Group #36474572 — subgroups

Subgroups — #36474572

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	#26018359	C₂	1	2	yes	maximal
3	#26018350	C₂	1	2	yes	maximal
4	#26018334	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1846	C₂	1	1	yes	maximal, characteristic
6	C₂×#15667952	C₂²	1	2	yes	—
7	C₂×#15667948	C₂²	1	1	yes	characteristic
8	#19151636	C₂²	1	1	yes	characteristic
9	#19151624	—	2	1	no	—
10	#19151358	—	2	1	no	—
11	C₂×#107×C₈₉	—	2	1	no	—
12	C₂×#106×C₈₉	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
14	#382×C₈₉	—	2	1	no	—
15	C₂×#13083558	—	2	1	no	—
16	#15667971	—	2	1	no	—
17	#15667952	—	2	2	no	—
18	#15667952	—	2	2	no	—
19	C₂×#13083549	—	2	2	no	—
20	#15667948	—	2	2	no	—
21	#15667946	Q₈	1	2	yes	—
22	#15667944	—	2	1	no	—
23	#15667941	—	4	2	no	—
24	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
26	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
27	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
28	#107×C₈₉	—	4	2	no	—
29	#106×C₈₉	—	2	4	no	—
30	C₄×C₈×C₈₉	—	2	2	no	—
31	#95×C₈₉	—	4	2	no	—
32	#13083558	—	4	1	no	—
33	#13083558	—	4	1	no	—
34	C₂×#10623494	—	4	1	no	—
35	C₄⋊Dic₈₉	—	2	2	no	—
36	C₂²⋊Dic₈₉	—	4	1	no	—
37	C₄×Dic₈₉	—	4	1	no	—
38	#13083549	—	4	2	no	—
39	C₂²×C₄×C₈₉	—	2	1	no	—
40	C₂²×C₄×C₈₉	—	4	2	no	—
41	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈×C₈₉	—	2	1	no	—
43	C₂×C₈×C₈₉	—	2	1	no	—
44	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
45	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₈×C₈₉	#41	1	2	yes	—
47	C₂×C₈×C₈₉	#41	1	2	yes	—
48	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
49	C₄⋊C₄×C₈₉	—	4	2	no	—
50	C₄⋊C₄×C₈₉	—	4	2	no	—
51	C₄²×C₈₉	—	2	1	no	—
52	C₄²×C₈₉	Q₁₆	1	2	yes	—
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	—	8	4	no	—
57	C₂×C₄×C₈₉	—	8	2	no	—
58	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
59	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
60	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
61	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
63	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
64	C₂×Dic₈₉	—	4	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	—	16	2	no	—
75	C₄×C₈₉	#400	1	1	yes	characteristic
76	C₄×C₈₉	#477	1	1	yes	characteristic
77	C₄×C₈₉	#481	1	1	yes	characteristic
78	C₄×C₈₉	#487	1	1	yes	characteristic
79	Dic₈₉	—	8	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	D₈₉	1	1	yes	characteristic
87	#1386	—	89	2	no	—
88	#1383	—	89	2	no	—
89	#1376	—	89	1	no	—
90	C₈₉	#13542	1	1	yes	characteristic, complemented, Sylow
91	C₂×M₃₂	—	89	2	no	—
92	C₂×#108	—	89	1	no	—
93	#427	—	89	1	no	—
94	C₂×#107	—	2	1	no	—
95	C₂×#106	C₂×D₈₉	1	2	yes	—
96	#424	—	178	1	no	—
97	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
98	#382	—	2	1	no	—
99	#339	—	178	1	no	—
100	C₂×M₁₆	—	178	1	no	—
101	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
102	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
103	#117	—	178	1	no	—
104	C₂×C₄⋊C₄	—	2	2	no	—
105	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
106	M₃₂	—	178	2	no	—
107	M₃₂	—	178	2	no	—
108	C₂×C₁₆	—	178	2	no	—
109	#108	—	178	2	no	—
110	#107	—	4	2	no	—
111	#107	—	89	2	no	—
112	#106	—	2	4	no	—
113	#106	—	178	1	no	—
114	C₄⋊C₈	—	356	2	no	—
115	C₄×C₈	—	2	2	no	—
116	#95	—	4	2	no	—
117	C₂²×C₄	—	2	1	no	—
118	C₂²×C₄	—	4	2	no	—
119	C₂²×C₄	C₂×C₂²⋊D₈₉	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	#13083561	1	2	yes	—
126	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
127	C₂×C₈	#13083573	1	2	yes	—
128	C₂×C₈	#13083574	1	2	yes	—
129	C₂×C₈	#13083575	1	1	yes	characteristic
130	C₄⋊C₄	—	4	2	no	—
131	C₄⋊C₄	—	4	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	#13083564	1	2	yes	—
137	C₁₆	—	356	2	no	—
138	C₂³	#15668038	1	1	yes	characteristic
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	8	4	no	—
142	C₂×C₄	—	8	2	no	—
143	C₂×C₄	—	356	2	no	—
144	C₂×C₄	#15667988	1	1	yes	characteristic
145	C₂×C₄	#15667995	1	1	yes	characteristic
146	C₂×C₄	#15667999	1	1	yes	characteristic
147	C₂×C₄	C₂×#13083561	1	1	yes	characteristic
148	C₂×C₄	C₂×#13083564	1	1	yes	characteristic
149	C₂×C₄	#15668038	1	1	yes	characteristic
150	C₈	—	2	2	no	—
151	C₈	—	712	1	no	—
152	C₈	#15668002	1	2	yes	—
153	C₈	#15668007	1	2	yes	—
154	C₂²	—	2	2	no	—
155	C₂²	#19151885	1	1	yes	characteristic
156	C₂²	#19151933	1	1	yes	characteristic
157	C₂²	#19151962	1	1	yes	characteristic
158	C₄	—	4	2	no	—
159	C₄	—	16	2	no	—
160	C₄	—	712	2	no	—
161	C₄	#19151536	1	1	yes	characteristic
162	C₄	#19151890	1	1	yes	characteristic
163	C₄	#19151916	1	1	yes	characteristic
164	C₄	#19151948	1	1	yes	characteristic
165	C₂	—	2	1	no	—
166	C₂	—	2	1	no	—
167	C₂	#26021814	1	1	yes	characteristic
168	C₂	#26022425	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26021000

C₂

yes

maximal, characteristic

#26018359

C₂

yes

maximal

#26018350

C₂

yes

maximal

#26018334

C₂

yes

maximal, characteristic

C₈₉×#1846

C₂

yes

maximal, characteristic

C₂×#15667952

C₂²

yes

—

C₂×#15667948

C₂²

yes

characteristic

#19151636

C₂²

yes

characteristic

#19151624

—

#19151358

—

C₂×#107×C₈₉

—

C₂×#106×C₈₉

C₂²

yes

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

#382×C₈₉

—

—

—

—

—

—

—

Q₈

yes

—

#15667944

—

#15667941

—

C₂²×C₈×C₈₉

D₄

yes

characteristic

C₂²×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄²×C₈₉

D₄

yes

characteristic

#107×C₈₉

—

#106×C₈₉

—

C₄×C₈×C₈₉

—

#95×C₈₉

—

#13083558

—

#13083558

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

#13083549

—

C₂²×C₄×C₈₉

—

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

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

Q₁₆

yes

—

C₂³×C₈₉

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

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

—

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

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

#13542

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#2100

yes

characteristic

C₂×C₈₉

#2231

yes

—

#1945

—

#1846

D₈₉

yes

—

—

—

C₈₉

#13542

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

—

—

—

C₂×D₈₉

yes

—

#424

—

178

—

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

#382

—

#339

—

178

—

100

C₂×M₁₆

—

178

—

101

C₂²×C₈

C₂²⋊D₈₉

yes

characteristic

102

C₂²×C₈

C₂²×D₈₉

yes

characteristic

103

#117

—

178

—

104

C₂×C₄⋊C₄

—

105

C₂×C₄²

C₂²⋊D₈₉

yes

characteristic

106

M₃₂

—

178

—

107

M₃₂

—

178

—

108

C₂×C₁₆

—

178

—

109

#108

—

178

—

110

#107

—

111

#107

—

112

#106

—

113

#106

—

178

—

114

C₄⋊C₈

—

356

—

115

C₄×C₈

—

116

#95

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

120

M₁₆

—

356

—

121

M₁₆

—

356

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

356

—

125

C₂×C₈

#13083561

yes

—

126

C₂×C₈

Q₈×D₈₉

yes

characteristic

127

C₂×C₈

#13083573

yes

—

128

C₂×C₈

#13083574

yes

—

129

C₂×C₈

#13083575

yes

characteristic

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

178

—

133

C₂²⋊C₄

—

356

—

134

C₄²

—

135

C₄²

—

356

—

136

C₄²

#13083564

yes

—

137

C₁₆

—

356

—

138

C₂³

#15668038

yes

characteristic

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

356

—

144

C₂×C₄

#15667988

yes

characteristic

145

C₂×C₄

#15667995

yes

characteristic

146

C₂×C₄

#15667999

yes

characteristic

147

C₂×C₄

C₂×#13083561

yes

characteristic

148

C₂×C₄

C₂×#13083564

yes

characteristic

149

C₂×C₄

#15668038

yes

characteristic

150

C₈

—

151

C₈

—

712

—

152

C₈

#15668002

yes

—

153

C₈

#15668007

yes

—

154

C₂²

—

155

C₂²

#19151885

yes

characteristic

156

C₂²

#19151933

yes

characteristic

157

C₂²

#19151962

yes

characteristic

158

C₄

—

159

C₄

—

160

C₄

—

712

—

161

C₄

#19151536

yes

characteristic

162

C₄

#19151890

yes

characteristic

163

C₄

#19151916

yes

characteristic

164

C₄

#19151948

yes

characteristic

165

C₂

—

166

C₂

—

167

C₂

#26021814

yes

characteristic

168

C₂

#26022425

yes

—

	subgroup	G/H
center	C₂²	#19151885
commutator, cosocle	C₂²×C₈×C₈₉	C₂³
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#1846	C₂
socle	C₂²×C₈₉	#476
prime radical	C₂³×C₈₉	AGL(1,ℤ/8)
hypercenter	#1846	D₈₉
hypocenter	C₈₉	#13542
LCS	G ▷ C₂²×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1846

subgroup

G/H

center

C₂²

#19151885

commutator, cosocle

C₂²×C₈×C₈₉

C₂³

Frattini

C₂²×C₈

C₂²×D₈₉

Fitting

C₈₉×#1846

C₂

C₂²×C₈₉

C₂³×C₈₉

D₈₉

C₈₉

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1846

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
#26018359	2
#26018350	2
#26018334	2
C₈₉×#1846	2
#13542	89

subgroup

index

C₈₉×#1846

Subgroups — #36474572

Conjugacy classes of subgroups

Complements

Maximal subgroups