Group #26021102 — subgroups

Subgroups — #26021102

Order 11392.

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

normal subgroups only

This group has 312 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	#19151622	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15667931	C₂	1	2	yes	maximal, complemented
3	C₂×#15667929	C₂	1	2	yes	maximal
4	C₄×#13083558	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#124×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13083558	C₂²	1	2	yes	—
7	C₂×#13083558	C₂²	1	2	yes	—
8	C₄⋊#10623494	—	2	2	no	—
9	#15667931	—	2	4	no	—
10	#15667929	—	2	4	no	—
11	#15667923	—	2	1	no	—
12	C₄×#10623494	—	2	1	no	—
13	C₂²×Q₈×C₈₉	C₂²	1	1	yes	characteristic
14	C₂²×D₄×C₈₉	C₂²	1	1	yes	characteristic
15	#124×C₈₉	—	2	1	no	—
16	#124×C₈₉	—	2	1	no	—
17	#124×C₈₉	—	2	2	no	—
18	C₂×C₂²⋊C₄×C₈₉	C₄	1	4	yes	—
19	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
20	#13083558	—	2	2	no	—
21	#13083558	—	2	2	no	—
22	#13083558	—	2	2	no	—
23	#13083558	D₄	1	4	yes	—
24	C₂×#10623494	—	2	2	no	—
25	C₂×#10623494	—	2	2	no	—
26	C₂⁴×C₈₉	C₂×C₄	1	2	yes	—
27	C₂×Q₈×C₈₉	—	2	1	no	—
28	C₂×Q₈×C₈₉	—	2	1	no	—
29	C₂×Q₈×C₈₉	—	2	2	no	—
30	C₂×Q₈×C₈₉	—	2	2	no	—
31	C₂×D₄×C₈₉	—	2	1	no	—
32	C₂×D₄×C₈₉	—	2	1	no	—
33	C₂×D₄×C₈₉	—	2	2	no	—
34	C₂×D₄×C₈₉	—	2	2	no	—
35	C₂²×C₄×C₈₉	C₂×C₄	1	2	yes	—
36	C₂²×C₄×C₈₉	C₂×C₄	1	2	yes	—
37	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
38	C₂²⋊C₄×C₈₉	—	2	4	no	—
39	C₂²⋊C₄×C₈₉	—	2	4	no	—
40	C₄²×C₈₉	D₄	1	1	yes	characteristic
41	C₄²×C₈₉	D₄	1	1	yes	characteristic
42	C₄²×C₈₉	D₄	1	2	yes	—
43	C₂³×C₈₉	—	2	2	no	—
44	C₂³×C₈₉	—	2	4	no	—
45	C₂³×C₈₉	—	4	4	no	—
46	C₂³×C₈₉	C₂²×C₄	1	1	yes	characteristic
47	Q₈×C₈₉	—	4	2	no	—
48	Q₈×C₈₉	—	4	1	no	—
49	Q₈×C₈₉	—	4	1	no	—
50	D₄×C₈₉	—	4	2	no	—
51	D₄×C₈₉	—	4	1	no	—
52	D₄×C₈₉	—	4	1	no	—
53	C₂×C₄×C₈₉	—	2	2	no	—
54	C₂×C₄×C₈₉	—	2	4	no	—
55	C₂×C₄×C₈₉	—	4	2	no	—
56	C₂×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
57	C₂×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
58	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₈₉	#41	1	2	yes	—
62	#10623494	—	2	4	no	—
63	#10623494	—	4	2	no	—
64	C₂²×C₈₉	—	2	2	no	—
65	C₂²×C₈₉	—	4	2	no	—
66	C₂²×C₈₉	—	4	4	no	—
67	C₂²×C₈₉	—	8	4	no	—
68	C₂²×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₈₉	C₄×D₄	1	2	yes	—
70	C₄×C₈₉	—	2	2	no	—
71	C₄×C₈₉	—	2	2	no	—
72	C₄×C₈₉	—	4	2	no	—
73	C₄×C₈₉	#121	1	2	yes	—
74	C₄×C₈₉	#124	1	1	yes	characteristic
75	C₄×C₈₉	#127	1	1	yes	characteristic
76	C₂×C₈₉	—	2	2	no	—
77	C₂×C₈₉	—	8	2	no	—
78	C₂×C₈₉	#390	1	1	yes	characteristic
79	C₂×C₈₉	#413	1	2	yes	—
80	#1967	—	89	1	no	maximal, Sylow
81	C₈₉	#1967	1	1	yes	characteristic, complemented, Sylow
82	C₂×#124	D₈₉	1	1	yes	characteristic
83	#423	—	89	1	no	—
84	C₂×#101	—	89	2	no	—
85	C₂×#100	—	89	2	no	—
86	C₄×M₁₆	—	89	1	no	—
87	C₂²×Q₈	C₂×D₈₉	1	1	yes	characteristic
88	C₂²×D₄	C₂×D₈₉	1	1	yes	characteristic
89	C₂×M₁₆	—	89	2	no	—
90	C₂×M₁₆	—	89	2	no	—
91	#124	—	2	1	no	—
92	#124	—	2	1	no	—
93	#124	—	2	2	no	—
94	C₂×C₂²⋊C₄	Dic₈₉	1	4	yes	—
95	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
96	C₄⋊C₈	—	178	2	no	—
97	#101	—	178	4	no	—
98	#100	—	178	4	no	—
99	#97	—	178	1	no	—
100	C₄×C₈	—	178	1	no	—
101	C₂⁴	C₂×Dic₈₉	1	2	yes	—
102	C₂×Q₈	—	2	2	no	—
103	C₂×Q₈	—	2	1	no	—
104	C₂×Q₈	—	2	1	no	—
105	C₂×Q₈	—	2	2	no	—
106	C₂×D₄	—	2	2	no	—
107	C₂×D₄	—	2	1	no	—
108	C₂×D₄	—	2	1	no	—
109	C₂×D₄	—	2	2	no	—
110	C₂²×C₄	C₂×Dic₈₉	1	2	yes	—
111	C₂²×C₄	C₂×Dic₈₉	1	2	yes	—
112	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
113	M₁₆	—	89	4	no	—
114	M₁₆	—	178	2	no	—
115	M₁₆	—	178	2	no	—
116	M₁₆	—	178	2	no	—
117	C₂×C₈	—	178	2	no	—
118	C₂×C₈	—	178	2	no	—
119	C₂²⋊C₄	—	2	4	no	—
120	C₂²⋊C₄	—	2	4	no	—
121	C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
122	C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
123	C₄²	C₂²⋊D₈₉	1	2	yes	—
124	C₂³	—	2	2	no	—
125	C₂³	—	2	4	no	—
126	C₂³	—	4	4	no	—
127	C₂³	C₂²×Dic₈₉	1	1	yes	characteristic
128	Q₈	—	4	2	no	—
129	Q₈	—	4	1	no	—
130	Q₈	—	4	1	no	—
131	D₄	—	4	2	no	—
132	D₄	—	4	1	no	—
133	D₄	—	4	1	no	—
134	C₂×C₄	—	2	4	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	4	2	no	—
137	C₂×C₄	C₂²⋊Dic₈₉	1	2	yes	—
138	C₂×C₄	C₂²⋊Dic₈₉	1	2	yes	—
139	C₂×C₄	D₄×D₈₉	1	2	yes	—
140	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
141	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
142	C₂×C₄	#13083573	1	2	yes	—
143	C₈	—	356	2	no	—
144	C₈	—	178	4	no	—
145	C₂²	—	2	2	no	—
146	C₂²	—	4	2	no	—
147	C₂²	—	4	4	no	—
148	C₂²	—	8	4	no	—
149	C₂²	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
150	C₂²	D₄×Dic₈₉	1	2	yes	—
151	C₄	—	2	2	no	—
152	C₄	—	2	2	no	—
153	C₄	—	4	2	no	—
154	C₄	#15667983	1	2	yes	—
155	C₄	#15667999	1	1	yes	characteristic
156	C₄	#15668008	1	1	yes	characteristic
157	C₂	—	2	2	no	—
158	C₂	—	8	2	no	—
159	C₂	#19151501	1	1	yes	characteristic
160	C₂	#19151582	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19151622

C₂

yes

maximal, characteristic, complemented

C₂×#15667931

C₂

yes

maximal, complemented

C₂×#15667929

C₂

yes

maximal

C₄×#13083558

C₂

yes

maximal, characteristic, complemented

C₂×#124×C₈₉

C₂

yes

maximal, characteristic

C₂×#13083558

C₂²

yes

—

C₂×#13083558

C₂²

yes

—

—

—

—

—

—

C₂²×Q₈×C₈₉

C₂²

yes

characteristic

C₂²×D₄×C₈₉

C₂²

yes

characteristic

#124×C₈₉

—

#124×C₈₉

—

#124×C₈₉

—

C₂×C₂²⋊C₄×C₈₉

C₄

yes

—

C₂×C₄²×C₈₉

C₂²

yes

—

—

—

D₄

yes

—

C₂×#10623494

—

C₂×#10623494

—

C₂⁴×C₈₉

C₂×C₄

yes

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×C₄

yes

—

C₂²×C₄×C₈₉

C₂×C₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂²⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

D₄

yes

characteristic

C₄²×C₈₉

D₄

yes

characteristic

C₄²×C₈₉

D₄

yes

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

C₂²×C₄

yes

characteristic

Q₈×C₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²⋊C₄

yes

—

C₂×C₄×C₈₉

C₂²⋊C₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

#41

yes

—

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₈₉

C₄×D₄

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#121

yes

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#390

yes

characteristic

C₂×C₈₉

#413

yes

—

#1967

—

maximal, Sylow

C₈₉

#1967

yes

characteristic, complemented, Sylow

C₂×#124

D₈₉

yes

—

—

—

C₄×M₁₆

—

C₂²×Q₈

C₂×D₈₉

yes

characteristic

C₂²×D₄

C₂×D₈₉

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

#124

—

#124

—

#124

—

C₂×C₂²⋊C₄

Dic₈₉

yes

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

C₄⋊C₈

—

178

—

#101

—

178

—

#100

—

178

—

#97

—

178

—

100

C₄×C₈

—

178

—

101

C₂⁴

C₂×Dic₈₉

yes

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂²×C₄

C₂×Dic₈₉

yes

—

111

C₂²×C₄

C₂×Dic₈₉

yes

—

112

C₂²×C₄

C₂²×D₈₉

yes

characteristic

113

M₁₆

—

114

M₁₆

—

178

—

115

M₁₆

—

178

—

116

M₁₆

—

178

—

117

C₂×C₈

—

178

—

118

C₂×C₈

—

178

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

121

C₄²

C₂²⋊D₈₉

yes

characteristic

122

C₄²

C₂²⋊D₈₉

yes

characteristic

123

C₄²

C₂²⋊D₈₉

yes

—

124

C₂³

—

125

C₂³

—

126

C₂³

—

127

C₂³

C₂²×Dic₈₉

yes

characteristic

128

Q₈

—

129

Q₈

—

130

Q₈

—

131

D₄

—

132

D₄

—

133

D₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

C₂²⋊Dic₈₉

yes

—

138

C₂×C₄

C₂²⋊Dic₈₉

yes

—

139

C₂×C₄

D₄×D₈₉

yes

—

140

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

141

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

142

C₂×C₄

#13083573

yes

—

143

C₈

—

356

—

144

C₈

—

178

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

C₂×C₂²⋊Dic₈₉

yes

characteristic

150

C₂²

D₄×Dic₈₉

yes

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

#15667983

yes

—

155

C₄

#15667999

yes

characteristic

156

C₄

#15668008

yes

characteristic

157

C₂

—

158

C₂

—

159

C₂

#19151501

yes

characteristic

160

C₂

#19151582

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₈₉
commutator	C₂³×C₈₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	C₂×#124×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₂²⋊C₄
prime radical	C₂²×D₄×C₈₉	C₂²
cosocle	C₂²×C₄×C₈₉	C₂³
hypercenter	C₂×#124	D₈₉
hypocenter	C₈₉	#1967
LCS	G ▷ C₂³×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#124

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₈₉

commutator

C₂³×C₈₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

C₂×#124×C₈₉

C₂

socle

C₂²×C₈₉

C₂×C₂²⋊C₄

prime radical

C₂²×D₄×C₈₉

C₂²

cosocle

C₂²×C₄×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#124

N	H ≅ G/N	type
1. #19151622	C₂	semidirect
2. C₂×#15667931	C₂	semidirect
4. C₄×#13083558	C₂	semidirect
81. C₈₉	#1967	semidirect

H ≅ G/N

type

1. #19151622

C₂

semidirect

2. C₂×#15667931

C₂

semidirect

4. C₄×#13083558

C₂

semidirect

81. C₈₉

#1967

semidirect

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

subgroup	index
#19151622	2
C₂×#15667931	2
C₂×#15667929	2
C₄×#13083558	2
C₂×#124×C₈₉	2
#1967	89

subgroup

index

C₂×#124×C₈₉

Subgroups — #26021102

Conjugacy classes of subgroups

Complements

Maximal subgroups