Group #36454278 — subgroups

Subgroups — #36454278

Order 22784.

542 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 199 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	#26021214	C₂	1	4	yes	maximal
2	#26020409	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151319	C₂	1	1	yes	maximal, characteristic
4	C₂×#494×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#15667944	—	2	2	no	—
6	C₂×C₄×#10623494	C₂²	1	2	yes	—
7	#19151468	C₂²	1	2	yes	—
8	#19151360	—	2	8	no	—
9	#19151319	—	2	2	no	—
10	C₂²×Q₁₆×C₈₉	—	2	2	no	—
11	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
12	#494×C₈₉	—	2	2	no	—
13	#494×C₈₉	C₄	1	4	yes	complemented
14	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₂²×#10623494	D₄	1	4	yes	—
16	C₂×C₄⋊Dic₈₉	—	2	4	no	—
17	C₂×C₄⋊Dic₈₉	—	2	2	no	—
18	#15667944	—	4	4	no	—
19	C₄×#10623494	—	2	2	no	—
20	C₄×#10623494	—	2	2	no	—
21	C₂²×Q₈×C₈₉	—	2	2	no	—
22	C₂×Q₁₆×C₈₉	—	2	8	no	—
23	C₂×Q₁₆×C₈₉	—	4	4	no	—
24	C₂×Q₁₆×C₈₉	—	4	4	no	—
25	C₂²×C₈×C₈₉	D₄	1	2	yes	—
26	C₄⋊Q₈×C₈₉	—	2	2	no	—
27	C₄⋊Q₈×C₈₉	—	2	4	no	—
28	C₄⋊Q₈×C₈₉	C₂×C₄	1	4	yes	—
29	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
30	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
31	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₈₉	D₄	1	2	yes	—
33	C₂²×Dic₈₉	—	4	2	no	—
34	C₂×#10623494	—	2	4	no	—
35	C₂×#10623494	—	2	8	no	—
36	C₄⋊Dic₈₉	—	4	4	no	—
37	C₄⋊Dic₈₉	—	4	8	no	—
38	C₂×Q₈×C₈₉	—	2	4	no	—
39	C₂×Q₈×C₈₉	—	4	2	no	—
40	C₂×Q₈×C₈₉	—	4	4	no	—
41	C₂×Q₈×C₈₉	—	4	4	no	—
42	C₂²×C₄×C₈₉	—	4	2	no	—
43	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
45	Q₁₆×C₈₉	—	4	16	no	—
46	C₂×C₈×C₈₉	C₂²⋊C₄	1	4	yes	—
47	C₂×C₈×C₈₉	D₈	1	4	yes	—
48	C₂×C₈×C₈₉	SD₁₆	1	4	yes	—
49	C₄⋊C₄×C₈₉	—	2	8	no	—
50	C₄⋊C₄×C₈₉	—	4	4	no	—
51	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
53	C₄²×C₈₉	#41	1	2	yes	—
54	C₂³×C₈₉	#127	1	1	yes	characteristic
55	Q₈×C₈₉	—	4	8	no	—
56	Q₈×C₈₉	—	8	4	no	—
57	C₂×C₄×C₈₉	—	4	4	no	—
58	C₂×C₄×C₈₉	—	8	2	no	—
59	C₂×C₄×C₈₉	—	8	2	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	#124	1	1	yes	characteristic
64	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
65	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
66	C₂×C₄×C₈₉	#137	1	8	yes	—
67	C₂×Dic₈₉	—	8	2	no	—
68	C₂×Dic₈₉	—	8	4	no	—
69	C₈×C₈₉	#102	1	8	yes	—
70	#10623494	—	4	4	no	—
71	C₂²×C₈₉	#390	1	1	yes	characteristic
72	C₂²×C₈₉	#492	1	1	yes	characteristic
73	C₂²×C₈₉	#493	1	1	yes	characteristic
74	C₂²×C₈₉	#497	1	4	yes	—
75	C₄×C₈₉	—	8	4	no	—
76	C₄×C₈₉	C₂×#102	1	2	yes	—
77	C₄×C₈₉	#441	1	4	yes	—
78	C₄×C₈₉	#470	1	4	yes	—
79	C₄×C₈₉	#486	1	2	yes	—
80	Dic₈₉	—	16	2	no	—
81	#9839	—	89	1	no	maximal, Sylow
82	C₂×C₈₉	#1716	1	2	yes	—
83	C₂×C₈₉	#1722	1	2	yes	—
84	C₂×C₈₉	#1957	1	1	yes	characteristic
85	C₂×C₈₉	#1971	1	2	yes	—
86	C₂×#494	D₈₉	1	1	yes	characteristic
87	#1990	—	89	4	no	—
88	#1844	—	89	1	no	—
89	C₂×#322	—	89	1	no	—
90	C₈₉	#9839	1	1	yes	characteristic, complemented, Sylow
91	C₂²×Q₁₆	—	2	2	no	—
92	C₂×C₄⋊Q₈	C₂×D₈₉	1	2	yes	—
93	#494	—	2	2	no	—
94	#494	Dic₈₉	1	4	yes	complemented
95	C₂×#106	—	178	2	no	—
96	C₂×C₄×C₈	—	89	2	no	—
97	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
98	#384	—	89	2	no	—
99	#340	—	178	8	no	—
100	#322	—	178	2	no	—
101	C₂²×Q₈	—	2	2	no	—
102	C₂×Q₁₆	—	2	8	no	—
103	C₂×Q₁₆	—	4	4	no	—
104	C₂×Q₁₆	—	4	4	no	—
105	C₂²×C₈	—	89	4	no	—
106	C₂²×C₈	C₂²⋊D₈₉	1	2	yes	—
107	C₄⋊Q₈	—	2	2	no	—
108	C₄⋊Q₈	—	2	4	no	—
109	C₄⋊Q₈	C₂×Dic₈₉	1	4	yes	—
110	C₂×C₄⋊C₄	—	2	4	no	—
111	C₂×C₄⋊C₄	—	178	4	no	—
112	C₂×C₄⋊C₄	—	178	2	no	—
113	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
114	#106	—	356	4	no	—
115	C₄×C₈	—	178	2	no	—
116	C₄×C₈	—	178	2	no	—
117	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
118	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
119	C₂×Q₈	—	2	4	no	—
120	C₂×Q₈	—	4	2	no	—
121	C₂×Q₈	—	4	4	no	—
122	C₂×Q₈	—	4	4	no	—
123	C₂²×C₄	—	4	2	no	—
124	C₂²×C₄	—	356	2	no	—
125	C₂²×C₄	D₄×D₈₉	1	2	yes	—
126	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
127	Q₁₆	—	4	16	no	—
128	C₂×C₈	—	178	4	no	—
129	C₂×C₈	—	178	8	no	—
130	C₂×C₈	C₂²⋊Dic₈₉	1	4	yes	—
131	C₂×C₈	#13083559	1	4	yes	—
132	C₂×C₈	#13083562	1	4	yes	—
133	C₄⋊C₄	—	2	8	no	—
134	C₄⋊C₄	—	4	4	no	—
135	C₄⋊C₄	—	356	4	no	—
136	C₄⋊C₄	—	356	8	no	—
137	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
138	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
139	C₄²	#13083573	1	2	yes	—
140	C₂³	#15668008	1	1	yes	characteristic
141	Q₈	—	4	8	no	—
142	Q₈	—	8	4	no	—
143	C₂×C₄	—	4	4	no	—
144	C₂×C₄	—	8	2	no	—
145	C₂×C₄	—	8	2	no	—
146	C₂×C₄	—	712	2	no	—
147	C₂×C₄	—	712	4	no	—
148	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
149	C₂×C₄	D₄×Dic₈₉	1	2	yes	—
150	C₂×C₄	#15667983	1	2	yes	—
151	C₂×C₄	#15667999	1	1	yes	characteristic
152	C₂×C₄	C₂×#13083559	1	2	yes	—
153	C₂×C₄	C₂×#13083562	1	2	yes	—
154	C₂×C₄	#15668044	1	8	yes	—
155	C₈	—	356	4	no	—
156	C₈	#15667933	1	8	yes	—
157	C₂²	#19151501	1	1	yes	characteristic
158	C₂²	#19151970	1	1	yes	characteristic
159	C₂²	#19151974	1	1	yes	characteristic
160	C₂²	#19151991	1	4	yes	—
161	C₄	—	8	4	no	—
162	C₄	—	1424	2	no	—
163	C₄	C₂×#15667933	1	2	yes	—
164	C₄	#19151705	1	4	yes	—
165	C₄	#19151853	1	4	yes	—
166	C₄	#19151944	1	2	yes	—
167	C₂	#26019883	1	2	yes	—
168	C₂	#26019924	1	2	yes	—
169	C₂	#26021059	1	1	yes	characteristic
170	C₂	#26021124	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26021214

C₂

yes

maximal

#26020409

C₂

yes

maximal, characteristic

C₂×#19151319

C₂

yes

maximal, characteristic

C₂×#494×C₈₉

C₂

yes

maximal, characteristic

C₂×#15667944

—

C₂×C₄×#10623494

C₂²

yes

—

#19151468

C₂²

yes

—

#19151360

—

#19151319

—

C₂²×Q₁₆×C₈₉

—

C₂×C₄⋊Q₈×C₈₉

C₂²

yes

—

#494×C₈₉

—

#494×C₈₉

C₄

yes

complemented

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂²×#10623494

D₄

yes

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

#15667944

—

C₄×#10623494

—

C₄×#10623494

—

C₂²×Q₈×C₈₉

—

C₂×Q₁₆×C₈₉

—

C₂×Q₁₆×C₈₉

—

C₂×Q₁₆×C₈₉

—

C₂²×C₈×C₈₉

D₄

yes

—

C₄⋊Q₈×C₈₉

—

C₄⋊Q₈×C₈₉

—

C₄⋊Q₈×C₈₉

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄×C₈×C₈₉

C₂×C₄

yes

—

C₄×C₈×C₈₉

D₄

yes

—

C₂²×Dic₈₉

—

C₂×#10623494

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×D₄

yes

—

Q₁₆×C₈₉

—

C₂×C₈×C₈₉

C₂²⋊C₄

yes

—

C₂×C₈×C₈₉

D₈

yes

—

C₂×C₈×C₈₉

SD₁₆

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂²×C₄

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₄²×C₈₉

#41

yes

—

C₂³×C₈₉

#127

yes

characteristic

Q₈×C₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

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

#124

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

—

C₂×C₄×C₈₉

C₂×SD₁₆

yes

—

C₂×C₄×C₈₉

#137

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₈×C₈₉

#102

yes

—

#10623494

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

—

C₄×C₈₉

—

C₄×C₈₉

C₂×#102

yes

—

C₄×C₈₉

#441

yes

—

C₄×C₈₉

#470

yes

—

C₄×C₈₉

#486

yes

—

Dic₈₉

—

#9839

—

maximal, Sylow

C₂×C₈₉

#1716

yes

—

C₂×C₈₉

#1722

yes

—

C₂×C₈₉

#1957

yes

characteristic

C₂×C₈₉

#1971

yes

—

C₂×#494

D₈₉

yes

—

—

—

C₈₉

#9839

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

C₂×D₈₉

yes

—

#494

—

#494

Dic₈₉

yes

complemented

C₂×#106

—

178

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

#384

—

#340

—

178

—

100

#322

—

178

—

101

C₂²×Q₈

—

102

C₂×Q₁₆

—

103

C₂×Q₁₆

—

104

C₂×Q₁₆

—

105

C₂²×C₈

—

106

C₂²×C₈

C₂²⋊D₈₉

yes

—

107

C₄⋊Q₈

—

108

C₄⋊Q₈

—

109

C₄⋊Q₈

C₂×Dic₈₉

yes

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

178

—

112

C₂×C₄⋊C₄

—

178

—

113

C₂×C₄²

C₂²×D₈₉

yes

characteristic

114

#106

—

356

—

115

C₄×C₈

—

178

—

116

C₄×C₈

—

178

—

117

C₄×C₈

C₂×Dic₈₉

yes

—

118

C₄×C₈

C₂²⋊D₈₉

yes

—

119

C₂×Q₈

—

120

C₂×Q₈

—

121

C₂×Q₈

—

122

C₂×Q₈

—

123

C₂²×C₄

—

124

C₂²×C₄

—

356

—

125

C₂²×C₄

D₄×D₈₉

yes

—

126

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

127

Q₁₆

—

128

C₂×C₈

—

178

—

129

C₂×C₈

—

178

—

130

C₂×C₈

C₂²⋊Dic₈₉

yes

—

131

C₂×C₈

#13083559

yes

—

132

C₂×C₈

#13083562

yes

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

356

—

136

C₄⋊C₄

—

356

—

137

C₄²

C₂²×Dic₈₉

yes

characteristic

138

C₄²

C₂×C₂²⋊D₈₉

yes

characteristic

139

C₄²

#13083573

yes

—

140

C₂³

#15668008

yes

characteristic

141

Q₈

—

142

Q₈

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

712

—

147

C₂×C₄

—

712

—

148

C₂×C₄

C₂×C₂²⋊Dic₈₉

yes

characteristic

149

C₂×C₄

D₄×Dic₈₉

yes

—

150

C₂×C₄

#15667983

yes

—

151

C₂×C₄

#15667999

yes

characteristic

152

C₂×C₄

C₂×#13083559

yes

—

153

C₂×C₄

C₂×#13083562

yes

—

154

C₂×C₄

#15668044

yes

—

155

C₈

—

356

—

156

C₈

#15667933

yes

—

157

C₂²

#19151501

yes

characteristic

158

C₂²

#19151970

yes

characteristic

159

C₂²

#19151974

yes

characteristic

160

C₂²

#19151991

yes

—

161

C₄

—

162

C₄

—

1424

—

163

C₄

C₂×#15667933

yes

—

164

C₄

#19151705

yes

—

165

C₄

#19151853

yes

—

166

C₄

#19151944

yes

—

167

C₂

#26019883

yes

—

168

C₂

#26019924

yes

—

169

C₂

#26021059

yes

characteristic

170

C₂

#26021124

yes

—

	subgroup	G/H
center	C₂³	#15668008
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₂×#494×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	#127
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₂×#494	D₈₉
hypocenter	C₈₉	#9839
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#494

subgroup

G/H

center

C₂³

#15668008

commutator

C₄²×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₂×#494×C₈₉

C₂

socle, prime radical

C₂³×C₈₉

#127

cosocle

C₂×C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#494

N	H ≅ G/N	type
13. #494×C₈₉	C₄	semidirect
90. C₈₉	#9839	semidirect
94. #494	Dic₈₉	semidirect

H ≅ G/N

type

13. #494×C₈₉

C₄

semidirect

90. C₈₉

#9839

semidirect

94. #494

Dic₈₉

semidirect

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

subgroup	index
#26021214	2
#26020409	2
C₂×#19151319	2
C₂×#494×C₈₉	2
#9839	89

subgroup

index

C₂×#494×C₈₉

Subgroups — #36454278

Conjugacy classes of subgroups

Complements

Maximal subgroups