Group #36457542 — subgroups

Subgroups — #36457542

Order 22784.

364 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 89 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	#26028639	C₂	1	1	yes	maximal, characteristic
2	#26021229	C₂	1	1	yes	maximal, characteristic
3	#26018817	C₂	1	2	yes	maximal
4	#26018111	C₂	1	2	yes	maximal
5	C₈₉×#1276	C₂	1	1	yes	maximal, characteristic
6	#19152155	C₂²	1	1	yes	characteristic
7	#19151622	C₂²	1	1	yes	characteristic
8	#19151496	—	2	2	no	—
9	#19151482	—	2	4	no	—
10	#19151464	C₂²	1	2	yes	—
11	#19151358	—	2	2	no	—
12	#19151336	—	2	4	no	—
13	#431×C₈₉	C₂²	1	2	yes	—
14	C₄³×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×#13083558	—	2	1	no	—
16	#15668012	Q₈	1	2	yes	—
17	#15668004	—	2	1	no	—
18	#15667991	—	2	2	no	—
19	C₄×Dic₁₇₈	—	2	2	no	—
20	#15667971	—	2	1	no	—
21	C₂×C₄⋊Dic₈₉	—	2	2	no	—
22	C₂×#13083553	—	2	4	no	—
23	C₄⋊#10623494	—	2	2	no	—
24	#15667919	—	2	4	no	—
25	#15667919	—	2	2	no	—
26	#15667919	D₄	1	4	yes	—
27	C₂×C₄²×C₈₉	—	2	2	no	—
28	C₂×C₄²×C₈₉	D₄	1	2	yes	—
29	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₈₉	—	2	2	no	—
31	C₂²⋊C₈×C₈₉	—	2	2	no	—
32	C₄×C₈×C₈₉	—	2	2	no	—
33	C₂×Dic₁₇₈	—	2	2	no	—
34	C₂²×Dic₈₉	—	2	4	no	—
35	#13083558	—	4	2	no	—
36	C₂×#10623494	—	4	1	no	—
37	C₄⋊Dic₈₉	—	2	2	no	—
38	C₄⋊Dic₈₉	—	2	2	no	—
39	C₄⋊Dic₈₉	—	4	2	no	—
40	C₄⋊Dic₈₉	—	4	2	no	—
41	#13083553	—	4	8	no	—
42	#13083553	—	4	2	no	—
43	C₂²⋊Dic₈₉	—	4	1	no	—
44	C₄×Dic₈₉	—	4	1	no	—
45	C₂²×C₄×C₈₉	—	2	2	no	—
46	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
47	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₈₉	—	4	2	no	—
49	C₄²×C₈₉	—	2	2	no	—
50	C₄²×C₈₉	—	2	2	no	—
51	C₄²×C₈₉	—	4	4	no	—
52	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
53	C₄²×C₈₉	#41	1	1	yes	characteristic
54	C₄²×C₈₉	#41	1	2	yes	—
55	C₂³×C₈₉	#120	1	1	yes	characteristic
56	C₂×C₄×C₈₉	—	2	4	no	—
57	C₂×C₄×C₈₉	—	4	2	no	—
58	C₂×C₄×C₈₉	—	4	2	no	—
59	C₂×C₄×C₈₉	—	4	2	no	—
60	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
61	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
62	C₂×C₄×C₈₉	#124	1	2	yes	—
63	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
64	C₂×C₄×C₈₉	#135	1	2	yes	—
65	C₂×C₄×C₈₉	#137	1	2	yes	—
66	C₂×Dic₈₉	—	4	8	no	—
67	C₂×Dic₈₉	—	4	2	no	—
68	C₂×Dic₈₉	—	4	4	no	—
69	Dic₁₇₈	—	8	2	no	—
70	C₈×C₈₉	—	8	2	no	—
71	#10623494	—	8	1	no	—
72	C₂²×C₈₉	—	2	2	no	—
73	C₂²×C₈₉	#393	1	1	yes	characteristic
74	C₂²×C₈₉	#452	1	2	yes	—
75	C₄×C₈₉	—	2	2	no	—
76	C₄×C₈₉	—	2	2	no	—
77	C₄×C₈₉	—	4	4	no	—
78	C₄×C₈₉	#479	1	2	yes	—
79	C₄×C₈₉	#488	1	1	yes	characteristic
80	C₄×C₈₉	#491	1	1	yes	characteristic
81	Dic₈₉	—	8	4	no	—
82	Dic₈₉	—	8	2	no	—
83	#10367	—	89	1	no	maximal, Sylow
84	C₂×C₈₉	—	2	2	no	—
85	C₂×C₈₉	#2026	1	1	yes	characteristic
86	C₂×C₈₉	#2031	1	2	yes	—
87	#2864	—	89	1	no	—
88	#1994	—	89	1	no	—
89	#1518	—	89	2	no	—
90	#1286	—	89	2	no	—
91	#1276	D₈₉	1	1	yes	characteristic
92	C₈₉	#10367	1	1	yes	characteristic, complemented, Sylow
93	#533	—	89	1	no	—
94	#431	C₂×D₈₉	1	2	yes	—
95	#423	—	89	1	no	—
96	#389	—	178	2	no	—
97	#387	—	178	4	no	—
98	#382	—	89	2	no	—
99	C₄³	C₂×D₈₉	1	1	yes	characteristic
100	#339	—	178	2	no	—
101	#328	—	178	4	no	—
102	C₂×M₁₆	—	178	1	no	—
103	C₄⋊Q₈	—	89	2	no	—
104	#125	—	178	1	no	—
105	C₂²⋊Q₈	—	178	2	no	—
106	C₄×Q₈	—	178	2	no	—
107	#117	—	178	1	no	—
108	C₂×C₄⋊C₄	—	178	4	no	—
109	C₂×C₄⋊C₄	—	178	2	no	—
110	C₂×C₄²	—	2	2	no	—
111	C₂×C₄²	D₃₅₆	1	2	yes	—
112	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
113	C₄⋊C₈	—	2	2	no	—
114	C₄⋊C₈	—	178	2	no	—
115	C₂²⋊C₈	—	2	2	no	—
116	C₄×C₈	—	2	2	no	—
117	#95	—	89	4	no	—
118	#95	—	178	4	no	—
119	#95	—	178	2	no	—
120	C₂×Q₈	—	178	2	no	—
121	C₂²×C₄	—	2	2	no	—
122	C₂²×C₄	—	178	4	no	—
123	C₂²×C₄	D₄×D₈₉	1	2	yes	—
124	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
125	M₁₆	—	356	2	no	—
126	C₂×C₈	—	4	2	no	—
127	C₂×C₈	—	356	1	no	—
128	C₄⋊C₄	—	356	8	no	—
129	C₄⋊C₄	—	356	2	no	—
130	C₄⋊C₄	—	356	2	no	—
131	C₄⋊C₄	—	356	2	no	—
132	C₄⋊C₄	—	178	2	no	—
133	C₄⋊C₄	—	178	2	no	—
134	C₂²⋊C₄	—	356	1	no	—
135	C₄²	—	2	2	no	—
136	C₄²	—	2	2	no	—
137	C₄²	—	4	4	no	—
138	C₄²	—	356	1	no	—
139	C₄²	Q₈×D₈₉	1	1	yes	characteristic
140	C₄²	#13083573	1	1	yes	characteristic
141	C₄²	#13083574	1	2	yes	—
142	C₂³	#15667982	1	1	yes	characteristic
143	Q₈	—	712	2	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	—	356	8	no	—
149	C₂×C₄	—	356	4	no	—
150	C₂×C₄	—	356	2	no	—
151	C₂×C₄	#15667990	1	2	yes	—
152	C₂×C₄	#15667992	1	1	yes	characteristic
153	C₂×C₄	#15667998	1	2	yes	—
154	C₂×C₄	#15668000	1	1	yes	characteristic
155	C₂×C₄	#15668034	1	2	yes	—
156	C₂×C₄	#15668045	1	2	yes	—
157	C₈	—	8	2	no	—
158	C₈	—	712	1	no	—
159	C₂²	—	2	2	no	—
160	C₂²	#19151511	1	1	yes	characteristic
161	C₂²	#19151762	1	2	yes	—
162	C₄	—	2	2	no	—
163	C₄	—	2	2	no	—
164	C₄	—	4	4	no	—
165	C₄	—	712	4	no	—
166	C₄	—	712	2	no	—
167	C₄	#19151906	1	2	yes	—
168	C₄	#19151953	1	1	yes	characteristic
169	C₄	#19151967	1	1	yes	characteristic
170	C₂	—	2	2	no	—
171	C₂	#26021404	1	1	yes	characteristic
172	C₂	#26021425	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26028639

C₂

yes

maximal, characteristic

#26021229

C₂

yes

maximal, characteristic

#26018817

C₂

yes

maximal

#26018111

C₂

yes

maximal

C₈₉×#1276

C₂

yes

maximal, characteristic

#19152155

C₂²

yes

characteristic

#19151622

C₂²

yes

—

—

C₂²

yes

—

#19151358

—

#19151336

—

#431×C₈₉

C₂²

yes

—

C₄³×C₈₉

C₂²

yes

characteristic

C₂×#13083558

—

#15668012

Q₈

yes

—

#15668004

—

#15667991

—

C₄×Dic₁₇₈

—

#15667971

—

C₂×C₄⋊Dic₈₉

—

—

—

—

—

D₄

yes

—

C₂×C₄²×C₈₉

—

C₂×C₄²×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄⋊C₈×C₈₉

—

C₂²⋊C₈×C₈₉

—

C₄×C₈×C₈₉

—

C₂×Dic₁₇₈

—

C₂²×Dic₈₉

—

#13083558

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083553

—

#13083553

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

C₂×Q₈

yes

C₄²×C₈₉

yes

C₄²×C₈₉

yes

—

C₂³×C₈₉

#120

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₈₉

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₈₉

#124

yes

—

C₂×C₄×C₈₉

#124

yes

characteristic

C₂×C₄×C₈₉

#135

yes

—

C₂×C₄×C₈₉

#137

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

#393

yes

characteristic

C₂²×C₈₉

#452

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#479

yes

—

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

Dic₈₉

—

#10367

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

#2026

yes

characteristic

C₂×C₈₉

#2031

yes

—

—

—

—

—

D₈₉

yes

characteristic

C₈₉

#10367

yes

characteristic, complemented, Sylow

#533

—

#431

C₂×D₈₉

yes

—

#423

—

#389

—

178

—

#387

—

178

—

#382

—

C₄³

C₂×D₈₉

yes

characteristic

100

#339

—

178

—

101

#328

—

178

—

102

C₂×M₁₆

—

178

—

103

C₄⋊Q₈

—

104

#125

—

178

—

105

C₂²⋊Q₈

—

178

—

106

C₄×Q₈

—

178

—

107

#117

—

178

—

108

C₂×C₄⋊C₄

—

178

—

109

C₂×C₄⋊C₄

—

178

—

110

C₂×C₄²

—

111

C₂×C₄²

D₃₅₆

yes

—

112

C₂×C₄²

C₂²×D₈₉

yes

characteristic

113

C₄⋊C₈

—

114

C₄⋊C₈

—

178

—

115

C₂²⋊C₈

—

116

C₄×C₈

—

117

#95

—

118

#95

—

178

—

119

#95

—

178

—

120

C₂×Q₈

—

178

—

121

C₂²×C₄

—

122

C₂²×C₄

—

178

—

123

C₂²×C₄

D₄×D₈₉

yes

—

124

C₂²×C₄

C₂×D₃₅₆

yes

characteristic

125

M₁₆

—

356

—

126

C₂×C₈

—

127

C₂×C₈

—

356

—

128

C₄⋊C₄

—

356

—

129

C₄⋊C₄

—

356

—

130

C₄⋊C₄

—

356

—

131

C₄⋊C₄

—

356

—

132

C₄⋊C₄

—

178

—

133

C₄⋊C₄

—

178

—

134

C₂²⋊C₄

—

356

—

135

C₄²

—

136

C₄²

—

137

C₄²

—

138

C₄²

—

356

—

139

C₄²

Q₈×D₈₉

yes

characteristic

140

C₄²

#13083573

yes

characteristic

141

C₄²

#13083574

yes

—

142

C₂³

#15667982

yes

characteristic

143

Q₈

—

712

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

356

—

149

C₂×C₄

—

356

—

150

C₂×C₄

—

356

—

151

C₂×C₄

#15667990

yes

—

152

C₂×C₄

#15667992

yes

characteristic

153

C₂×C₄

#15667998

yes

—

154

C₂×C₄

#15668000

yes

characteristic

155

C₂×C₄

#15668034

yes

—

156

C₂×C₄

#15668045

yes

—

157

C₈

—

158

C₈

—

712

—

159

C₂²

—

160

C₂²

#19151511

yes

characteristic

161

C₂²

#19151762

yes

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

—

712

—

166

C₄

—

712

—

167

C₄

#19151906

yes

—

168

C₄

#19151953

yes

characteristic

169

C₄

#19151967

yes

characteristic

170

C₂

—

171

C₂

#26021404

yes

characteristic

172

C₂

#26021425

yes

—

	subgroup	G/H
center	C₂²	#19151511
commutator, cosocle	C₂×C₄²×C₈₉	C₂³
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1276	C₂
socle	C₂²×C₈₉	#393
prime radical	C₂³×C₈₉	#120
hypercenter	#1276	D₈₉
hypocenter	C₈₉	#10367
LCS	G ▷ C₂×C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1276

subgroup

G/H

center

C₂²

#19151511

commutator, cosocle

C₂×C₄²×C₈₉

C₂³

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1276

C₂

C₂²×C₈₉

C₂³×C₈₉

D₈₉

C₈₉

G ▷ C₂×C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1276

N	H ≅ G/N	type
92. C₈₉	#10367	semidirect

H ≅ G/N

type

92. C₈₉

#10367

semidirect

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

subgroup	index
#26028639	2
#26021229	2
#26018817	2
#26018111	2
C₈₉×#1276	2
#10367	89

subgroup

index

C₈₉×#1276

Subgroups — #36457542

Conjugacy classes of subgroups

Complements

Maximal subgroups