Group #36441120 — subgroups

Subgroups — #36441120

Order 22784.

204 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 45 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	#26021575	C₂	1	1	yes	maximal, characteristic
2	#26018122	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#1322	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₃₅₆	C₄	1	1	yes	characteristic
5	#19151655	—	2	1	no	—
6	C₂×#15667937	—	2	1	no	—
7	#19151504	—	2	1	no	—
8	#19151361	C₄	1	1	yes	characteristic
9	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
10	C₂²⋊C₁₆×C₈₉	—	2	1	no	—
11	C₂²×Dic₁₇₈	—	2	1	no	—
12	C₂×Dic₃₅₆	—	4	1	no	—
13	C₂×Dic₃₅₆	—	4	2	no	—
14	C₂×C₄⋊Dic₈₉	—	2	1	no	—
15	C₄⋊#10623494	—	4	1	no	—
16	#15667937	—	4	2	no	—
17	C₂²⋊#10623494	—	4	1	no	—
18	#15667923	—	2	2	no	—
19	#15667919	—	4	1	no	—
20	#15667919	—	4	1	no	—
21	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
23	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
24	C₂×C₁₆×C₈₉	—	4	1	no	—
25	C₄⋊C₈×C₈₉	—	4	1	no	—
26	C₂×Dic₁₇₈	—	2	2	no	—
27	C₂×Dic₁₇₈	—	4	1	no	—
28	C₂×Dic₁₇₈	—	4	1	no	—
29	C₂×Dic₁₇₈	—	4	1	no	—
30	C₂×Dic₁₇₈	—	4	1	no	—
31	C₂×Dic₁₇₈	—	4	1	no	—
32	C₂²×Dic₈₉	—	4	1	no	—
33	C₂²×Dic₈₉	—	4	1	no	—
34	Dic₃₅₆	—	4	2	no	—
35	Dic₃₅₆	—	4	2	no	—
36	C₂×#10623494	—	4	2	no	—
37	C₄⋊Dic₈₉	—	4	1	no	—
38	C₄⋊Dic₈₉	—	4	1	no	—
39	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
41	C₂²×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
42	C₂×C₈×C₈₉	—	2	1	no	—
43	C₂×C₈×C₈₉	—	2	1	no	—
44	C₂×C₈×C₈₉	—	4	1	no	—
45	C₂×C₈×C₈₉	—	4	1	no	—
46	C₄²×C₈₉	—	2	1	no	—
47	C₄²×C₈₉	—	2	1	no	—
48	C₁₆×C₈₉	—	8	1	no	—
49	C₂³×C₈₉	#102	1	1	yes	characteristic
50	C₂×C₄×C₈₉	—	2	1	no	—
51	C₂×C₄×C₈₉	—	2	1	no	—
52	C₂×C₄×C₈₉	—	2	1	no	—
53	C₂×C₄×C₈₉	—	2	1	no	—
54	C₂×C₄×C₈₉	—	2	1	no	—
55	C₂×C₄×C₈₉	—	2	1	no	—
56	C₂×C₄×C₈₉	—	4	1	no	—
57	C₂×C₄×C₈₉	#99	1	1	yes	characteristic
58	C₂×C₄×C₈₉	#104	1	1	yes	characteristic
59	C₂×Dic₈₉	—	4	2	no	—
60	C₂×Dic₈₉	—	8	1	no	—
61	C₂×Dic₈₉	—	8	1	no	—
62	C₂×Dic₈₉	—	8	1	no	—
63	C₂×Dic₈₉	—	8	1	no	—
64	C₂×Dic₈₉	—	8	1	no	—
65	Dic₁₇₈	—	4	2	no	—
66	Dic₁₇₈	—	4	2	no	—
67	Dic₁₇₈	—	8	1	no	—
68	Dic₁₇₈	—	8	1	no	—
69	C₈×C₈₉	—	4	1	no	—
70	C₈×C₈₉	—	4	1	no	—
71	#10623494	—	8	2	no	—
72	C₂²×C₈₉	—	2	1	no	—
73	C₂²×C₈₉	—	2	1	no	—
74	C₂²×C₈₉	#327	1	1	yes	characteristic
75	C₂²×C₈₉	#359	1	1	yes	characteristic
76	C₂²×C₈₉	#362	1	1	yes	characteristic
77	C₄×C₈₉	—	2	1	no	—
78	C₄×C₈₉	—	4	1	no	—
79	C₄×C₈₉	—	4	1	no	—
80	C₄×C₈₉	AΣL(1,𝔽₁₆)	1	1	yes	characteristic
81	C₄×C₈₉	#355	1	1	yes	characteristic
82	Dic₈₉	—	8	2	no	—
83	Dic₈₉	—	16	1	no	—
84	#7082	—	89	1	no	maximal, Sylow
85	C₂×C₈₉	—	2	1	no	—
86	C₂×C₈₉	—	2	1	no	—
87	C₂×C₈₉	#1340	1	1	yes	characteristic
88	C₂×C₈₉	#1345	1	1	yes	characteristic
89	C₂×C₈₉	#1350	1	1	yes	characteristic
90	#2056	—	89	1	no	—
91	#1322	D₈₉	1	1	yes	characteristic
92	#1290	—	89	1	no	—
93	C₈₉	#7082	1	1	yes	characteristic, complemented, Sylow
94	C₂²×Q₁₆	—	89	1	no	—
95	#432	—	178	1	no	—
96	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
97	C₂×#103	—	178	1	no	—
98	#391	—	178	1	no	—
99	C₂²⋊C₁₆	—	2	1	no	—
100	#340	—	89	1	no	—
101	C₂²×Q₈	—	178	1	no	—
102	C₂×Q₁₆	—	356	1	no	—
103	C₂×Q₁₆	—	356	2	no	—
104	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
105	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
106	C₂×C₄⋊C₄	—	178	1	no	—
107	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
108	C₂×C₁₆	—	4	1	no	—
109	C₄⋊C₈	—	4	1	no	—
110	C₄⋊C₈	—	356	1	no	—
111	#103	—	356	2	no	—
112	C₂²⋊C₈	—	356	1	no	—
113	#97	—	178	2	no	—
114	#95	—	356	1	no	—
115	#95	—	356	1	no	—
116	C₂×Q₈	—	356	1	no	—
117	C₂×Q₈	—	356	1	no	—
118	C₂×Q₈	—	356	1	no	—
119	C₂×Q₈	—	356	1	no	—
120	C₂×Q₈	—	356	1	no	—
121	C₂×Q₈	—	178	2	no	—
122	C₂²×C₄	—	356	1	no	—
123	C₂²×C₄	—	356	1	no	—
124	C₂²×C₄	#13083551	1	1	yes	characteristic
125	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
126	C₂²×C₄	#13083563	1	1	yes	characteristic
127	Q₁₆	—	356	2	no	—
128	Q₁₆	—	356	2	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	4	1	no	—
132	C₂×C₈	—	4	1	no	—
133	C₂×C₈	—	356	2	no	—
134	C₄⋊C₄	—	356	1	no	—
135	C₄⋊C₄	—	356	1	no	—
136	C₄²	—	2	1	no	—
137	C₄²	—	2	1	no	—
138	C₁₆	—	8	1	no	—
139	C₂³	#15667934	1	1	yes	characteristic
140	Q₈	—	356	2	no	—
141	Q₈	—	356	2	no	—
142	Q₈	—	712	1	no	—
143	Q₈	—	712	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	356	2	no	—
152	C₂×C₄	—	712	1	no	—
153	C₂×C₄	—	712	1	no	—
154	C₂×C₄	—	712	1	no	—
155	C₂×C₄	—	712	1	no	—
156	C₂×C₄	—	712	1	no	—
157	C₂×C₄	#15667926	1	1	yes	characteristic
158	C₂×C₄	#15667938	1	1	yes	characteristic
159	C₈	—	4	1	no	—
160	C₈	—	4	1	no	—
161	C₈	—	712	2	no	—
162	C₂²	—	2	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	#19151333	1	1	yes	characteristic
165	C₂²	#19151406	1	1	yes	characteristic
166	C₂²	#19151415	1	1	yes	characteristic
167	C₄	—	2	1	no	—
168	C₄	—	4	1	no	—
169	C₄	—	4	1	no	—
170	C₄	—	1424	1	no	—
171	C₄	—	712	2	no	—
172	C₄	#19151384	1	1	yes	characteristic
173	C₄	#19151394	1	1	yes	characteristic
174	C₂	—	2	1	no	—
175	C₂	—	2	1	no	—
176	C₂	#26018241	1	1	yes	characteristic
177	C₂	#26018256	1	1	yes	characteristic
178	C₂	#26018271	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26021575

C₂

yes

maximal, characteristic

#26018122

C₂

yes

maximal, characteristic

C₈₉×#1322

C₂

yes

maximal, characteristic

C₂²×Dic₃₅₆

C₄

yes

—

—

—

C₄

yes

characteristic

C₂×C₄⋊C₈×C₈₉

C₂²

yes

characteristic

C₂²⋊C₁₆×C₈₉

—

C₂²×Dic₁₇₈

—

C₂×Dic₃₅₆

—

C₂×Dic₃₅₆

—

C₂×C₄⋊Dic₈₉

—

—

—

—

—

—

—

C₂²×C₈×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₈₉

D₄

yes

characteristic

C₂×C₄²×C₈₉

D₄

yes

characteristic

C₂×C₁₆×C₈₉

—

C₄⋊C₈×C₈₉

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

Dic₃₅₆

—

Dic₃₅₆

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂²×C₄×C₈₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₈₉

D₈

yes

characteristic

C₂²×C₄×C₈₉

SD₁₆

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₁₆×C₈₉

—

C₂³×C₈₉

#102

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

AΣL(1,𝔽₁₆)

yes

characteristic

C₄×C₈₉

#355

yes

characteristic

Dic₈₉

—

Dic₈₉

—

#7082

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

D₈₉

yes

characteristic

#1290

—

C₈₉

#7082

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

#432

—

178

—

C₂×C₄⋊C₈

C₂×D₈₉

yes

characteristic

C₂×#103

—

178

—

#391

—

178

—

C₂²⋊C₁₆

—

100

#340

—

101

C₂²×Q₈

—

178

—

102

C₂×Q₁₆

—

356

—

103

C₂×Q₁₆

—

356

—

104

C₂²×C₈

C₄×D₈₉

yes

characteristic

105

C₂²×C₈

C₂²⋊D₈₉

yes

characteristic

106

C₂×C₄⋊C₄

—

178

—

107

C₂×C₄²

D₃₅₆

yes

characteristic

108

C₂×C₁₆

—

109

C₄⋊C₈

—

110

C₄⋊C₈

—

356

—

111

#103

—

356

—

112

C₂²⋊C₈

—

356

—

113

#97

—

178

—

114

#95

—

356

—

115

#95

—

356

—

116

C₂×Q₈

—

356

—

117

C₂×Q₈

—

356

—

118

C₂×Q₈

—

356

—

119

C₂×Q₈

—

356

—

120

C₂×Q₈

—

356

—

121

C₂×Q₈

—

178

—

122

C₂²×C₄

—

356

—

123

C₂²×C₄

—

356

—

124

C₂²×C₄

#13083551

yes

characteristic

125

C₂²×C₄

D₇₁₂

yes

characteristic

126

C₂²×C₄

#13083563

yes

characteristic

127

Q₁₆

—

356

—

128

Q₁₆

—

356

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

356

—

134

C₄⋊C₄

—

356

—

135

C₄⋊C₄

—

356

—

136

C₄²

—

137

C₄²

—

138

C₁₆

—

139

C₂³

#15667934

yes

characteristic

140

Q₈

—

356

—

141

Q₈

—

356

—

142

Q₈

—

712

—

143

Q₈

—

712

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

356

—

152

C₂×C₄

—

712

—

153

C₂×C₄

—

712

—

154

C₂×C₄

—

712

—

155

C₂×C₄

—

712

—

156

C₂×C₄

—

712

—

157

C₂×C₄

#15667926

yes

characteristic

158

C₂×C₄

#15667938

yes

characteristic

159

C₈

—

160

C₈

—

161

C₈

—

712

—

162

C₂²

—

163

C₂²

—

164

C₂²

#19151333

yes

characteristic

165

C₂²

#19151406

yes

characteristic

166

C₂²

#19151415

yes

characteristic

167

C₄

—

168

C₄

—

169

C₄

—

170

C₄

—

1424

—

171

C₄

—

712

—

172

C₄

#19151384

yes

characteristic

173

C₄

#19151394

yes

characteristic

174

C₂

—

175

C₂

—

176

C₂

#26018241

yes

characteristic

177

C₂

#26018256

yes

characteristic

178

C₂

#26018271

yes

characteristic

	subgroup	G/H
center	C₂²	#19151333
commutator	C₂²×C₈×C₈₉	C₂×C₄
Frattini	C₂×C₄⋊C₈	C₂×D₈₉
Fitting	C₈₉×#1322	C₂
socle	C₂²×C₈₉	#327
prime radical	C₂³×C₈₉	#102
cosocle	C₂×C₄⋊C₈×C₈₉	C₂²
hypercenter	#1322	D₈₉
hypocenter	C₈₉	#7082
LCS	G ▷ C₂²×C₈×C₈₉ ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈ ◁ #1322

subgroup

G/H

center

C₂²

#19151333

commutator

C₂²×C₈×C₈₉

C₂×C₄

Frattini

C₂×C₄⋊C₈

C₂×D₈₉

Fitting

C₈₉×#1322

C₂

C₂²×C₈₉

C₂³×C₈₉

C₂×C₄⋊C₈×C₈₉

C₂²

D₈₉

C₈₉

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈ ◁ #1322

N	H ≅ G/N	type
93. C₈₉	#7082	semidirect

H ≅ G/N

type

93. C₈₉

#7082

semidirect

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

subgroup	index
#26021575	2
#26018122	2
C₈₉×#1322	2
#7082	89

subgroup

index

#26021575

#26018122

C₈₉×#1322

#7082

Subgroups — #36441120

Conjugacy classes of subgroups

Complements

Maximal subgroups