Group #26021399 — subgroups

Subgroups — #26021399

Order 11392.

294 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 83 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×#15668004	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15667934	C₂	1	2	yes	maximal
3	#19151500	C₂	1	1	yes	maximal, characteristic
4	#19151361	C₂	1	2	yes	maximal, complemented
5	#336×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₃₅₆	C₂²	1	1	yes	characteristic
7	#15668004	—	2	4	no	—
8	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
10	C₂×#13083553	—	2	2	no	—
11	C₂×#13083551	—	2	2	no	—
12	#15667934	—	2	4	no	—
13	C₂²×C₈×C₈₉	C₂²	1	2	yes	—
14	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
15	C₂³×D₈₉	—	2	1	no	—
16	C₂×D₃₅₆	—	2	2	no	—
17	C₂×D₃₅₆	—	2	2	no	—
18	C₂×D₃₅₆	—	2	1	no	—
19	C₂×D₃₅₆	Q₈	1	2	yes	—
20	C₂²×Dic₈₉	—	2	2	no	—
21	C₂²×Dic₈₉	—	2	1	no	—
22	C₄⋊Dic₈₉	—	2	2	no	—
23	C₄⋊Dic₈₉	—	2	2	no	—
24	C₄⋊Dic₈₉	D₄	1	4	yes	—
25	#13083553	—	2	8	no	—
26	#13083551	—	4	4	no	—
27	C₂²×C₄×C₈₉	D₄	1	2	yes	—
28	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
29	C₂×C₈×C₈₉	—	2	2	no	—
30	C₂×C₈×C₈₉	—	2	2	no	—
31	C₂×C₈×C₈₉	—	2	2	no	—
32	C₄²×C₈₉	—	2	2	no	—
33	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
34	C₂²×D₈₉	—	2	2	no	—
35	C₂²×D₈₉	—	4	1	no	—
36	C₂²×D₈₉	—	4	1	no	—
37	C₂²×D₈₉	—	4	2	no	—
38	C₂²×D₈₉	—	4	2	no	—
39	C₂×C₄×C₈₉	—	2	2	no	—
40	C₂×C₄×C₈₉	—	2	2	no	—
41	C₂×C₄×C₈₉	—	2	2	no	—
42	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
45	C₂×C₄×C₈₉	#41	1	2	yes	—
46	C₂×Dic₈₉	—	2	4	no	—
47	C₂×Dic₈₉	—	4	2	no	—
48	C₂×Dic₈₉	—	4	2	no	—
49	C₂×Dic₈₉	—	4	1	no	—
50	C₂×Dic₈₉	—	4	2	no	—
51	D₃₅₆	—	2	4	no	—
52	D₃₅₆	—	4	2	no	—
53	C₈×C₈₉	—	4	2	no	—
54	C₂²×C₈₉	#120	1	1	yes	characteristic
55	C₂²×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
56	C₂²×C₈₉	#124	1	1	yes	characteristic
57	C₂²×C₈₉	#135	1	2	yes	—
58	C₂²×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
59	C₂×D₈₉	—	4	2	no	—
60	C₂×D₈₉	—	4	4	no	—
61	C₂×D₈₉	—	8	2	no	—
62	C₂×D₈₉	—	8	1	no	—
63	C₂×D₈₉	—	8	1	no	—
64	C₄×C₈₉	—	4	2	no	—
65	C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
66	C₄×C₈₉	#124	1	2	yes	—
67	Dic₈₉	—	4	4	no	—
68	Dic₈₉	—	8	1	no	—
69	C₂×C₈₉	#393	1	1	yes	characteristic
70	C₂×C₈₉	#449	1	2	yes	—
71	C₂×C₈₉	#478	1	2	yes	—
72	C₂×C₈₉	#488	1	1	yes	characteristic
73	C₂×C₈₉	#490	1	1	yes	characteristic
74	D₈₉	—	8	2	no	—
75	#2025	—	89	1	no	maximal, Sylow
76	C₈₉	#2025	1	1	yes	characteristic, complemented, Sylow
77	C₂×#125	—	89	1	no	—
78	C₂×#102	—	89	2	no	—
79	#390	—	89	1	no	—
80	#340	—	89	2	no	—
81	#336	D₈₉	1	1	yes	characteristic, complemented
82	C₂²×D₄	—	89	1	no	—
83	C₂²×C₈	C₂×D₈₉	1	2	yes	—
84	#125	—	178	4	no	—
85	C₂×C₄⋊C₄	—	89	2	no	—
86	C₂×C₄⋊C₄	—	89	1	no	—
87	C₂×C₄⋊C₄	—	178	2	no	—
88	C₂×C₂²⋊C₄	—	178	2	no	—
89	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
90	#102	—	178	4	no	—
91	C₂⁴	—	178	1	no	—
92	C₂×D₄	—	89	2	no	—
93	C₂×D₄	—	178	2	no	—
94	C₂×D₄	—	178	1	no	—
95	C₂×D₄	—	178	2	no	—
96	C₂²×C₄	—	178	1	no	—
97	C₂²×C₄	—	178	2	no	—
98	C₂²×C₄	D₃₅₆	1	2	yes	—
99	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	2	2	no	—
103	C₄⋊C₄	—	89	4	no	—
104	C₄⋊C₄	—	178	8	no	—
105	C₄⋊C₄	—	178	2	no	—
106	C₄⋊C₄	—	178	2	no	—
107	C₂²⋊C₄	—	356	4	no	—
108	C₄²	—	2	2	no	—
109	C₂³	—	356	1	no	—
110	C₂³	—	356	1	no	—
111	C₂³	—	356	2	no	—
112	C₂³	—	356	2	no	—
113	C₂³	—	178	2	no	—
114	C₂³	C₂×D₃₅₆	1	1	yes	characteristic
115	D₄	—	356	2	no	—
116	D₄	—	178	4	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	356	2	no	—
121	C₂×C₄	—	356	2	no	—
122	C₂×C₄	—	356	1	no	—
123	C₂×C₄	—	356	2	no	—
124	C₂×C₄	—	178	4	no	—
125	C₂×C₄	D₄×D₈₉	1	2	yes	—
126	C₂×C₄	Q₈×D₈₉	1	1	yes	characteristic
127	C₂×C₄	#13083573	1	1	yes	characteristic
128	C₂×C₄	#13083574	1	2	yes	—
129	C₈	—	4	2	no	—
130	C₂²	—	356	4	no	—
131	C₂²	—	356	2	no	—
132	C₂²	—	712	1	no	—
133	C₂²	—	712	1	no	—
134	C₂²	—	712	2	no	—
135	C₂²	#15667982	1	1	yes	characteristic
136	C₂²	#15667992	1	1	yes	characteristic
137	C₂²	#15668000	1	1	yes	characteristic
138	C₂²	#15668034	1	2	yes	—
139	C₂²	#15668040	1	2	yes	—
140	C₄	—	4	2	no	—
141	C₄	—	356	4	no	—
142	C₄	—	712	1	no	—
143	C₄	#15667990	1	2	yes	—
144	C₄	#15667998	1	2	yes	—
145	C₂	—	712	2	no	—
146	C₂	#19151511	1	1	yes	characteristic
147	C₂	#19151745	1	2	yes	—
148	C₂	#19151899	1	2	yes	—
149	C₂	#19151953	1	1	yes	characteristic
150	C₂	#19151963	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15668004

C₂

yes

maximal, characteristic, complemented

C₂×#15667934

C₂

yes

maximal

#19151500

C₂

yes

maximal, characteristic

#19151361

C₂

yes

maximal, complemented

#336×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₃₅₆

C₂²

yes

characteristic

#15668004

—

C₂×C₄⋊Dic₈₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×#13083553

—

C₂×#13083551

—

#15667934

—

C₂²×C₈×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₂³×D₈₉

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

Q₈

yes

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

D₄

yes

—

#13083553

—

#13083551

—

C₂²×C₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₄²×C₈₉

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×Q₈

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

C₈×C₈₉

—

C₂²×C₈₉

#120

yes

characteristic

C₂²×C₈₉

C₂²⋊Q₈

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

—

C₂²×C₈₉

AGL(1,ℤ/8)

yes

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂²⋊Q₈

yes

—

C₄×C₈₉

#124

yes

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

#393

yes

characteristic

C₂×C₈₉

#449

yes

—

C₂×C₈₉

#478

yes

—

C₂×C₈₉

yes

C₂×C₈₉

yes

D₈₉

—

#2025

—

maximal, Sylow

C₈₉

#2025

yes

characteristic, complemented, Sylow

—

—

—

—

D₈₉

yes

characteristic, complemented

C₂²×D₄

—

C₂²×C₈

C₂×D₈₉

yes

—

#125

—

178

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

178

—

C₂×C₂²⋊C₄

—

178

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

#102

—

178

—

C₂⁴

—

178

—

C₂×D₄

—

C₂×D₄

—

178

—

C₂×D₄

—

178

—

C₂×D₄

—

178

—

C₂²×C₄

—

178

—

C₂²×C₄

—

178

—

C₂²×C₄

D₃₅₆

yes

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

178

—

105

C₄⋊C₄

—

178

—

106

C₄⋊C₄

—

178

—

107

C₂²⋊C₄

—

356

—

108

C₄²

—

109

C₂³

—

356

—

110

C₂³

—

356

—

111

C₂³

—

356

—

112

C₂³

—

356

—

113

C₂³

—

178

—

114

C₂³

C₂×D₃₅₆

yes

characteristic

115

D₄

—

356

—

116

D₄

—

178

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

356

—

121

C₂×C₄

—

356

—

122

C₂×C₄

—

356

—

123

C₂×C₄

—

356

—

124

C₂×C₄

—

178

—

125

C₂×C₄

D₄×D₈₉

yes

—

126

C₂×C₄

Q₈×D₈₉

yes

characteristic

127

C₂×C₄

#13083573

yes

characteristic

128

C₂×C₄

#13083574

yes

—

129

C₈

—

130

C₂²

—

356

—

131

C₂²

—

356

—

132

C₂²

—

712

—

133

C₂²

—

712

—

134

C₂²

—

712

—

135

C₂²

#15667982

yes

characteristic

136

C₂²

#15667992

yes

characteristic

137

C₂²

#15668000

yes

characteristic

138

C₂²

#15668034

yes

—

139

C₂²

#15668040

yes

—

140

C₄

—

141

C₄

—

356

—

142

C₄

—

712

—

143

C₄

#15667990

yes

—

144

C₄

#15667998

yes

—

145

C₂

—

712

—

146

C₂

#19151511

yes

characteristic

147

C₂

#19151745

yes

—

148

C₂

#19151899

yes

—

149

C₂

#19151953

yes

characteristic

150

C₂

#19151963

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₃₅₆
commutator, cosocle	C₂²×C₄×C₈₉	C₂³
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	#336×C₈₉	C₂
socle	C₂³×C₈₉	C₂×D₄
prime radical	C₂²×D₃₅₆	C₂²
hypercenter	#336	D₈₉
hypocenter	C₈₉	#2025
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₃₅₆

commutator, cosocle

C₂²×C₄×C₈₉

C₂³

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

#336×C₈₉

C₂

socle

C₂³×C₈₉

C₂×D₄

prime radical

C₂²×D₃₅₆

C₂²

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
1. C₂×#15668004	C₂	semidirect
4. #19151361	C₂	semidirect
5. #336×C₈₉	C₂	semidirect
76. C₈₉	#2025	semidirect
81. #336	D₈₉	semidirect

H ≅ G/N

type

1. C₂×#15668004

C₂

semidirect

4. #19151361

C₂

semidirect

5. #336×C₈₉

C₂

semidirect

76. C₈₉

#2025

semidirect

81. #336

D₈₉

semidirect

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

subgroup	index
C₂×#15668004	2
C₂×#15667934	2
#19151500	2
#19151361	2
#336×C₈₉	2
#2025	89

subgroup

index

#336×C₈₉

Subgroups — #26021399

Conjugacy classes of subgroups

Complements

Maximal subgroups