Group #36467831 — subgroups

Subgroups — #36467831

Order 22784.

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

normal subgroups only

This group has 372 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	#26022333	C₂	1	2	yes	maximal, complemented
2	#26019857	C₂	1	1	yes	maximal, characteristic, complemented
3	#26018289	C₂	1	1	yes	maximal, characteristic, complemented
4	#26018225	C₂	1	2	yes	maximal, complemented
5	C₈₉×#3292	C₂	1	1	yes	maximal, characteristic
6	#19152038	—	2	2	no	—
7	#19151978	C₂²	1	1	yes	characteristic, complemented
8	#19151944	—	2	2	no	—
9	#19151419	C₂²	1	2	yes	complemented
10	#19151413	—	2	2	no	—
11	C₈×#10623494	C₂²	1	1	yes	characteristic, complemented
12	#576×C₈₉	—	2	2	no	—
13	#493×C₈₉	C₂²	1	1	yes	characteristic
14	#472×C₈₉	—	2	2	no	—
15	#453×C₈₉	—	2	2	no	—
16	#443×C₈₉	C₂²	1	2	yes	—
17	C₂×Dic₃₅₆	—	2	2	no	—
18	#15668012	—	2	1	no	—
19	#15667958	—	4	2	no	—
20	#15667956	—	4	2	no	—
21	#15667952	—	2	2	no	—
22	#15667933	—	4	4	no	—
23	C₄×#10623494	—	2	1	no	—
24	H(5,𝔽₂)×C₈₉	—	2	2	no	—
25	AGL(1,ℤ/8)×C₈₉	—	2	4	no	—
26	AGL(1,ℤ/8)×C₈₉	—	4	4	no	—
27	#135×C₈₉	—	4	2	no	—
28	#135×C₈₉	D₄	1	2	yes	—
29	C₂×D₈×C₈₉	—	2	2	no	—
30	C₂×D₈×C₈₉	—	4	2	no	—
31	#131×C₈₉	—	2	2	no	—
32	#127×C₈₉	D₄	1	2	yes	—
33	#108×C₈₉	D₄	1	2	yes	—
34	#104×C₈₉	—	2	2	no	—
35	#100×C₈₉	—	2	2	no	—
36	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
37	C₂×Dic₁₇₈	—	4	1	no	—
38	Dic₃₅₆	—	4	2	no	—
39	Dic₃₅₆	—	4	2	no	—
40	C₂×#10623494	—	2	2	no	—
41	C₄⋊Dic₈₉	—	4	2	no	—
42	#13083549	—	4	2	no	—
43	#41×C₈₉	—	2	2	no	—
44	#41×C₈₉	—	4	4	no	—
45	C₂×D₄×C₈₉	—	2	2	no	—
46	C₂×D₄×C₈₉	—	4	2	no	—
47	C₂×D₄×C₈₉	—	4	4	no	—
48	C₂×D₄×C₈₉	—	8	2	no	—
49	Q₁₆×C₈₉	D₈	1	2	yes	—
50	SD₁₆×C₈₉	—	2	2	no	—
51	SD₁₆×C₈₉	—	4	4	no	—
52	D₈×C₈₉	—	4	4	no	—
53	D₈×C₈₉	—	4	2	no	—
54	D₈×C₈₉	—	4	2	no	—
55	D₈×C₈₉	D₈	1	2	yes	—
56	M₁₆×C₈₉	—	2	2	no	—
57	M₁₆×C₈₉	—	4	2	no	—
58	C₂×C₈×C₈₉	—	4	2	no	—
59	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
60	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
61	C₂³×C₈₉	—	4	2	no	—
62	C₂³×C₈₉	—	8	2	no	—
63	Q₈×C₈₉	—	2	2	no	—
64	D₄×C₈₉	—	2	2	no	—
65	D₄×C₈₉	—	4	2	no	—
66	D₄×C₈₉	—	4	4	no	—
67	D₄×C₈₉	—	8	4	no	—
68	D₄×C₈₉	—	8	4	no	—
69	C₂×C₄×C₈₉	—	2	1	no	—
70	C₂×C₄×C₈₉	—	4	2	no	—
71	C₂×C₄×C₈₉	—	8	2	no	—
72	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
73	C₂×Dic₈₉	—	8	1	no	—
74	Dic₁₇₈	—	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	—	8	2	no	—
81	C₂²×C₈₉	—	16	2	no	—
82	C₂²×C₈₉	—	32	2	no	—
83	C₂²×C₈₉	#453	1	1	yes	characteristic
84	C₄×C₈₉	—	2	2	no	—
85	C₄×C₈₉	—	4	2	no	—
86	C₄×C₈₉	#447	1	2	yes	—
87	Dic₈₉	—	16	1	no	—
88	#12312	—	89	1	no	maximal, Sylow
89	C₂×C₈₉	—	2	1	no	—
90	C₂×C₈₉	—	8	2	no	—
91	C₂×C₈₉	—	16	2	no	—
92	C₂×C₈₉	#1619	1	1	yes	characteristic
93	#3292	D₈₉	1	1	yes	characteristic
94	#2214	—	89	2	no	—
95	#1709	—	89	1	no	—
96	#1357	—	89	1	no	—
97	#1335	—	89	2	no	—
98	C₈₉	#12312	1	1	yes	characteristic, complemented, Sylow
99	#576	—	2	2	no	—
100	#510	—	178	2	no	—
101	#494	—	89	1	no	—
102	#493	C₂×D₈₉	1	1	yes	characteristic
103	#486	—	178	2	no	—
104	#472	—	2	2	no	—
105	#453	—	2	2	no	—
106	#443	C₂×D₈₉	1	2	yes	—
107	#364	—	89	2	no	—
108	#362	—	178	2	no	—
109	C₈²	—	89	1	no	—
110	H(5,𝔽₂)	—	2	2	no	—
111	AGL(1,ℤ/8)	—	2	4	no	—
112	AGL(1,ℤ/8)	—	4	4	no	—
113	#135	—	4	2	no	—
114	#135	C₂²⋊D₈₉	1	2	yes	—
115	C₂×Q₁₆	—	178	2	no	—
116	C₂×D₈	—	2	2	no	—
117	C₂×D₈	—	4	2	no	—
118	#131	—	2	2	no	—
119	C₄⋊Q₈	—	178	1	no	—
120	#127	C₂²⋊D₈₉	1	2	yes	—
121	Q₃₂	—	356	2	no	—
122	SD₃₂	—	356	2	no	—
123	M₃₂	—	178	2	no	—
124	#108	C₂²⋊D₈₉	1	2	yes	—
125	#104	—	2	2	no	—
126	#102	—	356	4	no	—
127	#100	—	2	2	no	—
128	C₄×C₈	—	178	1	no	—
129	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
130	#41	—	2	2	no	—
131	#41	—	4	4	no	—
132	C₂×Q₈	—	356	1	no	—
133	C₂×D₄	—	2	2	no	—
134	C₂×D₄	—	4	2	no	—
135	C₂×D₄	—	4	4	no	—
136	C₂×D₄	—	8	2	no	—
137	Q₁₆	—	356	2	no	—
138	Q₁₆	—	356	2	no	—
139	Q₁₆	#13083559	1	2	yes	—
140	SD₁₆	—	2	2	no	—
141	SD₁₆	—	4	4	no	—
142	D₈	—	4	4	no	—
143	D₈	—	4	2	no	—
144	D₈	—	4	2	no	—
145	D₈	#13083559	1	2	yes	—
146	M₁₆	—	2	2	no	—
147	M₁₆	—	4	2	no	—
148	C₂×C₈	—	4	2	no	—
149	C₂×C₈	—	178	2	no	—
150	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
151	C₄⋊C₄	—	356	2	no	—
152	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
153	C₁₆	—	356	2	no	—
154	C₂³	—	4	2	no	—
155	C₂³	—	8	2	no	—
156	Q₈	—	2	2	no	—
157	Q₈	—	712	2	no	—
158	D₄	—	2	2	no	—
159	D₄	—	4	2	no	—
160	D₄	—	4	4	no	—
161	D₄	—	8	4	no	—
162	D₄	—	8	4	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	8	2	no	—
166	C₂×C₄	—	712	1	no	—
167	C₂×C₄	#15667981	1	1	yes	characteristic
168	C₈	—	4	2	no	—
169	C₈	—	356	2	no	—
170	C₈	C₂×#13083559	1	2	yes	—
171	C₈	#15668038	1	2	yes	—
172	C₂²	—	4	2	no	—
173	C₂²	—	8	2	no	—
174	C₂²	—	16	2	no	—
175	C₂²	—	32	2	no	—
176	C₂²	#19151766	1	1	yes	characteristic
177	C₄	—	2	2	no	—
178	C₄	—	4	2	no	—
179	C₄	—	1424	1	no	—
180	C₄	#19151732	1	2	yes	—
181	C₂	—	2	1	no	—
182	C₂	—	8	2	no	—
183	C₂	—	16	2	no	—
184	C₂	#26019306	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26022333

C₂

yes

maximal, complemented

#26019857

C₂

yes

maximal, characteristic, complemented

#26018289

C₂

yes

maximal, characteristic, complemented

#26018225

C₂

yes

maximal, complemented

C₈₉×#3292

C₂

yes

maximal, characteristic

#19152038

—

#19151978

C₂²

yes

characteristic, complemented

#19151944

—

#19151419

C₂²

yes

complemented

#19151413

—

C₈×#10623494

C₂²

yes

characteristic, complemented

#576×C₈₉

—

#493×C₈₉

C₂²

yes

characteristic

#472×C₈₉

—

#453×C₈₉

—

#443×C₈₉

C₂²

yes

—

C₂×Dic₃₅₆

—

—

—

—

—

—

—

H(5,𝔽₂)×C₈₉

—

AGL(1,ℤ/8)×C₈₉

—

AGL(1,ℤ/8)×C₈₉

—

#135×C₈₉

—

#135×C₈₉

D₄

yes

—

C₂×D₈×C₈₉

—

C₂×D₈×C₈₉

—

#131×C₈₉

—

#127×C₈₉

D₄

yes

—

#108×C₈₉

D₄

yes

—

#104×C₈₉

—

#100×C₈₉

—

C₄×C₈×C₈₉

C₂³

yes

characteristic

C₂×Dic₁₇₈

—

Dic₃₅₆

—

Dic₃₅₆

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

#13083549

—

#41×C₈₉

—

#41×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

Q₁₆×C₈₉

D₈

yes

—

SD₁₆×C₈₉

—

SD₁₆×C₈₉

—

D₈×C₈₉

—

D₈×C₈₉

—

D₈×C₈₉

—

D₈×C₈₉

D₈

yes

—

M₁₆×C₈₉

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₂³×C₈₉

—

C₂³×C₈₉

—

Q₈×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#120

yes

characteristic

C₂×Dic₈₉

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

C₂×D₈

yes

—

C₈×C₈₉

AGL(1,ℤ/8)

yes

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#453

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#447

yes

—

Dic₈₉

—

#12312

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#1619

yes

characteristic

#3292

D₈₉

yes

—

—

—

—

C₈₉

#12312

yes

characteristic, complemented, Sylow

#576

—

100

#510

—

178

—

101

#494

—

102

#493

C₂×D₈₉

yes

characteristic

103

#486

—

178

—

104

#472

—

105

#453

—

106

#443

C₂×D₈₉

yes

—

107

#364

—

108

#362

—

178

—

109

C₈²

—

110

H(5,𝔽₂)

—

111

AGL(1,ℤ/8)

—

112

AGL(1,ℤ/8)

—

113

#135

—

114

#135

C₂²⋊D₈₉

yes

—

115

C₂×Q₁₆

—

178

—

116

C₂×D₈

—

117

C₂×D₈

—

118

#131

—

119

C₄⋊Q₈

—

178

—

120

#127

C₂²⋊D₈₉

yes

—

121

Q₃₂

—

356

—

122

SD₃₂

—

356

—

123

M₃₂

—

178

—

124

#108

C₂²⋊D₈₉

yes

—

125

#104

—

126

#102

—

356

—

127

#100

—

128

C₄×C₈

—

178

—

129

C₄×C₈

C₂²×D₈₉

yes

characteristic

130

#41

—

131

#41

—

132

C₂×Q₈

—

356

—

133

C₂×D₄

—

134

C₂×D₄

—

135

C₂×D₄

—

136

C₂×D₄

—

137

Q₁₆

—

356

—

138

Q₁₆

—

356

—

139

Q₁₆

#13083559

yes

—

140

SD₁₆

—

141

SD₁₆

—

142

D₈

—

143

D₈

—

144

D₈

—

145

D₈

#13083559

yes

—

146

M₁₆

—

147

M₁₆

—

148

C₂×C₈

—

149

C₂×C₈

—

178

—

150

C₂×C₈

C₂×C₂²⋊D₈₉

yes

—

151

C₄⋊C₄

—

356

—

152

C₄²

C₂×C₂²⋊D₈₉

yes

characteristic

153

C₁₆

—

356

—

154

C₂³

—

155

C₂³

—

156

Q₈

—

157

Q₈

—

712

—

158

D₄

—

159

D₄

—

160

D₄

—

161

D₄

—

162

D₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

712

—

167

C₂×C₄

#15667981

yes

characteristic

168

C₈

—

169

C₈

—

356

—

170

C₈

C₂×#13083559

yes

—

171

C₈

#15668038

yes

—

172

C₂²

—

173

C₂²

—

174

C₂²

—

175

C₂²

—

176

C₂²

#19151766

yes

characteristic

177

C₄

—

178

C₄

—

179

C₄

—

1424

—

180

C₄

#19151732

yes

—

181

C₂

—

182

C₂

—

183

C₂

—

184

C₂

#26019306

yes

characteristic

	subgroup	G/H
center	C₂	#26019306
commutator, cosocle	C₄×C₈×C₈₉	C₂³
Frattini	C₄×C₈	C₂²×D₈₉
Fitting, prime radical	C₈₉×#3292	C₂
socle	C₂×C₈₉	#1619
hypercenter	#3292	D₈₉
hypocenter	C₈₉	#12312
LCS	G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #3292

subgroup

G/H

center

C₂

#26019306

commutator, cosocle

C₄×C₈×C₈₉

C₂³

Frattini

C₄×C₈

C₂²×D₈₉

Fitting, prime radical

C₈₉×#3292

C₂

C₂×C₈₉

D₈₉

C₈₉

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

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #3292

N	H ≅ G/N	type
1. #26022333	C₂	semidirect
2. #26019857	C₂	semidirect
3. #26018289	C₂	semidirect
4. #26018225	C₂	semidirect
7. #19151978	C₂²	semidirect
9. #19151419	C₂²	semidirect
11. C₈×#10623494	C₂²	semidirect
98. C₈₉	#12312	semidirect

H ≅ G/N

type

1. #26022333

C₂

semidirect

2. #26019857

C₂

semidirect

3. #26018289

C₂

semidirect

4. #26018225

C₂

semidirect

7. #19151978

C₂²

semidirect

9. #19151419

C₂²

semidirect

11. C₈×#10623494

C₂²

semidirect

98. C₈₉

#12312

semidirect

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

subgroup	index
#26022333	2
#26019857	2
#26018289	2
#26018225	2
C₈₉×#3292	2
#12312	89

subgroup

index

C₈₉×#3292

Subgroups — #36467831

Conjugacy classes of subgroups

Complements

Maximal subgroups