C₂³

C₂×#13083553

yes

characteristic

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

#15667943

yes

—

138

C₂×C₄

#15667943

yes

—

139

C₂×C₄

C₄⋊C₄×D₈₉

yes

—

140

C₂×C₄

#15668008

yes

characteristic

141

C₂×C₄

#15668010

yes

characteristic

142

C₂×C₄

#15668011

yes

—

143

C₂×C₄

#15668018

yes

—

144

C₂×C₄

#15668020

yes

—

145

C₂×C₄

C₂×#13083561

yes

—

146

C₂×C₄

C₂×#13083562

yes

—

147

C₈

—

148

C₈

—

178

—

149

C₈

—

712

—

150

C₂²

#19151467

yes

characteristic

151

C₂²

#19151624

yes

characteristic

152

C₂²

#19151626

yes

characteristic

153

C₂²

C₂×#15667943

yes

—

154

C₂²

#19151646

yes

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

#19151623

yes

—

159

C₄

#19151970

yes

characteristic

160

C₄

#19151972

yes

characteristic

161

C₄

#19151994

yes

—

162

C₂

#26019022

yes

—

163

C₂

#26019024

yes

—

164

C₂

#26020405

yes

characteristic

165

C₂

#26020408

yes

characteristic

166

C₂

#26020426

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₈₉
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1842	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1842	D₈₉
hypocenter	C₈₉	#8809
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

subgroup

G/H

center

C₂²×C₄

C₂×C₂²⋊D₈₉

commutator

C₂²×C₄×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1842

C₂

socle, prime radical

C₂³×C₈₉

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

N	H ≅ G/N	type
88. C₈₉	#8809	semidirect

H ≅ G/N

type

88. C₈₉

#8809

semidirect

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

subgroup	index
#26020404	2
#26019998	2
C₂×#19151348	2
C₈₉×#1842	2
#8809	89

subgroup

index

C₈₉×#1842

Subgroups — #36448180

Conjugacy classes of subgroups

Complements

Maximal subgroups