Group #36475011 — subgroups

Subgroups — #36475011

Order 22784.

190 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 67 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×#19151438	C₂	1	1	yes	maximal, characteristic
2	Q₃₂×Dic₈₉	C₂	1	1	yes	maximal, characteristic
3	#26022110	C₂	1	1	yes	maximal, characteristic
4	#26018451	C₂	1	1	yes	maximal, characteristic
5	#26018434	C₂	1	1	yes	maximal, characteristic
6	C₃₂×Dic₈₉	C₂	1	1	yes	maximal, characteristic
7	C₈₉×#1416	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₇₁₂	C₂²	1	1	yes	characteristic
9	Q₁₆×Dic₈₉	—	2	1	no	—
10	#19151692	—	2	1	no	—
11	#19151438	C₄	1	4	yes	complemented
12	C₂×#15667918	C₂²	1	1	yes	characteristic
13	#19151424	C₂²	1	1	yes	characteristic
14	#19151402	—	2	1	no	—
15	#19151401	—	2	1	no	—
16	C₁₆×Dic₈₉	C₂²	1	1	yes	characteristic
17	C₂×Q₃₂×C₈₉	C₂²	1	1	yes	characteristic
18	C₂×C₃₂×C₈₉	C₂²	1	1	yes	characteristic
19	#366×C₈₉	C₂²	1	1	yes	characteristic
20	C₂×Dic₃₅₆	—	2	1	no	—
21	Q₈×Dic₈₉	—	4	1	no	—
22	#15667977	—	4	1	no	—
23	Dic₇₁₂	—	2	1	no	—
24	Dic₇₁₂	C₂×C₄	1	2	yes	—
25	C₂×#13083549	D₄	1	1	yes	characteristic
26	#15667946	—	2	1	no	—
27	#15667936	—	4	1	no	—
28	#15667935	—	4	1	no	—
29	C₈×Dic₈₉	D₄	1	1	yes	characteristic
30	#15667918	C₂×C₄	1	2	yes	—
31	C₂×Q₁₆×C₈₉	—	2	1	no	—
32	Q₃₂×C₈₉	—	2	1	no	—
33	Q₃₂×C₈₉	C₂×C₄	1	2	yes	—
34	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
35	#107×C₈₉	—	2	1	no	—
36	C₃₂×C₈₉	—	2	1	no	—
37	C₂×Dic₁₇₈	—	4	1	no	—
38	Dic₃₅₆	—	2	2	no	—
39	Dic₃₅₆	—	4	1	no	—
40	C₂×#10623494	D₈	1	1	yes	characteristic
41	C₄⋊Dic₈₉	—	4	1	no	—
42	C₄⋊Dic₈₉	—	8	1	no	—
43	#13083553	—	8	1	no	—
44	C₄×Dic₈₉	—	8	1	no	—
45	C₄×Dic₈₉	—	8	1	no	—
46	C₄×Dic₈₉	D₈	1	1	yes	characteristic
47	#13083549	—	2	1	no	—
48	C₂×Q₈×C₈₉	—	4	1	no	—
49	Q₁₆×C₈₉	—	2	2	no	—
50	Q₁₆×C₈₉	—	4	1	no	—
51	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
52	C₄⋊C₄×C₈₉	—	4	1	no	—
53	C₁₆×C₈₉	C₂²×C₄	1	1	yes	characteristic
54	C₁₆×C₈₉	#41	1	1	yes	characteristic
55	Q₈×C₈₉	—	4	2	no	—
56	Q₈×C₈₉	—	8	1	no	—
57	C₂×C₄×C₈₉	—	8	1	no	—
58	C₂×C₄×C₈₉	—	8	1	no	—
59	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
60	C₂×Dic₈₉	—	8	1	no	—
61	C₂×Dic₈₉	—	8	1	no	—
62	C₂×Dic₈₉	D₁₆	1	1	yes	characteristic
63	C₂×Dic₈₉	D₁₆	1	1	yes	characteristic
64	Dic₁₇₈	—	4	2	no	—
65	Dic₁₇₈	—	8	1	no	—
66	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
67	C₈×C₈₉	#135	1	1	yes	characteristic
68	#10623494	—	2	1	no	—
69	C₂²×C₈₉	C₂×D₁₆	1	1	yes	characteristic
70	C₄×C₈₉	—	8	2	no	—
71	C₄×C₈₉	—	16	1	no	—
72	C₄×C₈₉	C₄×D₈	1	1	yes	characteristic
73	C₄×C₈₉	#508	1	1	yes	characteristic
74	Dic₈₉	—	2	1	no	—
75	Dic₈₉	—	8	2	no	—
76	Dic₈₉	—	16	1	no	—
77	Dic₈₉	Q₆₄	1	2	yes	complemented
78	C₄×Q₆₄	—	89	1	no	maximal, Sylow
79	C₂×C₈₉	C₄×D₁₆	1	1	yes	characteristic
80	C₂×C₈₉	C₂×Q₆₄	1	1	yes	characteristic
81	C₂×C₈₉	#2262	1	1	yes	characteristic
82	C₂×Q₆₄	—	89	1	no	—
83	C₄×Q₃₂	—	89	1	no	—
84	C₄×Q₃₂	—	89	1	no	—
85	#1423	—	89	1	no	—
86	#1416	—	89	1	no	—
87	#1416	D₈₉	1	1	yes	characteristic
88	C₄×C₃₂	—	89	1	no	—
89	C₈₉	C₄×Q₆₄	1	1	yes	characteristic, complemented, Sylow
90	C₂×Q₃₂	—	89	1	no	—
91	C₂×Q₃₂	C₂×D₈₉	1	1	yes	characteristic
92	C₄×Q₁₆	—	178	1	no	—
93	C₄×Q₁₆	—	178	1	no	—
94	Q₆₄	—	89	4	no	—
95	C₂×C₃₂	—	89	1	no	—
96	C₂×C₃₂	C₂×D₈₉	1	1	yes	characteristic
97	#366	—	89	1	no	—
98	#366	C₂×D₈₉	1	1	yes	characteristic
99	#358	—	178	1	no	—
100	#358	—	178	1	no	—
101	C₄×C₁₆	—	89	1	no	—
102	C₂×Q₁₆	—	2	1	no	—
103	C₂×Q₁₆	—	178	1	no	—
104	C₄×Q₈	—	356	1	no	—
105	C₄×Q₈	—	356	1	no	—
106	Q₃₂	—	2	1	no	—
107	Q₃₂	—	89	2	no	—
108	Q₃₂	—	178	1	no	—
109	Q₃₂	C₄×D₈₉	1	2	yes	—
110	C₂×C₁₆	—	89	1	no	—
111	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
112	#107	—	2	1	no	—
113	#107	—	178	1	no	—
114	#103	—	356	1	no	—
115	#103	—	356	1	no	—
116	C₄×C₈	—	89	1	no	—
117	C₃₂	—	2	1	no	—
118	C₃₂	—	89	2	no	—
119	C₂×Q₈	—	4	1	no	—
120	C₂×Q₈	—	356	1	no	—
121	Q₁₆	—	2	2	no	—
122	Q₁₆	—	4	1	no	—
123	Q₁₆	—	356	1	no	—
124	Q₁₆	—	178	2	no	—
125	C₂×C₈	—	89	1	no	—
126	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
127	C₄⋊C₄	—	4	1	no	—
128	C₄⋊C₄	—	356	1	no	—
129	C₄⋊C₄	—	712	1	no	—
130	C₄⋊C₄	—	712	1	no	—
131	C₄²	—	89	1	no	—
132	C₄²	—	712	1	no	—
133	C₄²	—	712	1	no	—
134	C₁₆	—	178	1	no	—
135	C₁₆	C₂×C₄×D₈₉	1	1	yes	characteristic
136	C₁₆	#13083573	1	1	yes	characteristic
137	Q₈	—	4	2	no	—
138	Q₈	—	8	1	no	—
139	Q₈	—	356	2	no	—
140	Q₈	—	712	1	no	—
141	C₂×C₄	—	8	1	no	—
142	C₂×C₄	—	8	1	no	—
143	C₂×C₄	—	89	1	no	—
144	C₂×C₄	—	89	1	no	—
145	C₂×C₄	—	712	1	no	—
146	C₂×C₄	—	712	1	no	—
147	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
148	C₈	—	178	1	no	—
149	C₈	#15667972	1	1	yes	characteristic
150	C₈	#15668033	1	1	yes	characteristic
151	C₂²	D₁₆×D₈₉	1	1	yes	characteristic
152	C₄	—	8	2	no	—
153	C₄	—	16	1	no	—
154	C₄	—	89	2	no	—
155	C₄	—	1424	1	no	—
156	C₄	—	178	1	no	—
157	C₄	—	712	2	no	—
158	C₄	#19151678	1	1	yes	characteristic
159	C₄	#19152029	1	1	yes	characteristic
160	C₂	#26022096	1	1	yes	characteristic
161	C₂	Q₆₄×D₈₉	1	1	yes	characteristic
162	C₂	#26022560	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19151438

C₂

yes

maximal, characteristic

Q₃₂×Dic₈₉

C₂

yes

maximal, characteristic

#26022110

C₂

yes

maximal, characteristic

#26018451

C₂

yes

maximal, characteristic

#26018434

C₂

yes

maximal, characteristic

C₃₂×Dic₈₉

C₂

yes

maximal, characteristic

C₈₉×#1416

C₂

yes

maximal, characteristic

C₂×Dic₇₁₂

C₂²

yes

characteristic

Q₁₆×Dic₈₉

—

#19151692

—

#19151438

C₄

yes

complemented

C₂×#15667918

C₂²

yes

characteristic

#19151424

C₂²

yes

characteristic

#19151402

—

#19151401

—

C₁₆×Dic₈₉

C₂²

yes

characteristic

C₂×Q₃₂×C₈₉

C₂²

yes

characteristic

C₂×C₃₂×C₈₉

C₂²

yes

characteristic

#366×C₈₉

C₂²

yes

characteristic

C₂×Dic₃₅₆

—

Q₈×Dic₈₉

—

#15667977

—

Dic₇₁₂

—

Dic₇₁₂

C₂×C₄

yes

—

C₂×#13083549

D₄

yes

—

—

—

C₈×Dic₈₉

D₄

yes

characteristic

#15667918

C₂×C₄

yes

—

C₂×Q₁₆×C₈₉

—

Q₃₂×C₈₉

—

Q₃₂×C₈₉

C₂×C₄

yes

—

C₂×C₁₆×C₈₉

C₂³

yes

characteristic

#107×C₈₉

—

C₃₂×C₈₉

—

C₂×Dic₁₇₈

—

Dic₃₅₆

—

Dic₃₅₆

—

C₂×#10623494

D₈

yes

characteristic

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083553

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

C₄×Dic₈₉

D₈

yes

characteristic

#13083549

—

C₂×Q₈×C₈₉

—

Q₁₆×C₈₉

—

Q₁₆×C₈₉

—

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₄⋊C₄×C₈₉

—

C₁₆×C₈₉

C₂²×C₄

yes

characteristic

C₁₆×C₈₉

#41

yes

characteristic

Q₈×C₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

D₁₆

yes

characteristic

C₂×Dic₈₉

D₁₆

yes

characteristic

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

C₄×D₄

yes

C₈×C₈₉

yes

—

C₂²×C₈₉

C₂×D₁₆

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₄×D₈

yes

characteristic

C₄×C₈₉

#508

yes

characteristic

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

Q₆₄

yes

complemented

C₄×Q₆₄

—

maximal, Sylow

C₂×C₈₉

C₄×D₁₆

yes

characteristic

C₂×C₈₉

C₂×Q₆₄

yes

characteristic

C₂×C₈₉

#2262

yes

characteristic

C₂×Q₆₄

—

C₄×Q₃₂

—

C₄×Q₃₂

—

#1423

—

#1416

—

#1416

D₈₉

yes

characteristic

C₄×C₃₂

—

C₈₉

C₄×Q₆₄

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×Q₃₂

C₂×D₈₉

yes

characteristic

C₄×Q₁₆

—

178

—

C₄×Q₁₆

—

178

—

Q₆₄

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₈₉

yes

characteristic

#366

—

#366

C₂×D₈₉

yes

characteristic

#358

—

178

—

100

#358

—

178

—

101

C₄×C₁₆

—

102

C₂×Q₁₆

—

103

C₂×Q₁₆

—

178

—

104

C₄×Q₈

—

356

—

105

C₄×Q₈

—

356

—

106

Q₃₂

—

107

Q₃₂

—

108

Q₃₂

—

178

—

109

Q₃₂

C₄×D₈₉

yes

—

110

C₂×C₁₆

—

111

C₂×C₁₆

C₂²×D₈₉

yes

characteristic

112

#107

—

113

#107

—

178

—

114

#103

—

356

—

115

#103

—

356

—

116

C₄×C₈

—

117

C₃₂

—

118

C₃₂

—

119

C₂×Q₈

—

120

C₂×Q₈

—

356

—

121

Q₁₆

—

122

Q₁₆

—

123

Q₁₆

—

356

—

124

Q₁₆

—

178

—

125

C₂×C₈

—

126

C₂×C₈

D₄×D₈₉

yes

characteristic

127

C₄⋊C₄

—

128

C₄⋊C₄

—

356

—

129

C₄⋊C₄

—

712

—

130

C₄⋊C₄

—

712

—

131

C₄²

—

132

C₄²

—

712

—

133

C₄²

—

712

—

134

C₁₆

—

178

—

135

C₁₆

C₂×C₄×D₈₉

yes

characteristic

136

C₁₆

#13083573

yes

characteristic

137

Q₈

—

138

Q₈

—

139

Q₈

—

356

—

140

Q₈

—

712

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

712

—

146

C₂×C₄

—

712

—

147

C₂×C₄

D₈×D₈₉

yes

characteristic

148

C₈

—

178

—

149

C₈

#15667972

yes

characteristic

150

C₈

#15668033

yes

characteristic

151

C₂²

D₁₆×D₈₉

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

1424

—

156

C₄

—

178

—

157

C₄

—

712

—

158

C₄

#19151678

yes

characteristic

159

C₄

#19152029

yes

characteristic

160

C₂

#26022096

yes

characteristic

161

C₂

Q₆₄×D₈₉

yes

characteristic

162

C₂

#26022560

yes

characteristic

	subgroup	G/H
center	C₂²	D₁₆×D₈₉
commutator	C₁₆×C₈₉	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₈₉
Fitting	C₈₉×#1416	C₂
socle, prime radical	C₂²×C₈₉	C₂×D₁₆
cosocle	C₂×C₁₆×C₈₉	C₂³
hypercenter	#1416	D₈₉
hypocenter	C₈₉	C₄×Q₆₄
LCS	G ▷ C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1416

subgroup

G/H

center

C₂²

D₁₆×D₈₉

commutator

C₁₆×C₈₉

C₂²×C₄

Frattini

C₂×C₁₆

C₂²×D₈₉

Fitting

C₈₉×#1416

C₂

socle, prime radical

C₂²×C₈₉

C₂×D₁₆

cosocle

C₂×C₁₆×C₈₉

C₂³

hypercenter

#1416

D₈₉

hypocenter

C₈₉

C₄×Q₆₄

LCS

G ▷ C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1416

N	H ≅ G/N	type
11. #19151438	C₄	semidirect
77. Dic₈₉	Q₆₄	semidirect
89. C₈₉	C₄×Q₆₄	semidirect

H ≅ G/N

type

11. #19151438

C₄

semidirect

77. Dic₈₉

Q₆₄

semidirect

89. C₈₉

C₄×Q₆₄

semidirect

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

subgroup	index
C₂×#19151438	2
Q₃₂×Dic₈₉	2
#26022110	2
#26018451	2
#26018434	2
C₃₂×Dic₈₉	2
C₈₉×#1416	2
C₄×Q₆₄	89

subgroup

index

Q₃₂×Dic₈₉

C₃₂×Dic₈₉

C₈₉×#1416

C₄×Q₆₄

Subgroups — #36475011

Conjugacy classes of subgroups

Complements

Maximal subgroups