Group #26021723 — subgroups

Subgroups — #26021723

Order 11392.

230 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 93 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×#15667945	C₂	1	2	yes	maximal
2	#19151468	C₂	1	1	yes	maximal, characteristic
3	#19151360	C₂	1	2	yes	maximal
4	#19151353	C₂	1	1	yes	maximal, characteristic
5	#389×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10623494	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₈₉	—	2	1	no	—
8	C₂×C₄⋊Dic₈₉	—	2	1	no	—
9	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
10	#15667945	—	2	4	no	—
11	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
12	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
15	#95×C₈₉	—	2	1	no	—
16	C₂²×Dic₈₉	—	2	2	no	—
17	C₂×#10623494	—	2	2	no	—
18	C₂×#10623494	—	2	2	no	—
19	C₂×#10623494	Q₈	1	4	yes	—
20	C₄⋊Dic₈₉	—	2	2	no	—
21	C₄⋊Dic₈₉	—	2	2	no	—
22	C₄⋊Dic₈₉	—	4	2	no	—
23	C₄⋊Dic₈₉	—	4	2	no	—
24	C₂²×C₄×C₈₉	—	2	1	no	—
25	C₂²×C₄×C₈₉	—	2	1	no	—
26	C₂²×C₄×C₈₉	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₈₉	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₈₉	—	2	1	no	—
30	C₄⋊C₄×C₈₉	—	2	2	no	—
31	C₄⋊C₄×C₈₉	—	4	2	no	—
32	C₄⋊C₄×C₈₉	D₄	1	2	yes	—
33	C₄²×C₈₉	—	2	2	no	—
34	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₈₉	—	2	1	no	—
36	C₂×C₄×C₈₉	—	2	1	no	—
37	C₂×C₄×C₈₉	—	2	1	no	—
38	C₂×C₄×C₈₉	—	2	1	no	—
39	C₂×C₄×C₈₉	—	2	2	no	—
40	C₂×C₄×C₈₉	—	4	1	no	—
41	C₂×C₄×C₈₉	—	4	1	no	—
42	C₂×C₄×C₈₉	—	4	1	no	—
43	C₂×C₄×C₈₉	—	4	1	no	—
44	C₂×C₄×C₈₉	—	4	1	no	—
45	C₂×C₄×C₈₉	D₈	1	2	yes	—
46	C₂×C₄×C₈₉	Q₁₆	1	2	yes	—
47	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₈₉	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₈₉	#41	1	2	yes	—
50	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
51	C₂×Dic₈₉	—	4	2	no	—
52	C₂×Dic₈₉	—	4	2	no	—
53	C₂×Dic₈₉	—	4	2	no	—
54	#10623494	—	2	4	no	—
55	C₂²×C₈₉	#121	1	1	yes	characteristic
56	C₂²×C₈₉	#123	1	1	yes	characteristic
57	C₂²×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
58	C₂²×C₈₉	C₂×D₈	1	1	yes	characteristic
59	C₂²×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
60	C₂²×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
61	C₂²×C₈₉	#137	1	1	yes	characteristic
62	C₄×C₈₉	—	2	2	no	—
63	C₄×C₈₉	—	2	2	no	—
64	C₄×C₈₉	—	4	2	no	—
65	C₄×C₈₉	—	8	1	no	—
66	C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
67	C₄×C₈₉	#125	1	2	yes	—
68	Dic₈₉	—	8	2	no	—
69	C₂×C₈₉	#398	1	1	yes	characteristic
70	C₂×C₈₉	#459	1	1	yes	characteristic
71	C₂×C₈₉	#462	1	1	yes	characteristic
72	C₂×C₈₉	#480	1	1	yes	characteristic
73	C₂×C₈₉	#484	1	1	yes	characteristic
74	C₂×C₈₉	#500	1	1	yes	characteristic
75	C₂×C₈₉	#501	1	1	yes	characteristic
76	#2083	—	89	1	no	maximal, Sylow
77	C₈₉	#2083	1	1	yes	characteristic, complemented, Sylow
78	C₂×#107	—	89	2	no	—
79	#389	D₈₉	1	1	yes	characteristic
80	#384	—	89	1	no	—
81	#340	—	89	2	no	—
82	#336	—	89	1	no	—
83	C₂²×C₈	—	89	2	no	—
84	C₂×C₄⋊C₄	—	2	1	no	—
85	C₂×C₄⋊C₄	—	89	2	no	—
86	C₂×C₄⋊C₄	—	178	1	no	—
87	C₂×C₄⋊C₄	—	178	1	no	—
88	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
89	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
90	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
91	#107	—	178	4	no	—
92	#95	—	2	1	no	—
93	C₂²×C₄	—	2	1	no	—
94	C₂²×C₄	—	2	1	no	—
95	C₂²×C₄	—	178	2	no	—
96	C₂²×C₄	C₂²⋊D₈₉	1	1	yes	characteristic
97	C₂²×C₄	C₂²⋊D₈₉	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
99	C₂×C₈	—	89	4	no	—
100	C₂×C₈	—	178	2	no	—
101	C₂×C₈	—	178	2	no	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	4	2	no	—
105	C₄⋊C₄	—	356	2	no	—
106	C₄⋊C₄	—	356	2	no	—
107	C₄⋊C₄	—	178	2	no	—
108	C₄⋊C₄	—	178	2	no	—
109	C₄⋊C₄	D₃₅₆	1	2	yes	—
110	C₄²	—	2	2	no	—
111	C₂³	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	356	2	no	—
123	C₂×C₄	—	356	2	no	—
124	C₂×C₄	—	356	2	no	—
125	C₂×C₄	#13083559	1	2	yes	—
126	C₂×C₄	#13083564	1	2	yes	—
127	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
128	C₂×C₄	Q₈×D₈₉	1	2	yes	—
129	C₂×C₄	#13083573	1	2	yes	—
130	C₂×C₄	#13083574	1	1	yes	characteristic
131	C₈	—	178	4	no	—
132	C₂²	#15667986	1	1	yes	characteristic
133	C₂²	#15667993	1	1	yes	characteristic
134	C₂²	#15668010	1	1	yes	characteristic
135	C₂²	C₂×#13083559	1	1	yes	characteristic
136	C₂²	C₂×#13083564	1	1	yes	characteristic
137	C₂²	#15668037	1	1	yes	characteristic
138	C₂²	#15668042	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	—	2	2	no	—
141	C₄	—	4	2	no	—
142	C₄	—	8	1	no	—
143	C₄	—	712	2	no	—
144	C₄	#15667992	1	2	yes	—
145	C₄	#15668002	1	2	yes	—
146	C₂	#19151528	1	1	yes	characteristic
147	C₂	#19151793	1	1	yes	characteristic
148	C₂	#19151808	1	1	yes	characteristic
149	C₂	#19151909	1	1	yes	characteristic
150	C₂	#19151931	1	1	yes	characteristic
151	C₂	#19152001	1	1	yes	characteristic
152	C₂	#19152008	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15667945

C₂

yes

maximal

#19151468

C₂

yes

maximal, characteristic

#19151360

C₂

yes

maximal

#19151353

C₂

yes

maximal, characteristic

#389×C₈₉

C₂

yes

maximal, characteristic

C₂²×#10623494

C₂²

yes

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

#15667945

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₈₉

C₂²

yes

characteristic

C₂×C₄²×C₈₉

C₂²

yes

characteristic

#95×C₈₉

—

C₂²×Dic₈₉

—

C₂×#10623494

—

C₂×#10623494

—

C₂×#10623494

Q₈

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

D₄

yes

characteristic

C₂²×C₄×C₈₉

D₄

yes

characteristic

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

D₄

yes

—

C₄²×C₈₉

—

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

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

D₈

yes

—

C₂×C₄×C₈₉

Q₁₆

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

#41

yes

—

C₂×C₄×C₈₉

#41

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

#10623494

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

C₄⋊Q₈

yes

characteristic

C₂²×C₈₉

C₂×D₈

yes

characteristic

C₂²×C₈₉

C₂×Q₁₆

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂²⋊Q₈

yes

—

C₄×C₈₉

#125

yes

—

Dic₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

C₂×#107

—

#389

D₈₉

yes

—

—

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

178

—

C₂×C₄⋊C₄

—

178

—

C₂×C₄⋊C₄

C₂×D₈₉

yes

characteristic

C₂×C₄⋊C₄

C₂×D₈₉

yes

characteristic

C₂×C₄²

C₂×D₈₉

yes

characteristic

#107

—

178

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

178

—

C₂²×C₄

C₂²⋊D₈₉

yes

characteristic

C₂²×C₄

C₂²⋊D₈₉

yes

characteristic

C₂²×C₄

C₂²×D₈₉

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

178

—

101

C₂×C₈

—

178

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

356

—

106

C₄⋊C₄

—

356

—

107

C₄⋊C₄

—

178

—

108

C₄⋊C₄

—

178

—

109

C₄⋊C₄

D₃₅₆

yes

—

110

C₄²

—

111

C₂³

C₂×C₂²⋊D₈₉

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

356

—

123

C₂×C₄

—

356

—

124

C₂×C₄

—

356

—

125

C₂×C₄

#13083559

yes

—

126

C₂×C₄

#13083564

yes

—

127

C₂×C₄

C₂×D₃₅₆

yes

characteristic

128

C₂×C₄

Q₈×D₈₉

yes

—

129

C₂×C₄

#13083573

yes

—

130

C₂×C₄

#13083574

yes

characteristic

131

C₈

—

178

—

132

C₂²

#15667986

yes

characteristic

133

C₂²

#15667993

yes

characteristic

134

C₂²

#15668010

yes

characteristic

135

C₂²

C₂×#13083559

yes

characteristic

136

C₂²

C₂×#13083564

yes

characteristic

137

C₂²

#15668037

yes

characteristic

138

C₂²

#15668042

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

712

—

144

C₄

#15667992

yes

—

145

C₄

#15668002

yes

—

146

C₂

#19151528

yes

characteristic

147

C₂

#19151793

yes

characteristic

148

C₂

#19151808

yes

characteristic

149

C₂

#19151909

yes

characteristic

150

C₂

#19151931

yes

characteristic

151

C₂

#19152001

yes

characteristic

152

C₂

#19152008

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₈₉
commutator, cosocle	C₂²×C₄×C₈₉	C₂³
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	#389×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₄
hypercenter	#389	D₈₉
hypocenter	C₈₉	#2083
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₈₉

commutator, cosocle

C₂²×C₄×C₈₉

C₂³

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

#389×C₈₉

C₂

socle, prime radical

C₂³×C₈₉

C₂×D₄

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

N	H ≅ G/N	type
77. C₈₉	#2083	semidirect

H ≅ G/N

type

77. C₈₉

#2083

semidirect

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

subgroup	index
C₂×#15667945	2
#19151468	2
#19151360	2
#19151353	2
#389×C₈₉	2
#2083	89

subgroup

index

#389×C₈₉

Subgroups — #26021723

Conjugacy classes of subgroups

Complements

Maximal subgroups