Group #26021714 — subgroups

Subgroups — #26021714

Order 11392.

204 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 57 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	#19151646	C₂	1	1	yes	maximal, characteristic, complemented
2	#19151602	C₂	1	1	yes	maximal, characteristic
3	#19151602	C₂	1	1	yes	maximal, characteristic
4	C₂×#15667928	C₂	1	1	yes	maximal, characteristic
5	#19151368	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151368	C₂	1	1	yes	maximal, characteristic, complemented
7	H(3,ℤ/4)×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₃₅₆	C₂²	1	1	yes	characteristic
9	C₂×#13083558	C₂²	1	1	yes	characteristic
10	C₂×#13083558	C₂²	1	1	yes	characteristic
11	C₂×#13083558	C₂²	1	1	yes	characteristic
12	#15667947	—	2	2	no	—
13	#15667932	—	2	2	no	—
14	#15667932	—	2	2	no	—
15	#15667928	—	2	2	no	—
16	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
17	#117×C₈₉	C₂²	1	1	yes	characteristic
18	#117×C₈₉	C₂²	1	1	yes	characteristic
19	C₂³×D₈₉	—	2	1	no	—
20	C₂×D₃₅₆	—	2	1	no	—
21	C₂×D₃₅₆	—	2	1	no	—
22	C₂×D₃₅₆	—	2	1	no	—
23	C₂×D₃₅₆	—	2	1	no	—
24	C₂×D₃₅₆	—	2	1	no	—
25	C₂×D₃₅₆	D₄	1	2	yes	—
26	#13083558	—	2	1	no	—
27	#13083558	—	2	1	no	—
28	#13083558	—	2	1	no	—
29	#13083558	—	2	1	no	—
30	#13083558	—	2	1	no	—
31	#13083558	—	2	1	no	—
32	C₂×#10623494	—	2	1	no	—
33	C₂×#10623494	—	2	1	no	—
34	C₂×#10623494	D₄	1	2	yes	—
35	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
36	M₁₆×C₈₉	—	2	1	no	—
37	M₁₆×C₈₉	—	2	1	no	—
38	C₂×C₈×C₈₉	—	2	1	no	—
39	C₄⋊C₄×C₈₉	—	2	1	no	—
40	C₄⋊C₄×C₈₉	—	2	1	no	—
41	C₂²⋊C₄×C₈₉	—	2	1	no	—
42	C₂²⋊C₄×C₈₉	—	2	1	no	—
43	C₄²×C₈₉	—	2	1	no	—
44	C₄²×C₈₉	—	2	1	no	—
45	C₂³×C₈₉	#41	1	1	yes	characteristic
46	C₂²×D₈₉	—	2	2	no	—
47	C₂²×D₈₉	—	4	1	no	—
48	C₂²×D₈₉	—	4	1	no	—
49	C₂²×D₈₉	—	8	2	no	—
50	C₂×C₄×C₈₉	—	4	1	no	—
51	C₂×C₄×C₈₉	—	4	1	no	—
52	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
55	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
56	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
57	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
58	D₃₅₆	—	2	2	no	—
59	D₃₅₆	—	2	2	no	—
60	D₃₅₆	—	4	1	no	—
61	D₃₅₆	—	4	1	no	—
62	C₈×C₈₉	—	4	1	no	—
63	#10623494	—	2	2	no	—
64	#10623494	—	4	1	no	—
65	#10623494	—	4	1	no	—
66	C₂²×C₈₉	—	4	1	no	—
67	C₂²×C₈₉	#123	1	1	yes	characteristic
68	C₂²×C₈₉	#124	1	1	yes	characteristic
69	C₂²×C₈₉	#126	1	1	yes	characteristic
70	C₂×D₈₉	—	4	2	no	—
71	C₂×D₈₉	—	8	2	no	—
72	C₂×D₈₉	—	8	2	no	—
73	C₂×D₈₉	—	8	2	no	—
74	C₄×C₈₉	—	8	1	no	—
75	C₄×C₈₉	—	8	1	no	—
76	C₄×C₈₉	#121	1	1	yes	characteristic
77	C₄×C₈₉	#123	1	1	yes	characteristic
78	C₄×C₈₉	#123	1	1	yes	characteristic
79	C₄×C₈₉	#126	1	1	yes	characteristic
80	C₂×C₈₉	—	2	1	no	—
81	C₂×C₈₉	—	2	1	no	—
82	C₂×C₈₉	—	2	1	no	—
83	C₂×C₈₉	#397	1	1	yes	characteristic
84	D₈₉	—	8	2	no	—
85	#2081	—	89	1	no	maximal, Sylow
86	C₈₉	#2081	1	1	yes	characteristic, complemented, Sylow
87	#430	—	89	1	no	—
88	#418	—	89	1	no	—
89	#418	—	89	1	no	—
90	C₂×#100	—	89	1	no	—
91	#344	—	89	1	no	—
92	#344	—	89	1	no	—
93	H(3,ℤ/4)	D₈₉	1	1	yes	characteristic, complemented
94	C₂²×D₄	—	89	1	no	—
95	C₂×M₁₆	—	89	1	no	—
96	C₂×M₁₆	—	89	1	no	—
97	C₂×M₁₆	—	89	1	no	—
98	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
99	#117	C₂×D₈₉	1	1	yes	characteristic
100	#117	C₂×D₈₉	1	1	yes	characteristic
101	#108	—	178	2	no	—
102	#102	—	178	2	no	—
103	#102	—	178	2	no	—
104	#100	—	178	2	no	—
105	C₂⁴	—	178	1	no	—
106	C₂×D₄	—	89	2	no	—
107	C₂×D₄	—	178	1	no	—
108	C₂×D₄	—	178	1	no	—
109	C₂×D₄	—	178	1	no	—
110	C₂×D₄	—	178	1	no	—
111	C₂×D₄	—	178	1	no	—
112	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
113	M₁₆	—	2	1	no	—
114	M₁₆	—	2	1	no	—
115	M₁₆	—	178	1	no	—
116	M₁₆	—	178	1	no	—
117	M₁₆	—	178	1	no	—
118	M₁₆	—	178	1	no	—
119	M₁₆	—	178	1	no	—
120	M₁₆	—	178	1	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	89	2	no	—
123	C₂×C₈	—	178	1	no	—
124	C₂×C₈	—	178	1	no	—
125	C₄⋊C₄	—	2	1	no	—
126	C₄⋊C₄	—	2	1	no	—
127	C₂²⋊C₄	—	2	1	no	—
128	C₂²⋊C₄	—	2	1	no	—
129	C₄²	—	2	1	no	—
130	C₄²	—	2	1	no	—
131	C₂³	—	356	1	no	—
132	C₂³	—	356	1	no	—
133	C₂³	—	178	2	no	—
134	C₂³	—	712	2	no	—
135	C₂³	#13083573	1	1	yes	characteristic
136	D₄	—	356	1	no	—
137	D₄	—	356	1	no	—
138	D₄	—	178	2	no	—
139	D₄	—	178	2	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	—	4	1	no	—
142	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
143	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
144	C₂×C₄	#13083574	1	1	yes	characteristic
145	C₂×C₄	#13083574	1	1	yes	characteristic
146	C₂×C₄	#13083574	1	1	yes	characteristic
147	C₂×C₄	#13083575	1	1	yes	characteristic
148	C₈	—	4	1	no	—
149	C₈	—	356	1	no	—
150	C₈	—	356	1	no	—
151	C₈	—	178	2	no	—
152	C₂²	—	4	1	no	—
153	C₂²	—	356	2	no	—
154	C₂²	—	712	2	no	—
155	C₂²	—	712	2	no	—
156	C₂²	—	712	2	no	—
157	C₂²	#15667994	1	1	yes	characteristic
158	C₂²	#15667998	1	1	yes	characteristic
159	C₂²	#15668007	1	1	yes	characteristic
160	C₄	—	8	1	no	—
161	C₄	—	8	1	no	—
162	C₄	#15667985	1	1	yes	characteristic
163	C₄	#15667997	1	1	yes	characteristic
164	C₄	#15667997	1	1	yes	characteristic
165	C₄	#15668006	1	1	yes	characteristic
166	C₂	—	2	1	no	—
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	—	712	2	no	—
170	C₂	#19151527	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19151646

C₂

yes

maximal, characteristic, complemented

#19151602

C₂

yes

maximal, characteristic

#19151602

C₂

yes

maximal, characteristic

C₂×#15667928

C₂

yes

maximal, characteristic

#19151368

C₂

yes

maximal, characteristic, complemented

#19151368

C₂

yes

maximal, characteristic, complemented

H(3,ℤ/4)×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₃₅₆

C₂²

yes

characteristic

C₂×#13083558

C₂²

yes

characteristic

C₂×#13083558

C₂²

yes

characteristic

C₂×#13083558

C₂²

yes

—

—

—

—

C₂×M₁₆×C₈₉

C₂²

yes

characteristic

#117×C₈₉

C₂²

yes

characteristic

#117×C₈₉

C₂²

yes

characteristic

C₂³×D₈₉

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

D₄

yes

—

—

—

—

—

—

—

—

—

D₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

M₁₆×C₈₉

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₂³×C₈₉

#41

yes

characteristic

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

C₈×C₈₉

—

#10623494

—

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#397

yes

characteristic

D₈₉

—

#2081

—

maximal, Sylow

C₈₉

#2081

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

D₈₉

yes

characteristic, complemented

C₂²×D₄

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₈₉

yes

characteristic

#117

C₂×D₈₉

yes

characteristic

100

#117

C₂×D₈₉

yes

characteristic

101

#108

—

178

—

102

#102

—

178

—

103

#102

—

178

—

104

#100

—

178

—

105

C₂⁴

—

178

—

106

C₂×D₄

—

107

C₂×D₄

—

178

—

108

C₂×D₄

—

178

—

109

C₂×D₄

—

178

—

110

C₂×D₄

—

178

—

111

C₂×D₄

—

178

—

112

C₂²×C₄

C₂²×D₈₉

yes

characteristic

113

M₁₆

—

114

M₁₆

—

115

M₁₆

—

178

—

116

M₁₆

—

178

—

117

M₁₆

—

178

—

118

M₁₆

—

178

—

119

M₁₆

—

178

—

120

M₁₆

—

178

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

178

—

124

C₂×C₈

—

178

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₂²⋊C₄

—

128

C₂²⋊C₄

—

129

C₄²

—

130

C₄²

—

131

C₂³

—

356

—

132

C₂³

—

356

—

133

C₂³

—

178

—

134

C₂³

—

712

—

135

C₂³

#13083573

yes

characteristic

136

D₄

—

356

—

137

D₄

—

356

—

138

D₄

—

178

—

139

D₄

—

178

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

D₄×D₈₉

yes

characteristic

143

C₂×C₄

D₄×D₈₉

yes

characteristic

144

C₂×C₄

#13083574

yes

characteristic

145

C₂×C₄

#13083574

yes

characteristic

146

C₂×C₄

#13083574

yes

characteristic

147

C₂×C₄

#13083575

yes

characteristic

148

C₈

—

149

C₈

—

356

—

150

C₈

—

356

—

151

C₈

—

178

—

152

C₂²

—

153

C₂²

—

356

—

154

C₂²

—

712

—

155

C₂²

—

712

—

156

C₂²

—

712

—

157

C₂²

#15667994

yes

characteristic

158

C₂²

#15667998

yes

characteristic

159

C₂²

#15668007

yes

characteristic

160

C₄

—

161

C₄

—

162

C₄

#15667985

yes

characteristic

163

C₄

#15667997

yes

characteristic

164

C₄

#15667997

yes

characteristic

165

C₄

#15668006

yes

characteristic

166

C₂

—

167

C₂

—

168

C₂

—

169

C₂

—

712

—

170

C₂

#19151527

yes

characteristic

	subgroup	G/H
center	C₂	#19151527
commutator, cosocle	C₂²×C₄×C₈₉	C₂³
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	H(3,ℤ/4)×C₈₉	C₂
socle	C₂×C₈₉	#397
prime radical	C₂²×D₃₅₆	C₂²
hypercenter	H(3,ℤ/4)	D₈₉
hypocenter	C₈₉	#2081
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₂

#19151527

commutator, cosocle

C₂²×C₄×C₈₉

C₂³

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

H(3,ℤ/4)×C₈₉

C₂

socle

C₂×C₈₉

#397

prime radical

C₂²×D₃₅₆

C₂²

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
1. #19151646	C₂	semidirect
5. #19151368	C₂	semidirect
6. #19151368	C₂	semidirect
7. H(3,ℤ/4)×C₈₉	C₂	semidirect
86. C₈₉	#2081	semidirect
93. H(3,ℤ/4)	D₈₉	semidirect

H ≅ G/N

type

1. #19151646

C₂

semidirect

5. #19151368

C₂

semidirect

6. #19151368

C₂

semidirect

7. H(3,ℤ/4)×C₈₉

C₂

semidirect

86. C₈₉

#2081

semidirect

93. H(3,ℤ/4)

D₈₉

semidirect

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

subgroup	index
#19151646	2
#19151602	2
C₂×#15667928	2
#19151368	2
H(3,ℤ/4)×C₈₉	2
#2081	89

subgroup

index

H(3,ℤ/4)×C₈₉

Subgroups — #26021714

Conjugacy classes of subgroups

Complements

Maximal subgroups