Group #26021682 — subgroups

Subgroups — #26021682

Order 11392.

256 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 81 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×#15667946	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15667924	C₂	1	1	yes	maximal, characteristic
3	#19151455	C₂	1	2	yes	maximal
4	#19151360	C₂	1	2	yes	maximal, complemented
5	#340×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₈₉	C₂²	1	1	yes	characteristic
7	C₂²×#10623494	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
9	C₂×#13083551	—	2	2	no	—
10	#15667946	—	2	2	no	—
11	#15667924	—	2	2	no	—
12	#15667919	—	2	2	no	—
13	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
15	C₂³×D₈₉	D₄	1	1	yes	characteristic
16	C₂²×Dic₈₉	—	2	2	no	—
17	C₂²×Dic₈₉	D₄	1	1	yes	characteristic
18	C₂×C₄×D₈₉	—	2	1	no	—
19	C₂×C₄×D₈₉	—	2	2	no	—
20	C₂×C₄×D₈₉	—	2	2	no	—
21	C₂×C₄×D₈₉	—	2	1	no	—
22	C₂×#10623494	—	2	1	no	—
23	C₂×#10623494	—	2	2	no	—
24	C₄⋊Dic₈₉	—	2	2	no	—
25	C₄⋊Dic₈₉	—	2	2	no	—
26	#13083551	—	4	4	no	—
27	C₂²×C₄×C₈₉	—	2	2	no	—
28	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
29	C₂×C₈×C₈₉	—	2	2	no	—
30	C₂×C₈×C₈₉	D₄	1	2	yes	—
31	C₄⋊C₄×C₈₉	—	2	2	no	—
32	C₄⋊C₄×C₈₉	—	2	2	no	—
33	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
34	C₂²×D₈₉	—	2	1	no	—
35	C₂²×D₈₉	—	2	2	no	—
36	C₂²×D₈₉	—	2	1	no	—
37	C₂²×D₈₉	—	2	2	no	—
38	C₂²×D₈₉	Q₁₆	1	2	yes	—
39	C₂×C₄×C₈₉	—	4	2	no	—
40	C₂×C₄×C₈₉	—	4	2	no	—
41	C₂×C₄×C₈₉	—	4	2	no	—
42	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₈₉	#41	1	2	yes	—
44	C₂×C₄×C₈₉	#41	1	2	yes	—
45	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
46	C₂×Dic₈₉	—	2	1	no	—
47	C₂×Dic₈₉	—	2	2	no	—
48	C₂×Dic₈₉	—	4	2	no	—
49	C₂×Dic₈₉	—	4	2	no	—
50	C₂×Dic₈₉	—	4	2	no	—
51	C₄×D₈₉	—	4	1	no	—
52	C₄×D₈₉	—	4	1	no	—
53	C₄×D₈₉	—	4	2	no	—
54	C₈×C₈₉	—	2	2	no	—
55	#10623494	—	4	1	no	—
56	C₂²×C₈₉	#121	1	1	yes	characteristic
57	C₂²×C₈₉	#123	1	2	yes	—
58	C₂²×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
59	C₂²×C₈₉	#135	1	1	yes	characteristic
60	C₂²×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
61	C₂×D₈₉	—	2	2	no	—
62	C₂×D₈₉	—	2	4	no	—
63	C₂×D₈₉	—	4	1	no	—
64	C₂×D₈₉	—	4	1	no	—
65	C₂×D₈₉	—	4	2	no	—
66	C₄×C₈₉	—	8	2	no	—
67	C₄×C₈₉	#123	1	1	yes	characteristic
68	C₄×C₈₉	#124	1	1	yes	characteristic
69	C₄×C₈₉	#126	1	2	yes	—
70	Dic₈₉	—	4	1	no	—
71	Dic₈₉	—	8	2	no	—
72	C₂×C₈₉	#397	1	1	yes	characteristic
73	C₂×C₈₉	#467	1	1	yes	characteristic
74	C₂×C₈₉	#469	1	1	yes	characteristic
75	C₂×C₈₉	#482	1	2	yes	—
76	C₂×C₈₉	#484	1	2	yes	—
77	D₈₉	—	4	2	no	—
78	#2076	—	89	1	no	maximal, Sylow
79	C₈₉	#2076	1	1	yes	characteristic, complemented, Sylow
80	C₂×#107	—	89	1	no	—
81	C₂×C₂²⋊C₈	—	89	1	no	—
82	#380	—	89	2	no	—
83	#340	—	89	2	no	—
84	#340	D₈₉	1	1	yes	characteristic, complemented
85	C₂³×C₄	—	89	1	no	—
86	C₂²×C₈	—	89	1	no	—
87	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
88	C₂×C₄⋊C₄	—	89	2	no	—
89	C₂×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
90	C₂×C₂²⋊C₄	—	178	2	no	—
91	#107	—	178	2	no	—
92	C₂²⋊C₈	—	178	2	no	—
93	#95	—	178	2	no	—
94	C₂⁴	—	89	1	no	—
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	—	89	1	no	—
97	C₂²×C₄	—	178	2	no	—
98	C₂²×C₄	—	178	1	no	—
99	C₂²×C₄	—	178	2	no	—
100	C₂²×C₄	—	178	2	no	—
101	C₂²×C₄	—	178	1	no	—
102	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	—	178	1	no	—
105	C₂×C₈	—	178	2	no	—
106	C₂×C₈	D₃₅₆	1	2	yes	—
107	C₄⋊C₄	—	2	2	no	—
108	C₄⋊C₄	—	2	2	no	—
109	C₄⋊C₄	—	178	2	no	—
110	C₄⋊C₄	—	178	2	no	—
111	C₂²⋊C₄	—	356	4	no	—
112	C₂³	—	89	2	no	—
113	C₂³	—	178	1	no	—
114	C₂³	—	178	2	no	—
115	C₂³	—	178	1	no	—
116	C₂³	—	178	2	no	—
117	C₂³	D₄×D₈₉	1	1	yes	characteristic
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	356	1	no	—
122	C₂×C₄	—	356	1	no	—
123	C₂×C₄	—	356	2	no	—
124	C₂×C₄	—	356	2	no	—
125	C₂×C₄	—	356	2	no	—
126	C₂×C₄	—	356	2	no	—
127	C₂×C₄	—	178	1	no	—
128	C₂×C₄	—	178	2	no	—
129	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
130	C₂×C₄	#13083573	1	2	yes	—
131	C₂×C₄	#13083574	1	2	yes	—
132	C₂×C₄	#13083575	1	1	yes	characteristic
133	C₈	—	2	2	no	—
134	C₈	—	356	1	no	—
135	C₂²	—	356	2	no	—
136	C₂²	—	356	1	no	—
137	C₂²	—	356	1	no	—
138	C₂²	—	178	4	no	—
139	C₂²	—	178	2	no	—
140	C₂²	#15667987	1	1	yes	characteristic
141	C₂²	#15667994	1	2	yes	—
142	C₂²	Q₁₆×D₈₉	1	1	yes	characteristic
143	C₂²	#15668031	1	1	yes	characteristic
144	C₂²	#15668039	1	2	yes	—
145	C₄	—	8	2	no	—
146	C₄	—	356	1	no	—
147	C₄	—	712	2	no	—
148	C₄	#15667996	1	1	yes	characteristic
149	C₄	#15668000	1	1	yes	characteristic
150	C₄	#15668007	1	2	yes	—
151	C₂	—	356	2	no	—
152	C₂	#19151525	1	1	yes	characteristic
153	C₂	#19151840	1	1	yes	characteristic
154	C₂	#19151852	1	1	yes	characteristic
155	C₂	#19151920	1	2	yes	—
156	C₂	#19151932	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15667946

C₂

yes

maximal, characteristic, complemented

C₂×#15667924

C₂

yes

maximal, characteristic

#19151455

C₂

yes

maximal

#19151360

C₂

yes

maximal, complemented

#340×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₈₉

C₂²

yes

characteristic

C₂²×#10623494

C₂²

yes

characteristic

C₂×C₄⋊Dic₈₉

C₂²

yes

—

—

—

—

—

C₂²×C₈×C₈₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₈₉

C₂²

yes

—

C₂³×D₈₉

D₄

yes

characteristic

C₂²×Dic₈₉

—

C₂²×Dic₈₉

D₄

yes

characteristic

C₂×C₄×D₈₉

—

C₂×C₄×D₈₉

—

C₂×C₄×D₈₉

—

C₂×C₄×D₈₉

—

C₂×#10623494

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083551

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

D₄

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

Q₁₆

yes

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

#41

yes

—

C₂×C₄×C₈₉

#41

yes

—

C₂×C₄×C₈₉

#41

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₄×D₈₉

—

C₄×D₈₉

—

C₄×D₈₉

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

#121

yes

characteristic

C₂²×C₈₉

#123

yes

—

C₂²×C₈₉

C₂×Q₁₆

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₂×C₈₉

#484

yes

—

D₈₉

—

#2076

—

maximal, Sylow

C₈₉

#2076

yes

characteristic, complemented, Sylow

C₂×#107

—

C₂×C₂²⋊C₈

—

#380

—

#340

—

#340

D₈₉

yes

characteristic, complemented

C₂³×C₄

—

C₂²×C₈

—

C₂²×C₈

C₂×D₈₉

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₈₉

yes

—

C₂×C₂²⋊C₄

—

178

—

#107

—

178

—

C₂²⋊C₈

—

178

—

#95

—

178

—

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

178

—

C₂²×C₄

—

178

—

C₂²×C₄

—

178

—

100

C₂²×C₄

—

178

—

101

C₂²×C₄

—

178

—

102

C₂²×C₄

C₂²×D₈₉

yes

characteristic

103

C₂×C₈

—

104

C₂×C₈

—

178

—

105

C₂×C₈

—

178

—

106

C₂×C₈

D₃₅₆

yes

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

178

—

110

C₄⋊C₄

—

178

—

111

C₂²⋊C₄

—

356

—

112

C₂³

—

113

C₂³

—

178

—

114

C₂³

—

178

—

115

C₂³

—

178

—

116

C₂³

—

178

—

117

C₂³

D₄×D₈₉

yes

characteristic

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

356

—

122

C₂×C₄

—

356

—

123

C₂×C₄

—

356

—

124

C₂×C₄

—

356

—

125

C₂×C₄

—

356

—

126

C₂×C₄

—

356

—

127

C₂×C₄

—

178

—

128

C₂×C₄

—

178

—

129

C₂×C₄

C₂×D₃₅₆

yes

characteristic

130

C₂×C₄

#13083573

yes

—

131

C₂×C₄

#13083574

yes

—

132

C₂×C₄

#13083575

yes

characteristic

133

C₈

—

134

C₈

—

356

—

135

C₂²

—

356

—

136

C₂²

—

356

—

137

C₂²

—

356

—

138

C₂²

—

178

—

139

C₂²

—

178

—

140

C₂²

#15667987

yes

characteristic

141

C₂²

#15667994

yes

—

142

C₂²

Q₁₆×D₈₉

yes

characteristic

143

C₂²

#15668031

yes

characteristic

144

C₂²

#15668039

yes

—

145

C₄

—

146

C₄

—

356

—

147

C₄

—

712

—

148

C₄

#15667996

yes

characteristic

149

C₄

#15668000

yes

characteristic

150

C₄

#15668007

yes

—

151

C₂

—

356

—

152

C₂

#19151525

yes

characteristic

153

C₂

#19151840

yes

characteristic

154

C₂

#19151852

yes

characteristic

155

C₂

#19151920

yes

—

156

C₂

#19151932

yes

—

	subgroup	G/H
center	C₂³	D₄×D₈₉
commutator, cosocle	C₂²×C₄×C₈₉	C₂³
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	#340×C₈₉	C₂
socle	C₂³×C₈₉	C₂×D₄
prime radical	C₂³×D₈₉	D₄
hypercenter	#340	D₈₉
hypocenter	C₈₉	#2076
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×D₈₉

commutator, cosocle

C₂²×C₄×C₈₉

C₂³

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

#340×C₈₉

C₂

socle

C₂³×C₈₉

C₂×D₄

prime radical

C₂³×D₈₉

D₄

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
1. C₂×#15667946	C₂	semidirect
4. #19151360	C₂	semidirect
5. #340×C₈₉	C₂	semidirect
79. C₈₉	#2076	semidirect
84. #340	D₈₉	semidirect

H ≅ G/N

type

1. C₂×#15667946

C₂

semidirect

4. #19151360

C₂

semidirect

5. #340×C₈₉

C₂

semidirect

79. C₈₉

#2076

semidirect

84. #340

D₈₉

semidirect

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

subgroup	index
C₂×#15667946	2
C₂×#15667924	2
#19151455	2
#19151360	2
#340×C₈₉	2
#2076	89

subgroup

index

#340×C₈₉

Subgroups — #26021682

Conjugacy classes of subgroups

Complements

Maximal subgroups