Group #36463381 — subgroups

Subgroups — #36463381

Order 22784.

842 conjugacy classes of subgroups; only the 235 normal subgroups are shown.

normal subgroups only

This group has 842 conjugacy classes of subgroups — too many to draw. Only its 235 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	#26019991	C₂	1	3	yes	maximal
2	#26019949	C₂	1	3	yes	maximal
3	C₈₉×#3458	C₂	1	1	yes	maximal, characteristic
4	C₂²×#13083558	C₂²	1	3	yes	—
5	C₂³×#10623494	C₂²	1	1	yes	characteristic
6	#19151367	—	2	6	no	—
7	#19151353	—	2	6	no	—
8	#544×C₈₉	—	2	6	no	—
9	#543×C₈₉	C₄	1	4	yes	—
10	#533×C₈₉	—	2	6	no	—
11	C₂×#117×C₈₉	C₂²	1	3	yes	—
12	C₂×#13083558	—	2	6	no	—
13	C₂×#13083558	—	2	3	no	—
14	C₂×#13083558	—	2	6	no	—
15	C₂×#13083558	Q₈	1	6	yes	complemented
16	C₂²×#10623494	—	2	3	no	—
17	C₂²×#10623494	—	2	3	no	—
18	C₂²×#10623494	—	2	3	no	—
19	C₂²×#10623494	Q₈	1	2	yes	complemented
20	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
21	C₄⋊Q₈×C₈₉	—	2	12	no	—
22	#125×C₈₉	—	2	12	no	—
23	C₂²⋊Q₈×C₈₉	—	2	24	no	—
24	C₄×Q₈×C₈₉	—	2	8	no	—
25	#117×C₈₉	—	2	6	no	—
26	#117×C₈₉	—	2	6	no	—
27	C₂×C₄⋊C₄×C₈₉	D₄	1	6	yes	—
28	C₂×C₂²⋊C₄×C₈₉	C₂×C₄	1	6	yes	—
29	C₂×C₄²×C₈₉	D₄	1	6	yes	—
30	#13083558	—	2	12	no	—
31	#13083558	—	4	6	no	—
32	C₂×#10623494	—	2	6	no	—
33	C₂×#10623494	—	2	6	no	—
34	C₂×#10623494	—	4	1	no	—
35	C₂×#10623494	—	4	3	no	—
36	C₂×#10623494	—	4	3	no	—
37	C₂×#10623494	—	4	3	no	—
38	C₂⁴×C₈₉	C₂²×C₄	1	1	yes	characteristic
39	C₂×Q₈×C₈₉	—	2	8	no	—
40	C₂²×C₄×C₈₉	C₂²⋊C₄	1	12	yes	—
41	C₂²×C₄×C₈₉	C₂×D₄	1	3	yes	—
42	C₂²×C₄×C₈₉	C₂×D₄	1	3	yes	—
43	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
44	C₂²×C₄×C₈₉	C₂×Q₈	1	3	yes	—
45	C₂²×C₄×C₈₉	#41	1	3	yes	—
46	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
47	C₄⋊C₄×C₈₉	—	2	24	no	—
48	C₄⋊C₄×C₈₉	—	2	12	no	—
49	C₂²⋊C₄×C₈₉	—	2	12	no	—
50	C₄²×C₈₉	—	2	6	no	—
51	C₄²×C₈₉	—	2	6	no	—
52	C₂³×C₈₉	—	4	2	no	—
53	C₂³×C₈₉	C₂×C₂²⋊C₄	1	3	yes	—
54	C₂³×C₈₉	C₄×Q₈	1	1	yes	characteristic
55	C₂³×C₈₉	C₄×Q₈	1	3	yes	—
56	Q₈×C₈₉	—	8	8	no	—
57	C₂×C₄×C₈₉	—	2	12	no	—
58	C₂×C₄×C₈₉	—	4	12	no	—
59	C₂×C₄×C₈₉	#120	1	3	yes	—
60	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
61	C₂×C₄×C₈₉	C₂²⋊Q₈	1	3	yes	—
62	C₂×C₄×C₈₉	C₂²⋊Q₈	1	3	yes	—
63	C₂×C₄×C₈₉	C₂²⋊Q₈	1	6	yes	—
64	C₂×C₄×C₈₉	#124	1	3	yes	—
65	C₂×C₄×C₈₉	#124	1	3	yes	—
66	C₂×C₄×C₈₉	C₄⋊Q₈	1	3	yes	—
67	C₂×C₄×C₈₉	C₄⋊Q₈	1	3	yes	—
68	#10623494	—	4	2	no	—
69	#10623494	—	4	6	no	—
70	C₂²×C₈₉	—	2	6	no	—
71	C₂²×C₈₉	—	4	1	no	—
72	C₂²×C₈₉	—	4	3	no	—
73	C₂²×C₈₉	#379	1	1	yes	characteristic
74	C₂²×C₈₉	#391	1	3	yes	—
75	C₂²×C₈₉	#391	1	3	yes	—
76	C₄×C₈₉	—	4	12	no	—
77	C₄×C₈₉	#393	1	3	yes	—
78	C₄×C₈₉	#393	1	1	yes	characteristic
79	C₄×C₈₉	#395	1	3	yes	—
80	C₄×C₈₉	#395	1	1	yes	characteristic
81	#11391	—	89	1	no	maximal, Sylow
82	C₂×C₈₉	—	2	3	no	—
83	C₂×C₈₉	—	2	3	no	—
84	C₂×C₈₉	#1446	1	1	yes	characteristic
85	C₂×C₈₉	#1791	1	2	yes	—
86	#3458	D₈₉	1	1	yes	characteristic
87	#1742	—	89	3	no	—
88	#1729	—	89	3	no	—
89	C₈₉	#11391	1	1	yes	characteristic, complemented, Sylow
90	C₂²×M₁₆	—	89	3	no	—
91	C₂³×C₈	—	89	1	no	—
92	#544	—	2	6	no	—
93	#543	Dic₈₉	1	4	yes	—
94	#533	—	2	6	no	—
95	C₂×#117	C₂×D₈₉	1	3	yes	—
96	#343	—	178	6	no	—
97	#336	—	178	6	no	—
98	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
99	C₂×M₁₆	—	89	6	no	—
100	C₂×M₁₆	—	178	6	no	—
101	C₂×M₁₆	—	178	6	no	—
102	C₂×M₁₆	—	178	3	no	—
103	C₂²×C₈	—	89	2	no	—
104	C₂²×C₈	—	178	3	no	—
105	C₂²×C₈	—	178	3	no	—
106	C₂²×C₈	—	178	3	no	—
107	C₄⋊Q₈	—	2	12	no	—
108	#125	—	2	12	no	—
109	C₂²⋊Q₈	—	2	24	no	—
110	C₄×Q₈	—	2	8	no	—
111	#117	—	2	6	no	—
112	#117	—	2	6	no	—
113	C₂×C₄⋊C₄	C₂²⋊D₈₉	1	6	yes	—
114	C₂×C₂²⋊C₄	C₂×Dic₈₉	1	6	yes	—
115	C₂×C₄²	C₂²⋊D₈₉	1	6	yes	—
116	C₂⁴	C₂²×Dic₈₉	1	1	yes	characteristic
117	C₂×Q₈	—	2	8	no	—
118	C₂²×C₄	C₂²⋊Dic₈₉	1	12	yes	—
119	C₂²×C₄	C₂×C₂²⋊D₈₉	1	3	yes	—
120	C₂²×C₄	C₂×C₂²⋊D₈₉	1	3	yes	—
121	C₂²×C₄	Q₈×D₈₉	1	1	yes	characteristic
122	C₂²×C₄	Q₈×D₈₉	1	3	yes	—
123	C₂²×C₄	#13083575	1	3	yes	—
124	C₂²×C₄	#13083575	1	1	yes	characteristic
125	M₁₆	—	356	6	no	—
126	M₁₆	—	178	12	no	—
127	C₂×C₈	—	356	3	no	—
128	C₂×C₈	—	356	1	no	—
129	C₂×C₈	—	356	3	no	—
130	C₂×C₈	—	356	3	no	—
131	C₂×C₈	—	178	6	no	—
132	C₂×C₈	—	178	6	no	—
133	C₄⋊C₄	—	2	24	no	—
134	C₄⋊C₄	—	2	12	no	—
135	C₂²⋊C₄	—	2	12	no	—
136	C₄²	—	2	6	no	—
137	C₄²	—	2	6	no	—
138	C₂³	—	4	2	no	—
139	C₂³	C₂×C₂²⋊Dic₈₉	1	3	yes	—
140	C₂³	Q₈×Dic₈₉	1	1	yes	characteristic
141	C₂³	Q₈×Dic₈₉	1	3	yes	—
142	Q₈	—	8	8	no	—
143	C₂×C₄	—	2	12	no	—
144	C₂×C₄	—	4	12	no	—
145	C₂×C₄	#15667981	1	3	yes	—
146	C₂×C₄	#15667981	1	1	yes	characteristic
147	C₂×C₄	#15667988	1	3	yes	—
148	C₂×C₄	#15667988	1	3	yes	—
149	C₂×C₄	#15667988	1	6	yes	—
150	C₂×C₄	#15668001	1	3	yes	—
151	C₂×C₄	#15668001	1	3	yes	—
152	C₂×C₄	#15668010	1	3	yes	—
153	C₂×C₄	#15668010	1	3	yes	—
154	C₈	—	356	2	no	—
155	C₈	—	356	6	no	—
156	C₂²	—	2	6	no	—
157	C₂²	—	4	1	no	—
158	C₂²	—	4	3	no	—
159	C₂²	#19151454	1	1	yes	characteristic
160	C₂²	#19151505	1	3	yes	—
161	C₂²	#19151505	1	3	yes	—
162	C₄	—	4	12	no	—
163	C₄	#19151509	1	3	yes	—
164	C₄	#19151509	1	1	yes	characteristic
165	C₄	#19151517	1	3	yes	—
166	C₄	#19151517	1	1	yes	characteristic
167	C₂	—	2	3	no	—
168	C₂	—	2	3	no	—
169	C₂	#26018523	1	1	yes	characteristic
170	C₂	#26020203	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26019991

C₂

yes

maximal

#26019949

C₂

yes

maximal

C₈₉×#3458

C₂

yes

maximal, characteristic

C₂²×#13083558

C₂²

yes

—

C₂³×#10623494

C₂²

yes

characteristic

#19151367

—

#19151353

—

#544×C₈₉

—

#543×C₈₉

C₄

yes

—

#533×C₈₉

—

C₂×#117×C₈₉

C₂²

yes

—

—

—

—

Q₈

yes

—

—

—

Q₈

yes

complemented

C₂³×C₄×C₈₉

C₂³

yes

characteristic

C₄⋊Q₈×C₈₉

—

#125×C₈₉

—

C₂²⋊Q₈×C₈₉

—

C₄×Q₈×C₈₉

—

#117×C₈₉

—

#117×C₈₉

—

C₂×C₄⋊C₄×C₈₉

D₄

yes

—

C₂×C₂²⋊C₄×C₈₉

C₂×C₄

yes

—

C₂×C₄²×C₈₉

D₄

yes

—

—

—

—

—

—

—

—

—

C₂⁴×C₈₉

C₂²×C₄

yes

characteristic

C₂×Q₈×C₈₉

—

C₂²×C₄×C₈₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₈₉

C₂×D₄

yes

—

C₂²×C₄×C₈₉

C₂×D₄

yes

—

C₂²×C₄×C₈₉

C₂×Q₈

yes

characteristic

C₂²×C₄×C₈₉

C₂×Q₈

yes

—

C₂²×C₄×C₈₉

#41

yes

—

C₂²×C₄×C₈₉

#41

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

C₂×C₂²⋊C₄

yes

—

C₂³×C₈₉

C₄×Q₈

yes

characteristic

C₂³×C₈₉

C₄×Q₈

yes

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#120

yes

—

C₂×C₄×C₈₉

#120

yes

characteristic

C₂×C₄×C₈₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₈₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₈₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₈₉

#124

yes

—

C₂×C₄×C₈₉

#124

yes

—

C₂×C₄×C₈₉

C₄⋊Q₈

yes

—

C₂×C₄×C₈₉

C₄⋊Q₈

yes

—

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#379

yes

characteristic

C₂²×C₈₉

#391

yes

—

C₂²×C₈₉

#391

yes

—

C₄×C₈₉

—

C₄×C₈₉

#393

yes

—

C₄×C₈₉

#393

yes

characteristic

C₄×C₈₉

#395

yes

—

C₄×C₈₉

yes

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#1446

yes

characteristic

C₂×C₈₉

#1791

yes

—

#3458

D₈₉

yes

characteristic

#1742

—

#1729

—

C₈₉

#11391

yes

characteristic, complemented, Sylow

C₂²×M₁₆

—

C₂³×C₈

—

#544

—

#543

Dic₈₉

yes

—

#533

—

C₂×#117

C₂×D₈₉

yes

—

#343

—

178

—

#336

—

178

—

C₂³×C₄

C₂²×D₈₉

yes

characteristic

C₂×M₁₆

—

100

C₂×M₁₆

—

178

—

101

C₂×M₁₆

—

178

—

102

C₂×M₁₆

—

178

—

103

C₂²×C₈

—

104

C₂²×C₈

—

178

—

105

C₂²×C₈

—

178

—

106

C₂²×C₈

—

178

—

107

C₄⋊Q₈

—

108

#125

—

109

C₂²⋊Q₈

—

110

C₄×Q₈

—

111

#117

—

112

#117

—

113

C₂×C₄⋊C₄

C₂²⋊D₈₉

yes

—

114

C₂×C₂²⋊C₄

C₂×Dic₈₉

yes

—

115

C₂×C₄²

C₂²⋊D₈₉

yes

—

116

C₂⁴

C₂²×Dic₈₉

yes

characteristic

117

C₂×Q₈

—

118

C₂²×C₄

C₂²⋊Dic₈₉

yes

—

119

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

—

120

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

—

121

C₂²×C₄

Q₈×D₈₉

yes

characteristic

122

C₂²×C₄

Q₈×D₈₉

yes

—

123

C₂²×C₄

#13083575

yes

—

124

C₂²×C₄

#13083575

yes

characteristic

125

M₁₆

—

356

—

126

M₁₆

—

178

—

127

C₂×C₈

—

356

—

128

C₂×C₈

—

356

—

129

C₂×C₈

—

356

—

130

C₂×C₈

—

356

—

131

C₂×C₈

—

178

—

132

C₂×C₈

—

178

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₂²⋊C₄

—

136

C₄²

—

137

C₄²

—

138

C₂³

—

139

C₂³

C₂×C₂²⋊Dic₈₉

yes

—

140

C₂³

Q₈×Dic₈₉

yes

characteristic

141

C₂³

Q₈×Dic₈₉

yes

—

142

Q₈

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

#15667981

yes

—

146

C₂×C₄

#15667981

yes

characteristic

147

C₂×C₄

#15667988

yes

—

148

C₂×C₄

#15667988

yes

—

149

C₂×C₄

#15667988

yes

—

150

C₂×C₄

#15668001

yes

—

151

C₂×C₄

#15668001

yes

—

152

C₂×C₄

#15668010

yes

—

153

C₂×C₄

#15668010

yes

—

154

C₈

—

356

—

155

C₈

—

356

—

156

C₂²

—

157

C₂²

—

158

C₂²

—

159

C₂²

#19151454

yes

characteristic

160

C₂²

#19151505

yes

—

161

C₂²

#19151505

yes

—

162

C₄

—

163

C₄

#19151509

yes

—

164

C₄

#19151509

yes

characteristic

165

C₄

#19151517

yes

—

166

C₄

#19151517

yes

characteristic

167

C₂

—

168

C₂

—

169

C₂

#26018523

yes

characteristic

170

C₂

#26020203

yes

—

	subgroup	G/H
center	C₂×C₄	#15667981
commutator, prime radical	C₂⁴×C₈₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₈₉
Fitting	C₈₉×#3458	C₂
socle	C₂²×C₈₉	#379
cosocle	C₂³×C₄×C₈₉	C₂³
hypercenter	#3458	D₈₉
hypocenter	C₈₉	#11391
LCS	G ▷ C₂⁴×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #3458

subgroup

G/H

center

C₂×C₄

#15667981

commutator, prime radical

C₂⁴×C₈₉

C₂²×C₄

Frattini

C₂³×C₄

C₂²×D₈₉

Fitting

C₈₉×#3458

C₂

socle

C₂²×C₈₉

#379

cosocle

C₂³×C₄×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂⁴×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #3458

N	H ≅ G/N	type
15. C₂×#13083558	Q₈	semidirect
19. C₂²×#10623494	Q₈	semidirect
89. C₈₉	#11391	semidirect

H ≅ G/N

type

15. C₂×#13083558

Q₈

semidirect

19. C₂²×#10623494

Q₈

semidirect

89. C₈₉

#11391

semidirect

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

subgroup	index
#26019991	2
#26019949	2
C₈₉×#3458	2
#11391	89

subgroup

index

#26019991

#26019949

C₈₉×#3458

#11391

Subgroups — #36463381

Conjugacy classes of subgroups

Complements

Maximal subgroups