Group #36459240 — subgroups

Subgroups — #36459240

Order 22784.

294 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 85 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	#26021072	C₂	1	1	yes	maximal, characteristic
2	#26020393	C₂	1	1	yes	maximal, characteristic
3	#26018923	C₂	1	2	yes	maximal
4	#26018111	C₂	1	2	yes	maximal
5	C₈₉×#1281	C₂	1	1	yes	maximal, characteristic
6	C₂×#15668012	C₂²	1	1	yes	characteristic
7	#19151622	C₂²	1	2	yes	—
8	C₂×#15667937	—	2	2	no	—
9	#19151464	C₂²	1	1	yes	characteristic
10	#19151363	—	2	2	no	—
11	#19151343	—	2	4	no	—
12	#431×C₈₉	C₂²	1	2	yes	—
13	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₇₈	—	2	1	no	—
15	C₂×#13083558	—	2	2	no	—
16	#15668012	—	2	2	no	—
17	#15668012	—	2	1	no	—
18	#15668012	D₄	1	2	yes	—
19	C₂×C₄⋊Dic₈₉	—	2	1	no	—
20	C₂×C₄⋊Dic₈₉	—	2	2	no	—
21	C₄⋊#10623494	—	2	2	no	—
22	C₄⋊#10623494	—	2	2	no	—
23	#15667937	—	4	4	no	—
24	#15667919	—	2	2	no	—
25	C₂²×C₈×C₈₉	D₄	1	2	yes	—
26	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
27	C₄⋊C₈×C₈₉	—	2	2	no	—
28	C₂²⋊C₈×C₈₉	—	2	2	no	—
29	#97×C₈₉	—	2	2	no	—
30	C₄×C₈×C₈₉	—	2	2	no	—
31	C₂×Dic₁₇₈	—	2	2	no	—
32	C₂×Dic₁₇₈	—	4	2	no	—
33	C₂×Dic₁₇₈	—	4	2	no	—
34	C₂×Dic₁₇₈	—	4	1	no	—
35	C₂²×Dic₈₉	—	4	1	no	—
36	C₂²×Dic₈₉	—	4	1	no	—
37	#13083558	—	4	2	no	—
38	#13083558	—	4	2	no	—
39	C₂×#10623494	—	4	2	no	—
40	C₄⋊Dic₈₉	—	2	4	no	—
41	C₄⋊Dic₈₉	—	4	2	no	—
42	C₄⋊Dic₈₉	—	4	2	no	—
43	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₈₉	#41	1	2	yes	—
45	C₂×C₈×C₈₉	—	2	2	no	—
46	C₂×C₈×C₈₉	—	2	2	no	—
47	C₂×C₈×C₈₉	—	2	2	no	—
48	C₂×C₈×C₈₉	—	4	2	no	—
49	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
50	C₄²×C₈₉	#41	1	1	yes	characteristic
51	C₄²×C₈₉	#41	1	2	yes	—
52	C₂³×C₈₉	#124	1	1	yes	characteristic
53	C₂×C₄×C₈₉	—	4	2	no	—
54	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
55	C₂×C₄×C₈₉	#123	1	2	yes	—
56	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
57	C₂×C₄×C₈₉	#126	1	2	yes	—
58	C₂×C₄×C₈₉	#135	1	2	yes	—
59	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
60	C₂×Dic₈₉	—	4	2	no	—
61	C₂×Dic₈₉	—	8	1	no	—
62	C₂×Dic₈₉	—	8	1	no	—
63	C₂×Dic₈₉	—	8	2	no	—
64	C₂×Dic₈₉	—	8	1	no	—
65	Dic₁₇₈	—	4	4	no	—
66	Dic₁₇₈	—	8	2	no	—
67	C₈×C₈₉	—	4	2	no	—
68	C₈×C₈₉	—	8	2	no	—
69	#10623494	—	8	2	no	—
70	C₂²×C₈₉	—	2	2	no	—
71	C₂²×C₈₉	#397	1	1	yes	characteristic
72	C₂²×C₈₉	#488	1	1	yes	characteristic
73	C₂²×C₈₉	#490	1	1	yes	characteristic
74	C₄×C₈₉	—	2	2	no	—
75	C₄×C₈₉	—	2	2	no	—
76	C₄×C₈₉	#463	1	2	yes	—
77	C₄×C₈₉	#482	1	2	yes	—
78	Dic₈₉	—	8	2	no	—
79	Dic₈₉	—	16	1	no	—
80	#10635	—	89	1	no	maximal, Sylow
81	C₂×C₈₉	—	2	1	no	—
82	C₂×C₈₉	—	2	1	no	—
83	C₂×C₈₉	#2073	1	1	yes	characteristic
84	C₂×C₈₉	#2082	1	2	yes	—
85	#1960	—	89	1	no	—
86	#1838	—	89	1	no	—
87	#1546	—	89	2	no	—
88	#1286	—	89	2	no	—
89	#1281	D₈₉	1	1	yes	characteristic
90	C₈₉	#10635	1	1	yes	characteristic, complemented, Sylow
91	C₂×C₄⋊Q₈	—	89	1	no	—
92	#431	C₂×D₈₉	1	2	yes	—
93	#423	—	89	2	no	—
94	C₂×#103	—	178	2	no	—
95	C₂×#97	C₂×D₈₉	1	1	yes	characteristic
96	#382	—	89	1	no	—
97	#341	—	178	2	no	—
98	#332	—	178	4	no	—
99	C₂²×Q₈	—	178	1	no	—
100	C₂×M₁₆	—	178	2	no	—
101	C₂²×C₈	D₃₅₆	1	2	yes	—
102	C₄⋊Q₈	—	89	2	no	—
103	C₄⋊Q₈	—	178	2	no	—
104	C₄⋊Q₈	—	178	1	no	—
105	C₂×C₄⋊C₄	—	178	1	no	—
106	C₂×C₄⋊C₄	—	178	2	no	—
107	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
108	C₄⋊C₈	—	2	2	no	—
109	C₄⋊C₈	—	178	2	no	—
110	C₄⋊C₈	—	178	2	no	—
111	#103	—	356	4	no	—
112	C₂²⋊C₈	—	2	2	no	—
113	#97	—	2	2	no	—
114	C₄×C₈	—	2	2	no	—
115	#95	—	178	2	no	—
116	C₂×Q₈	—	356	1	no	—
117	C₂×Q₈	—	356	2	no	—
118	C₂×Q₈	—	356	2	no	—
119	C₂×Q₈	—	178	2	no	—
120	C₂²×C₄	—	356	1	no	—
121	C₂²×C₄	—	356	1	no	—
122	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
123	C₂²×C₄	#13083574	1	2	yes	—
124	M₁₆	—	356	2	no	—
125	M₁₆	—	356	2	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	4	2	no	—
130	C₂×C₈	—	356	2	no	—
131	C₄⋊C₄	—	356	2	no	—
132	C₄⋊C₄	—	356	2	no	—
133	C₄⋊C₄	—	178	4	no	—
134	C₄²	D₄×D₈₉	1	1	yes	characteristic
135	C₄²	#13083573	1	2	yes	—
136	C₄²	#13083575	1	1	yes	characteristic
137	C₂³	#15668000	1	1	yes	characteristic
138	Q₈	—	356	4	no	—
139	Q₈	—	712	2	no	—
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	356	2	no	—
142	C₂×C₄	—	712	2	no	—
143	C₂×C₄	—	712	1	no	—
144	C₂×C₄	—	712	1	no	—
145	C₂×C₄	—	712	1	no	—
146	C₂×C₄	#15667987	1	1	yes	characteristic
147	C₂×C₄	#15667994	1	2	yes	—
148	C₂×C₄	#15667996	1	1	yes	characteristic
149	C₂×C₄	#15668007	1	2	yes	—
150	C₂×C₄	#15668034	1	2	yes	—
151	C₂×C₄	#15668040	1	2	yes	—
152	C₈	—	4	2	no	—
153	C₈	—	8	2	no	—
154	C₈	—	712	2	no	—
155	C₂²	—	2	2	no	—
156	C₂²	#19151525	1	1	yes	characteristic
157	C₂²	#19151953	1	1	yes	characteristic
158	C₂²	#19151963	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	2	2	no	—
161	C₄	—	1424	1	no	—
162	C₄	—	712	2	no	—
163	C₄	#19151814	1	2	yes	—
164	C₄	#19151923	1	2	yes	—
165	C₂	—	2	1	no	—
166	C₂	—	2	1	no	—
167	C₂	#26021667	1	1	yes	characteristic
168	C₂	#26021718	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26021072

C₂

yes

maximal, characteristic

#26020393

C₂

yes

maximal, characteristic

#26018923

C₂

yes

maximal

#26018111

C₂

yes

maximal

C₈₉×#1281

C₂

yes

maximal, characteristic

C₂×#15668012

C₂²

yes

characteristic

#19151622

C₂²

yes

—

C₂×#15667937

—

#19151464

C₂²

yes

characteristic

#19151363

—

#19151343

—

#431×C₈₉

C₂²

yes

—

C₂×#97×C₈₉

C₂²

yes

characteristic

C₂²×Dic₁₇₈

—

—

—

—

D₄

yes

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

—

—

—

—

C₂²×C₈×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄⋊C₈×C₈₉

—

C₂²⋊C₈×C₈₉

—

#97×C₈₉

—

C₄×C₈×C₈₉

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

#13083558

—

#13083558

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

#41

yes

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₄²×C₈₉

C₂×D₄

yes

C₄²×C₈₉

yes

C₄²×C₈₉

yes

—

C₂³×C₈₉

#124

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#121

yes

characteristic

C₂×C₄×C₈₉

#123

yes

—

C₂×C₄×C₈₉

#123

yes

characteristic

C₂×C₄×C₈₉

#126

yes

—

C₂×C₄×C₈₉

#135

yes

—

C₂×C₄×C₈₉

AGL(1,ℤ/8)

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#463

yes

—

C₄×C₈₉

#482

yes

—

Dic₈₉

—

Dic₈₉

—

#10635

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#2073

yes

characteristic

C₂×C₈₉

#2082

yes

—

—

—

—

—

D₈₉

yes

characteristic

C₈₉

#10635

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

#431

C₂×D₈₉

yes

—

#423

—

C₂×#103

—

178

—

C₂×#97

C₂×D₈₉

yes

characteristic

#382

—

#341

—

178

—

#332

—

178

—

C₂²×Q₈

—

178

—

100

C₂×M₁₆

—

178

—

101

C₂²×C₈

D₃₅₆

yes

—

102

C₄⋊Q₈

—

103

C₄⋊Q₈

—

178

—

104

C₄⋊Q₈

—

178

—

105

C₂×C₄⋊C₄

—

178

—

106

C₂×C₄⋊C₄

—

178

—

107

C₂×C₄²

C₂²×D₈₉

yes

characteristic

108

C₄⋊C₈

—

109

C₄⋊C₈

—

178

—

110

C₄⋊C₈

—

178

—

111

#103

—

356

—

112

C₂²⋊C₈

—

113

#97

—

114

C₄×C₈

—

115

#95

—

178

—

116

C₂×Q₈

—

356

—

117

C₂×Q₈

—

356

—

118

C₂×Q₈

—

356

—

119

C₂×Q₈

—

178

—

120

C₂²×C₄

—

356

—

121

C₂²×C₄

—

356

—

122

C₂²×C₄

C₂×D₃₅₆

yes

characteristic

123

C₂²×C₄

#13083574

yes

—

124

M₁₆

—

356

—

125

M₁₆

—

356

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

356

—

131

C₄⋊C₄

—

356

—

132

C₄⋊C₄

—

356

—

133

C₄⋊C₄

—

178

—

134

C₄²

D₄×D₈₉

yes

characteristic

135

C₄²

#13083573

yes

—

136

C₄²

#13083575

yes

characteristic

137

C₂³

#15668000

yes

characteristic

138

Q₈

—

356

—

139

Q₈

—

712

—

140

C₂×C₄

—

141

C₂×C₄

—

356

—

142

C₂×C₄

—

712

—

143

C₂×C₄

—

712

—

144

C₂×C₄

—

712

—

145

C₂×C₄

—

712

—

146

C₂×C₄

#15667987

yes

characteristic

147

C₂×C₄

#15667994

yes

—

148

C₂×C₄

#15667996

yes

characteristic

149

C₂×C₄

#15668007

yes

—

150

C₂×C₄

#15668034

yes

—

151

C₂×C₄

#15668040

yes

—

152

C₈

—

153

C₈

—

154

C₈

—

712

—

155

C₂²

—

156

C₂²

#19151525

yes

characteristic

157

C₂²

#19151953

yes

characteristic

158

C₂²

#19151963

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

1424

—

162

C₄

—

712

—

163

C₄

#19151814

yes

—

164

C₄

#19151923

yes

—

165

C₂

—

166

C₂

—

167

C₂

#26021667

yes

characteristic

168

C₂

#26021718

yes

—

	subgroup	G/H
center	C₂²	#19151525
commutator, cosocle	C₂×C₄²×C₈₉	C₂³
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1281	C₂
socle	C₂²×C₈₉	#397
prime radical	C₂³×C₈₉	#124
hypercenter	#1281	D₈₉
hypocenter	C₈₉	#10635
LCS	G ▷ C₂×C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1281

subgroup

G/H

center

C₂²

#19151525

commutator, cosocle

C₂×C₄²×C₈₉

C₂³

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1281

C₂

C₂²×C₈₉

C₂³×C₈₉

D₈₉

C₈₉

G ▷ C₂×C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1281

N	H ≅ G/N	type
90. C₈₉	#10635	semidirect

H ≅ G/N

type

90. C₈₉

#10635

semidirect

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

subgroup	index
#26021072	2
#26020393	2
#26018923	2
#26018111	2
C₈₉×#1281	2
#10635	89

subgroup

index

C₈₉×#1281

Subgroups — #36459240

Conjugacy classes of subgroups

Complements

Maximal subgroups