Group #36453912 — subgroups

Subgroups — #36453912

Order 22784.

390 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 119 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	C₂×#19151996	C₂	1	1	yes	maximal, characteristic
2	#26018938	C₂	1	2	yes	maximal
3	#26018934	C₂	1	2	yes	maximal
4	#26018544	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1837	C₂	1	1	yes	maximal, characteristic
6	C₂×#15668012	C₂²	1	2	yes	—
7	#19151996	—	2	2	no	—
8	#19151996	C₄	1	4	yes	—
9	#19151651	C₂²	1	2	yes	—
10	C₂×#15667944	—	2	2	no	—
11	#19151346	—	2	4	no	—
12	#19151343	—	2	4	no	—
13	#19151320	—	2	2	no	—
14	#423×C₈₉	C₂²	1	1	yes	characteristic
15	#423×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	#341×C₈₉	—	2	2	no	—
18	C₂²×Dic₁₇₈	—	2	2	no	—
19	#15668012	—	2	2	no	—
20	#15668012	—	2	2	no	—
21	#15668012	—	2	2	no	—
22	#15668012	C₂×C₄	1	4	yes	—
23	C₂×C₄⋊Dic₈₉	—	2	2	no	—
24	C₂×C₄⋊Dic₈₉	—	2	2	no	—
25	#15667944	—	2	8	no	—
26	C₄⋊#10623494	D₄	1	4	yes	—
27	C₂²⋊#10623494	—	2	2	no	—
28	C₄×#10623494	—	2	2	no	—
29	C₂×M₁₆×C₈₉	—	2	1	no	—
30	C₂×M₁₆×C₈₉	—	2	1	no	—
31	C₂²×C₈×C₈₉	D₄	1	2	yes	—
32	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₈₉	—	2	2	no	—
34	C₄⋊C₈×C₈₉	—	2	2	no	—
35	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₈₉	D₄	1	2	yes	—
37	C₂×Dic₁₇₈	—	2	4	no	—
38	C₂×Dic₁₇₈	—	4	2	no	—
39	C₂×Dic₁₇₈	—	8	4	no	—
40	C₂²×Dic₈₉	—	4	2	no	—
41	C₂×#10623494	—	2	4	no	—
42	C₄⋊Dic₈₉	—	2	4	no	—
43	C₄⋊Dic₈₉	—	2	4	no	—
44	C₄⋊Dic₈₉	—	4	2	no	—
45	C₄⋊Dic₈₉	—	4	2	no	—
46	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₈₉	#41	1	2	yes	—
48	M₁₆×C₈₉	—	4	2	no	—
49	M₁₆×C₈₉	—	4	2	no	—
50	C₂×C₈×C₈₉	—	2	2	no	—
51	C₂×C₈×C₈₉	—	2	2	no	—
52	C₂×C₈×C₈₉	—	4	1	no	—
53	C₂×C₈×C₈₉	—	4	1	no	—
54	C₂×C₈×C₈₉	C₂²⋊C₄	1	4	yes	—
55	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₈₉	C₂×D₄	1	2	yes	—
58	C₂³×C₈₉	#124	1	1	yes	characteristic
59	C₂×C₄×C₈₉	—	2	4	no	—
60	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
62	C₂×C₄×C₈₉	#121	1	2	yes	—
63	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
64	C₂×C₄×C₈₉	#137	1	2	yes	—
65	C₂×C₄×C₈₉	#137	1	2	yes	—
66	C₂×Dic₈₉	—	4	4	no	—
67	C₂×Dic₈₉	—	8	4	no	—
68	Dic₁₇₈	—	8	4	no	—
69	Dic₁₇₈	—	8	4	no	—
70	C₈×C₈₉	—	2	4	no	—
71	C₈×C₈₉	—	8	1	no	—
72	C₈×C₈₉	—	8	1	no	—
73	#10623494	—	4	4	no	—
74	C₂²×C₈₉	—	2	2	no	—
75	C₂²×C₈₉	#390	1	1	yes	characteristic
76	C₂²×C₈₉	#491	1	2	yes	—
77	C₄×C₈₉	—	2	2	no	—
78	C₄×C₈₉	#419	1	2	yes	—
79	C₄×C₈₉	#471	1	4	yes	—
80	C₄×C₈₉	#497	1	1	yes	characteristic
81	C₄×C₈₉	#497	1	1	yes	characteristic
82	Dic₈₉	—	8	4	no	—
83	#9776	—	89	1	no	maximal, Sylow
84	C₂×C₈₉	—	2	2	no	—
85	C₂×C₈₉	#1960	1	1	yes	characteristic
86	C₂×C₈₉	#1977	1	2	yes	—
87	C₂×#498	—	89	1	no	—
88	#1837	D₈₉	1	1	yes	characteristic
89	#1549	—	89	2	no	—
90	#1548	—	89	2	no	—
91	#1452	—	89	1	no	—
92	C₈₉	#9776	1	1	yes	characteristic, complemented, Sylow
93	C₂×C₄⋊Q₈	—	89	2	no	—
94	#498	—	89	4	no	—
95	#498	—	178	2	no	—
96	#431	—	89	2	no	—
97	C₂×#106	—	178	2	no	—
98	#423	C₂×D₈₉	1	1	yes	characteristic
99	#423	C₂×D₈₉	1	1	yes	characteristic
100	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
101	#341	—	2	2	no	—
102	#333	—	178	4	no	—
103	#332	—	178	4	no	—
104	#322	—	178	2	no	—
105	C₂²×Q₈	—	178	2	no	—
106	C₂×M₁₆	—	2	1	no	—
107	C₂×M₁₆	—	2	1	no	—
108	C₂²×C₈	C₂²⋊D₈₉	1	2	yes	—
109	C₄⋊Q₈	—	89	4	no	—
110	C₄⋊Q₈	—	178	2	no	—
111	C₄⋊Q₈	—	178	2	no	—
112	C₄⋊Q₈	—	178	2	no	—
113	C₂×C₄⋊C₄	—	178	2	no	—
114	C₂×C₄⋊C₄	—	178	2	no	—
115	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
116	#106	—	178	8	no	—
117	C₄⋊C₈	—	2	2	no	—
118	C₄⋊C₈	—	2	2	no	—
119	C₄⋊C₈	—	89	4	no	—
120	C₂²⋊C₈	—	178	2	no	—
121	C₄×C₈	—	178	2	no	—
122	C₄×C₈	C₄×D₈₉	1	2	yes	—
123	C₄×C₈	D₃₅₆	1	2	yes	—
124	C₂×Q₈	—	356	2	no	—
125	C₂×Q₈	—	178	4	no	—
126	C₂×Q₈	—	712	4	no	—
127	C₂²×C₄	—	356	2	no	—
128	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
129	C₂²×C₄	#13083575	1	2	yes	—
130	M₁₆	—	4	2	no	—
131	M₁₆	—	4	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	4	1	no	—
135	C₂×C₈	—	4	1	no	—
136	C₂×C₈	—	178	4	no	—
137	C₂×C₈	#13083551	1	4	yes	—
138	C₄⋊C₄	—	356	2	no	—
139	C₄⋊C₄	—	356	2	no	—
140	C₄⋊C₄	—	178	4	no	—
141	C₄⋊C₄	—	178	4	no	—
142	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
143	C₄²	D₄×D₈₉	1	2	yes	—
144	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
145	C₂³	#15668001	1	1	yes	characteristic
146	Q₈	—	712	4	no	—
147	Q₈	—	712	4	no	—
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	356	4	no	—
150	C₂×C₄	—	712	4	no	—
151	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
152	C₂×C₄	#15667974	1	2	yes	—
153	C₂×C₄	#15667987	1	2	yes	—
154	C₂×C₄	#15668008	1	1	yes	characteristic
155	C₂×C₄	#15668042	1	2	yes	—
156	C₂×C₄	#15668042	1	2	yes	—
157	C₈	—	2	4	no	—
158	C₈	—	8	1	no	—
159	C₈	—	8	1	no	—
160	C₈	—	356	4	no	—
161	C₂²	—	2	2	no	—
162	C₂²	#19151499	1	1	yes	characteristic
163	C₂²	#19151968	1	2	yes	—
164	C₄	—	2	2	no	—
165	C₄	—	712	4	no	—
166	C₄	#19151606	1	2	yes	—
167	C₄	#19151862	1	4	yes	—
168	C₄	#19151993	1	1	yes	characteristic
169	C₄	#19151993	1	1	yes	characteristic
170	C₂	—	2	2	no	—
171	C₂	#26021071	1	1	yes	characteristic
172	C₂	#26021153	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19151996

C₂

yes

maximal, characteristic

#26018938

C₂

yes

maximal

#26018934

C₂

yes

maximal

#26018544

C₂

yes

maximal, characteristic

C₈₉×#1837

C₂

yes

maximal, characteristic

C₂×#15668012

C₂²

yes

—

#19151996

—

#19151996

C₄

yes

—

#19151651

C₂²

yes

—

—

—

—

—

#423×C₈₉

C₂²

yes

characteristic

#423×C₈₉

C₂²

yes

characteristic

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

#341×C₈₉

—

C₂²×Dic₁₇₈

—

—

—

—

C₂×C₄

yes

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

#15667944

—

C₄⋊#10623494

D₄

yes

—

C₂²⋊#10623494

—

C₄×#10623494

—

C₂×M₁₆×C₈₉

—

C₂×M₁₆×C₈₉

—

C₂²×C₈×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄⋊C₈×C₈₉

—

C₄⋊C₈×C₈₉

—

C₄×C₈×C₈₉

C₂×C₄

yes

—

C₄×C₈×C₈₉

D₄

yes

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂²×Dic₈₉

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

#41

yes

—

M₁₆×C₈₉

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂²⋊C₄

yes

—

C₄²×C₈₉

C₂²×C₄

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

—

C₂³×C₈₉

#124

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₄×D₄

yes

—

C₂×C₄×C₈₉

#121

yes

—

C₂×C₄×C₈₉

#127

yes

characteristic

C₂×C₄×C₈₉

#137

yes

—

C₂×C₄×C₈₉

#137

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

#390

yes

characteristic

C₂²×C₈₉

#491

yes

—

C₄×C₈₉

—

C₄×C₈₉

#419

yes

—

C₄×C₈₉

#471

yes

—

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

#9776

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

#1960

yes

characteristic

C₂×C₈₉

#1977

yes

—

C₂×#498

—

#1837

D₈₉

yes

—

—

—

C₈₉

#9776

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

#498

—

#498

—

178

—

#431

—

C₂×#106

—

178

—

#423

C₂×D₈₉

yes

characteristic

#423

C₂×D₈₉

yes

characteristic

100

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

101

#341

—

102

#333

—

178

—

103

#332

—

178

—

104

#322

—

178

—

105

C₂²×Q₈

—

178

—

106

C₂×M₁₆

—

107

C₂×M₁₆

—

108

C₂²×C₈

C₂²⋊D₈₉

yes

—

109

C₄⋊Q₈

—

110

C₄⋊Q₈

—

178

—

111

C₄⋊Q₈

—

178

—

112

C₄⋊Q₈

—

178

—

113

C₂×C₄⋊C₄

—

178

—

114

C₂×C₄⋊C₄

—

178

—

115

C₂×C₄²

C₂²×D₈₉

yes

characteristic

116

#106

—

178

—

117

C₄⋊C₈

—

118

C₄⋊C₈

—

119

C₄⋊C₈

—

120

C₂²⋊C₈

—

178

—

121

C₄×C₈

—

178

—

122

C₄×C₈

C₄×D₈₉

yes

—

123

C₄×C₈

D₃₅₆

yes

—

124

C₂×Q₈

—

356

—

125

C₂×Q₈

—

178

—

126

C₂×Q₈

—

712

—

127

C₂²×C₄

—

356

—

128

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

129

C₂²×C₄

#13083575

yes

—

130

M₁₆

—

131

M₁₆

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

178

—

137

C₂×C₈

#13083551

yes

—

138

C₄⋊C₄

—

356

—

139

C₄⋊C₄

—

356

—

140

C₄⋊C₄

—

178

—

141

C₄⋊C₄

—

178

—

142

C₄²

C₂×C₄×D₈₉

yes

characteristic

143

C₄²

D₄×D₈₉

yes

—

144

C₄²

C₂×D₃₅₆

yes

characteristic

145

C₂³

#15668001

yes

characteristic

146

Q₈

—

712

—

147

Q₈

—

712

—

148

C₂×C₄

—

149

C₂×C₄

—

356

—

150

C₂×C₄

—

712

—

151

C₂×C₄

C₂×#13083551

yes

characteristic

152

C₂×C₄

#15667974

yes

—

153

C₂×C₄

#15667987

yes

—

154

C₂×C₄

#15668008

yes

characteristic

155

C₂×C₄

#15668042

yes

—

156

C₂×C₄

#15668042

yes

—

157

C₈

—

158

C₈

—

159

C₈

—

160

C₈

—

356

—

161

C₂²

—

162

C₂²

#19151499

yes

characteristic

163

C₂²

#19151968

yes

—

164

C₄

—

165

C₄

—

712

—

166

C₄

#19151606

yes

—

167

C₄

#19151862

yes

—

168

C₄

#19151993

yes

characteristic

169

C₄

#19151993

yes

characteristic

170

C₂

—

171

C₂

#26021071

yes

characteristic

172

C₂

#26021153

yes

—

	subgroup	G/H
center	C₂²	#19151499
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1837	C₂
socle	C₂²×C₈₉	#390
prime radical	C₂³×C₈₉	#124
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1837	D₈₉
hypocenter	C₈₉	#9776
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1837

subgroup

G/H

center

C₂²

#19151499

commutator

C₄²×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1837

C₂

C₂²×C₈₉

C₂³×C₈₉

C₂×C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1837

N	H ≅ G/N	type
92. C₈₉	#9776	semidirect

H ≅ G/N

type

92. C₈₉

#9776

semidirect

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

subgroup	index
C₂×#19151996	2
#26018938	2
#26018934	2
#26018544	2
C₈₉×#1837	2
#9776	89

subgroup

index

C₈₉×#1837

Subgroups — #36453912

Conjugacy classes of subgroups

Complements

Maximal subgroups