Group #36448215 — subgroups

Subgroups — #36448215

Order 22784.

438 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 203 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	#26020412	C₂	1	1	yes	maximal, characteristic
2	#26020411	C₂	1	2	yes	maximal
3	#26020379	C₂	1	2	yes	maximal
4	C₂×#19151319	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1844	C₂	1	1	yes	maximal, characteristic
6	C₂×#15667946	—	2	2	no	—
7	C₂×#15667945	—	2	4	no	—
8	C₂×C₄×#10623494	C₂²	1	2	yes	—
9	#19151468	C₂²	1	2	yes	—
10	#19151360	—	2	4	no	—
11	#19151319	C₄	1	4	yes	complemented
12	C₂×#106×C₈₉	—	2	2	no	—
13	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
14	#384×C₈₉	C₂²	1	1	yes	characteristic
15	#384×C₈₉	C₂²	1	1	yes	characteristic
16	C₂²×#10623494	D₄	1	4	yes	—
17	C₂×C₄⋊Dic₈₉	—	2	2	no	—
18	C₂×C₄⋊Dic₈₉	—	2	2	no	—
19	C₂×C₄⋊Dic₈₉	—	2	2	no	—
20	#15667946	—	4	4	no	—
21	#15667945	—	4	8	no	—
22	C₄×#10623494	C₂×C₄	1	4	yes	—
23	C₄×#10623494	Q₈	1	4	yes	—
24	C₂²×C₈×C₈₉	D₄	1	2	yes	—
25	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
26	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
27	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
28	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
29	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
30	#106×C₈₉	—	4	4	no	—
31	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₈₉	Q₈	1	2	yes	—
33	C₂²×Dic₈₉	—	4	2	no	—
34	C₂×#10623494	—	2	4	no	—
35	C₂×#10623494	—	2	4	no	—
36	C₂×#10623494	C₄⋊C₄	1	8	yes	—
37	C₄⋊Dic₈₉	—	4	4	no	—
38	C₄⋊Dic₈₉	—	4	4	no	—
39	C₄⋊Dic₈₉	—	4	4	no	—
40	C₂²×C₄×C₈₉	—	4	1	no	—
41	C₂²×C₄×C₈₉	—	4	1	no	—
42	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
44	C₂×C₈×C₈₉	C₄⋊C₄	1	4	yes	—
45	C₂×C₈×C₈₉	D₈	1	4	yes	—
46	C₂×C₈×C₈₉	Q₁₆	1	4	yes	—
47	C₄⋊C₄×C₈₉	—	4	2	no	—
48	C₄⋊C₄×C₈₉	—	4	2	no	—
49	C₄⋊C₄×C₈₉	—	4	4	no	—
50	C₄⋊C₄×C₈₉	—	4	4	no	—
51	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
53	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
54	C₂³×C₈₉	#127	1	1	yes	characteristic
55	C₂×C₄×C₈₉	—	8	1	no	—
56	C₂×C₄×C₈₉	—	8	1	no	—
57	C₂×C₄×C₈₉	—	8	2	no	—
58	C₂×C₄×C₈₉	—	8	2	no	—
59	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
60	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
62	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
63	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
64	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
65	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	4	yes	—
66	C₂×C₄×C₈₉	#137	1	4	yes	—
67	C₂×Dic₈₉	—	8	2	no	—
68	C₂×Dic₈₉	—	8	2	no	—
69	C₂×Dic₈₉	—	8	2	no	—
70	C₈×C₈₉	#107	1	8	yes	complemented
71	#10623494	—	2	8	no	—
72	C₂²×C₈₉	#384	1	1	yes	characteristic
73	C₂²×C₈₉	#493	1	1	yes	characteristic
74	C₂²×C₈₉	#494	1	1	yes	characteristic
75	C₂²×C₈₉	#496	1	2	yes	—
76	C₂²×C₈₉	#497	1	2	yes	—
77	C₄×C₈₉	—	16	1	no	—
78	C₄×C₈₉	—	16	1	no	—
79	C₄×C₈₉	C₂×#107	1	2	yes	—
80	C₄×C₈₉	#428	1	4	yes	—
81	C₄×C₈₉	#500	1	2	yes	—
82	C₄×C₈₉	#501	1	4	yes	—
83	Dic₈₉	—	16	2	no	—
84	#8820	—	89	1	no	maximal, Sylow
85	C₂×C₈₉	#1721	1	1	yes	characteristic
86	C₂×C₈₉	#1722	1	1	yes	characteristic
87	C₂×C₈₉	#1723	1	2	yes	—
88	C₂×C₈₉	#1845	1	1	yes	characteristic
89	C₂×C₈₉	#1847	1	2	yes	—
90	#1845	—	89	1	no	—
91	#1845	—	89	2	no	—
92	#1844	D₈₉	1	1	yes	characteristic
93	#1834	—	89	2	no	—
94	C₂×#322	—	89	1	no	—
95	C₈₉	#8820	1	1	yes	characteristic, complemented, Sylow
96	C₂×#107	—	178	4	no	—
97	C₂×#107	—	178	2	no	—
98	C₂×#106	—	2	2	no	—
99	C₂×C₄×C₈	—	89	2	no	—
100	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
101	#384	—	89	2	no	—
102	#384	C₂×D₈₉	1	1	yes	characteristic
103	#384	C₂×D₈₉	1	1	yes	characteristic
104	#340	—	178	4	no	—
105	#322	—	89	4	no	—
106	C₂²×C₈	—	89	4	no	—
107	C₂²×C₈	C₂²⋊D₈₉	1	2	yes	—
108	C₂×C₄⋊C₄	—	2	2	no	—
109	C₂×C₄⋊C₄	—	2	2	no	—
110	C₂×C₄⋊C₄	—	2	1	no	—
111	C₂×C₄⋊C₄	—	2	1	no	—
112	C₂×C₄⋊C₄	—	178	2	no	—
113	C₂×C₄⋊C₄	—	178	2	no	—
114	C₂×C₄⋊C₄	—	178	2	no	—
115	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
116	#107	—	356	8	no	—
117	#107	—	356	4	no	—
118	#106	—	4	4	no	—
119	C₄×C₈	—	89	4	no	—
120	C₄×C₈	—	89	4	no	—
121	C₄×C₈	Dic₁₇₈	1	2	yes	—
122	C₄×C₈	C₄×D₈₉	1	2	yes	—
123	C₂²×C₄	—	4	1	no	—
124	C₂²×C₄	—	4	1	no	—
125	C₂²×C₄	—	356	2	no	—
126	C₂²×C₄	D₄×D₈₉	1	2	yes	—
127	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
128	C₂×C₈	—	89	8	no	—
129	C₂×C₈	—	178	4	no	—
130	C₂×C₈	—	178	4	no	—
131	C₂×C₈	#13083553	1	4	yes	—
132	C₂×C₈	#13083559	1	4	yes	—
133	C₂×C₈	#13083564	1	4	yes	—
134	C₄⋊C₄	—	4	2	no	—
135	C₄⋊C₄	—	4	2	no	—
136	C₄⋊C₄	—	4	4	no	—
137	C₄⋊C₄	—	4	4	no	—
138	C₄⋊C₄	—	356	4	no	—
139	C₄⋊C₄	—	356	4	no	—
140	C₄⋊C₄	—	356	4	no	—
141	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
142	C₄²	Q₈×D₈₉	1	2	yes	—
143	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
144	C₂³	#15668008	1	1	yes	characteristic
145	C₂×C₄	—	8	1	no	—
146	C₂×C₄	—	8	1	no	—
147	C₂×C₄	—	8	2	no	—
148	C₂×C₄	—	8	2	no	—
149	C₂×C₄	—	712	2	no	—
150	C₂×C₄	—	712	2	no	—
151	C₂×C₄	—	712	2	no	—
152	C₂×C₄	C₄⋊C₄×D₈₉	1	2	yes	—
153	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
154	C₂×C₄	#15668010	1	1	yes	characteristic
155	C₂×C₄	#15668011	1	2	yes	—
156	C₂×C₄	C₂×#13083559	1	2	yes	—
157	C₂×C₄	C₂×#13083564	1	2	yes	—
158	C₂×C₄	#15668039	1	4	yes	—
159	C₂×C₄	#15668041	1	4	yes	—
160	C₈	—	178	8	no	—
161	C₈	#15667945	1	8	yes	complemented
162	C₂²	#19151467	1	1	yes	characteristic
163	C₂²	#19151974	1	1	yes	characteristic
164	C₂²	#19151977	1	1	yes	characteristic
165	C₂²	#19151984	1	2	yes	—
166	C₂²	#19151989	1	2	yes	—
167	C₄	—	16	1	no	—
168	C₄	—	16	1	no	—
169	C₄	—	1424	2	no	—
170	C₄	C₂×#15667945	1	2	yes	—
171	C₄	#19151637	1	4	yes	—
172	C₄	#19152001	1	2	yes	—
173	C₄	#19152005	1	4	yes	—
174	C₂	#26019916	1	1	yes	characteristic
175	C₂	#26019921	1	1	yes	characteristic
176	C₂	#26019926	1	2	yes	—
177	C₂	#26020411	1	1	yes	characteristic
178	C₂	#26020417	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26020412

C₂

yes

maximal, characteristic

#26020411

C₂

yes

maximal

#26020379

C₂

yes

maximal

C₂×#19151319

C₂

yes

maximal, characteristic

C₈₉×#1844

C₂

yes

maximal, characteristic

C₂×#15667946

—

C₂×#15667945

—

C₂×C₄×#10623494

C₂²

yes

—

#19151468

C₂²

yes

—

#19151360

—

#19151319

C₄

yes

complemented

C₂×#106×C₈₉

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

#384×C₈₉

C₂²

yes

characteristic

#384×C₈₉

C₂²

yes

characteristic

C₂²×#10623494

D₄

yes

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

#15667946

—

#15667945

—

C₄×#10623494

C₂×C₄

yes

—

C₄×#10623494

Q₈

yes

—

C₂²×C₈×C₈₉

D₄

yes

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

#106×C₈₉

—

C₄×C₈×C₈₉

C₂×C₄

yes

—

C₄×C₈×C₈₉

Q₈

yes

—

C₂²×Dic₈₉

—

C₂×#10623494

—

C₂×#10623494

—

C₂×#10623494

C₄⋊C₄

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₈×C₈₉

C₄⋊C₄

yes

—

C₂×C₈×C₈₉

D₈

yes

—

C₂×C₈×C₈₉

Q₁₆

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂²×C₄

yes

characteristic

C₄²×C₈₉

C₂×Q₈

yes

—

C₄²×C₈₉

C₂×Q₈

yes

characteristic

C₂³×C₈₉

#127

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₈₉

C₄⋊Q₈

yes

—

C₂×C₄×C₈₉

C₂×D₈

yes

—

C₂×C₄×C₈₉

C₂×Q₁₆

yes

—

C₂×C₄×C₈₉

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₈₉

#137

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₈×C₈₉

#107

yes

complemented

#10623494

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

—

C₂²×C₈₉

#497

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂×#107

yes

—

C₄×C₈₉

#428

yes

—

C₄×C₈₉

#500

yes

—

C₄×C₈₉

#501

yes

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₂×C₈₉

#1845

yes

characteristic

C₂×C₈₉

#1847

yes

—

#1845

—

#1845

—

#1844

D₈₉

yes

characteristic

#1834

—

C₂×#322

—

C₈₉

#8820

yes

characteristic, complemented, Sylow

C₂×#107

—

178

—

C₂×#107

—

178

—

C₂×#106

—

C₂×C₄×C₈

—

100

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

101

#384

—

102

#384

C₂×D₈₉

yes

characteristic

103

#384

C₂×D₈₉

yes

characteristic

104

#340

—

178

—

105

#322

—

106

C₂²×C₈

—

107

C₂²×C₈

C₂²⋊D₈₉

yes

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

178

—

113

C₂×C₄⋊C₄

—

178

—

114

C₂×C₄⋊C₄

—

178

—

115

C₂×C₄²

C₂²×D₈₉

yes

characteristic

116

#107

—

356

—

117

#107

—

356

—

118

#106

—

119

C₄×C₈

—

120

C₄×C₈

—

121

C₄×C₈

Dic₁₇₈

yes

—

122

C₄×C₈

C₄×D₈₉

yes

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

356

—

126

C₂²×C₄

D₄×D₈₉

yes

—

127

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

128

C₂×C₈

—

129

C₂×C₈

—

178

—

130

C₂×C₈

—

178

—

131

C₂×C₈

#13083553

yes

—

132

C₂×C₈

#13083559

yes

—

133

C₂×C₈

#13083564

yes

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

356

—

139

C₄⋊C₄

—

356

—

140

C₄⋊C₄

—

356

—

141

C₄²

C₂×C₄×D₈₉

yes

characteristic

142

C₄²

Q₈×D₈₉

yes

—

143

C₄²

C₂×Dic₁₇₈

yes

characteristic

144

C₂³

#15668008

yes

characteristic

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

712

—

150

C₂×C₄

—

712

—

151

C₂×C₄

—

712

—

152

C₂×C₄

C₄⋊C₄×D₈₉

yes

—

153

C₂×C₄

C₂×#13083553

yes

characteristic

154

C₂×C₄

#15668010

yes

characteristic

155

C₂×C₄

#15668011

yes

—

156

C₂×C₄

C₂×#13083559

yes

—

157

C₂×C₄

C₂×#13083564

yes

—

158

C₂×C₄

#15668039

yes

—

159

C₂×C₄

#15668041

yes

—

160

C₈

—

178

—

161

C₈

#15667945

yes

complemented

162

C₂²

#19151467

yes

characteristic

163

C₂²

#19151974

yes

characteristic

164

C₂²

#19151977

yes

characteristic

165

C₂²

#19151984

yes

—

166

C₂²

#19151989

yes

—

167

C₄

—

168

C₄

—

169

C₄

—

1424

—

170

C₄

C₂×#15667945

yes

—

171

C₄

#19151637

yes

—

172

C₄

#19152001

yes

—

173

C₄

#19152005

yes

—

174

C₂

#26019916

yes

characteristic

175

C₂

#26019921

yes

characteristic

176

C₂

#26019926

yes

—

177

C₂

#26020411

yes

characteristic

178

C₂

#26020417

yes

—

	subgroup	G/H
center	C₂³	#15668008
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1844	C₂
socle, prime radical	C₂³×C₈₉	#127
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1844	D₈₉
hypocenter	C₈₉	#8820
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

subgroup

G/H

center

C₂³

#15668008

commutator

C₄²×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1844

C₂

socle, prime radical

C₂³×C₈₉

#127

cosocle

C₂×C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

N	H ≅ G/N	type
11. #19151319	C₄	semidirect
70. C₈×C₈₉	#107	semidirect
95. C₈₉	#8820	semidirect
161. C₈	#15667945	semidirect

H ≅ G/N

type

11. #19151319

C₄

semidirect

70. C₈×C₈₉

#107

semidirect

95. C₈₉

#8820

semidirect

161. C₈

#15667945

semidirect

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

subgroup	index
#26020412	2
#26020411	2
#26020379	2
C₂×#19151319	2
C₈₉×#1844	2
#8820	89

subgroup

index

C₈₉×#1844

Subgroups — #36448215

Conjugacy classes of subgroups

Complements

Maximal subgroups