Group #36469180 — subgroups

Subgroups — #36469180

Order 22784.

340 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 91 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	#26022502	C₂	1	2	yes	maximal, complemented
2	#26022476	C₂	1	2	yes	maximal, complemented
3	#26019867	C₂	1	1	yes	maximal, characteristic, complemented
4	#26019844	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#1312	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₄₂₄	—	2	4	no	—
7	#19152003	C₂²	1	1	yes	characteristic
8	#19151975	—	2	1	no	—
9	#19151975	—	2	1	no	—
10	#19151975	C₂²	1	1	yes	characteristic
11	#19151975	C₂²	1	2	yes	—
12	#19151971	—	2	1	no	—
13	#19151945	—	2	1	no	—
14	#19151400	—	2	4	no	—
15	C₄×C₁₆×C₈₉	C₂²	1	2	yes	—
16	C₈²×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×#13083563	—	4	2	no	—
18	C₂×D₇₁₂	—	2	2	no	—
19	C₂×D₇₁₂	—	2	2	no	—
20	C₂×D₇₁₂	—	2	2	no	—
21	C₂×D₇₁₂	—	4	2	no	—
22	C₂×D₇₁₂	—	4	2	no	—
23	#15668012	—	2	1	no	—
24	#15668009	—	2	1	no	—
25	#15668009	—	2	2	no	—
26	D₁₄₂₄	—	4	8	no	—
27	#15667946	—	2	2	no	—
28	#15667934	—	4	2	no	—
29	C₂×C₁₆×C₈₉	D₄	1	4	yes	—
30	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
31	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
32	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
33	C₂×Dic₁₇₈	—	4	1	no	—
34	C₂×D₃₅₆	—	4	1	no	—
35	C₂×D₃₅₆	—	4	2	no	—
36	C₂×D₃₅₆	—	4	2	no	—
37	C₂×D₃₅₆	—	4	2	no	—
38	C₂×D₃₅₆	—	4	1	no	—
39	C₂×D₃₅₆	—	4	1	no	—
40	#13083563	—	8	4	no	—
41	D₇₁₂	—	4	4	no	—
42	D₇₁₂	—	4	4	no	—
43	D₇₁₂	—	4	4	no	—
44	D₇₁₂	—	8	4	no	—
45	D₇₁₂	—	8	4	no	—
46	C₄⋊Dic₈₉	—	4	1	no	—
47	C₄⋊Dic₈₉	—	4	1	no	—
48	C₂×C₈×C₈₉	D₈	1	2	yes	—
49	C₂×C₈×C₈₉	D₈	1	2	yes	—
50	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
51	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
52	C₁₆×C₈₉	—	2	4	no	—
53	C₂²×D₈₉	—	8	2	no	—
54	C₂²×D₈₉	—	8	1	no	—
55	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
56	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
57	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
58	C₂×Dic₈₉	—	8	1	no	—
59	D₃₅₆	—	8	2	no	—
60	D₃₅₆	—	8	2	no	—
61	D₃₅₆	—	8	2	no	—
62	D₃₅₆	—	8	4	no	—
63	D₃₅₆	—	8	2	no	—
64	D₃₅₆	—	8	4	no	—
65	D₃₅₆	—	8	2	no	—
66	Dic₁₇₈	—	8	2	no	—
67	C₈×C₈₉	—	2	2	no	—
68	C₈×C₈₉	D₁₆	1	4	yes	complemented
69	C₈×C₈₉	AGL(1,ℤ/8)	1	4	yes	—
70	C₂²×C₈₉	#493	1	1	yes	characteristic
71	C₂×D₈₉	—	16	2	no	—
72	C₂×D₈₉	—	16	1	no	—
73	C₂×D₈₉	—	16	2	no	—
74	C₂×D₈₉	—	16	1	no	—
75	C₂×D₈₉	—	16	2	no	—
76	C₂×D₈₉	—	16	1	no	—
77	C₄×C₈₉	#496	1	2	yes	—
78	C₄×C₈₉	C₂×D₁₆	1	2	yes	—
79	C₄×C₈₉	#509	1	2	yes	—
80	Dic₈₉	—	16	1	no	—
81	#12525	—	89	1	no	maximal, Sylow
82	C₂×C₈₉	#1721	1	1	yes	characteristic
83	C₂×C₈₉	#2246	1	1	yes	characteristic
84	C₂×C₈₉	#2250	1	1	yes	characteristic
85	D₈₉	—	32	2	no	—
86	D₈₉	—	32	1	no	—
87	#2246	—	89	2	no	—
88	#2240	—	89	2	no	—
89	#1712	—	89	1	no	—
90	#1706	—	89	1	no	—
91	#1312	D₈₉	1	1	yes	characteristic, complemented
92	C₈₉	#12525	1	1	yes	characteristic, complemented, Sylow
93	C₂×D₁₆	—	178	4	no	—
94	#500	—	89	1	no	—
95	#493	—	89	1	no	—
96	#493	—	89	2	no	—
97	#493	—	178	1	no	—
98	#493	—	178	1	no	—
99	#492	—	178	1	no	—
100	#486	—	178	1	no	—
101	#357	—	178	4	no	—
102	C₄×C₁₆	C₂×D₈₉	1	2	yes	—
103	C₈²	C₂×D₈₉	1	1	yes	characteristic
104	C₂×SD₁₆	—	356	2	no	—
105	C₂×D₈	—	356	2	no	—
106	C₂×D₈	—	356	2	no	—
107	C₂×D₈	—	178	2	no	—
108	C₂×D₈	—	178	2	no	—
109	C₂×D₈	—	178	2	no	—
110	C₄⋊Q₈	—	178	1	no	—
111	#127	—	178	1	no	—
112	#127	—	178	2	no	—
113	D₁₆	—	356	8	no	—
114	C₂×C₁₆	D₃₅₆	1	4	yes	—
115	#107	—	178	2	no	—
116	#102	—	356	2	no	—
117	C₄×C₈	D₃₅₆	1	1	yes	characteristic
118	C₄×C₈	D₃₅₆	1	1	yes	characteristic
119	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
120	C₂×Q₈	—	356	1	no	—
121	C₂×D₄	—	356	1	no	—
122	C₂×D₄	—	356	2	no	—
123	C₂×D₄	—	356	1	no	—
124	C₂×D₄	—	356	2	no	—
125	C₂×D₄	—	356	2	no	—
126	C₂×D₄	—	356	1	no	—
127	SD₁₆	—	712	4	no	—
128	D₈	—	356	4	no	—
129	D₈	—	356	4	no	—
130	D₈	—	356	4	no	—
131	D₈	—	712	4	no	—
132	D₈	—	712	4	no	—
133	C₂×C₈	D₇₁₂	1	2	yes	—
134	C₂×C₈	D₇₁₂	1	2	yes	—
135	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
136	C₄⋊C₄	—	356	1	no	—
137	C₄⋊C₄	—	356	1	no	—
138	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
139	C₁₆	—	2	4	no	—
140	C₂³	—	712	1	no	—
141	C₂³	—	712	2	no	—
142	Q₈	—	712	2	no	—
143	D₄	—	712	2	no	—
144	D₄	—	712	2	no	—
145	D₄	—	712	2	no	—
146	D₄	—	712	4	no	—
147	D₄	—	712	2	no	—
148	D₄	—	712	4	no	—
149	D₄	—	712	2	no	—
150	C₂×C₄	—	712	1	no	—
151	C₂×C₄	#15668009	1	1	yes	characteristic
152	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
153	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
154	C₈	—	2	2	no	—
155	C₈	D₁₄₂₄	1	4	yes	complemented
156	C₈	#15668040	1	4	yes	—
157	C₂²	—	1424	1	no	—
158	C₂²	—	1424	2	no	—
159	C₂²	—	1424	2	no	—
160	C₂²	—	1424	1	no	—
161	C₂²	—	1424	2	no	—
162	C₂²	—	1424	1	no	—
163	C₂²	#19151975	1	1	yes	characteristic
164	C₄	—	1424	1	no	—
165	C₄	#19151987	1	2	yes	—
166	C₄	C₂×D₁₄₂₄	1	2	yes	—
167	C₄	#19152036	1	2	yes	—
168	C₂	—	2848	2	no	—
169	C₂	—	2848	1	no	—
170	C₂	#26019918	1	1	yes	characteristic
171	C₂	#26022502	1	1	yes	characteristic
172	C₂	#26022518	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26022502

C₂

yes

maximal, complemented

#26022476

C₂

yes

maximal, complemented

#26019867

C₂

yes

maximal, characteristic, complemented

#26019844

C₂

yes

maximal, characteristic, complemented

C₈₉×#1312

C₂

yes

maximal, characteristic, complemented

C₂×D₁₄₂₄

—

#19152003

C₂²

yes

—

—

C₂²

yes

characteristic

#19151975

C₂²

yes

—

#19151971

—

#19151945

—

#19151400

—

C₄×C₁₆×C₈₉

C₂²

yes

—

C₈²×C₈₉

C₂²

yes

characteristic

C₂×#13083563

—

C₂×D₇₁₂

—

C₂×D₇₁₂

—

C₂×D₇₁₂

—

C₂×D₇₁₂

—

C₂×D₇₁₂

—

#15668012

—

#15668009

—

#15668009

—

D₁₄₂₄

—

#15667946

—

#15667934

—

C₂×C₁₆×C₈₉

D₄

yes

—

C₄×C₈×C₈₉

D₄

yes

characteristic

C₄×C₈×C₈₉

D₄

yes

characteristic

C₄×C₈×C₈₉

C₂³

yes

characteristic

C₂×Dic₁₇₈

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

#13083563

—

D₇₁₂

—

D₇₁₂

—

D₇₁₂

—

D₇₁₂

—

D₇₁₂

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂×C₈×C₈₉

D₈

yes

—

C₂×C₈×C₈₉

D₈

yes

—

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₁₆×C₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂×C₄×C₈₉

#127

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

D₁₆

yes

complemented

C₈×C₈₉

AGL(1,ℤ/8)

yes

—

C₂²×C₈₉

#493

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

#496

yes

—

C₄×C₈₉

C₂×D₁₆

yes

—

C₄×C₈₉

#509

yes

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

D₈₉

—

D₈₉

—

—

—

—

—

D₈₉

yes

characteristic, complemented

C₈₉

#12525

yes

characteristic, complemented, Sylow

C₂×D₁₆

—

178

—

—

—

—

—

178

—

#493

—

178

—

#492

—

178

—

100

#486

—

178

—

101

#357

—

178

—

102

C₄×C₁₆

C₂×D₈₉

yes

—

103

C₈²

C₂×D₈₉

yes

characteristic

104

C₂×SD₁₆

—

356

—

105

C₂×D₈

—

356

—

106

C₂×D₈

—

356

—

107

C₂×D₈

—

178

—

108

C₂×D₈

—

178

—

109

C₂×D₈

—

178

—

110

C₄⋊Q₈

—

178

—

111

#127

—

178

—

112

#127

—

178

—

113

D₁₆

—

356

—

114

C₂×C₁₆

D₃₅₆

yes

—

115

#107

—

178

—

116

#102

—

356

—

117

C₄×C₈

D₃₅₆

yes

characteristic

118

C₄×C₈

D₃₅₆

yes

characteristic

119

C₄×C₈

C₂²×D₈₉

yes

characteristic

120

C₂×Q₈

—

356

—

121

C₂×D₄

—

356

—

122

C₂×D₄

—

356

—

123

C₂×D₄

—

356

—

124

C₂×D₄

—

356

—

125

C₂×D₄

—

356

—

126

C₂×D₄

—

356

—

127

SD₁₆

—

712

—

128

D₈

—

356

—

129

D₈

—

356

—

130

D₈

—

356

—

131

D₈

—

712

—

132

D₈

—

712

—

133

C₂×C₈

D₇₁₂

yes

—

134

C₂×C₈

D₇₁₂

yes

—

135

C₂×C₈

C₂×D₃₅₆

yes

—

136

C₄⋊C₄

—

356

—

137

C₄⋊C₄

—

356

—

138

C₄²

C₂×D₃₅₆

yes

characteristic

139

C₁₆

—

140

C₂³

—

712

—

141

C₂³

—

712

—

142

Q₈

—

712

—

143

D₄

—

712

—

144

D₄

—

712

—

145

D₄

—

712

—

146

D₄

—

712

—

147

D₄

—

712

—

148

D₄

—

712

—

149

D₄

—

712

—

150

C₂×C₄

—

712

—

151

C₂×C₄

#15668009

yes

characteristic

152

C₂×C₄

C₂×D₇₁₂

yes

characteristic

153

C₂×C₄

C₂×D₇₁₂

yes

characteristic

154

C₈

—

155

C₈

D₁₄₂₄

yes

complemented

156

C₈

#15668040

yes

—

157

C₂²

—

1424

—

158

C₂²

—

1424

—

159

C₂²

—

1424

—

160

C₂²

—

1424

—

161

C₂²

—

1424

—

162

C₂²

—

1424

—

163

C₂²

#19151975

yes

characteristic

164

C₄

—

1424

—

165

C₄

#19151987

yes

—

166

C₄

C₂×D₁₄₂₄

yes

—

167

C₄

#19152036

yes

—

168

C₂

—

2848

—

169

C₂

—

2848

—

170

C₂

#26019918

yes

characteristic

171

C₂

#26022502

yes

characteristic

172

C₂

#26022518

yes

characteristic

	subgroup	G/H
center	C₂²	#19151975
commutator, cosocle	C₄×C₈×C₈₉	C₂³
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈₉×#1312	C₂
socle	C₂²×C₈₉	#493
hypercenter	#1312	D₈₉
hypocenter	C₈₉	#12525
LCS	G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

subgroup

G/H

center

C₂²

#19151975

commutator, cosocle

C₄×C₈×C₈₉

C₂³

Frattini

C₄×C₈

C₂²×D₈₉

Fitting

C₈₉×#1312

C₂

C₂²×C₈₉

D₈₉

C₈₉

G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

N	H ≅ G/N	type
1. #26022502	C₂	semidirect
2. #26022476	C₂	semidirect
3. #26019867	C₂	semidirect
4. #26019844	C₂	semidirect
5. C₈₉×#1312	C₂	semidirect
68. C₈×C₈₉	D₁₆	semidirect
91. #1312	D₈₉	semidirect
92. C₈₉	#12525	semidirect
155. C₈	D₁₄₂₄	semidirect

H ≅ G/N

type

1. #26022502

C₂

semidirect

2. #26022476

C₂

semidirect

3. #26019867

C₂

semidirect

4. #26019844

C₂

semidirect

5. C₈₉×#1312

C₂

semidirect

68. C₈×C₈₉

D₁₆

semidirect

91. #1312

D₈₉

semidirect

92. C₈₉

#12525

semidirect

155. C₈

D₁₄₂₄

semidirect

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

subgroup	index
#26022502	2
#26022476	2
#26019867	2
#26019844	2
C₈₉×#1312	2
#12525	89

subgroup

index

C₈₉×#1312

Subgroups — #36469180

Conjugacy classes of subgroups

Complements

Maximal subgroups