Group #36473020 — subgroups

Subgroups — #36473020

Order 22784.

334 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 127 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	C₄×#15667948	C₂	1	1	yes	maximal, characteristic
3	#26018361	C₂	1	4	yes	maximal
4	C₈₉×#2150	C₂	1	1	yes	maximal, characteristic
5	C₂×#15668004	—	2	1	no	—
6	#19152000	—	2	2	no	—
7	#19152000	C₄	1	4	yes	—
8	C₂×#15667948	C₂²	1	2	yes	—
9	C₂×#15667946	C₂²	1	2	yes	—
10	C₂×#15667944	—	2	1	no	—
11	C₄×#13083558	—	2	1	no	—
12	#19151363	—	2	1	no	—
13	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₄⋊C₁₆×C₈₉	—	2	2	no	—
16	C₂×#13083558	—	2	2	no	—
17	#15668004	—	2	4	no	—
18	#15668004	—	4	1	no	—
19	#15668004	—	4	1	no	—
20	C₂×C₄⋊Dic₈₉	—	2	2	no	—
21	C₂×#13083553	—	4	2	no	—
22	#15667948	—	2	2	no	—
23	#15667948	Q₈	1	4	yes	—
24	#15667946	—	2	2	no	—
25	#15667946	C₂×C₄	1	4	yes	—
26	#15667944	—	2	4	no	—
27	#15667923	—	4	1	no	—
28	C₄×#10623494	—	4	1	no	—
29	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
32	M₃₂×C₈₉	—	2	4	no	—
33	C₂×C₁₆×C₈₉	—	2	2	no	—
34	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₈₉	D₄	1	2	yes	—
36	C₂²×Dic₈₉	—	4	2	no	—
37	#13083558	—	2	4	no	—
38	#13083558	—	4	2	no	—
39	C₂×#10623494	—	4	2	no	—
40	C₄⋊Dic₈₉	—	2	4	no	—
41	C₄⋊Dic₈₉	—	4	2	no	—
42	#13083553	—	4	4	no	—
43	#13083553	—	4	4	no	—
44	C₂²×C₄×C₈₉	SD₁₆	1	2	yes	—
45	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₈₉	—	2	2	no	—
47	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
48	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
51	C₂×C₈×C₈₉	#41	1	2	yes	—
52	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
53	C₄²×C₈₉	D₈	1	1	yes	characteristic
54	C₄²×C₈₉	D₈	1	1	yes	characteristic
55	C₁₆×C₈₉	—	4	2	no	—
56	C₂³×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
57	C₂×C₄×C₈₉	—	4	2	no	—
58	C₂×C₄×C₈₉	#102	1	2	yes	—
59	C₂×C₄×C₈₉	#102	1	2	yes	—
60	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
62	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
63	C₂×C₄×C₈₉	#137	1	2	yes	—
64	C₂×Dic₈₉	—	4	4	no	—
65	C₂×Dic₈₉	—	8	4	no	—
66	C₈×C₈₉	—	2	2	no	—
67	C₈×C₈₉	C₄×Q₈	1	2	yes	—
68	C₈×C₈₉	#124	1	1	yes	characteristic
69	C₈×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
70	#10623494	—	4	4	no	—
71	C₂²×C₈₉	—	2	2	no	—
72	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
73	C₂²×C₈₉	#475	1	2	yes	—
74	C₄×C₈₉	—	2	2	no	—
75	C₄×C₈₉	—	2	2	no	—
76	C₄×C₈₉	#391	1	1	yes	characteristic
77	C₄×C₈₉	#419	1	1	yes	characteristic
78	C₄×C₈₉	#474	1	2	yes	—
79	Dic₈₉	—	8	4	no	—
80	#13280	—	89	1	no	maximal, Sylow
81	C₂×C₈₉	—	2	2	no	—
82	C₂×C₈₉	#1988	1	1	yes	characteristic
83	C₂×C₈₉	#2148	1	2	yes	—
84	C₂×#499	—	89	1	no	—
85	#2150	D₈₉	1	1	yes	characteristic
86	C₄×#108	—	89	1	no	—
87	#1387	—	89	4	no	—
88	C₈₉	#13280	1	1	yes	characteristic, complemented, Sylow
89	C₂×#125	—	178	1	no	—
90	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
91	#499	—	89	4	no	—
92	#499	—	178	2	no	—
93	C₂×#108	—	89	2	no	—
94	C₂×#107	—	89	2	no	—
95	C₂×#106	—	178	1	no	—
96	C₄×M₁₆	—	178	1	no	—
97	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
98	C₄⋊C₁₆	—	2	2	no	—
99	#341	—	178	1	no	—
100	C₂×M₁₆	—	178	2	no	—
101	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
102	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
103	#125	—	356	1	no	—
104	#125	—	356	1	no	—
105	#125	—	178	4	no	—
106	C₂×C₄⋊C₄	—	356	2	no	—
107	C₂×C₄⋊C₄	—	178	2	no	—
108	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
109	M₃₂	—	2	4	no	—
110	C₂×C₁₆	—	2	2	no	—
111	#108	—	89	4	no	—
112	#108	—	178	2	no	—
113	#107	—	89	4	no	—
114	#107	—	178	2	no	—
115	#106	—	178	4	no	—
116	#97	—	356	1	no	—
117	C₄×C₈	—	356	1	no	—
118	C₄×C₈	C₄×D₈₉	1	2	yes	—
119	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
120	C₂²×C₄	—	356	2	no	—
121	C₂²×C₄	#13083563	1	2	yes	—
122	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
123	M₁₆	—	356	2	no	—
124	M₁₆	—	178	4	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	356	2	no	—
127	C₂×C₈	#13083551	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	Q₈×D₈₉	1	2	yes	—
131	C₂×C₈	#13083573	1	2	yes	—
132	C₄⋊C₄	—	356	2	no	—
133	C₄⋊C₄	—	356	4	no	—
134	C₄⋊C₄	—	356	4	no	—
135	C₄⋊C₄	—	178	4	no	—
136	C₄²	#13083551	1	2	yes	—
137	C₄²	D₇₁₂	1	1	yes	characteristic
138	C₄²	D₇₁₂	1	1	yes	characteristic
139	C₁₆	—	4	2	no	—
140	C₂³	C₂×#13083563	1	1	yes	characteristic
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	356	4	no	—
143	C₂×C₄	—	712	4	no	—
144	C₂×C₄	#15667934	1	2	yes	—
145	C₂×C₄	#15667934	1	2	yes	—
146	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
147	C₂×C₄	#15667992	1	2	yes	—
148	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
149	C₂×C₄	#15668045	1	2	yes	—
150	C₈	—	2	2	no	—
151	C₈	—	356	4	no	—
152	C₈	#15667977	1	2	yes	—
153	C₈	#15667999	1	1	yes	characteristic
154	C₈	#15668010	1	1	yes	characteristic
155	C₂²	—	2	2	no	—
156	C₂²	C₂×#15667934	1	1	yes	characteristic
157	C₂²	#19151884	1	2	yes	—
158	C₄	—	2	2	no	—
159	C₄	—	2	2	no	—
160	C₄	—	712	4	no	—
161	C₄	#19151503	1	1	yes	characteristic
162	C₄	#19151608	1	1	yes	characteristic
163	C₄	#19151879	1	2	yes	—
164	C₂	—	2	2	no	—
165	C₂	#26021203	1	1	yes	characteristic
166	C₂	#26022002	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19152000

C₂

yes

maximal, characteristic

C₄×#15667948

C₂

yes

maximal, characteristic

#26018361

C₂

yes

maximal

C₈₉×#2150

C₂

yes

maximal, characteristic

C₂×#15668004

—

#19152000

—

#19152000

C₄

yes

—

C₂×#15667948

C₂²

yes

—

C₂×#15667946

C₂²

yes

—

C₂×#15667944

—

C₄×#13083558

—

#19151363

—

C₂×M₃₂×C₈₉

C₂²

yes

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

C₄⋊C₁₆×C₈₉

—

—

—

—

—

C₂×C₄⋊Dic₈₉

—

C₂×#13083553

—

#15667948

—

#15667948

Q₈

yes

—

#15667946

—

#15667946

C₂×C₄

yes

—

#15667944

—

#15667923

—

C₄×#10623494

—

C₂²×C₈×C₈₉

D₄

yes

characteristic

C₂²×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₄²×C₈₉

D₄

yes

characteristic

M₃₂×C₈₉

—

C₂×C₁₆×C₈₉

—

C₄×C₈×C₈₉

C₂×C₄

yes

—

C₄×C₈×C₈₉

D₄

yes

—

C₂²×Dic₈₉

—

#13083558

—

#13083558

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083553

—

#13083553

—

C₂²×C₄×C₈₉

SD₁₆

yes

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂²⋊C₄

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

D₈

yes

characteristic

C₄²×C₈₉

D₈

yes

characteristic

C₁₆×C₈₉

—

C₂³×C₈₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#102

yes

—

C₂×C₄×C₈₉

#102

yes

—

C₂×C₄×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

#137

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₈×C₈₉

—

C₈×C₈₉

C₄×Q₈

yes

—

C₈×C₈₉

#124

yes

characteristic

C₈×C₈₉

C₄⋊Q₈

yes

characteristic

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×#102

yes

characteristic

C₂²×C₈₉

#475

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

Dic₈₉

—

#13280

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

#1988

yes

characteristic

C₂×C₈₉

#2148

yes

—

C₂×#499

—

#2150

D₈₉

yes

characteristic

C₄×#108

—

#1387

—

C₈₉

#13280

yes

characteristic, complemented, Sylow

C₂×#125

—

178

—

C₂×M₃₂

C₂×D₈₉

yes

—

#499

—

#499

—

178

—

C₂×#108

—

C₂×#107

—

C₂×#106

—

178

—

C₄×M₁₆

—

178

—

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

C₄⋊C₁₆

—

#341

—

178

—

100

C₂×M₁₆

—

178

—

101

C₂²×C₈

D₃₅₆

yes

characteristic

102

C₂²×C₈

C₂²×D₈₉

yes

characteristic

103

#125

—

356

—

104

#125

—

356

—

105

#125

—

178

—

106

C₂×C₄⋊C₄

—

356

—

107

C₂×C₄⋊C₄

—

178

—

108

C₂×C₄²

D₃₅₆

yes

characteristic

109

M₃₂

—

110

C₂×C₁₆

—

111

#108

—

112

#108

—

178

—

113

#107

—

114

#107

—

178

—

115

#106

—

178

—

116

#97

—

356

—

117

C₄×C₈

—

356

—

118

C₄×C₈

C₄×D₈₉

yes

—

119

C₄×C₈

C₂²⋊D₈₉

yes

—

120

C₂²×C₄

—

356

—

121

C₂²×C₄

#13083563

yes

—

122

C₂²×C₄

C₂×D₃₅₆

yes

characteristic

123

M₁₆

—

356

—

124

M₁₆

—

178

—

125

C₂×C₈

—

126

C₂×C₈

—

356

—

127

C₂×C₈

#13083551

yes

—

128

C₂×C₈

C₂×C₄×D₈₉

yes

characteristic

129

C₂×C₈

C₂×C₂²⋊D₈₉

yes

characteristic

130

C₂×C₈

Q₈×D₈₉

yes

—

131

C₂×C₈

#13083573

yes

—

132

C₄⋊C₄

—

356

—

133

C₄⋊C₄

—

356

—

134

C₄⋊C₄

—

356

—

135

C₄⋊C₄

—

178

—

136

C₄²

#13083551

yes

—

137

C₄²

D₇₁₂

yes

characteristic

138

C₄²

D₇₁₂

yes

characteristic

139

C₁₆

—

140

C₂³

C₂×#13083563

yes

characteristic

141

C₂×C₄

—

142

C₂×C₄

—

356

—

143

C₂×C₄

—

712

—

144

C₂×C₄

#15667934

yes

—

145

C₂×C₄

#15667934

yes

—

146

C₂×C₄

C₂×#13083551

yes

characteristic

147

C₂×C₄

#15667992

yes

—

148

C₂×C₄

C₂×D₇₁₂

yes

characteristic

149

C₂×C₄

#15668045

yes

—

150

C₈

—

151

C₈

—

356

—

152

C₈

#15667977

yes

—

153

C₈

#15667999

yes

characteristic

154

C₈

#15668010

yes

characteristic

155

C₂²

—

156

C₂²

C₂×#15667934

yes

characteristic

157

C₂²

#19151884

yes

—

158

C₄

—

159

C₄

—

160

C₄

—

712

—

161

C₄

#19151503

yes

characteristic

162

C₄

#19151608

yes

characteristic

163

C₄

#19151879

yes

—

164

C₂

—

165

C₂

#26021203

yes

characteristic

166

C₂

#26022002

yes

—

	subgroup	G/H
center	C₂²	C₂×#15667934
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#2150	C₂
socle	C₂²×C₈₉	C₂×#102
prime radical	C₂³×C₈₉	C₂×SD₁₆
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#2150	D₈₉
hypocenter	C₈₉	#13280
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2150

subgroup

G/H

center

C₂²

C₂×#15667934

commutator

C₂×C₈×C₈₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₈₉

Fitting

C₈₉×#2150

C₂

socle

C₂²×C₈₉

C₂×#102

prime radical

C₂³×C₈₉

C₂×SD₁₆

cosocle

C₂²×C₈×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2150

N	H ≅ G/N	type
88. C₈₉	#13280	semidirect

H ≅ G/N

type

88. C₈₉

#13280

semidirect

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

subgroup	index
C₂×#19152000	2
C₄×#15667948	2
#26018361	2
C₈₉×#2150	2
#13280	89

subgroup

index

C₈₉×#2150

Subgroups — #36473020

Conjugacy classes of subgroups

Complements

Maximal subgroups