Group #36467826 — subgroups

Subgroups — #36467826

Order 22784.

366 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 67 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	#26022328	C₂	1	2	yes	maximal, complemented
2	#26019859	C₂	1	1	yes	maximal, characteristic, complemented
3	#26018291	C₂	1	1	yes	maximal, characteristic, complemented
4	#26018225	C₂	1	2	yes	maximal, complemented
5	C₈₉×#3293	C₂	1	1	yes	maximal, characteristic, complemented
6	#19152033	—	2	2	no	—
7	#19151975	C₂²	1	1	yes	characteristic
8	#19151946	—	2	2	no	—
9	#19151419	C₂²	1	2	yes	—
10	#19151410	—	2	2	no	—
11	C₈×#10623494	C₂²	1	1	yes	characteristic
12	#578×C₈₉	—	2	2	no	—
13	#494×C₈₉	C₂²	1	1	yes	characteristic
14	#473×C₈₉	—	2	2	no	—
15	#456×C₈₉	—	2	2	no	—
16	#443×C₈₉	C₂²	1	2	yes	—
17	C₂×D₇₁₂	—	2	2	no	—
18	#15668009	—	2	1	no	—
19	#15667955	—	4	2	no	—
20	#15667953	—	4	2	no	—
21	#15667952	—	2	2	no	—
22	#15667932	—	4	4	no	—
23	C₄×#10623494	—	2	1	no	—
24	2₋¹⁺⁴×C₈₉	—	2	2	no	—
25	#137×C₈₉	—	2	4	no	—
26	#137×C₈₉	—	4	4	no	—
27	#135×C₈₉	—	4	2	no	—
28	#135×C₈₉	D₄	1	2	yes	—
29	C₂×Q₁₆×C₈₉	—	2	2	no	—
30	C₂×Q₁₆×C₈₉	—	4	2	no	—
31	#131×C₈₉	—	2	2	no	—
32	C₄⋊Q₈×C₈₉	D₄	1	2	yes	—
33	#108×C₈₉	D₄	1	2	yes	—
34	#104×C₈₉	—	2	2	no	—
35	#101×C₈₉	—	2	2	no	—
36	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
37	C₂×D₃₅₆	—	4	2	no	—
38	C₂×D₃₅₆	—	4	1	no	—
39	D₇₁₂	—	4	2	no	—
40	D₇₁₂	—	4	2	no	—
41	C₂×#10623494	—	2	2	no	—
42	#13083549	—	4	2	no	—
43	#41×C₈₉	—	2	2	no	—
44	#41×C₈₉	—	4	4	no	—
45	#41×C₈₉	—	8	2	no	—
46	C₂×Q₈×C₈₉	—	2	2	no	—
47	C₂×Q₈×C₈₉	—	4	4	no	—
48	Q₁₆×C₈₉	—	4	4	no	—
49	Q₁₆×C₈₉	—	4	2	no	—
50	Q₁₆×C₈₉	—	4	2	no	—
51	Q₁₆×C₈₉	D₈	1	2	yes	—
52	SD₁₆×C₈₉	—	2	2	no	—
53	SD₁₆×C₈₉	—	4	4	no	—
54	D₈×C₈₉	D₈	1	2	yes	—
55	M₁₆×C₈₉	—	2	2	no	—
56	M₁₆×C₈₉	—	4	2	no	—
57	C₂×C₈×C₈₉	—	4	2	no	—
58	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
59	C₄⋊C₄×C₈₉	—	4	2	no	—
60	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
61	C₂²×D₈₉	—	8	1	no	—
62	Q₈×C₈₉	—	2	2	no	—
63	Q₈×C₈₉	—	4	4	no	—
64	Q₈×C₈₉	—	8	2	no	—
65	D₄×C₈₉	—	2	2	no	—
66	D₄×C₈₉	—	4	2	no	—
67	D₄×C₈₉	—	8	2	no	—
68	C₂×C₄×C₈₉	—	2	1	no	—
69	C₂×C₄×C₈₉	—	4	2	no	—
70	C₂×C₄×C₈₉	—	4	2	no	—
71	C₂×C₄×C₈₉	—	8	4	no	—
72	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
73	D₃₅₆	—	8	4	no	—
74	D₃₅₆	—	8	2	no	—
75	C₈×C₈₉	—	4	2	no	—
76	C₈×C₈₉	C₂×D₈	1	2	yes	—
77	C₈×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
78	#10623494	—	4	2	no	—
79	C₂²×C₈₉	—	4	2	no	—
80	C₂²×C₈₉	#453	1	1	yes	characteristic
81	C₂×D₈₉	—	16	2	no	—
82	C₂×D₈₉	—	16	1	no	—
83	C₄×C₈₉	—	2	2	no	—
84	C₄×C₈₉	—	4	2	no	—
85	C₄×C₈₉	—	8	2	no	—
86	C₄×C₈₉	#447	1	2	yes	—
87	#12311	—	89	1	no	maximal, Sylow
88	C₂×C₈₉	—	2	1	no	—
89	C₂×C₈₉	—	8	2	no	—
90	C₂×C₈₉	#1619	1	1	yes	characteristic
91	D₈₉	—	32	1	no	—
92	#3293	D₈₉	1	1	yes	characteristic, complemented
93	#2213	—	89	2	no	—
94	#1709	—	89	1	no	—
95	#1358	—	89	1	no	—
96	#1335	—	89	2	no	—
97	C₈₉	#12311	1	1	yes	characteristic, complemented, Sylow
98	#578	—	2	2	no	—
99	#509	—	178	2	no	—
100	#494	C₂×D₈₉	1	1	yes	characteristic
101	#493	—	89	1	no	—
102	#486	—	178	2	no	—
103	#473	—	2	2	no	—
104	#456	—	2	2	no	—
105	#443	C₂×D₈₉	1	2	yes	—
106	#364	—	89	2	no	—
107	#361	—	178	2	no	—
108	C₈²	—	89	1	no	—
109	2₋¹⁺⁴	—	2	2	no	—
110	#137	—	2	4	no	—
111	#137	—	4	4	no	—
112	#135	—	4	2	no	—
113	#135	C₂²⋊D₈₉	1	2	yes	—
114	C₂×Q₁₆	—	2	2	no	—
115	C₂×Q₁₆	—	4	2	no	—
116	C₂×D₈	—	178	2	no	—
117	#131	—	2	2	no	—
118	C₄⋊Q₈	C₂²⋊D₈₉	1	2	yes	—
119	#127	—	178	1	no	—
120	SD₃₂	—	356	2	no	—
121	D₁₆	—	356	2	no	—
122	M₃₂	—	178	2	no	—
123	#108	C₂²⋊D₈₉	1	2	yes	—
124	#104	—	2	2	no	—
125	#102	—	356	4	no	—
126	#101	—	2	2	no	—
127	C₄×C₈	—	178	1	no	—
128	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
129	#41	—	2	2	no	—
130	#41	—	4	4	no	—
131	#41	—	8	2	no	—
132	C₂×Q₈	—	2	2	no	—
133	C₂×Q₈	—	4	4	no	—
134	C₂×D₄	—	356	2	no	—
135	C₂×D₄	—	356	1	no	—
136	Q₁₆	—	4	4	no	—
137	Q₁₆	—	4	2	no	—
138	Q₁₆	—	4	2	no	—
139	Q₁₆	#13083559	1	2	yes	—
140	SD₁₆	—	2	2	no	—
141	SD₁₆	—	4	4	no	—
142	D₈	—	356	2	no	—
143	D₈	—	356	2	no	—
144	D₈	#13083559	1	2	yes	—
145	M₁₆	—	2	2	no	—
146	M₁₆	—	4	2	no	—
147	C₂×C₈	—	4	2	no	—
148	C₂×C₈	—	178	2	no	—
149	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
150	C₄⋊C₄	—	4	2	no	—
151	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
152	C₁₆	—	356	2	no	—
153	C₂³	—	712	1	no	—
154	Q₈	—	2	2	no	—
155	Q₈	—	4	4	no	—
156	Q₈	—	8	2	no	—
157	D₄	—	2	2	no	—
158	D₄	—	4	2	no	—
159	D₄	—	8	2	no	—
160	D₄	—	712	4	no	—
161	D₄	—	712	2	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	8	4	no	—
166	C₂×C₄	#15667981	1	1	yes	characteristic
167	C₈	—	4	2	no	—
168	C₈	—	356	2	no	—
169	C₈	C₂×#13083559	1	2	yes	—
170	C₈	#15668038	1	2	yes	—
171	C₂²	—	4	2	no	—
172	C₂²	—	1424	2	no	—
173	C₂²	—	1424	1	no	—
174	C₂²	#19151766	1	1	yes	characteristic
175	C₄	—	2	2	no	—
176	C₄	—	4	2	no	—
177	C₄	—	8	2	no	—
178	C₄	#19151732	1	2	yes	—
179	C₂	—	2	1	no	—
180	C₂	—	8	2	no	—
181	C₂	—	2848	1	no	—
182	C₂	#26019306	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26022328

C₂

yes

maximal, complemented

#26019859

C₂

yes

maximal, characteristic, complemented

#26018291

C₂

yes

maximal, characteristic, complemented

#26018225

C₂

yes

maximal, complemented

C₈₉×#3293

C₂

yes

maximal, characteristic, complemented

#19152033

—

#19151975

C₂²

yes

characteristic

#19151946

—

#19151419

C₂²

yes

—

#19151410

—

C₈×#10623494

C₂²

yes

characteristic

#578×C₈₉

—

#494×C₈₉

C₂²

yes

characteristic

#473×C₈₉

—

#456×C₈₉

—

#443×C₈₉

C₂²

yes

—

C₂×D₇₁₂

—

—

—

—

—

—

—

2₋¹⁺⁴×C₈₉

—

#137×C₈₉

—

#137×C₈₉

—

#135×C₈₉

—

#135×C₈₉

D₄

yes

—

C₂×Q₁₆×C₈₉

—

C₂×Q₁₆×C₈₉

—

#131×C₈₉

—

C₄⋊Q₈×C₈₉

D₄

yes

—

#108×C₈₉

D₄

yes

—

#104×C₈₉

—

#101×C₈₉

—

C₄×C₈×C₈₉

C₂³

yes

characteristic

C₂×D₃₅₆

—

C₂×D₃₅₆

—

D₇₁₂

—

D₇₁₂

—

C₂×#10623494

—

#13083549

—

#41×C₈₉

—

#41×C₈₉

—

#41×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

Q₁₆×C₈₉

—

Q₁₆×C₈₉

—

Q₁₆×C₈₉

—

Q₁₆×C₈₉

D₈

yes

—

SD₁₆×C₈₉

—

SD₁₆×C₈₉

—

D₈×C₈₉

D₈

yes

—

M₁₆×C₈₉

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₂²×D₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#120

yes

characteristic

D₃₅₆

—

D₃₅₆

—

C₈×C₈₉

—

C₈×C₈₉

C₂×D₈

yes

—

C₈×C₈₉

AGL(1,ℤ/8)

yes

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

#453

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#447

yes

—

#12311

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#1619

yes

characteristic

D₈₉

—

#3293

D₈₉

yes

characteristic, complemented

—

—

—

—

C₈₉

#12311

yes

characteristic, complemented, Sylow

#578

—

#509

—

178

—

100

#494

C₂×D₈₉

yes

characteristic

101

#493

—

102

#486

—

178

—

103

#473

—

104

#456

—

105

#443

C₂×D₈₉

yes

—

106

#364

—

107

#361

—

178

—

108

C₈²

—

109

2₋¹⁺⁴

—

110

#137

—

111

#137

—

112

#135

—

113

#135

C₂²⋊D₈₉

yes

—

114

C₂×Q₁₆

—

115

C₂×Q₁₆

—

116

C₂×D₈

—

178

—

117

#131

—

118

C₄⋊Q₈

C₂²⋊D₈₉

yes

—

119

#127

—

178

—

120

SD₃₂

—

356

—

121

D₁₆

—

356

—

122

M₃₂

—

178

—

123

#108

C₂²⋊D₈₉

yes

—

124

#104

—

125

#102

—

356

—

126

#101

—

127

C₄×C₈

—

178

—

128

C₄×C₈

C₂²×D₈₉

yes

characteristic

129

#41

—

130

#41

—

131

#41

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×D₄

—

356

—

135

C₂×D₄

—

356

—

136

Q₁₆

—

137

Q₁₆

—

138

Q₁₆

—

139

Q₁₆

#13083559

yes

—

140

SD₁₆

—

141

SD₁₆

—

142

D₈

—

356

—

143

D₈

—

356

—

144

D₈

#13083559

yes

—

145

M₁₆

—

146

M₁₆

—

147

C₂×C₈

—

148

C₂×C₈

—

178

—

149

C₂×C₈

C₂×C₂²⋊D₈₉

yes

—

150

C₄⋊C₄

—

151

C₄²

C₂×C₂²⋊D₈₉

yes

characteristic

152

C₁₆

—

356

—

153

C₂³

—

712

—

154

Q₈

—

155

Q₈

—

156

Q₈

—

157

D₄

—

158

D₄

—

159

D₄

—

160

D₄

—

712

—

161

D₄

—

712

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

#15667981

yes

characteristic

167

C₈

—

168

C₈

—

356

—

169

C₈

C₂×#13083559

yes

—

170

C₈

#15668038

yes

—

171

C₂²

—

172

C₂²

—

1424

—

173

C₂²

—

1424

—

174

C₂²

#19151766

yes

characteristic

175

C₄

—

176

C₄

—

177

C₄

—

178

C₄

#19151732

yes

—

179

C₂

—

180

C₂

—

181

C₂

—

2848

—

182

C₂

#26019306

yes

characteristic

	subgroup	G/H
center	C₂	#26019306
commutator, cosocle	C₄×C₈×C₈₉	C₂³
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈₉×#3293	C₂
socle	C₂×C₈₉	#1619
hypercenter	#3293	D₈₉
hypocenter	C₈₉	#12311
LCS	G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #3293

subgroup

G/H

center

C₂

#26019306

commutator, cosocle

C₄×C₈×C₈₉

C₂³

Frattini

C₄×C₈

C₂²×D₈₉

Fitting

C₈₉×#3293

C₂

C₂×C₈₉

D₈₉

C₈₉

G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #3293

N	H ≅ G/N	type
1. #26022328	C₂	semidirect
2. #26019859	C₂	semidirect
3. #26018291	C₂	semidirect
4. #26018225	C₂	semidirect
5. C₈₉×#3293	C₂	semidirect
92. #3293	D₈₉	semidirect
97. C₈₉	#12311	semidirect

H ≅ G/N

type

1. #26022328

C₂

semidirect

2. #26019859

C₂

semidirect

3. #26018291

C₂

semidirect

4. #26018225

C₂

semidirect

5. C₈₉×#3293

C₂

semidirect

92. #3293

D₈₉

semidirect

97. C₈₉

#12311

semidirect

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

subgroup	index
#26022328	2
#26019859	2
#26018291	2
#26018225	2
C₈₉×#3293	2
#12311	89

subgroup

index

C₈₉×#3293

Subgroups — #36467826

Conjugacy classes of subgroups

Complements

Maximal subgroups