Group #26020276 — subgroups

Subgroups — #26020276

Order 11392.

336 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 109 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	C₂²×C₈×D₈₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#15667943	C₂	1	2	yes	maximal, complemented
3	#19151455	C₂	1	2	yes	maximal
4	#19151361	C₂	1	1	yes	maximal, characteristic, 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×C₈×D₈₉	—	2	4	no	—
9	C₂×C₈×D₈₉	C₄	1	4	yes	complemented
10	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
11	C₂×#13083551	—	2	2	no	—
12	#15667943	—	2	4	no	—
13	#15667919	—	2	2	no	—
14	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
16	C₂³×D₈₉	D₄	1	1	yes	characteristic
17	C₂²×Dic₈₉	—	2	2	no	—
18	C₂²×Dic₈₉	D₄	1	1	yes	characteristic
19	C₂×C₄×D₈₉	—	2	2	no	—
20	C₂×C₄×D₈₉	—	2	2	no	—
21	C₂×C₄×D₈₉	C₂×C₄	1	2	yes	—
22	C₂×C₄×D₈₉	Q₈	1	2	yes	—
23	C₂×#10623494	C₂×C₄	1	2	yes	—
24	C₂×#10623494	D₄	1	4	yes	—
25	C₈×D₈₉	—	2	8	no	—
26	C₄⋊Dic₈₉	—	2	2	no	—
27	C₄⋊Dic₈₉	—	2	2	no	—
28	#13083551	—	4	4	no	—
29	C₂²×C₄×C₈₉	—	2	2	no	—
30	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₈₉	—	2	2	no	—
32	C₂×C₈×C₈₉	C₂×C₄	1	2	yes	—
33	C₄⋊C₄×C₈₉	—	2	2	no	—
34	C₄⋊C₄×C₈₉	—	2	2	no	—
35	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
36	C₂²×D₈₉	—	2	2	no	—
37	C₂²×D₈₉	—	2	1	no	—
38	C₂²×D₈₉	—	2	1	no	—
39	C₂²×D₈₉	C₄⋊C₄	1	2	yes	—
40	C₂²×D₈₉	SD₁₆	1	4	yes	—
41	C₂×C₄×C₈₉	—	4	2	no	—
42	C₂×C₄×C₈₉	—	4	2	no	—
43	C₂×C₄×C₈₉	—	4	2	no	—
44	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₈₉	#41	1	2	yes	—
48	C₂×Dic₈₉	—	2	2	no	—
49	C₂×Dic₈₉	—	4	2	no	—
50	C₂×Dic₈₉	—	4	2	no	—
51	C₂×Dic₈₉	—	4	2	no	—
52	C₂×Dic₈₉	C₄⋊C₄	1	2	yes	—
53	C₄×D₈₉	—	2	2	no	—
54	C₄×D₈₉	—	2	2	no	—
55	C₄×D₈₉	—	2	4	no	—
56	C₈×C₈₉	—	2	2	no	—
57	#10623494	C₂²⋊C₄	1	4	yes	complemented
58	C₂²×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₈₉	#121	1	2	yes	—
60	C₂²×C₈₉	C₂×SD₁₆	1	2	yes	—
61	C₂²×C₈₉	#135	1	1	yes	characteristic
62	C₂²×C₈₉	#135	1	1	yes	characteristic
63	C₂×D₈₉	—	2	2	no	—
64	C₂×D₈₉	—	2	2	no	—
65	C₂×D₈₉	—	2	4	no	—
66	C₂×D₈₉	—	2	4	no	—
67	C₂×D₈₉	#106	1	4	yes	complemented
68	C₄×C₈₉	—	8	2	no	—
69	C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₄×C₈₉	#117	1	1	yes	characteristic
71	C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
72	Dic₈₉	—	2	2	no	—
73	Dic₈₉	—	8	2	no	—
74	C₂×C₈₉	#380	1	1	yes	characteristic
75	C₂×C₈₉	C₂×#106	1	1	yes	characteristic
76	C₂×C₈₉	#427	1	1	yes	characteristic
77	C₂×C₈₉	#465	1	2	yes	—
78	C₂×C₈₉	#465	1	2	yes	—
79	D₈₉	—	2	4	no	—
80	#1807	—	89	1	no	maximal, Sylow
81	C₈₉	#1807	1	1	yes	characteristic, complemented, Sylow
82	C₂³×C₈	—	89	1	no	—
83	C₂×#106	—	89	2	no	—
84	#380	—	89	2	no	—
85	#340	—	89	1	no	—
86	#340	D₈₉	1	1	yes	characteristic, complemented
87	C₂³×C₄	—	89	1	no	—
88	C₂²×C₈	—	89	1	no	—
89	C₂²×C₈	—	89	4	no	—
90	C₂²×C₈	—	178	4	no	—
91	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
92	C₂×C₄⋊C₄	—	89	2	no	—
93	C₂×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
94	C₂×C₂²⋊C₄	—	178	2	no	—
95	#106	—	178	4	no	—
96	#95	—	178	2	no	—
97	C₂⁴	—	89	1	no	—
98	C₂²×C₄	—	2	2	no	—
99	C₂²×C₄	—	89	1	no	—
100	C₂²×C₄	—	89	2	no	—
101	C₂²×C₄	—	89	2	no	—
102	C₂²×C₄	—	178	2	no	—
103	C₂²×C₄	—	178	2	no	—
104	C₂²×C₄	—	178	2	no	—
105	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	89	2	no	—
108	C₂×C₈	—	89	4	no	—
109	C₂×C₈	—	178	8	no	—
110	C₂×C₈	C₄×D₈₉	1	2	yes	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	2	2	no	—
113	C₄⋊C₄	—	178	2	no	—
114	C₄⋊C₄	—	178	2	no	—
115	C₂²⋊C₄	—	356	4	no	—
116	C₂³	—	89	4	no	—
117	C₂³	—	89	2	no	—
118	C₂³	—	178	2	no	—
119	C₂³	—	178	1	no	—
120	C₂³	—	178	1	no	—
121	C₂³	D₄×D₈₉	1	1	yes	characteristic
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	89	2	no	—
126	C₂×C₄	—	356	2	no	—
127	C₂×C₄	—	356	2	no	—
128	C₂×C₄	—	356	2	no	—
129	C₂×C₄	—	178	2	no	—
130	C₂×C₄	—	178	2	no	—
131	C₂×C₄	—	178	4	no	—
132	C₂×C₄	—	178	2	no	—
133	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
134	C₂×C₄	D₄×D₈₉	1	2	yes	—
135	C₂×C₄	Q₈×D₈₉	1	1	yes	characteristic
136	C₂×C₄	#13083574	1	2	yes	—
137	C₈	—	2	2	no	—
138	C₈	—	89	4	no	—
139	C₂²	—	89	4	no	—
140	C₂²	—	178	2	no	—
141	C₂²	—	178	2	no	—
142	C₂²	—	178	4	no	—
143	C₂²	—	178	4	no	—
144	C₂²	C₄⋊C₄×D₈₉	1	1	yes	characteristic
145	C₂²	#15667985	1	2	yes	—
146	C₂²	SD₁₆×D₈₉	1	2	yes	—
147	C₂²	#15668031	1	1	yes	characteristic
148	C₂²	#15668035	1	1	yes	characteristic
149	C₄	—	8	2	no	—
150	C₄	—	178	2	no	—
151	C₄	—	712	2	no	—
152	C₄	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
153	C₄	#15667969	1	1	yes	characteristic
154	C₄	#15667990	1	2	yes	—
155	C₂	—	178	4	no	—
156	C₂	#19151457	1	1	yes	characteristic
157	C₂	#106×D₈₉	1	1	yes	characteristic
158	C₂	#19151635	1	1	yes	characteristic
159	C₂	#19151826	1	2	yes	—
160	C₂	#19151830	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₈×D₈₉

C₂

yes

maximal, characteristic

C₂×#15667943

C₂

yes

maximal, complemented

#19151455

C₂

yes

maximal

#19151361

C₂

yes

maximal, characteristic, complemented

#340×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₈₉

C₂²

yes

characteristic

C₂²×#10623494

C₂²

yes

characteristic

C₂×C₈×D₈₉

—

C₂×C₈×D₈₉

C₄

yes

complemented

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×#13083551

—

#15667943

—

#15667919

—

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

yes

—

C₂×C₄×D₈₉

Q₈

yes

—

C₂×#10623494

C₂×C₄

yes

—

C₂×#10623494

D₄

yes

—

C₈×D₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083551

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂×C₄

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

C₄⋊C₄

yes

—

C₂²×D₈₉

SD₁₆

yes

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₈₉

#41

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

C₄⋊C₄

yes

—

C₄×D₈₉

—

C₄×D₈₉

—

C₄×D₈₉

—

C₈×C₈₉

—

#10623494

C₂²⋊C₄

yes

complemented

C₂²×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₈₉

#121

yes

—

C₂²×C₈₉

C₂×SD₁₆

yes

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

#106

yes

complemented

C₄×C₈₉

—

C₄×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₈₉

#117

yes

characteristic

C₄×C₈₉

C₂²⋊Q₈

yes

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₂×C₈₉

#465

yes

—

D₈₉

—

#1807

—

maximal, Sylow

C₈₉

#1807

yes

characteristic, complemented, Sylow

C₂³×C₈

—

—

—

—

D₈₉

yes

characteristic, complemented

C₂³×C₄

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

178

—

C₂²×C₈

C₂×D₈₉

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₈₉

yes

—

C₂×C₂²⋊C₄

—

178

—

#106

—

178

—

#95

—

178

—

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

178

—

103

C₂²×C₄

—

178

—

104

C₂²×C₄

—

178

—

105

C₂²×C₄

C₂²×D₈₉

yes

characteristic

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

178

—

110

C₂×C₈

C₄×D₈₉

yes

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

178

—

114

C₄⋊C₄

—

178

—

115

C₂²⋊C₄

—

356

—

116

C₂³

—

117

C₂³

—

118

C₂³

—

178

—

119

C₂³

—

178

—

120

C₂³

—

178

—

121

C₂³

D₄×D₈₉

yes

characteristic

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

356

—

127

C₂×C₄

—

356

—

128

C₂×C₄

—

356

—

129

C₂×C₄

—

178

—

130

C₂×C₄

—

178

—

131

C₂×C₄

—

178

—

132

C₂×C₄

—

178

—

133

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

134

C₂×C₄

D₄×D₈₉

yes

—

135

C₂×C₄

Q₈×D₈₉

yes

characteristic

136

C₂×C₄

#13083574

yes

—

137

C₈

—

138

C₈

—

139

C₂²

—

140

C₂²

—

178

—

141

C₂²

—

178

—

142

C₂²

—

178

—

143

C₂²

—

178

—

144

C₂²

C₄⋊C₄×D₈₉

yes

characteristic

145

C₂²

#15667985

yes

—

146

C₂²

SD₁₆×D₈₉

yes

—

147

C₂²

#15668031

yes

characteristic

148

C₂²

#15668035

yes

characteristic

149

C₄

—

150

C₄

—

178

—

151

C₄

—

712

—

152

C₄

C₂²⋊C₄×D₈₉

yes

characteristic

153

C₄

#15667969

yes

characteristic

154

C₄

#15667990

yes

—

155

C₂

—

178

—

156

C₂

#19151457

yes

characteristic

157

C₂

#106×D₈₉

yes

characteristic

158

C₂

#19151635

yes

characteristic

159

C₂

#19151826

yes

—

160

C₂

#19151830

yes

—

	subgroup	G/H
center	C₂³	D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	#340×C₈₉	C₂
socle	C₂³×C₈₉	C₂×D₄
prime radical	C₂³×D₈₉	D₄
cosocle	C₂²×C₄×C₈₉	C₂³
hypercenter	#340	D₈₉
hypocenter	C₈₉	#1807
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×D₈₉

commutator

C₂×C₄×C₈₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

#340×C₈₉

C₂

socle

C₂³×C₈₉

C₂×D₄

prime radical

C₂³×D₈₉

D₄

cosocle

C₂²×C₄×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
2. C₂×#15667943	C₂	semidirect
4. #19151361	C₂	semidirect
5. #340×C₈₉	C₂	semidirect
9. C₂×C₈×D₈₉	C₄	semidirect
57. #10623494	C₂²⋊C₄	semidirect
67. C₂×D₈₉	#106	semidirect
81. C₈₉	#1807	semidirect
86. #340	D₈₉	semidirect

H ≅ G/N

type

2. C₂×#15667943

C₂

semidirect

4. #19151361

C₂

semidirect

5. #340×C₈₉

C₂

semidirect

9. C₂×C₈×D₈₉

C₄

semidirect

57. #10623494

C₂²⋊C₄

semidirect

67. C₂×D₈₉

#106

semidirect

81. C₈₉

#1807

semidirect

86. #340

D₈₉

semidirect

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

subgroup	index
C₂²×C₈×D₈₉	2
C₂×#15667943	2
#19151455	2
#19151361	2
#340×C₈₉	2
#1807	89

subgroup

index

C₂²×C₈×D₈₉

#340×C₈₉

Subgroups — #26020276

Conjugacy classes of subgroups

Complements

Maximal subgroups