Group #26020787 — subgroups

Subgroups — #26020787

Order 11392.

314 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 85 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	#19152172	C₂	1	1	yes	maximal, characteristic
2	#19152076	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151614	C₂	1	2	yes	maximal
4	#19151355	C₂	1	2	yes	maximal, complemented
5	C₂×#101×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13083574	C₂²	1	1	yes	characteristic
7	#15668010	C₄	1	2	yes	complemented
8	#15668001	C₄	1	2	yes	complemented
9	#15667995	—	2	2	no	—
10	#15667988	—	2	2	no	—
11	Q₈×Dic₈₉	—	2	4	no	—
12	#15667971	—	2	2	no	—
13	#15667971	C₂²	1	1	yes	characteristic
14	#15667971	C₂²	1	2	yes	—
15	#15667938	—	2	4	no	—
16	C₂²×Q₈×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×M₁₆×C₈₉	C₂²	1	2	yes	—
18	#101×C₈₉	—	2	2	no	—
19	#13083574	—	2	2	no	—
20	#13083574	—	2	2	no	—
21	C₂×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
22	C₂×D₃₅₆	C₂×C₄	1	1	yes	characteristic
23	C₂×C₂²⋊D₈₉	—	2	1	no	—
24	C₂×C₄×D₈₉	—	2	1	no	—
25	C₄⋊Dic₈₉	—	2	4	no	—
26	C₄⋊Dic₈₉	—	2	2	no	—
27	C₄⋊Dic₈₉	—	2	1	no	—
28	#13083553	—	2	4	no	—
29	C₂²⋊Dic₈₉	—	2	2	no	—
30	C₂²⋊Dic₈₉	—	2	1	no	—
31	#13083551	—	2	4	no	—
32	C₄×Dic₈₉	—	2	4	no	—
33	C₄×Dic₈₉	C₂×C₄	1	2	yes	—
34	C₄×Dic₈₉	D₄	1	4	yes	—
35	C₂×Q₈×C₈₉	—	2	2	no	—
36	C₂×Q₈×C₈₉	—	2	2	no	—
37	C₂×Q₈×C₈₉	C₂×C₄	1	2	yes	—
38	C₂×Q₈×C₈₉	D₄	1	2	yes	—
39	C₂²×C₄×C₈₉	D₄	1	2	yes	—
40	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
41	M₁₆×C₈₉	—	2	2	no	—
42	M₁₆×C₈₉	—	2	2	no	—
43	C₂×C₈×C₈₉	—	2	2	no	—
44	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
45	C₂²×D₈₉	—	2	1	no	—
46	Q₈×C₈₉	—	2	4	no	—
47	Q₈×C₈₉	—	2	4	no	—
48	C₂×C₄×C₈₉	—	2	2	no	—
49	C₂×C₄×C₈₉	—	2	2	no	—
50	C₂×C₄×C₈₉	C₂²⋊C₄	1	4	yes	—
51	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₈₉	#41	1	2	yes	—
55	C₂²⋊D₈₉	—	4	1	no	—
56	C₂²⋊D₈₉	—	4	1	no	—
57	C₂×Dic₈₉	—	2	2	no	—
58	C₂×Dic₈₉	—	2	1	no	—
59	C₂×Dic₈₉	—	2	4	no	—
60	D₃₅₆	—	2	2	no	—
61	C₄×D₈₉	—	4	2	no	—
62	Dic₁₇₈	—	2	2	no	—
63	C₈×C₈₉	—	4	2	no	—
64	C₂²×C₈₉	—	4	1	no	—
65	C₂²×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₈₉	#120	1	1	yes	characteristic
67	C₂²×C₈₉	#123	1	1	yes	characteristic
68	C₂×D₈₉	—	4	2	no	—
69	C₂×D₈₉	—	4	1	no	—
70	C₄×C₈₉	—	2	4	no	—
71	C₄×C₈₉	C₄×D₄	1	2	yes	—
72	C₄×C₈₉	#121	1	2	yes	—
73	Dic₈₉	—	4	4	no	—
74	Dic₈₉	—	4	2	no	—
75	Dic₈₉	—	4	1	no	—
76	C₂×C₈₉	—	2	1	no	—
77	C₂×C₈₉	—	2	1	no	—
78	C₂×C₈₉	—	2	1	no	—
79	C₂×C₈₉	#386	1	1	yes	characteristic
80	D₈₉	—	8	1	no	—
81	#1912	—	89	1	no	maximal, Sylow
82	C₈₉	#1912	1	1	yes	characteristic, complemented, Sylow
83	#536	—	89	1	no	—
84	#519	—	89	1	no	—
85	#421	—	89	2	no	—
86	C₂×#101	D₈₉	1	1	yes	characteristic, complemented
87	H(3,ℤ/4)	—	89	2	no	—
88	C₂×#41	—	89	1	no	—
89	C₂²×Q₈	C₂×D₈₉	1	1	yes	characteristic
90	C₂×M₁₆	C₂×D₈₉	1	2	yes	—
91	C₄⋊Q₈	—	89	2	no	—
92	#124	—	89	2	no	—
93	#123	—	178	2	no	—
94	C₂²⋊Q₈	—	178	2	no	—
95	C₄×Q₈	—	178	4	no	—
96	#117	—	89	2	no	—
97	#117	—	89	1	no	—
98	#117	—	178	2	no	—
99	#104	—	178	4	no	—
100	#101	—	2	2	no	—
101	#41	—	178	2	no	—
102	#41	—	178	2	no	—
103	C₂×Q₈	—	2	2	no	—
104	C₂×Q₈	—	2	2	no	—
105	C₂×Q₈	—	89	1	no	—
106	C₂×Q₈	C₄×D₈₉	1	2	yes	—
107	C₂×Q₈	C₂²⋊D₈₉	1	2	yes	—
108	C₂×D₄	—	89	1	no	—
109	C₂×D₄	—	178	1	no	—
110	C₂²×C₄	—	178	1	no	—
111	C₂²×C₄	D₃₅₆	1	2	yes	—
112	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
113	M₁₆	—	2	2	no	—
114	M₁₆	—	2	2	no	—
115	C₂×C₈	—	2	2	no	—
116	C₄⋊C₄	—	178	4	no	—
117	C₄⋊C₄	—	178	4	no	—
118	C₄⋊C₄	—	178	2	no	—
119	C₄⋊C₄	—	178	1	no	—
120	C₂²⋊C₄	—	178	2	no	—
121	C₂²⋊C₄	—	178	1	no	—
122	C₂²⋊C₄	—	178	4	no	—
123	C₄²	—	89	2	no	—
124	C₄²	—	89	4	no	—
125	C₄²	—	178	4	no	—
126	C₂³	—	178	1	no	—
127	C₂³	C₂×D₃₅₆	1	1	yes	characteristic
128	Q₈	—	2	4	no	—
129	Q₈	—	2	4	no	—
130	Q₈	—	178	2	no	—
131	D₄	—	356	1	no	—
132	D₄	—	356	1	no	—
133	D₄	—	178	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	356	2	no	—
137	C₂×C₄	—	178	4	no	—
138	C₂×C₄	—	178	2	no	—
139	C₂×C₄	—	178	1	no	—
140	C₂×C₄	#13083551	1	4	yes	—
141	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
142	C₂×C₄	D₄×D₈₉	1	2	yes	—
143	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
144	C₂×C₄	#13083573	1	2	yes	—
145	C₈	—	4	2	no	—
146	C₂²	—	4	1	no	—
147	C₂²	—	356	2	no	—
148	C₂²	—	356	1	no	—
149	C₂²	C₂×#13083551	1	1	yes	characteristic
150	C₂²	#15667982	1	1	yes	characteristic
151	C₂²	#15667996	1	1	yes	characteristic
152	C₄	—	2	4	no	—
153	C₄	—	356	4	no	—
154	C₄	—	356	2	no	—
155	C₄	—	356	1	no	—
156	C₄	#15667972	1	2	yes	—
157	C₄	#15667984	1	2	yes	—
158	C₂	—	2	1	no	—
159	C₂	—	2	1	no	—
160	C₂	—	2	1	no	—
161	C₂	—	712	1	no	—
162	C₂	#19151475	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19152172

C₂

yes

maximal, characteristic

#19152076

C₂

yes

maximal, characteristic, complemented

#19151614

C₂

yes

maximal

#19151355

C₂

yes

maximal, complemented

C₂×#101×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×#13083574

C₂²

yes

characteristic

#15668010

C₄

yes

complemented

#15668001

C₄

yes

complemented

#15667995

—

#15667988

—

Q₈×Dic₈₉

—

#15667971

—

#15667971

C₂²

yes

characteristic

#15667971

C₂²

yes

—

#15667938

—

C₂²×Q₈×C₈₉

C₂²

yes

characteristic

C₂×M₁₆×C₈₉

C₂²

yes

—

#101×C₈₉

—

#13083574

—

#13083574

—

C₂×Dic₁₇₈

C₂×C₄

yes

characteristic

C₂×D₃₅₆

C₂×C₄

yes

characteristic

C₂×C₂²⋊D₈₉

—

C₂×C₄×D₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083553

—

C₂²⋊Dic₈₉

—

C₂²⋊Dic₈₉

—

#13083551

—

C₄×Dic₈₉

—

C₄×Dic₈₉

C₂×C₄

yes

—

C₄×Dic₈₉

D₄

yes

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

C₂×C₄

yes

—

C₂×Q₈×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

M₁₆×C₈₉

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂²×D₈₉

—

Q₈×C₈₉

—

Q₈×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

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

#41

yes

—

C₂²⋊D₈₉

—

C₂²⋊D₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

C₄×D₈₉

—

Dic₁₇₈

—

C₈×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×C₂²⋊C₄

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₄×D₄

yes

—

C₄×C₈₉

#121

yes

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#386

yes

characteristic

D₈₉

—

#1912

—

maximal, Sylow

C₈₉

#1912

yes

characteristic, complemented, Sylow

—

—

—

D₈₉

yes

characteristic, complemented

H(3,ℤ/4)

—

C₂×#41

—

C₂²×Q₈

C₂×D₈₉

yes

characteristic

C₂×M₁₆

C₂×D₈₉

yes

—

C₄⋊Q₈

—

#124

—

#123

—

178

—

C₂²⋊Q₈

—

178

—

C₄×Q₈

—

178

—

#117

—

#117

—

#117

—

178

—

#104

—

178

—

100

#101

—

101

#41

—

178

—

102

#41

—

178

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

C₄×D₈₉

yes

—

107

C₂×Q₈

C₂²⋊D₈₉

yes

—

108

C₂×D₄

—

109

C₂×D₄

—

178

—

110

C₂²×C₄

—

178

—

111

C₂²×C₄

D₃₅₆

yes

—

112

C₂²×C₄

C₂²×D₈₉

yes

characteristic

113

M₁₆

—

114

M₁₆

—

115

C₂×C₈

—

116

C₄⋊C₄

—

178

—

117

C₄⋊C₄

—

178

—

118

C₄⋊C₄

—

178

—

119

C₄⋊C₄

—

178

—

120

C₂²⋊C₄

—

178

—

121

C₂²⋊C₄

—

178

—

122

C₂²⋊C₄

—

178

—

123

C₄²

—

124

C₄²

—

125

C₄²

—

178

—

126

C₂³

—

178

—

127

C₂³

C₂×D₃₅₆

yes

characteristic

128

Q₈

—

129

Q₈

—

130

Q₈

—

178

—

131

D₄

—

356

—

132

D₄

—

356

—

133

D₄

—

178

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

356

—

137

C₂×C₄

—

178

—

138

C₂×C₄

—

178

—

139

C₂×C₄

—

178

—

140

C₂×C₄

#13083551

yes

—

141

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

142

C₂×C₄

D₄×D₈₉

yes

—

143

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

144

C₂×C₄

#13083573

yes

—

145

C₈

—

146

C₂²

—

147

C₂²

—

356

—

148

C₂²

—

356

—

149

C₂²

C₂×#13083551

yes

characteristic

150

C₂²

#15667982

yes

characteristic

151

C₂²

#15667996

yes

characteristic

152

C₄

—

153

C₄

—

356

—

154

C₄

—

356

—

155

C₄

—

356

—

156

C₄

#15667972

yes

—

157

C₄

#15667984

yes

—

158

C₂

—

159

C₂

—

160

C₂

—

161

C₂

—

712

—

162

C₂

#19151475

yes

characteristic

	subgroup	G/H
center	C₂	#19151475
commutator	C₂×C₄×C₈₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	C₂×#101×C₈₉	C₂
socle	C₂×C₈₉	#386
prime radical	C₂×#13083574	C₂²
cosocle	C₂²×C₄×C₈₉	C₂³
hypercenter	C₂×#101	D₈₉
hypocenter	C₈₉	#1912
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#101

subgroup

G/H

center

C₂

#19151475

commutator

C₂×C₄×C₈₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

C₂×#101×C₈₉

C₂

C₂×C₈₉

C₂²

C₂²×C₄×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#101

N	H ≅ G/N	type
2. #19152076	C₂	semidirect
4. #19151355	C₂	semidirect
5. C₂×#101×C₈₉	C₂	semidirect
7. #15668010	C₄	semidirect
8. #15668001	C₄	semidirect
82. C₈₉	#1912	semidirect
86. C₂×#101	D₈₉	semidirect

H ≅ G/N

type

2. #19152076

C₂

semidirect

4. #19151355

C₂

semidirect

5. C₂×#101×C₈₉

C₂

semidirect

7. #15668010

C₄

semidirect

8. #15668001

C₄

semidirect

82. C₈₉

#1912

semidirect

86. C₂×#101

D₈₉

semidirect

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

subgroup	index
#19152172	2
#19152076	2
#19151614	2
#19151355	2
C₂×#101×C₈₉	2
#1912	89

subgroup

index

C₂×#101×C₈₉

Subgroups — #26020787

Conjugacy classes of subgroups

Complements

Maximal subgroups