Group #36473025 — subgroups

Subgroups — #36473025

Order 22784.

366 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 163 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	C₂×#19152000	C₂	1	1	yes	maximal, characteristic
2	#26020416	C₂	1	1	yes	maximal, characteristic
3	#26018344	C₂	1	4	yes	maximal
4	C₂×C₄×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#15668004	—	2	1	no	—
6	#19152000	—	2	2	no	—
7	#19152000	C₄	1	4	yes	complemented
8	C₂×#15667946	C₂²	1	2	yes	—
9	C₂×#15667946	C₂²	1	2	yes	—
10	C₂×#15667944	—	2	1	no	—
11	C₂×#15667944	—	2	1	no	—
12	#19151464	—	2	1	no	—
13	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₄×C₁₆×C₈₉	—	2	2	no	—
16	#15668004	—	2	4	no	—
17	#15668004	—	4	2	no	—
18	C₂×C₄⋊Dic₈₉	—	2	2	no	—
19	C₂×C₄⋊Dic₈₉	—	2	2	no	—
20	C₂×#13083553	—	4	2	no	—
21	#15667946	—	2	2	no	—
22	#15667946	—	2	4	no	—
23	#15667946	C₂×C₄	1	4	yes	—
24	#15667944	—	2	4	no	—
25	#15667944	—	4	2	no	—
26	#15667919	—	4	2	no	—
27	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
29	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
30	C₂×C₁₆×C₈₉	—	2	2	no	—
31	C₂×C₁₆×C₈₉	—	2	2	no	—
32	C₂×C₁₆×C₈₉	Q₈	1	4	yes	—
33	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₈₉	D₄	1	2	yes	—
35	C₂²×Dic₈₉	—	4	2	no	—
36	C₂²×Dic₈₉	—	4	2	no	—
37	C₄⋊Dic₈₉	—	2	4	no	—
38	C₄⋊Dic₈₉	—	4	2	no	—
39	C₄⋊Dic₈₉	—	4	2	no	—
40	C₄⋊Dic₈₉	—	4	2	no	—
41	#13083553	—	4	8	no	—
42	C₂²×C₄×C₈₉	D₈	1	2	yes	—
43	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
45	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
46	C₂×C₈×C₈₉	Q₁₆	1	2	yes	—
47	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
50	C₂×C₈×C₈₉	#41	1	2	yes	—
51	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
52	C₄²×C₈₉	SD₁₆	1	2	yes	—
53	C₁₆×C₈₉	—	2	4	no	—
54	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
55	C₂×C₄×C₈₉	—	2	4	no	—
56	C₂×C₄×C₈₉	#102	1	4	yes	—
57	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
59	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
60	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
61	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
63	C₂×Dic₈₉	—	4	4	no	—
64	C₂×Dic₈₉	—	8	4	no	—
65	C₂×Dic₈₉	—	8	2	no	—
66	C₂×Dic₈₉	—	8	4	no	—
67	C₈×C₈₉	#103	1	4	yes	—
68	C₈×C₈₉	C₄×Q₈	1	2	yes	—
69	C₈×C₈₉	C₂²⋊Q₈	1	2	yes	—
70	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
71	C₂²×C₈₉	#486	1	1	yes	characteristic
72	C₂²×C₈₉	#500	1	1	yes	characteristic
73	C₂²×C₈₉	#508	1	4	yes	—
74	C₄×C₈₉	—	2	4	no	—
75	C₄×C₈₉	#391	1	1	yes	characteristic
76	C₄×C₈₉	C₂×#103	1	1	yes	characteristic
77	C₄×C₈₉	#487	1	1	yes	characteristic
78	C₄×C₈₉	#498	1	1	yes	characteristic
79	Dic₈₉	—	8	4	no	—
80	Dic₈₉	—	16	2	no	—
81	#13281	—	89	1	no	maximal, Sylow
82	C₂×C₈₉	#1990	1	1	yes	characteristic
83	C₂×C₈₉	#2138	1	2	yes	—
84	C₂×C₈₉	#2242	1	2	yes	—
85	C₂×C₈₉	#2253	1	2	yes	—
86	C₂×#499	—	89	1	no	—
87	C₂×C₄×C₁₆	D₈₉	1	1	yes	characteristic
88	#1846	—	89	1	no	—
89	#1380	—	89	4	no	—
90	C₈₉	#13281	1	1	yes	characteristic, complemented, Sylow
91	C₂×#125	—	178	1	no	—
92	C₂²×C₁₆	C₂×D₈₉	1	2	yes	—
93	#499	—	89	4	no	—
94	#499	—	178	2	no	—
95	C₂×#107	—	89	2	no	—
96	C₂×#107	—	89	2	no	—
97	C₂×#106	—	178	1	no	—
98	C₂×#106	—	178	1	no	—
99	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
100	#382	—	178	1	no	—
101	C₄×C₁₆	—	2	2	no	—
102	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
103	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
104	#125	—	356	2	no	—
105	#125	—	178	4	no	—
106	C₂×C₄⋊C₄	—	356	2	no	—
107	C₂×C₄⋊C₄	—	178	2	no	—
108	C₂×C₄⋊C₄	—	178	2	no	—
109	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
110	C₂×C₁₆	—	2	2	no	—
111	C₂×C₁₆	—	2	2	no	—
112	C₂×C₁₆	Dic₁₇₈	1	4	yes	—
113	#107	—	89	4	no	—
114	#107	—	178	4	no	—
115	#107	—	178	2	no	—
116	#106	—	356	2	no	—
117	#106	—	178	4	no	—
118	C₄×C₈	C₄×D₈₉	1	2	yes	—
119	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
120	#95	—	356	2	no	—
121	C₂²×C₄	—	356	2	no	—
122	C₂²×C₄	—	356	2	no	—
123	C₂²×C₄	D₇₁₂	1	2	yes	—
124	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
125	C₂×C₈	#13083551	1	2	yes	—
126	C₂×C₈	#13083563	1	2	yes	—
127	C₂×C₈	Dic₃₅₆	1	2	yes	—
128	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
129	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
130	C₂×C₈	C₂×Dic₁₇₈	1	2	yes	—
131	C₂×C₈	#13083574	1	2	yes	—
132	C₄⋊C₄	—	356	2	no	—
133	C₄⋊C₄	—	356	8	no	—
134	C₄⋊C₄	—	356	2	no	—
135	C₄⋊C₄	—	356	2	no	—
136	C₄⋊C₄	—	178	4	no	—
137	C₄²	#13083551	1	2	yes	—
138	C₄²	#13083563	1	2	yes	—
139	C₁₆	—	2	4	no	—
140	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	356	4	no	—
143	C₂×C₄	—	712	2	no	—
144	C₂×C₄	—	712	4	no	—
145	C₂×C₄	—	712	4	no	—
146	C₂×C₄	#15667934	1	4	yes	—
147	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
148	C₂×C₄	#15668000	1	1	yes	characteristic
149	C₂×C₄	#15668012	1	1	yes	characteristic
150	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
151	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
152	C₂×C₄	C₂×Dic₃₅₆	1	1	yes	characteristic
153	C₈	#15667937	1	4	yes	—
154	C₈	C₄×Dic₁₇₈	1	2	yes	—
155	C₈	#15667991	1	2	yes	—
156	C₂²	C₂×#15667934	1	1	yes	characteristic
157	C₂²	#19151945	1	1	yes	characteristic
158	C₂²	#19152003	1	1	yes	characteristic
159	C₂²	#19152030	1	4	yes	—
160	C₄	—	2	4	no	—
161	C₄	—	1424	2	no	—
162	C₄	—	712	4	no	—
163	C₄	#19151504	1	1	yes	characteristic
164	C₄	C₂×#15667937	1	1	yes	characteristic
165	C₄	#19151949	1	1	yes	characteristic
166	C₄	#19151996	1	1	yes	characteristic
167	C₂	#26021213	1	1	yes	characteristic
168	C₂	#26021960	1	2	yes	—
169	C₂	#26022484	1	2	yes	—
170	C₂	#26022531	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19152000

C₂

yes

maximal, characteristic

#26020416

C₂

yes

maximal, characteristic

#26018344

C₂

yes

maximal

C₂×C₄×C₁₆×C₈₉

C₂

yes

maximal, characteristic

C₂×#15668004

—

#19152000

—

#19152000

C₄

yes

complemented

C₂×#15667946

C₂²

yes

—

C₂×#15667946

C₂²

yes

—

C₂×#15667944

—

C₂×#15667944

—

#19151464

—

C₂²×C₁₆×C₈₉

C₂²

yes

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

C₄×C₁₆×C₈₉

—

#15668004

—

#15668004

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

—

—

—

C₂×C₄

yes

—

#15667944

—

#15667944

—

#15667919

—

C₂²×C₈×C₈₉

D₄

yes

characteristic

C₂²×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₄²×C₈₉

D₄

yes

characteristic

C₂×C₁₆×C₈₉

—

C₂×C₁₆×C₈₉

—

C₂×C₁₆×C₈₉

Q₈

yes

—

C₄×C₈×C₈₉

C₂×C₄

yes

—

C₄×C₈×C₈₉

D₄

yes

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083553

—

C₂²×C₄×C₈₉

D₈

yes

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

C₂²⋊C₄

yes

—

C₂×C₈×C₈₉

SD₁₆

yes

—

C₂×C₈×C₈₉

Q₁₆

yes

—

C₂×C₈×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

C₂×Q₈

yes

—

C₂×C₈×C₈₉

#41

yes

—

C₄²×C₈₉

C₂²⋊C₄

yes

—

C₄²×C₈₉

SD₁₆

yes

—

C₁₆×C₈₉

—

C₂³×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#102

yes

—

C₂×C₄×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₉

#124

yes

characteristic

C₂×C₄×C₈₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₈₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₈₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₁₆

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₈×C₈₉

#103

yes

—

C₈×C₈₉

C₄×Q₈

yes

—

C₈×C₈₉

C₂²⋊Q₈

yes

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₂×C₈₉

#2242

yes

—

C₂×C₈₉

#2253

yes

—

C₂×#499

—

C₂×C₄×C₁₆

D₈₉

yes

characteristic

#1846

—

#1380

—

C₈₉

#13281

yes

characteristic, complemented, Sylow

C₂×#125

—

178

—

C₂²×C₁₆

C₂×D₈₉

yes

—

#499

—

#499

—

178

—

C₂×#107

—

C₂×#107

—

C₂×#106

—

178

—

C₂×#106

—

178

—

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

100

#382

—

178

—

101

C₄×C₁₆

—

102

C₂²×C₈

D₃₅₆

yes

characteristic

103

C₂²×C₈

C₂²×D₈₉

yes

characteristic

104

#125

—

356

—

105

#125

—

178

—

106

C₂×C₄⋊C₄

—

356

—

107

C₂×C₄⋊C₄

—

178

—

108

C₂×C₄⋊C₄

—

178

—

109

C₂×C₄²

D₃₅₆

yes

characteristic

110

C₂×C₁₆

—

111

C₂×C₁₆

—

112

C₂×C₁₆

Dic₁₇₈

yes

—

113

#107

—

114

#107

—

178

—

115

#107

—

178

—

116

#106

—

356

—

117

#106

—

178

—

118

C₄×C₈

C₄×D₈₉

yes

—

119

C₄×C₈

C₂²⋊D₈₉

yes

—

120

#95

—

356

—

121

C₂²×C₄

—

356

—

122

C₂²×C₄

—

356

—

123

C₂²×C₄

D₇₁₂

yes

—

124

C₂²×C₄

C₂×D₃₅₆

yes

characteristic

125

C₂×C₈

#13083551

yes

—

126

C₂×C₈

#13083563

yes

—

127

C₂×C₈

Dic₃₅₆

yes

—

128

C₂×C₈

C₂×C₄×D₈₉

yes

characteristic

129

C₂×C₈

C₂×C₂²⋊D₈₉

yes

characteristic

130

C₂×C₈

C₂×Dic₁₇₈

yes

—

131

C₂×C₈

#13083574

yes

—

132

C₄⋊C₄

—

356

—

133

C₄⋊C₄

—

356

—

134

C₄⋊C₄

—

356

—

135

C₄⋊C₄

—

356

—

136

C₄⋊C₄

—

178

—

137

C₄²

#13083551

yes

—

138

C₄²

#13083563

yes

—

139

C₁₆

—

140

C₂³

C₂×D₇₁₂

yes

characteristic

141

C₂×C₄

—

142

C₂×C₄

—

356

—

143

C₂×C₄

—

712

—

144

C₂×C₄

—

712

—

145

C₂×C₄

—

712

—

146

C₂×C₄

#15667934

yes

—

147

C₂×C₄

C₂×#13083551

yes

characteristic

148

C₂×C₄

#15668000

yes

characteristic

149

C₂×C₄

#15668012

yes

characteristic

150

C₂×C₄

C₂×#13083563

yes

characteristic

151

C₂×C₄

C₂×#13083563

yes

characteristic

152

C₂×C₄

C₂×Dic₃₅₆

yes

characteristic

153

C₈

#15667937

yes

—

154

C₈

C₄×Dic₁₇₈

yes

—

155

C₈

#15667991

yes

—

156

C₂²

C₂×#15667934

yes

characteristic

157

C₂²

#19151945

yes

characteristic

158

C₂²

#19152003

yes

characteristic

159

C₂²

#19152030

yes

—

160

C₄

—

161

C₄

—

1424

—

162

C₄

—

712

—

163

C₄

#19151504

yes

characteristic

164

C₄

C₂×#15667937

yes

characteristic

165

C₄

#19151949

yes

characteristic

166

C₄

#19151996

yes

characteristic

167

C₂

#26021213

yes

characteristic

168

C₂

#26021960

yes

—

169

C₂

#26022484

yes

—

170

C₂

#26022531

yes

—

	subgroup	G/H
center	C₂³	C₂×D₇₁₂
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂×C₄×C₁₆×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₈
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×C₄×C₁₆	D₈₉
hypocenter	C₈₉	#13281
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

subgroup

G/H

center

C₂³

C₂×D₇₁₂

commutator

C₂×C₈×C₈₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₈₉

Fitting

C₂×C₄×C₁₆×C₈₉

C₂

socle, prime radical

C₂³×C₈₉

C₂×D₈

cosocle

C₂²×C₈×C₈₉

C₂³

hypercenter

C₂×C₄×C₁₆

D₈₉

hypocenter

C₈₉

#13281

LCS

G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

N	H ≅ G/N	type
7. #19152000	C₄	semidirect
90. C₈₉	#13281	semidirect

H ≅ G/N

type

7. #19152000

C₄

semidirect

90. C₈₉

#13281

semidirect

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

subgroup	index
C₂×#19152000	2
#26020416	2
#26018344	2
C₂×C₄×C₁₆×C₈₉	2
#13281	89

subgroup

index

C₂×C₄×C₁₆×C₈₉

Subgroups — #36473025

Conjugacy classes of subgroups

Complements

Maximal subgroups