Group #26020774 — subgroups

Subgroups — #26020774

Order 11392.

312 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 89 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	#19152169	C₂	1	1	yes	maximal, characteristic
2	#19151608	C₂	1	2	yes	maximal, complemented
3	#19151572	C₂	1	1	yes	maximal, characteristic
4	#19151355	C₂	1	2	yes	maximal, complemented
5	#568×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₇₈	C₂²	1	1	yes	characteristic
7	#15668010	C₄	1	2	yes	complemented
8	#15667999	C₄	1	2	yes	complemented
9	#15667993	—	2	2	no	—
10	#15667991	—	2	2	no	—
11	#15667971	C₂²	1	2	yes	—
12	#15667971	C₂²	1	1	yes	characteristic
13	#15667937	—	2	4	no	—
14	#15667927	—	2	2	no	—
15	C₂×#41×C₈₉	C₂²	1	1	yes	characteristic
16	#131×C₈₉	—	2	4	no	—
17	C₂×M₁₆×C₈₉	—	2	2	no	—
18	C₂×M₁₆×C₈₉	C₂²	1	2	yes	—
19	C₂×Dic₁₇₈	—	2	2	no	—
20	C₂×Dic₁₇₈	—	2	2	no	—
21	C₂×Dic₁₇₈	—	2	1	no	—
22	C₂×Dic₁₇₈	C₂×C₄	1	2	yes	—
23	C₂²×Dic₈₉	—	2	1	no	—
24	C₄⋊Dic₈₉	—	2	1	no	—
25	C₄⋊Dic₈₉	—	2	2	no	—
26	#13083553	—	2	4	no	—
27	C₂²⋊Dic₈₉	—	2	4	no	—
28	C₂²⋊Dic₈₉	—	2	1	no	—
29	C₂²⋊Dic₈₉	—	2	2	no	—
30	C₄×Dic₈₉	—	2	2	no	—
31	C₄×Dic₈₉	C₂×C₄	1	2	yes	—
32	#41×C₈₉	—	2	2	no	—
33	#41×C₈₉	—	2	2	no	—
34	C₂×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
35	C₂×D₄×C₈₉	C₂×C₄	1	1	yes	characteristic
36	C₂×D₄×C₈₉	D₄	1	2	yes	—
37	C₂²×C₄×C₈₉	D₄	1	2	yes	—
38	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
39	M₁₆×C₈₉	—	2	4	no	—
40	M₁₆×C₈₉	—	2	2	no	—
41	M₁₆×C₈₉	—	2	2	no	—
42	C₂×C₈×C₈₉	—	2	4	no	—
43	C₂×C₈×C₈₉	D₄	1	4	yes	—
44	C₂³×C₈₉	C₂²⋊C₄	1	2	yes	—
45	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
46	Q₈×C₈₉	—	2	2	no	—
47	D₄×C₈₉	—	2	4	no	—
48	D₄×C₈₉	—	2	2	no	—
49	C₂×C₄×C₈₉	—	2	2	no	—
50	C₂×C₄×C₈₉	—	2	2	no	—
51	C₂×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
52	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₈₉	#41	1	2	yes	—
56	C₂×Dic₈₉	—	2	2	no	—
57	C₂×Dic₈₉	—	2	2	no	—
58	C₂×Dic₈₉	—	4	2	no	—
59	C₂×Dic₈₉	—	4	2	no	—
60	Dic₁₇₈	—	2	4	no	—
61	Dic₁₇₈	—	4	1	no	—
62	Dic₁₇₈	—	4	1	no	—
63	C₈×C₈₉	—	2	4	no	—
64	C₂²×C₈₉	—	2	2	no	—
65	C₂²×C₈₉	—	4	1	no	—
66	C₂²×C₈₉	—	4	2	no	—
67	C₂²×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₈₉	#121	1	2	yes	—
69	C₄×C₈₉	—	2	2	no	—
70	C₄×C₈₉	C₄×D₄	1	2	yes	—
71	C₄×C₈₉	#120	1	1	yes	characteristic
72	C₄×C₈₉	#123	1	1	yes	characteristic
73	Dic₈₉	—	4	2	no	—
74	Dic₈₉	—	4	2	no	—
75	Dic₈₉	—	8	2	no	—
76	C₂×C₈₉	—	2	1	no	—
77	C₂×C₈₉	—	2	2	no	—
78	C₂×C₈₉	—	4	2	no	—
79	C₂×C₈₉	#386	1	1	yes	characteristic
80	#1909	—	89	1	no	maximal, Sylow
81	C₈₉	#1909	1	1	yes	characteristic, complemented, Sylow
82	#568	D₈₉	1	1	yes	characteristic
83	#536	—	89	1	no	—
84	#419	—	89	2	no	—
85	#410	—	89	1	no	—
86	H(3,ℤ/4)	—	89	2	no	—
87	C₂×#41	C₂×D₈₉	1	1	yes	characteristic
88	C₂²×Q₈	—	89	1	no	—
89	#131	—	2	4	no	—
90	C₂×M₁₆	—	2	2	no	—
91	C₂×M₁₆	C₂×D₈₉	1	2	yes	—
92	C₄⋊Q₈	—	89	2	no	—
93	#124	—	89	2	no	—
94	#123	—	178	2	no	—
95	C₂²⋊Q₈	—	178	2	no	—
96	#117	—	89	1	no	—
97	#117	—	89	2	no	—
98	#103	—	178	4	no	—
99	#99	—	178	2	no	—
100	#41	—	2	2	no	—
101	#41	—	2	2	no	—
102	C₂×Q₈	—	89	2	no	—
103	C₂×Q₈	—	178	2	no	—
104	C₂×Q₈	—	178	2	no	—
105	C₂×Q₈	—	178	1	no	—
106	C₂×Q₈	C₄×D₈₉	1	1	yes	characteristic
107	C₂×D₄	C₄×D₈₉	1	1	yes	characteristic
108	C₂×D₄	D₃₅₆	1	2	yes	—
109	C₂²×C₄	—	178	1	no	—
110	C₂²×C₄	C₂²⋊D₈₉	1	2	yes	—
111	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
112	M₁₆	—	2	4	no	—
113	M₁₆	—	2	2	no	—
114	M₁₆	—	2	2	no	—
115	C₂×C₈	—	2	4	no	—
116	C₂×C₈	C₂²⋊D₈₉	1	4	yes	—
117	C₄⋊C₄	—	178	2	no	—
118	C₄⋊C₄	—	178	4	no	—
119	C₄⋊C₄	—	178	1	no	—
120	C₂²⋊C₄	—	178	2	no	—
121	C₂²⋊C₄	—	178	4	no	—
122	C₂²⋊C₄	—	178	1	no	—
123	C₄²	—	89	2	no	—
124	C₄²	—	178	2	no	—
125	C₂³	#13083551	1	2	yes	—
126	C₂³	C₂×D₃₅₆	1	1	yes	characteristic
127	Q₈	—	2	2	no	—
128	Q₈	—	356	1	no	—
129	Q₈	—	356	1	no	—
130	Q₈	—	178	4	no	—
131	D₄	—	2	4	no	—
132	D₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	356	2	no	—
136	C₂×C₄	—	356	2	no	—
137	C₂×C₄	—	178	2	no	—
138	C₂×C₄	—	178	2	no	—
139	C₂×C₄	#13083551	1	2	yes	—
140	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
141	C₂×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
142	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
143	C₂×C₄	#13083574	1	2	yes	—
144	C₈	—	2	4	no	—
145	C₂²	—	2	2	no	—
146	C₂²	—	4	1	no	—
147	C₂²	—	4	2	no	—
148	C₂²	C₂×#13083551	1	1	yes	characteristic
149	C₂²	#15667986	1	2	yes	—
150	C₄	—	2	2	no	—
151	C₄	—	356	2	no	—
152	C₄	—	356	2	no	—
153	C₄	—	712	2	no	—
154	C₄	C₄×C₂²⋊D₈₉	1	2	yes	—
155	C₄	#15667981	1	1	yes	characteristic
156	C₄	#15667995	1	1	yes	characteristic
157	C₂	—	2	1	no	—
158	C₂	—	2	2	no	—
159	C₂	—	4	2	no	—
160	C₂	#19151477	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19152169

C₂

yes

maximal, characteristic

#19151608

C₂

yes

maximal, complemented

#19151572

C₂

yes

maximal, characteristic

#19151355

C₂

yes

maximal, complemented

#568×C₈₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₈

C₂²

yes

characteristic

#15668010

C₄

yes

complemented

#15667999

C₄

yes

—

—

C₂²

yes

—

#15667971

C₂²

yes

characteristic

#15667937

—

#15667927

—

C₂×#41×C₈₉

C₂²

yes

characteristic

#131×C₈₉

—

C₂×M₁₆×C₈₉

—

C₂×M₁₆×C₈₉

C₂²

yes

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

C₂×C₄

yes

—

C₂²×Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083553

—

C₂²⋊Dic₈₉

—

C₂²⋊Dic₈₉

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

C₄×Dic₈₉

C₂×C₄

yes

—

#41×C₈₉

—

#41×C₈₉

—

C₂×Q₈×C₈₉

C₂×C₄

yes

characteristic

C₂×D₄×C₈₉

C₂×C₄

yes

characteristic

C₂×D₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

M₁₆×C₈₉

—

M₁₆×C₈₉

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

D₄

yes

—

C₂³×C₈₉

C₂²⋊C₄

yes

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

Q₈×C₈₉

—

D₄×C₈₉

—

D₄×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

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₈₉

#121

yes

—

C₄×C₈₉

—

C₄×C₈₉

C₄×D₄

yes

—

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

#568

D₈₉

yes

—

—

—

—

C₂×#41

C₂×D₈₉

yes

characteristic

C₂²×Q₈

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₈₉

yes

—

C₄⋊Q₈

—

#124

—

#123

—

178

—

C₂²⋊Q₈

—

178

—

#117

—

#117

—

#103

—

178

—

#99

—

178

—

100

#41

—

101

#41

—

102

C₂×Q₈

—

103

C₂×Q₈

—

178

—

104

C₂×Q₈

—

178

—

105

C₂×Q₈

—

178

—

106

C₂×Q₈

C₄×D₈₉

yes

characteristic

107

C₂×D₄

C₄×D₈₉

yes

characteristic

108

C₂×D₄

D₃₅₆

yes

—

109

C₂²×C₄

—

178

—

110

C₂²×C₄

C₂²⋊D₈₉

yes

—

111

C₂²×C₄

C₂²×D₈₉

yes

characteristic

112

M₁₆

—

113

M₁₆

—

114

M₁₆

—

115

C₂×C₈

—

116

C₂×C₈

C₂²⋊D₈₉

yes

—

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

—

178

—

125

C₂³

#13083551

yes

—

126

C₂³

C₂×D₃₅₆

yes

characteristic

127

Q₈

—

128

Q₈

—

356

—

129

Q₈

—

356

—

130

Q₈

—

178

—

131

D₄

—

132

D₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

356

—

136

C₂×C₄

—

356

—

137

C₂×C₄

—

178

—

138

C₂×C₄

—

178

—

139

C₂×C₄

#13083551

yes

—

140

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

141

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

142

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

143

C₂×C₄

#13083574

yes

—

144

C₈

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

C₂×#13083551

yes

characteristic

149

C₂²

#15667986

yes

—

150

C₄

—

151

C₄

—

356

—

152

C₄

—

356

—

153

C₄

—

712

—

154

C₄

C₄×C₂²⋊D₈₉

yes

—

155

C₄

#15667981

yes

characteristic

156

C₄

#15667995

yes

characteristic

157

C₂

—

158

C₂

—

159

C₂

—

160

C₂

#19151477

yes

characteristic

	subgroup	G/H
center	C₂	#19151477
commutator	C₂×C₄×C₈₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	#568×C₈₉	C₂
socle	C₂×C₈₉	#386
prime radical	C₂×#41×C₈₉	C₂²
cosocle	C₂²×C₄×C₈₉	C₂³
hypercenter	#568	D₈₉
hypocenter	C₈₉	#1909
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #568

subgroup

G/H

center

C₂

#19151477

commutator

C₂×C₄×C₈₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

#568×C₈₉

C₂

socle

C₂×C₈₉

#386

prime radical

C₂×#41×C₈₉

C₂²

cosocle

C₂²×C₄×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #568

N	H ≅ G/N	type
2. #19151608	C₂	semidirect
4. #19151355	C₂	semidirect
7. #15668010	C₄	semidirect
8. #15667999	C₄	semidirect
81. C₈₉	#1909	semidirect

H ≅ G/N

type

2. #19151608

C₂

semidirect

4. #19151355

C₂

semidirect

7. #15668010

C₄

semidirect

8. #15667999

C₄

semidirect

81. C₈₉

#1909

semidirect

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

subgroup	index
#19152169	2
#19151608	2
#19151572	2
#19151355	2
#568×C₈₉	2
#1909	89

subgroup

index

#568×C₈₉

Subgroups — #26020774

Conjugacy classes of subgroups

Complements

Maximal subgroups