Group #26020344 — subgroups

Subgroups — #26020344

Order 11392.

300 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 101 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×#15667971	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊#10623494	C₂	1	1	yes	maximal, characteristic
3	#19151360	C₂	1	2	yes	maximal, complemented
4	#19151360	C₂	1	2	yes	maximal, complemented
5	#380×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10623494	C₂²	1	2	yes	—
7	#15667971	—	2	2	no	—
8	#15667971	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊Dic₈₉	—	2	1	no	—
12	C₂×C₄×Dic₈₉	—	2	1	no	—
13	C₂²⋊#10623494	—	2	2	no	—
14	C₂³×C₄×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
16	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
17	#95×C₈₉	—	2	1	no	—
18	C₂²×Dic₈₉	—	2	1	no	—
19	C₂²×Dic₈₉	—	2	1	no	—
20	C₂×#10623494	—	2	2	no	—
21	C₂×#10623494	—	2	4	no	—
22	C₄⋊Dic₈₉	—	2	1	no	—
23	C₄⋊Dic₈₉	—	2	1	no	—
24	C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
25	C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
26	C₂²⋊Dic₈₉	—	2	4	no	—
27	C₄×Dic₈₉	—	2	4	no	—
28	C₂⁴×C₈₉	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₈₉	—	2	4	no	—
30	C₂²×C₄×C₈₉	—	2	2	no	—
31	C₂²×C₄×C₈₉	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₈₉	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₈₉	D₄	1	2	yes	—
34	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₈₉	—	2	4	no	—
36	C₂²⋊C₄×C₈₉	—	4	2	no	—
37	C₂³×C₈₉	—	2	2	no	—
38	C₂³×C₈₉	—	2	2	no	—
39	C₂³×C₈₉	—	2	1	no	—
40	C₂³×C₈₉	—	2	1	no	—
41	C₂³×C₈₉	C₂²⋊C₄	1	2	yes	—
42	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₈₉	—	2	2	no	—
44	C₂×C₄×C₈₉	—	2	4	no	—
45	C₂×C₄×C₈₉	—	2	4	no	—
46	C₂×C₄×C₈₉	—	4	2	no	—
47	C₂×C₄×C₈₉	—	4	2	no	—
48	C₂×C₄×C₈₉	—	4	2	no	—
49	C₂×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
50	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₈₉	#41	1	4	yes	—
53	C₂×Dic₈₉	—	2	2	no	—
54	C₂×Dic₈₉	—	2	2	no	—
55	C₂×Dic₈₉	—	4	2	no	—
56	C₂×Dic₈₉	—	4	1	no	—
57	C₂×Dic₈₉	—	4	1	no	—
58	#10623494	—	4	2	no	—
59	C₂²×C₈₉	—	2	2	no	—
60	C₂²×C₈₉	—	2	2	no	—
61	C₂²×C₈₉	—	2	2	no	—
62	C₂²×C₈₉	—	4	2	no	—
63	C₂²×C₈₉	—	4	2	no	—
64	C₂²×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₈₉	#123	1	2	yes	—
66	C₂²×C₈₉	#135	1	2	yes	—
67	C₂²×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
68	C₂²×C₈₉	#137	1	1	yes	characteristic
69	C₄×C₈₉	—	2	2	no	—
70	C₄×C₈₉	—	8	2	no	—
71	C₄×C₈₉	#117	1	2	yes	—
72	C₄×C₈₉	#123	1	2	yes	—
73	Dic₈₉	—	4	2	no	—
74	Dic₈₉	—	4	2	no	—
75	C₂×C₈₉	—	4	2	no	—
76	C₂×C₈₉	#381	1	1	yes	characteristic
77	C₂×C₈₉	#416	1	1	yes	characteristic
78	C₂×C₈₉	#417	1	1	yes	characteristic
79	C₂×C₈₉	#482	1	2	yes	—
80	C₂×C₈₉	#485	1	2	yes	—
81	#1825	—	89	1	no	maximal, Sylow
82	C₈₉	#1825	1	1	yes	characteristic, complemented, Sylow
83	C₂×#117	—	89	1	no	—
84	C₂×C₂²⋊C₈	—	89	1	no	—
85	#380	D₈₉	1	1	yes	characteristic
86	#340	—	89	2	no	—
87	#340	—	89	2	no	—
88	C₂³×C₄	C₂×D₈₉	1	1	yes	characteristic
89	C₂²×C₈	—	89	2	no	—
90	#117	—	89	4	no	—
91	#117	—	178	2	no	—
92	C₂×C₄⋊C₄	—	89	1	no	—
93	C₂×C₄⋊C₄	—	89	1	no	—
94	C₂×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
95	C₂×C₂²⋊C₄	—	2	1	no	—
96	C₂×C₂²⋊C₄	—	178	1	no	—
97	C₂×C₄²	—	178	1	no	—
98	C₂²⋊C₈	—	178	2	no	—
99	#95	—	2	1	no	—
100	C₂⁴	C₂²⋊D₈₉	1	1	yes	characteristic
101	C₂²×C₄	—	2	4	no	—
102	C₂²×C₄	—	2	2	no	—
103	C₂²×C₄	—	178	1	no	—
104	C₂²×C₄	—	178	1	no	—
105	C₂²×C₄	C₄×D₈₉	1	2	yes	—
106	C₂²×C₄	D₃₅₆	1	2	yes	—
107	C₂²×C₄	C₂²⋊D₈₉	1	1	yes	characteristic
108	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
109	C₂×C₈	—	178	2	no	—
110	C₂×C₈	—	178	4	no	—
111	C₄⋊C₄	—	2	4	no	—
112	C₄⋊C₄	—	89	2	no	—
113	C₄⋊C₄	—	89	2	no	—
114	C₄⋊C₄	—	178	1	no	—
115	C₄⋊C₄	—	178	1	no	—
116	C₂²⋊C₄	—	4	2	no	—
117	C₂²⋊C₄	—	178	4	no	—
118	C₄²	—	178	4	no	—
119	C₂³	—	2	2	no	—
120	C₂³	—	2	2	no	—
121	C₂³	—	2	1	no	—
122	C₂³	—	2	1	no	—
123	C₂³	#13083551	1	2	yes	—
124	C₂³	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	4	2	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	356	2	no	—
132	C₂×C₄	—	356	1	no	—
133	C₂×C₄	—	356	1	no	—
134	C₂×C₄	—	178	2	no	—
135	C₂×C₄	—	178	2	no	—
136	C₂×C₄	#13083551	1	2	yes	—
137	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
138	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
139	C₂×C₄	#13083573	1	4	yes	—
140	C₈	—	356	2	no	—
141	C₂²	—	2	2	no	—
142	C₂²	—	2	2	no	—
143	C₂²	—	2	2	no	—
144	C₂²	—	4	2	no	—
145	C₂²	—	4	2	no	—
146	C₂²	C₂×#13083551	1	1	yes	characteristic
147	C₂²	#15667993	1	2	yes	—
148	C₂²	#15668032	1	2	yes	—
149	C₂²	#15668038	1	1	yes	characteristic
150	C₂²	#15668043	1	1	yes	characteristic
151	C₄	—	2	2	no	—
152	C₄	—	8	2	no	—
153	C₄	—	356	2	no	—
154	C₄	—	356	2	no	—
155	C₄	#15667968	1	2	yes	—
156	C₄	#15667996	1	2	yes	—
157	C₂	—	4	2	no	—
158	C₂	#19151459	1	1	yes	characteristic
159	C₂	#19151592	1	1	yes	characteristic
160	C₂	#19151597	1	1	yes	characteristic
161	C₂	#19151919	1	2	yes	—
162	C₂	#19151936	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15667971

C₂

yes

maximal, characteristic

C₂×C₂²⋊#10623494

C₂

yes

maximal, characteristic

#19151360

C₂

yes

maximal, complemented

#19151360

C₂

yes

maximal, complemented

#380×C₈₉

C₂

yes

maximal, characteristic

C₂²×#10623494

C₂²

yes

—

#15667971

—

#15667971

C₄

yes

complemented

C₂×C₄⋊Dic₈₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₈₉

C₂²

yes

characteristic

C₂×C₂²⋊Dic₈₉

—

C₂×C₄×Dic₈₉

—

C₂²⋊#10623494

—

C₂³×C₄×C₈₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₈₉

C₂²

yes

—

C₂×C₂²⋊C₄×C₈₉

—

#95×C₈₉

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

C₂×#10623494

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

C₂×C₄

yes

—

C₄⋊Dic₈₉

C₂×C₄

yes

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

C₂⁴×C₈₉

D₄

yes

characteristic

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

—

C₂²⋊C₄×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

C₂²⋊C₄

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

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

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₈₉

#123

yes

—

C₂²×C₈₉

#135

yes

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#117

yes

—

C₄×C₈₉

#123

yes

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₂×C₈₉

#485

yes

—

#1825

—

maximal, Sylow

C₈₉

#1825

yes

characteristic, complemented, Sylow

C₂×#117

—

C₂×C₂²⋊C₈

—

#380

D₈₉

yes

characteristic

#340

—

#340

—

C₂³×C₄

C₂×D₈₉

yes

characteristic

C₂²×C₈

—

#117

—

#117

—

178

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₈₉

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

178

—

C₂×C₄²

—

178

—

C₂²⋊C₈

—

178

—

#95

—

100

C₂⁴

C₂²⋊D₈₉

yes

characteristic

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

178

—

104

C₂²×C₄

—

178

—

105

C₂²×C₄

C₄×D₈₉

yes

—

106

C₂²×C₄

D₃₅₆

yes

—

107

C₂²×C₄

C₂²⋊D₈₉

yes

characteristic

108

C₂²×C₄

C₂²×D₈₉

yes

characteristic

109

C₂×C₈

—

178

—

110

C₂×C₈

—

178

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

178

—

115

C₄⋊C₄

—

178

—

116

C₂²⋊C₄

—

117

C₂²⋊C₄

—

178

—

118

C₄²

—

178

—

119

C₂³

—

120

C₂³

—

121

C₂³

—

122

C₂³

—

123

C₂³

#13083551

yes

—

124

C₂³

C₂×C₂²⋊D₈₉

yes

characteristic

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

356

—

132

C₂×C₄

—

356

—

133

C₂×C₄

—

356

—

134

C₂×C₄

—

178

—

135

C₂×C₄

—

178

—

136

C₂×C₄

#13083551

yes

—

137

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

138

C₂×C₄

C₂×D₃₅₆

yes

characteristic

139

C₂×C₄

#13083573

yes

—

140

C₈

—

356

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

C₂×#13083551

yes

characteristic

147

C₂²

#15667993

yes

—

148

C₂²

#15668032

yes

—

149

C₂²

#15668038

yes

characteristic

150

C₂²

#15668043

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

356

—

154

C₄

—

356

—

155

C₄

#15667968

yes

—

156

C₄

#15667996

yes

—

157

C₂

—

158

C₂

#19151459

yes

characteristic

159

C₂

#19151592

yes

characteristic

160

C₂

#19151597

yes

characteristic

161

C₂

#19151919

yes

—

162

C₂

#19151936

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₈₉
commutator	C₂×C₄×C₈₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	#380×C₈₉	C₂
socle	C₂³×C₈₉	C₂×D₄
prime radical	C₂⁴×C₈₉	D₄
cosocle	C₂²×C₄×C₈₉	C₂³
hypercenter	#380	D₈₉
hypocenter	C₈₉	#1825
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #380

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₈₉

commutator

C₂×C₄×C₈₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

#380×C₈₉

C₂

socle

C₂³×C₈₉

C₂×D₄

prime radical

C₂⁴×C₈₉

D₄

cosocle

C₂²×C₄×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #380

N	H ≅ G/N	type
3. #19151360	C₂	semidirect
4. #19151360	C₂	semidirect
8. #15667971	C₄	semidirect
82. C₈₉	#1825	semidirect

H ≅ G/N

type

3. #19151360

C₂

semidirect

4. #19151360

C₂

semidirect

8. #15667971

C₄

semidirect

82. C₈₉

#1825

semidirect

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

subgroup	index
C₂×#15667971	2
C₂×C₂²⋊#10623494	2
#19151360	2
#380×C₈₉	2
#1825	89

subgroup

index

C₂×#15667971

C₂×C₂²⋊#10623494

#19151360

#380×C₈₉

#1825

Subgroups — #26020344

Conjugacy classes of subgroups

Complements

Maximal subgroups