Group #36474574 — subgroups

Subgroups — #36474574

Order 22784.

252 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 93 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	#26021000	C₂	1	1	yes	maximal, characteristic
2	#26020416	C₂	1	1	yes	maximal, characteristic
3	#26018358	C₂	1	2	yes	maximal
4	#26018351	C₂	1	2	yes	maximal
5	C₈₉×#1376	C₂	1	1	yes	maximal, characteristic
6	C₂×#15667948	C₂²	1	1	yes	characteristic
7	#19151636	C₂²	1	1	yes	characteristic
8	C₂×#15667946	—	2	1	no	—
9	C₂×#15667944	C₂²	1	2	yes	—
10	#19151624	—	2	1	no	—
11	#19151464	—	2	1	no	—
12	#19151358	—	2	1	no	—
13	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×#13083558	—	2	1	no	—
16	#15667971	—	2	1	no	—
17	C₂×C₄⋊Dic₈₉	—	2	2	no	—
18	#15667948	—	2	2	no	—
19	#15667946	—	4	2	no	—
20	#15667946	Q₈	1	2	yes	—
21	#15667944	—	2	1	no	—
22	#15667944	—	2	4	no	—
23	#15667941	—	4	2	no	—
24	#15667919	—	4	2	no	—
25	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
27	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
28	M₃₂×C₈₉	—	2	2	no	—
29	M₃₂×C₈₉	—	2	2	no	—
30	C₂×C₁₆×C₈₉	—	2	2	no	—
31	C₄×C₈×C₈₉	—	2	2	no	—
32	C₂²×Dic₈₉	—	4	2	no	—
33	#13083558	—	4	1	no	—
34	#13083558	—	4	1	no	—
35	C₂×#10623494	—	4	1	no	—
36	C₄⋊Dic₈₉	—	2	2	no	—
37	C₄⋊Dic₈₉	—	4	2	no	—
38	C₄⋊Dic₈₉	—	4	2	no	—
39	C₂²⋊Dic₈₉	—	4	1	no	—
40	C₄×Dic₈₉	—	4	1	no	—
41	C₂²×C₄×C₈₉	—	2	1	no	—
42	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₈₉	—	2	1	no	—
44	C₂×C₈×C₈₉	—	2	1	no	—
45	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
46	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₈₉	#41	1	2	yes	—
48	C₂×C₈×C₈₉	#41	1	2	yes	—
49	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
50	C₄²×C₈₉	—	2	1	no	—
51	C₄²×C₈₉	Q₁₆	1	2	yes	—
52	C₁₆×C₈₉	—	4	2	no	—
53	C₂³×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
54	C₂×C₄×C₈₉	—	2	2	no	—
55	C₂×C₄×C₈₉	—	4	2	no	—
56	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
57	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
58	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
59	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
60	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
61	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
62	C₂×Dic₈₉	—	4	2	no	—
63	C₂×Dic₈₉	—	8	4	no	—
64	C₂×Dic₈₉	—	8	2	no	—
65	C₈×C₈₉	—	2	2	no	—
66	C₈×C₈₉	#125	1	2	yes	—
67	C₈×C₈₉	#126	1	2	yes	—
68	#10623494	—	8	1	no	—
69	C₂²×C₈₉	—	2	2	no	—
70	C₂²×C₈₉	#476	1	1	yes	characteristic
71	C₂²×C₈₉	#484	1	1	yes	characteristic
72	C₂²×C₈₉	#490	1	1	yes	characteristic
73	C₄×C₈₉	—	4	2	no	—
74	C₄×C₈₉	#400	1	1	yes	characteristic
75	C₄×C₈₉	#477	1	1	yes	characteristic
76	C₄×C₈₉	#481	1	1	yes	characteristic
77	C₄×C₈₉	#487	1	1	yes	characteristic
78	Dic₈₉	—	8	2	no	—
79	Dic₈₉	—	16	2	no	—
80	#13542	—	89	1	no	maximal, Sylow
81	C₂×C₈₉	—	2	1	no	—
82	C₂×C₈₉	—	2	1	no	—
83	C₂×C₈₉	#2100	1	1	yes	characteristic
84	C₂×C₈₉	#2231	1	2	yes	—
85	#1945	—	89	1	no	—
86	#1846	—	89	1	no	—
87	#1386	—	89	2	no	—
88	#1383	—	89	2	no	—
89	#1376	D₈₉	1	1	yes	characteristic
90	C₈₉	#13542	1	1	yes	characteristic, complemented, Sylow
91	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
92	C₂×#108	—	89	1	no	—
93	#427	—	89	1	no	—
94	C₂×#107	—	178	1	no	—
95	C₂×#106	—	89	2	no	—
96	#424	—	178	1	no	—
97	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
98	#382	—	178	1	no	—
99	#339	—	178	1	no	—
100	C₂×M₁₆	—	178	1	no	—
101	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
102	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
103	#117	—	178	1	no	—
104	C₂×C₄⋊C₄	—	178	2	no	—
105	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
106	M₃₂	—	2	2	no	—
107	M₃₂	—	2	2	no	—
108	C₂×C₁₆	—	2	2	no	—
109	#108	—	178	2	no	—
110	#107	—	89	2	no	—
111	#107	—	356	2	no	—
112	#106	—	178	4	no	—
113	#106	—	178	1	no	—
114	C₄⋊C₈	—	356	2	no	—
115	C₄×C₈	—	2	2	no	—
116	#95	—	356	2	no	—
117	C₂²×C₄	—	2	1	no	—
118	C₂²×C₄	—	356	2	no	—
119	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
120	M₁₆	—	356	1	no	—
121	M₁₆	—	356	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	356	1	no	—
125	C₂×C₈	#13083563	1	2	yes	—
126	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
127	C₂×C₈	#13083573	1	2	yes	—
128	C₂×C₈	#13083573	1	1	yes	characteristic
129	C₂×C₈	#13083574	1	2	yes	—
130	C₄⋊C₄	—	356	2	no	—
131	C₄⋊C₄	—	356	2	no	—
132	C₄⋊C₄	—	178	2	no	—
133	C₂²⋊C₄	—	356	1	no	—
134	C₄²	—	2	1	no	—
135	C₄²	—	356	1	no	—
136	C₄²	Dic₃₅₆	1	2	yes	—
137	C₁₆	—	4	2	no	—
138	C₂³	#15668040	1	1	yes	characteristic
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	356	2	no	—
142	C₂×C₄	—	712	4	no	—
143	C₂×C₄	—	712	2	no	—
144	C₂×C₄	#15667992	1	1	yes	characteristic
145	C₂×C₄	#15667996	1	1	yes	characteristic
146	C₂×C₄	#15668000	1	1	yes	characteristic
147	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
148	C₂×C₄	C₂×Dic₃₅₆	1	1	yes	characteristic
149	C₂×C₄	#15668040	1	1	yes	characteristic
150	C₈	—	2	2	no	—
151	C₈	—	712	1	no	—
152	C₈	#15668002	1	2	yes	—
153	C₈	#15668005	1	2	yes	—
154	C₂²	—	2	2	no	—
155	C₂²	#19151889	1	1	yes	characteristic
156	C₂²	#19151934	1	1	yes	characteristic
157	C₂²	#19151963	1	1	yes	characteristic
158	C₄	—	4	2	no	—
159	C₄	—	1424	2	no	—
160	C₄	—	712	2	no	—
161	C₄	#19151538	1	1	yes	characteristic
162	C₄	#19151894	1	1	yes	characteristic
163	C₄	#19151917	1	1	yes	characteristic
164	C₄	#19151949	1	1	yes	characteristic
165	C₂	—	2	1	no	—
166	C₂	—	2	1	no	—
167	C₂	#26021816	1	1	yes	characteristic
168	C₂	#26022429	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26021000

C₂

yes

maximal, characteristic

#26020416

C₂

yes

maximal, characteristic

#26018358

C₂

yes

maximal

#26018351

C₂

yes

maximal

C₈₉×#1376

C₂

yes

maximal, characteristic

C₂×#15667948

C₂²

yes

characteristic

#19151636

C₂²

yes

characteristic

C₂×#15667946

—

C₂×#15667944

C₂²

yes

—

#19151624

—

#19151464

—

#19151358

—

C₂×M₃₂×C₈₉

C₂²

yes

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂×#13083558

—

#15667971

—

C₂×C₄⋊Dic₈₉

—

#15667948

—

#15667946

—

#15667946

Q₈

yes

—

—

—

—

—

C₂²×C₈×C₈₉

D₄

yes

characteristic

C₂²×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₄²×C₈₉

D₄

yes

characteristic

M₃₂×C₈₉

—

M₃₂×C₈₉

—

C₂×C₁₆×C₈₉

—

C₄×C₈×C₈₉

—

C₂²×Dic₈₉

—

#13083558

—

#13083558

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

SD₁₆

yes

—

C₂×C₈×C₈₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₈₉

#41

yes

—

C₂×C₈×C₈₉

#41

yes

—

C₂×C₈×C₈₉

#41

yes

characteristic

C₄²×C₈₉

—

C₄²×C₈₉

Q₁₆

yes

—

C₁₆×C₈₉

—

C₂³×C₈₉

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²⋊Q₈

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₈₉

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₈×C₈₉

—

C₈×C₈₉

#125

yes

—

C₈×C₈₉

#126

yes

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

Dic₈₉

—

#13542

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#2100

yes

characteristic

C₂×C₈₉

#2231

yes

—

—

—

—

—

D₈₉

yes

characteristic

C₈₉

#13542

yes

characteristic, complemented, Sylow

C₂×M₃₂

C₂×D₈₉

yes

—

C₂×#108

—

#427

—

C₂×#107

—

178

—

C₂×#106

—

#424

—

178

—

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

#382

—

178

—

#339

—

178

—

100

C₂×M₁₆

—

178

—

101

C₂²×C₈

D₃₅₆

yes

characteristic

102

C₂²×C₈

C₂²×D₈₉

yes

characteristic

103

#117

—

178

—

104

C₂×C₄⋊C₄

—

178

—

105

C₂×C₄²

D₃₅₆

yes

characteristic

106

M₃₂

—

107

M₃₂

—

108

C₂×C₁₆

—

109

#108

—

178

—

110

#107

—

111

#107

—

356

—

112

#106

—

178

—

113

#106

—

178

—

114

C₄⋊C₈

—

356

—

115

C₄×C₈

—

116

#95

—

356

—

117

C₂²×C₄

—

118

C₂²×C₄

—

356

—

119

C₂²×C₄

C₂×D₃₅₆

yes

characteristic

120

M₁₆

—

356

—

121

M₁₆

—

356

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

356

—

125

C₂×C₈

#13083563

yes

—

126

C₂×C₈

Q₈×D₈₉

yes

characteristic

127

C₂×C₈

#13083573

yes

—

128

C₂×C₈

#13083573

yes

characteristic

129

C₂×C₈

#13083574

yes

—

130

C₄⋊C₄

—

356

—

131

C₄⋊C₄

—

356

—

132

C₄⋊C₄

—

178

—

133

C₂²⋊C₄

—

356

—

134

C₄²

—

135

C₄²

—

356

—

136

C₄²

Dic₃₅₆

yes

—

137

C₁₆

—

138

C₂³

#15668040

yes

characteristic

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

356

—

142

C₂×C₄

—

712

—

143

C₂×C₄

—

712

—

144

C₂×C₄

#15667992

yes

characteristic

145

C₂×C₄

#15667996

yes

characteristic

146

C₂×C₄

#15668000

yes

characteristic

147

C₂×C₄

C₂×#13083563

yes

characteristic

148

C₂×C₄

C₂×Dic₃₅₆

yes

characteristic

149

C₂×C₄

#15668040

yes

characteristic

150

C₈

—

151

C₈

—

712

—

152

C₈

#15668002

yes

—

153

C₈

#15668005

yes

—

154

C₂²

—

155

C₂²

#19151889

yes

characteristic

156

C₂²

#19151934

yes

characteristic

157

C₂²

#19151963

yes

characteristic

158

C₄

—

159

C₄

—

1424

—

160

C₄

—

712

—

161

C₄

#19151538

yes

characteristic

162

C₄

#19151894

yes

characteristic

163

C₄

#19151917

yes

characteristic

164

C₄

#19151949

yes

characteristic

165

C₂

—

166

C₂

—

167

C₂

#26021816

yes

characteristic

168

C₂

#26022429

yes

—

	subgroup	G/H
center	C₂²	#19151889
commutator, cosocle	C₂²×C₈×C₈₉	C₂³
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#1376	C₂
socle	C₂²×C₈₉	#476
prime radical	C₂³×C₈₉	AGL(1,ℤ/8)
hypercenter	#1376	D₈₉
hypocenter	C₈₉	#13542
LCS	G ▷ C₂²×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1376

subgroup

G/H

center

C₂²

#19151889

commutator, cosocle

C₂²×C₈×C₈₉

C₂³

Frattini

C₂²×C₈

C₂²×D₈₉

Fitting

C₈₉×#1376

C₂

C₂²×C₈₉

C₂³×C₈₉

D₈₉

C₈₉

G ▷ C₂²×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1376

N	H ≅ G/N	type
90. C₈₉	#13542	semidirect

H ≅ G/N

type

90. C₈₉

#13542

semidirect

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

subgroup	index
#26021000	2
#26020416	2
#26018358	2
#26018351	2
C₈₉×#1376	2
#13542	89

subgroup

index

C₈₉×#1376

Subgroups — #36474574

Conjugacy classes of subgroups

Complements

Maximal subgroups