Group #26021163 — subgroups

Subgroups — #26021163

Order 11392.

316 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 97 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×#15667948	C₂	1	1	yes	maximal, characteristic, complemented
2	#19151603	C₂	1	2	yes	maximal, complemented
3	C₂×#15667929	C₂	1	2	yes	maximal, complemented
4	#19151553	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₈×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13083558	C₂²	1	2	yes	—
7	C₂×#13083558	C₂²	1	1	yes	characteristic
8	#15667971	C₂²	1	1	yes	characteristic
9	#15667948	—	2	2	no	—
10	#15667933	—	2	4	no	—
11	#15667929	—	2	4	no	—
12	#15667922	—	2	1	no	—
13	C₈×Dic₈₉	—	2	1	no	—
14	C₂²×D₄×C₈₉	C₂²	1	2	yes	—
15	C₂×D₈×C₈₉	—	2	1	no	—
16	C₂×D₈×C₈₉	—	2	1	no	—
17	C₂×D₈×C₈₉	—	2	2	no	—
18	C₂×D₈×C₈₉	C₄	1	4	yes	complemented
19	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
20	#13083558	—	2	1	no	—
21	#13083558	—	2	2	no	—
22	#13083558	—	2	2	no	—
23	#13083558	D₄	1	2	yes	—
24	C₂×#10623494	—	2	1	no	—
25	C₂×#10623494	—	2	2	no	—
26	C₄⋊Dic₈₉	D₄	1	2	yes	—
27	C₂²⋊Dic₈₉	—	2	1	no	—
28	C₄×Dic₈₉	—	2	1	no	—
29	C₂⁴×C₈₉	—	2	2	no	—
30	C₂×D₄×C₈₉	—	2	2	no	—
31	C₂×D₄×C₈₉	—	2	2	no	—
32	C₂×D₄×C₈₉	—	2	2	no	—
33	C₂×D₄×C₈₉	—	2	2	no	—
34	C₂×D₄×C₈₉	—	2	2	no	—
35	C₂×D₄×C₈₉	C₂×C₄	1	4	yes	—
36	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
37	D₈×C₈₉	—	2	4	no	—
38	D₈×C₈₉	—	2	4	no	—
39	C₂×C₈×C₈₉	C₂×C₄	1	2	yes	—
40	C₂×C₈×C₈₉	D₄	1	1	yes	characteristic
41	C₂×C₈×C₈₉	D₄	1	1	yes	characteristic
42	C₂×C₈×C₈₉	D₄	1	2	yes	—
43	C₂³×C₈₉	—	2	4	no	—
44	C₂³×C₈₉	—	4	2	no	—
45	C₂³×C₈₉	—	4	2	no	—
46	C₂³×C₈₉	—	8	4	no	—
47	C₂³×C₈₉	#41	1	1	yes	characteristic
48	D₄×C₈₉	—	2	4	no	—
49	D₄×C₈₉	—	2	4	no	—
50	D₄×C₈₉	—	4	2	no	—
51	D₄×C₈₉	—	4	2	no	—
52	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
58	C₂×Dic₈₉	—	2	2	no	—
59	C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
60	C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
61	#10623494	—	2	2	no	—
62	#10623494	—	4	2	no	—
63	C₂²×C₈₉	—	2	2	no	—
64	C₂²×C₈₉	—	4	4	no	—
65	C₂²×C₈₉	—	8	4	no	—
66	C₂²×C₈₉	—	8	4	no	—
67	C₂²×C₈₉	—	8	4	no	—
68	C₂²×C₈₉	C₄×D₄	1	1	yes	characteristic
69	C₂²×C₈₉	#121	1	1	yes	characteristic
70	C₂²×C₈₉	#124	1	1	yes	characteristic
71	C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
73	C₄×C₈₉	#121	1	1	yes	characteristic
74	C₄×C₈₉	#127	1	1	yes	characteristic
75	Dic₈₉	—	4	2	no	—
76	C₂×C₈₉	—	2	1	no	—
77	C₂×C₈₉	—	2	1	no	—
78	C₂×C₈₉	—	8	4	no	—
79	C₂×C₈₉	#390	1	1	yes	characteristic
80	C₂×C₈₉	#472	1	2	yes	—
81	#1978	—	89	1	no	maximal, Sylow
82	C₈₉	#1978	1	1	yes	characteristic, complemented, Sylow
83	C₂²×D₈	D₈₉	1	1	yes	characteristic
84	C₂×#108	—	89	1	no	—
85	#418	—	89	2	no	—
86	C₂×#100	—	89	2	no	—
87	#405	—	89	1	no	—
88	C₂²×D₄	C₂×D₈₉	1	2	yes	—
89	C₂×D₈	—	2	1	no	—
90	C₂×D₈	—	2	1	no	—
91	C₂×D₈	—	2	2	no	—
92	C₂×D₈	Dic₈₉	1	4	yes	complemented
93	C₂×M₁₆	—	89	1	no	—
94	C₂×M₁₆	—	89	2	no	—
95	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
96	#117	—	89	1	no	—
97	#108	—	178	2	no	—
98	#102	—	178	4	no	—
99	#100	—	178	4	no	—
100	#97	—	178	1	no	—
101	C₄×C₈	—	178	1	no	—
102	C₂⁴	—	2	2	no	—
103	C₂×D₄	—	2	2	no	—
104	C₂×D₄	—	2	2	no	—
105	C₂×D₄	—	2	2	no	—
106	C₂×D₄	—	2	2	no	—
107	C₂×D₄	—	2	2	no	—
108	C₂×D₄	C₂×Dic₈₉	1	4	yes	—
109	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
110	D₈	—	2	4	no	—
111	D₈	—	2	4	no	—
112	M₁₆	—	89	2	no	—
113	M₁₆	—	178	2	no	—
114	M₁₆	—	178	2	no	—
115	M₁₆	—	178	1	no	—
116	C₂×C₈	—	178	2	no	—
117	C₂×C₈	—	178	1	no	—
118	C₂×C₈	C₂×Dic₈₉	1	2	yes	—
119	C₂×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
120	C₂×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
121	C₂×C₈	C₂²⋊D₈₉	1	2	yes	—
122	C₄⋊C₄	—	89	2	no	—
123	C₂²⋊C₄	—	178	1	no	—
124	C₄²	—	178	1	no	—
125	C₂³	—	2	4	no	—
126	C₂³	—	4	2	no	—
127	C₂³	—	4	2	no	—
128	C₂³	—	8	4	no	—
129	C₂³	#13083573	1	1	yes	characteristic
130	D₄	—	2	4	no	—
131	D₄	—	2	4	no	—
132	D₄	—	4	2	no	—
133	D₄	—	4	2	no	—
134	C₂×C₄	—	178	2	no	—
135	C₂×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
136	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
137	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
138	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
139	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
140	C₂×C₄	#13083573	1	1	yes	characteristic
141	C₈	—	356	2	no	—
142	C₈	—	178	2	no	—
143	C₈	C₂²⋊Dic₈₉	1	2	yes	—
144	C₈	C₂²⋊Dic₈₉	1	2	yes	—
145	C₂²	—	2	2	no	—
146	C₂²	—	4	4	no	—
147	C₂²	—	8	4	no	—
148	C₂²	—	8	4	no	—
149	C₂²	—	8	4	no	—
150	C₂²	D₄×Dic₈₉	1	1	yes	characteristic
151	C₂²	#15667983	1	1	yes	characteristic
152	C₂²	#15667999	1	1	yes	characteristic
153	C₄	—	356	2	no	—
154	C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
155	C₄	D₄×Dic₈₉	1	1	yes	characteristic
156	C₄	#15667983	1	1	yes	characteristic
157	C₄	#15668008	1	1	yes	characteristic
158	C₂	—	2	1	no	—
159	C₂	—	2	1	no	—
160	C₂	—	8	4	no	—
161	C₂	#19151501	1	1	yes	characteristic
162	C₂	#19151865	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15667948

C₂

yes

maximal, characteristic, complemented

#19151603

C₂

yes

maximal, complemented

C₂×#15667929

C₂

yes

maximal, complemented

#19151553

C₂

yes

maximal, characteristic, complemented

C₂²×D₈×C₈₉

C₂

yes

maximal, characteristic

C₂×#13083558

C₂²

yes

—

C₂×#13083558

C₂²

yes

characteristic

#15667971

C₂²

yes

—

—

—

—

C₈×Dic₈₉

—

C₂²×D₄×C₈₉

C₂²

yes

—

C₂×D₈×C₈₉

—

C₂×D₈×C₈₉

—

C₂×D₈×C₈₉

—

C₂×D₈×C₈₉

C₄

yes

complemented

C₂²×C₈×C₈₉

C₂²

yes

—

—

—

D₄

yes

—

C₂×#10623494

—

C₂×#10623494

—

C₄⋊Dic₈₉

D₄

yes

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

C₂⁴×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

C₂×C₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

D₈×C₈₉

—

D₈×C₈₉

—

C₂×C₈×C₈₉

C₂×C₄

yes

—

C₂×C₈×C₈₉

D₄

yes

characteristic

C₂×C₈×C₈₉

D₄

yes

characteristic

C₂×C₈×C₈₉

D₄

yes

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

#41

yes

characteristic

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

#41

yes

characteristic

C₂×Dic₈₉

—

C₈×C₈₉

C₂²⋊C₄

yes

—

C₈×C₈₉

C₂²⋊C₄

yes

—

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₄×D₄

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₄×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₈₉

C₄×D₄

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#390

yes

characteristic

C₂×C₈₉

#472

yes

—

#1978

—

maximal, Sylow

C₈₉

#1978

yes

characteristic, complemented, Sylow

C₂²×D₈

D₈₉

yes

—

—

—

—

C₂²×D₄

C₂×D₈₉

yes

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

Dic₈₉

yes

complemented

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₈₉

yes

characteristic

#117

—

#108

—

178

—

#102

—

178

—

#100

—

178

—

100

#97

—

178

—

101

C₄×C₈

—

178

—

102

C₂⁴

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

C₂×Dic₈₉

yes

—

109

C₂²×C₄

C₂²×D₈₉

yes

characteristic

110

D₈

—

111

D₈

—

112

M₁₆

—

113

M₁₆

—

178

—

114

M₁₆

—

178

—

115

M₁₆

—

178

—

116

C₂×C₈

—

178

—

117

C₂×C₈

—

178

—

118

C₂×C₈

C₂×Dic₈₉

yes

—

119

C₂×C₈

C₂²⋊D₈₉

yes

characteristic

120

C₂×C₈

C₂²⋊D₈₉

yes

characteristic

121

C₂×C₈

C₂²⋊D₈₉

yes

—

122

C₄⋊C₄

—

123

C₂²⋊C₄

—

178

—

124

C₄²

—

178

—

125

C₂³

—

126

C₂³

—

127

C₂³

—

128

C₂³

—

129

C₂³

#13083573

yes

characteristic

130

D₄

—

131

D₄

—

132

D₄

—

133

D₄

—

134

C₂×C₄

—

178

—

135

C₂×C₄

C₂²×Dic₈₉

yes

characteristic

136

C₂×C₄

D₄×D₈₉

yes

characteristic

137

C₂×C₄

D₄×D₈₉

yes

characteristic

138

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

139

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

140

C₂×C₄

#13083573

yes

characteristic

141

C₈

—

356

—

142

C₈

—

178

—

143

C₈

C₂²⋊Dic₈₉

yes

—

144

C₈

C₂²⋊Dic₈₉

yes

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

D₄×Dic₈₉

yes

characteristic

151

C₂²

#15667983

yes

characteristic

152

C₂²

#15667999

yes

characteristic

153

C₄

—

356

—

154

C₄

C₂×C₂²⋊Dic₈₉

yes

characteristic

155

C₄

D₄×Dic₈₉

yes

characteristic

156

C₄

#15667983

yes

characteristic

157

C₄

#15668008

yes

characteristic

158

C₂

—

159

C₂

—

160

C₂

—

161

C₂

#19151501

yes

characteristic

162

C₂

#19151865

yes

—

	subgroup	G/H
center	C₂²	#15667983
commutator	C₂×C₄×C₈₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting, prime radical	C₂²×D₈×C₈₉	C₂
socle	C₂²×C₈₉	#121
cosocle	C₂²×C₄×C₈₉	C₂³
hypercenter	C₂²×D₈	D₈₉
hypocenter	C₈₉	#1978
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×D₈

subgroup

G/H

center

C₂²

#15667983

commutator

C₂×C₄×C₈₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting, prime radical

C₂²×D₈×C₈₉

C₂

socle

C₂²×C₈₉

#121

cosocle

C₂²×C₄×C₈₉

C₂³

hypercenter

C₂²×D₈

D₈₉

hypocenter

C₈₉

#1978

LCS

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×D₈

N	H ≅ G/N	type
1. C₂×#15667948	C₂	semidirect
2. #19151603	C₂	semidirect
3. C₂×#15667929	C₂	semidirect
4. #19151553	C₂	semidirect
18. C₂×D₈×C₈₉	C₄	semidirect
82. C₈₉	#1978	semidirect
92. C₂×D₈	Dic₈₉	semidirect

H ≅ G/N

type

1. C₂×#15667948

C₂

semidirect

2. #19151603

C₂

semidirect

3. C₂×#15667929

C₂

semidirect

4. #19151553

C₂

semidirect

18. C₂×D₈×C₈₉

C₄

semidirect

82. C₈₉

#1978

semidirect

92. C₂×D₈

Dic₈₉

semidirect

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

subgroup	index
C₂×#15667948	2
#19151603	2
C₂×#15667929	2
#19151553	2
C₂²×D₈×C₈₉	2
#1978	89

subgroup

index

C₂²×D₈×C₈₉

Subgroups — #26021163

Conjugacy classes of subgroups

Complements

Maximal subgroups