Group #26020338 — subgroups

Subgroups — #26020338

Order 11392.

344 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 125 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×C₂²⋊#10623494	C₂	1	1	yes	maximal, characteristic
2	#19151458	C₂	1	1	yes	maximal, characteristic
3	#19151360	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄⋊C₄×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#10623494	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
7	C₂×C₂²⋊Dic₈₉	—	2	1	no	—
8	C₂²⋊#10623494	—	2	2	no	—
9	#15667919	—	2	1	no	—
10	C₂³×C₄×C₈₉	—	2	1	no	—
11	C₂³×C₄×C₈₉	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
13	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
14	C₂×C₄⋊C₄×C₈₉	C₄	1	4	yes	complemented
15	C₂×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
16	C₂²×Dic₈₉	—	2	2	no	—
17	C₂×#10623494	—	2	2	no	—
18	C₂×#10623494	—	2	4	no	—
19	C₄⋊Dic₈₉	—	2	4	no	—
20	C₂²⋊Dic₈₉	—	4	2	no	—
21	C₂⁴×C₈₉	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₈₉	—	2	4	no	—
23	C₂²×C₄×C₈₉	—	2	4	no	—
24	C₂²×C₄×C₈₉	—	2	2	no	—
25	C₂²×C₄×C₈₉	—	4	2	no	—
26	C₂²×C₄×C₈₉	—	4	2	no	—
27	C₂²×C₄×C₈₉	C₂×C₄	1	2	yes	—
28	C₂²×C₄×C₈₉	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₈₉	D₄	1	2	yes	—
30	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
31	C₄⋊C₄×C₈₉	—	2	4	no	—
32	C₄⋊C₄×C₈₉	—	2	2	no	—
33	C₄⋊C₄×C₈₉	C₂×C₄	1	4	yes	—
34	C₂³×C₈₉	—	2	2	no	—
35	C₂³×C₈₉	—	2	2	no	—
36	C₂³×C₈₉	C₂²⋊C₄	1	2	yes	—
37	C₂³×C₈₉	D₈	1	2	yes	—
38	C₂³×C₈₉	SD₁₆	1	2	yes	—
39	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₈₉	—	2	2	no	—
41	C₂×C₄×C₈₉	—	2	4	no	—
42	C₂×C₄×C₈₉	—	2	4	no	—
43	C₂×C₄×C₈₉	—	2	4	no	—
44	C₂×C₄×C₈₉	—	4	8	no	—
45	C₂×C₄×C₈₉	—	4	2	no	—
46	C₂×C₄×C₈₉	—	4	2	no	—
47	C₂×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
48	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₈₉	#41	1	4	yes	—
51	C₂×Dic₈₉	—	4	2	no	—
52	C₂×Dic₈₉	—	4	2	no	—
53	C₂×Dic₈₉	—	4	2	no	—
54	#10623494	—	4	2	no	—
55	C₂²×C₈₉	—	2	2	no	—
56	C₂²×C₈₉	—	2	2	no	—
57	C₂²×C₈₉	—	2	4	no	—
58	C₂²×C₈₉	—	2	4	no	—
59	C₂²×C₈₉	#102	1	4	yes	—
60	C₂²×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₈₉	#123	1	2	yes	—
62	C₂²×C₈₉	C₂×D₈	1	1	yes	characteristic
63	C₂²×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
64	C₂²×C₈₉	#137	1	2	yes	—
65	C₄×C₈₉	—	2	2	no	—
66	C₄×C₈₉	—	4	4	no	—
67	C₄×C₈₉	#117	1	2	yes	—
68	C₄×C₈₉	#123	1	2	yes	—
69	Dic₈₉	—	8	2	no	—
70	C₂×C₈₉	—	2	4	no	—
71	C₂×C₈₉	#381	1	1	yes	characteristic
72	C₂×C₈₉	C₂×#102	1	1	yes	characteristic
73	C₂×C₈₉	#419	1	1	yes	characteristic
74	C₂×C₈₉	#480	1	2	yes	—
75	C₂×C₈₉	#483	1	2	yes	—
76	#1823	—	89	1	no	maximal, Sylow
77	C₈₉	#1823	1	1	yes	characteristic, complemented, Sylow
78	C₂²×C₄⋊C₄	D₈₉	1	1	yes	characteristic
79	C₂×C₂²⋊C₈	—	89	1	no	—
80	#380	—	89	1	no	—
81	#340	—	89	4	no	—
82	C₂³×C₄	—	2	1	no	—
83	C₂³×C₄	C₂×D₈₉	1	1	yes	characteristic
84	C₂²×C₈	—	89	2	no	—
85	C₂×C₄⋊C₄	—	2	1	no	—
86	C₂×C₄⋊C₄	—	2	2	no	—
87	C₂×C₄⋊C₄	—	89	2	no	—
88	C₂×C₄⋊C₄	Dic₈₉	1	4	yes	complemented
89	C₂×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
90	C₂×C₂²⋊C₄	—	178	1	no	—
91	C₂²⋊C₈	—	178	2	no	—
92	#95	—	178	1	no	—
93	C₂⁴	C₂²⋊D₈₉	1	1	yes	characteristic
94	C₂²×C₄	—	2	4	no	—
95	C₂²×C₄	—	2	4	no	—
96	C₂²×C₄	—	2	2	no	—
97	C₂²×C₄	—	4	2	no	—
98	C₂²×C₄	—	4	2	no	—
99	C₂²×C₄	—	178	2	no	—
100	C₂²×C₄	C₂×Dic₈₉	1	2	yes	—
101	C₂²×C₄	C₂²⋊D₈₉	1	1	yes	characteristic
102	C₂²×C₄	C₂²⋊D₈₉	1	2	yes	—
103	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
104	C₂×C₈	—	178	2	no	—
105	C₂×C₈	—	178	4	no	—
106	C₄⋊C₄	—	2	4	no	—
107	C₄⋊C₄	—	2	2	no	—
108	C₄⋊C₄	—	178	4	no	—
109	C₄⋊C₄	C₂×Dic₈₉	1	4	yes	—
110	C₂²⋊C₄	—	356	2	no	—
111	C₂³	—	2	2	no	—
112	C₂³	—	2	2	no	—
113	C₂³	C₂²⋊Dic₈₉	1	2	yes	—
114	C₂³	#13083559	1	2	yes	—
115	C₂³	#13083562	1	2	yes	—
116	C₂³	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	4	8	no	—
124	C₂×C₄	—	356	2	no	—
125	C₂×C₄	—	356	2	no	—
126	C₂×C₄	—	356	2	no	—
127	C₂×C₄	C₂²⋊Dic₈₉	1	2	yes	—
128	C₂×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
129	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
130	C₂×C₄	#13083574	1	4	yes	—
131	C₈	—	356	2	no	—
132	C₂²	—	2	4	no	—
133	C₂²	—	2	4	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	2	2	no	—
136	C₂²	#15667933	1	4	yes	—
137	C₂²	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
138	C₂²	#15667995	1	2	yes	—
139	C₂²	C₂×#13083559	1	1	yes	characteristic
140	C₂²	C₂×#13083562	1	1	yes	characteristic
141	C₂²	#15668043	1	2	yes	—
142	C₄	—	2	2	no	—
143	C₄	—	4	4	no	—
144	C₄	—	712	2	no	—
145	C₄	#15667971	1	2	yes	—
146	C₄	#15667995	1	2	yes	—
147	C₂	—	2	4	no	—
148	C₂	#19151461	1	1	yes	characteristic
149	C₂	C₂×#15667933	1	1	yes	characteristic
150	C₂	#19151607	1	1	yes	characteristic
151	C₂	#19151911	1	2	yes	—
152	C₂	#19151928	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂²⋊#10623494

C₂

yes

maximal, characteristic

#19151458

C₂

yes

maximal, characteristic

#19151360

C₂

yes

maximal, complemented

C₂²×C₄⋊C₄×C₈₉

C₂

yes

maximal, characteristic

C₂²×#10623494

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×C₂²⋊Dic₈₉

—

C₂²⋊#10623494

—

#15667919

—

C₂³×C₄×C₈₉

—

C₂³×C₄×C₈₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

C₄

yes

complemented

C₂×C₄⋊C₄×C₈₉

C₂²

yes

—

C₂²×Dic₈₉

—

C₂×#10623494

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₂²⋊Dic₈₉

—

C₂⁴×C₈₉

D₄

yes

characteristic

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×C₄

yes

—

C₂²×C₄×C₈₉

D₄

yes

characteristic

C₂²×C₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

C₂×C₄

yes

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

C₂²⋊C₄

yes

—

C₂³×C₈₉

D₈

yes

—

C₂³×C₈₉

SD₁₆

yes

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²⋊C₄

yes

—

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

#41

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#102

yes

—

C₂²×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₈₉

#123

yes

—

C₂²×C₈₉

C₂×D₈

yes

characteristic

C₂²×C₈₉

C₂×SD₁₆

yes

characteristic

C₂²×C₈₉

#137

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#117

yes

—

C₄×C₈₉

#123

yes

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₂×C₈₉

#483

yes

—

#1823

—

maximal, Sylow

C₈₉

#1823

yes

characteristic, complemented, Sylow

C₂²×C₄⋊C₄

D₈₉

yes

characteristic

C₂×C₂²⋊C₈

—

#380

—

#340

—

C₂³×C₄

—

C₂³×C₄

C₂×D₈₉

yes

characteristic

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

Dic₈₉

yes

complemented

C₂×C₄⋊C₄

C₂×D₈₉

yes

—

C₂×C₂²⋊C₄

—

178

—

C₂²⋊C₈

—

178

—

#95

—

178

—

C₂⁴

C₂²⋊D₈₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

178

—

100

C₂²×C₄

C₂×Dic₈₉

yes

—

101

C₂²×C₄

C₂²⋊D₈₉

yes

characteristic

102

C₂²×C₄

C₂²⋊D₈₉

yes

—

103

C₂²×C₄

C₂²×D₈₉

yes

characteristic

104

C₂×C₈

—

178

—

105

C₂×C₈

—

178

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

178

—

109

C₄⋊C₄

C₂×Dic₈₉

yes

—

110

C₂²⋊C₄

—

356

—

111

C₂³

—

112

C₂³

—

113

C₂³

C₂²⋊Dic₈₉

yes

—

114

C₂³

#13083559

yes

—

115

C₂³

#13083562

yes

—

116

C₂³

C₂×C₂²⋊D₈₉

yes

characteristic

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

356

—

125

C₂×C₄

—

356

—

126

C₂×C₄

—

356

—

127

C₂×C₄

C₂²⋊Dic₈₉

yes

—

128

C₂×C₄

C₂²×Dic₈₉

yes

characteristic

129

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

130

C₂×C₄

#13083574

yes

—

131

C₈

—

356

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

#15667933

yes

—

137

C₂²

C₂×C₂²⋊Dic₈₉

yes

characteristic

138

C₂²

#15667995

yes

—

139

C₂²

C₂×#13083559

yes

characteristic

140

C₂²

C₂×#13083562

yes

characteristic

141

C₂²

#15668043

yes

—

142

C₄

—

143

C₄

—

144

C₄

—

712

—

145

C₄

#15667971

yes

—

146

C₄

#15667995

yes

—

147

C₂

—

148

C₂

#19151461

yes

characteristic

149

C₂

C₂×#15667933

yes

characteristic

150

C₂

#19151607

yes

characteristic

151

C₂

#19151911

yes

—

152

C₂

#19151928

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₈₉
commutator	C₂×C₄×C₈₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	C₂²×C₄⋊C₄×C₈₉	C₂
socle	C₂³×C₈₉	C₂×D₄
prime radical	C₂⁴×C₈₉	D₄
cosocle	C₂²×C₄×C₈₉	C₂³
hypercenter	C₂²×C₄⋊C₄	D₈₉
hypocenter	C₈₉	#1823
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₈₉

commutator

C₂×C₄×C₈₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

C₂²×C₄⋊C₄×C₈₉

C₂

socle

C₂³×C₈₉

C₂×D₄

prime radical

C₂⁴×C₈₉

D₄

cosocle

C₂²×C₄×C₈₉

C₂³

hypercenter

C₂²×C₄⋊C₄

D₈₉

hypocenter

C₈₉

#1823

LCS

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₄

N	H ≅ G/N	type
3. #19151360	C₂	semidirect
14. C₂×C₄⋊C₄×C₈₉	C₄	semidirect
77. C₈₉	#1823	semidirect
88. C₂×C₄⋊C₄	Dic₈₉	semidirect

H ≅ G/N

type

3. #19151360

C₂

semidirect

14. C₂×C₄⋊C₄×C₈₉

C₄

semidirect

77. C₈₉

#1823

semidirect

88. C₂×C₄⋊C₄

Dic₈₉

semidirect

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

subgroup	index
C₂×C₂²⋊#10623494	2
#19151458	2
#19151360	2
C₂²×C₄⋊C₄×C₈₉	2
#1823	89

subgroup

index

C₂×C₂²⋊#10623494

#19151458

#19151360

C₂²×C₄⋊C₄×C₈₉

#1823

Subgroups — #26020338

Conjugacy classes of subgroups

Complements

Maximal subgroups