Group #36467794 — subgroups

Subgroups — #36467794

Order 22784.

262 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 55 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	#26031706	C₂	1	1	yes	maximal, characteristic
2	#26022472	C₂	1	2	yes	maximal
3	#26021832	C₂	1	1	yes	maximal, characteristic
4	#26018247	C₂	1	2	yes	maximal
5	C₈₉×#1328	C₂	1	1	yes	maximal, characteristic
6	#19152210	—	2	1	no	—
7	#19151906	—	2	2	no	—
8	#19151856	C₂²	1	2	yes	—
9	#19151839	—	2	1	no	—
10	#19151809	—	2	2	no	—
11	#19151647	C₂²	1	1	yes	characteristic
12	#19151421	—	2	2	no	—
13	#19151415	—	2	2	no	—
14	#19151368	—	2	1	no	—
15	#19151363	—	2	1	no	—
16	#19151356	C₂²	1	1	yes	characteristic
17	#424×C₈₉	C₂²	1	1	yes	characteristic
18	#349×C₈₉	C₂²	1	2	yes	—
19	C₂×Dic₃₅₆	—	2	1	no	—
20	C₂×Dic₃₅₆	D₄	1	2	yes	—
21	C₂×#13083558	—	2	1	no	—
22	C₂×#13083558	—	2	1	no	—
23	#15668012	—	2	1	no	—
24	#15668005	—	4	2	no	—
25	#15668004	—	2	1	no	—
26	#15668002	—	4	2	no	—
27	#15667991	—	2	2	no	—
28	C₄×Dic₁₇₈	—	2	2	no	—
29	#15667948	—	2	2	no	—
30	#15667946	—	2	1	no	—
31	#15667944	D₄	1	2	yes	—
32	#15667937	—	2	2	no	—
33	#15667937	—	2	2	no	—
34	C₂×M₁₆×C₈₉	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
36	#117×C₈₉	D₄	1	1	yes	characteristic
37	M₃₂×C₈₉	—	2	2	no	—
38	C₄⋊C₈×C₈₉	—	2	2	no	—
39	C₂×Dic₁₇₈	—	2	2	no	—
40	Dic₃₅₆	—	4	2	no	—
41	Dic₃₅₆	—	8	1	no	—
42	#13083558	—	4	1	no	—
43	#13083558	—	4	1	no	—
44	#13083558	—	4	1	no	—
45	#13083558	—	4	1	no	—
46	C₂×#10623494	—	4	1	no	—
47	C₂×#10623494	—	4	1	no	—
48	C₄⋊Dic₈₉	—	2	2	no	—
49	C₄⋊Dic₈₉	—	4	2	no	—
50	#13083553	—	4	2	no	—
51	#13083553	—	4	2	no	—
52	#13083553	—	4	2	no	—
53	C₂²⋊Dic₈₉	—	4	2	no	—
54	C₄×Dic₈₉	—	4	2	no	—
55	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
56	M₁₆×C₈₉	—	4	1	no	—
57	C₂×C₈×C₈₉	—	2	1	no	—
58	C₂×C₈×C₈₉	—	4	1	no	—
59	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
60	C₄⋊C₄×C₈₉	—	2	1	no	—
61	C₂²⋊C₄×C₈₉	SD₁₆	1	2	yes	—
62	C₄²×C₈₉	—	2	1	no	—
63	C₁₆×C₈₉	—	4	2	no	—
64	C₂³×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
65	C₂×C₄×C₈₉	—	2	1	no	—
66	C₂×C₄×C₈₉	—	2	1	no	—
67	C₂×C₄×C₈₉	—	2	2	no	—
68	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
69	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
70	C₂×Dic₈₉	—	4	2	no	—
71	C₂×Dic₈₉	—	4	2	no	—
72	Dic₁₇₈	—	4	2	no	—
73	Dic₁₇₈	—	4	2	no	—
74	C₈×C₈₉	—	2	2	no	—
75	C₈×C₈₉	—	4	1	no	—
76	#10623494	—	8	1	no	—
77	#10623494	—	8	1	no	—
78	C₂²×C₈₉	—	2	1	no	—
79	C₂²×C₈₉	—	4	1	no	—
80	C₂²×C₈₉	#450	1	1	yes	characteristic
81	C₄×C₈₉	—	2	1	no	—
82	C₄×C₈₉	—	4	2	no	—
83	C₄×C₈₉	#454	1	1	yes	characteristic
84	C₄×C₈₉	#458	1	1	yes	characteristic
85	Dic₈₉	—	8	2	no	—
86	Dic₈₉	—	8	2	no	—
87	#12305	—	89	1	no	maximal, Sylow
88	C₂×C₈₉	—	2	1	no	—
89	C₂×C₈₉	—	4	1	no	—
90	C₂×C₈₉	#1616	1	1	yes	characteristic
91	#3261	—	89	1	no	—
92	#2239	—	89	2	no	—
93	#2103	—	89	1	no	—
94	#1342	—	89	2	no	—
95	#1328	D₈₉	1	1	yes	characteristic
96	C₈₉	#12305	1	1	yes	characteristic, complemented, Sylow
97	#541	—	178	1	no	—
98	#479	—	178	2	no	—
99	#470	—	89	2	no	—
100	#467	—	178	1	no	—
101	#462	—	178	2	no	—
102	#430	—	89	1	no	—
103	#424	C₂×D₈₉	1	1	yes	characteristic
104	#365	—	178	2	no	—
105	#362	—	178	2	no	—
106	#349	C₂×D₈₉	1	2	yes	—
107	#344	—	178	1	no	—
108	#341	—	178	1	no	—
109	#338	—	89	1	no	—
110	C₂×Q₁₆	—	89	2	no	—
111	C₂×Q₁₆	—	178	1	no	—
112	C₂×M₁₆	—	178	1	no	—
113	C₂×M₁₆	—	178	1	no	—
114	C₂×M₁₆	C₂²×D₈₉	1	1	yes	characteristic
115	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
116	C₄⋊Q₈	—	178	1	no	—
117	#126	—	356	2	no	—
118	#125	—	356	2	no	—
119	#125	—	178	1	no	—
120	C₂²⋊Q₈	—	178	2	no	—
121	C₄×Q₈	—	178	2	no	—
122	#117	D₃₅₆	1	1	yes	characteristic
123	M₃₂	—	2	2	no	—
124	#108	—	178	2	no	—
125	#107	—	178	1	no	—
126	#106	—	89	2	no	—
127	C₄⋊C₈	—	2	2	no	—
128	#103	—	178	2	no	—
129	#103	—	178	2	no	—
130	C₂×Q₈	—	178	2	no	—
131	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
132	Q₁₆	—	356	2	no	—
133	Q₁₆	—	712	1	no	—
134	M₁₆	—	4	1	no	—
135	M₁₆	—	356	1	no	—
136	M₁₆	—	356	1	no	—
137	M₁₆	—	356	1	no	—
138	M₁₆	—	356	1	no	—
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	—	4	1	no	—
141	C₂×C₈	—	356	1	no	—
142	C₂×C₈	—	356	1	no	—
143	C₂×C₈	D₄×D₈₉	1	2	yes	—
144	C₄⋊C₄	—	2	1	no	—
145	C₄⋊C₄	—	356	2	no	—
146	C₄⋊C₄	—	356	2	no	—
147	C₄⋊C₄	—	356	2	no	—
148	C₄⋊C₄	—	356	2	no	—
149	C₄⋊C₄	—	178	2	no	—
150	C₂²⋊C₄	—	356	2	no	—
151	C₂²⋊C₄	#13083563	1	2	yes	—
152	C₄²	—	2	1	no	—
153	C₄²	—	356	2	no	—
154	C₁₆	—	4	2	no	—
155	C₂³	C₂×#13083563	1	1	yes	characteristic
156	Q₈	—	356	2	no	—
157	Q₈	—	356	2	no	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	356	2	no	—
162	C₂×C₄	—	356	2	no	—
163	C₂×C₄	#15667982	1	1	yes	characteristic
164	C₂×C₄	#15668040	1	1	yes	characteristic
165	C₈	—	2	2	no	—
166	C₈	—	4	1	no	—
167	C₈	—	712	1	no	—
168	C₈	—	712	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	—	4	1	no	—
171	C₂²	#19151751	1	1	yes	characteristic
172	C₄	—	2	1	no	—
173	C₄	—	4	2	no	—
174	C₄	—	712	2	no	—
175	C₄	—	712	2	no	—
176	C₄	#19151773	1	1	yes	characteristic
177	C₄	#19151789	1	1	yes	characteristic
178	C₂	—	2	1	no	—
179	C₂	—	4	1	no	—
180	C₂	#26019291	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26031706

C₂

yes

maximal, characteristic

#26022472

C₂

yes

maximal

#26021832

C₂

yes

maximal, characteristic

#26018247

C₂

yes

maximal

C₈₉×#1328

C₂

yes

maximal, characteristic

#19152210

—

#19151906

—

#19151856

C₂²

yes

—

#19151839

—

#19151809

—

#19151647

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

#424×C₈₉

C₂²

yes

characteristic

#349×C₈₉

C₂²

yes

—

C₂×Dic₃₅₆

—

C₂×Dic₃₅₆

D₄

yes

—

—

—

—

—

—

—

—

C₄×Dic₁₇₈

—

#15667948

—

#15667946

—

#15667944

D₄

yes

—

#15667937

—

#15667937

—

C₂×M₁₆×C₈₉

C₂³

yes

characteristic

C₂²×C₈×C₈₉

D₄

yes

characteristic

#117×C₈₉

D₄

yes

characteristic

M₃₂×C₈₉

—

C₄⋊C₈×C₈₉

—

C₂×Dic₁₇₈

—

Dic₃₅₆

—

Dic₃₅₆

—

—

—

—

—

—

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083553

—

#13083553

—

#13083553

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₄⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

SD₁₆

yes

—

C₄²×C₈₉

—

C₁₆×C₈₉

—

C₂³×C₈₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

—

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#450

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

Dic₈₉

—

#12305

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

yes

—

—

—

—

D₈₉

yes

characteristic

C₈₉

#12305

yes

characteristic, complemented, Sylow

#541

—

178

—

#479

—

178

—

#470

—

100

#467

—

178

—

101

#462

—

178

—

102

#430

—

103

#424

C₂×D₈₉

yes

characteristic

104

#365

—

178

—

105

#362

—

178

—

106

#349

C₂×D₈₉

yes

—

107

#344

—

178

—

108

#341

—

178

—

109

#338

—

110

C₂×Q₁₆

—

111

C₂×Q₁₆

—

178

—

112

C₂×M₁₆

—

178

—

113

C₂×M₁₆

—

178

—

114

C₂×M₁₆

C₂²×D₈₉

yes

characteristic

115

C₂²×C₈

D₃₅₆

yes

characteristic

116

C₄⋊Q₈

—

178

—

117

#126

—

356

—

118

#125

—

356

—

119

#125

—

178

—

120

C₂²⋊Q₈

—

178

—

121

C₄×Q₈

—

178

—

122

#117

D₃₅₆

yes

characteristic

123

M₃₂

—

124

#108

—

178

—

125

#107

—

178

—

126

#106

—

127

C₄⋊C₈

—

128

#103

—

178

—

129

#103

—

178

—

130

C₂×Q₈

—

178

—

131

C₂²×C₄

C₂×D₃₅₆

yes

characteristic

132

Q₁₆

—

356

—

133

Q₁₆

—

712

—

134

M₁₆

—

135

M₁₆

—

356

—

136

M₁₆

—

356

—

137

M₁₆

—

356

—

138

M₁₆

—

356

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

356

—

142

C₂×C₈

—

356

—

143

C₂×C₈

D₄×D₈₉

yes

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

356

—

146

C₄⋊C₄

—

356

—

147

C₄⋊C₄

—

356

—

148

C₄⋊C₄

—

356

—

149

C₄⋊C₄

—

178

—

150

C₂²⋊C₄

—

356

—

151

C₂²⋊C₄

#13083563

yes

—

152

C₄²

—

153

C₄²

—

356

—

154

C₁₆

—

155

C₂³

C₂×#13083563

yes

characteristic

156

Q₈

—

356

—

157

Q₈

—

356

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

356

—

162

C₂×C₄

—

356

—

163

C₂×C₄

#15667982

yes

characteristic

164

C₂×C₄

#15668040

yes

characteristic

165

C₈

—

166

C₈

—

167

C₈

—

712

—

168

C₈

—

712

—

169

C₂²

—

170

C₂²

—

171

C₂²

#19151751

yes

characteristic

172

C₄

—

173

C₄

—

174

C₄

—

712

—

175

C₄

—

712

—

176

C₄

#19151773

yes

characteristic

177

C₄

#19151789

yes

characteristic

178

C₂

—

179

C₂

—

180

C₂

#26019291

yes

characteristic

	subgroup	G/H
center	C₂	#26019291
commutator, cosocle	C₂×M₁₆×C₈₉	C₂³
Frattini	C₂×M₁₆	C₂²×D₈₉
Fitting	C₈₉×#1328	C₂
socle	C₂×C₈₉	#1616
prime radical	C₂³×C₈₉	C₂×SD₁₆
hypercenter	#1328	D₈₉
hypocenter	C₈₉	#12305
LCS	G ▷ C₂×M₁₆×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #1328

subgroup

G/H

center

C₂

#26019291

commutator, cosocle

C₂×M₁₆×C₈₉

C₂³

Frattini

C₂×M₁₆

C₂²×D₈₉

Fitting

C₈₉×#1328

C₂

socle

C₂×C₈₉

#1616

prime radical

C₂³×C₈₉

C₂×SD₁₆

D₈₉

C₈₉

G ▷ C₂×M₁₆×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #1328

N	H ≅ G/N	type
96. C₈₉	#12305	semidirect

H ≅ G/N

type

96. C₈₉

#12305

semidirect

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

subgroup	index
#26031706	2
#26022472	2
#26021832	2
#26018247	2
C₈₉×#1328	2
#12305	89

subgroup

index

C₈₉×#1328

Subgroups — #36467794

Conjugacy classes of subgroups

Complements

Maximal subgroups