Group #36441349 — subgroups

Subgroups — #36441349

Order 22784.

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

normal subgroups only

This group has 246 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	C₄×#15667948	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151349	C₂	1	1	yes	maximal, characteristic
3	C₄×M₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#15667948	C₄	1	2	yes	—
5	C₂×C₄⋊#10623494	—	2	1	no	—
6	C₄×#13083558	—	2	1	no	—
7	#19151363	—	2	1	no	—
8	#19151349	—	2	1	no	—
9	#19151349	C₄	1	2	yes	—
10	C₂×M₃₂×C₈₉	—	2	1	no	—
11	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
12	#346×C₈₉	—	2	1	no	—
13	C₄×C₁₆×C₈₉	C₄	1	2	yes	—
14	C₂×#13083558	—	2	2	no	—
15	C₂²×#10623494	—	4	1	no	—
16	#15667948	—	2	2	no	—
17	#15667948	C₈	1	4	yes	complemented
18	C₄⋊#10623494	—	2	2	no	—
19	C₄⋊#10623494	—	4	1	no	—
20	#15667923	—	4	1	no	—
21	C₄×#10623494	—	4	1	no	—
22	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
24	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
25	M₃₂×C₈₉	—	2	4	no	—
26	C₂×C₁₆×C₈₉	—	2	2	no	—
27	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
30	C₄×C₈×C₈₉	Q₈	1	1	yes	characteristic
31	#13083558	—	2	4	no	—
32	#13083558	—	4	2	no	—
33	C₂×#10623494	—	4	2	no	—
34	C₂×#10623494	—	4	2	no	—
35	C₂×#10623494	—	8	2	no	—
36	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
38	C₂²×C₄×C₈₉	Q₁₆	1	1	yes	characteristic
39	C₂×C₈×C₈₉	—	2	2	no	—
40	C₂×C₈×C₈₉	C₄²	1	1	yes	characteristic
41	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
45	C₂×C₈×C₈₉	M₁₆	1	2	yes	—
46	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
47	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
48	C₄²×C₈₉	D₈	1	1	yes	characteristic
49	C₄²×C₈₉	SD₁₆	1	1	yes	characteristic
50	C₁₆×C₈₉	—	2	4	no	—
51	C₂³×C₈₉	#103	1	1	yes	characteristic
52	C₂×C₄×C₈₉	—	2	2	no	—
53	C₂×C₄×C₈₉	—	4	1	no	—
54	C₂×C₄×C₈₉	#95	1	1	yes	characteristic
55	C₂×C₄×C₈₉	C₂²⋊C₈	1	2	yes	—
56	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
57	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
58	C₂×C₄×C₈₉	#103	1	1	yes	characteristic
59	C₂×C₄×C₈₉	#104	1	2	yes	—
60	C₂×C₄×C₈₉	#106	1	1	yes	characteristic
61	C₂×C₄×C₈₉	#107	1	1	yes	characteristic
62	C₈×C₈₉	—	2	1	no	—
63	C₈×C₈₉	—	2	1	no	—
64	C₈×C₈₉	C₄×C₈	1	1	yes	characteristic
65	C₈×C₈₉	#97	1	1	yes	characteristic
66	C₈×C₈₉	C₄⋊C₈	1	2	yes	—
67	#10623494	—	4	4	no	—
68	#10623494	—	8	2	no	—
69	C₂²×C₈₉	—	2	1	no	—
70	C₂²×C₈₉	—	2	1	no	—
71	C₂²×C₈₉	Q₈⋊C₈	1	2	yes	—
72	C₂²×C₈₉	#340	1	1	yes	characteristic
73	C₄×C₈₉	—	2	1	no	—
74	C₄×C₈₉	—	2	1	no	—
75	C₄×C₈₉	—	2	1	no	—
76	C₄×C₈₉	D₄⋊C₈	1	2	yes	—
77	C₄×C₈₉	#336	1	1	yes	characteristic
78	C₄×C₈₉	#339	1	1	yes	characteristic
79	C₄×C₈₉	#365	1	2	yes	—
80	#7173	—	89	1	no	maximal, Sylow
81	C₂×C₈₉	—	2	2	no	—
82	C₂×C₈₉	#1279	1	1	yes	characteristic
83	C₂×C₈₉	#1383	1	1	yes	characteristic
84	C₂×C₈₉	#1385	1	1	yes	characteristic
85	C₄×M₃₂	D₈₉	1	1	yes	characteristic
86	C₄×#108	—	89	1	no	—
87	C₂×#334	—	89	1	no	—
88	C₈₉	#7173	1	1	yes	characteristic, complemented, Sylow
89	C₂×M₃₂	—	2	1	no	—
90	C₂×#108	—	89	2	no	—
91	C₂×C₄⋊C₈	—	178	1	no	—
92	C₄×M₁₆	—	178	1	no	—
93	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
94	#346	—	2	1	no	—
95	C₄×C₁₆	Dic₈₉	1	2	yes	—
96	#341	—	178	1	no	—
97	#334	—	89	2	no	—
98	#334	—	178	1	no	—
99	C₂×M₁₆	—	178	2	no	—
100	C₂²×C₈	—	356	1	no	—
101	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
102	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
103	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
104	M₃₂	—	2	4	no	—
105	C₂×C₁₆	—	2	2	no	—
106	#108	—	89	4	no	—
107	#108	—	178	2	no	—
108	C₄⋊C₈	—	356	1	no	—
109	C₄⋊C₈	—	178	2	no	—
110	#97	—	356	1	no	—
111	C₄×C₈	—	356	1	no	—
112	C₄×C₈	Dic₁₇₈	1	1	yes	characteristic
113	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
114	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
115	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
116	C₂²×C₄	#13083551	1	1	yes	characteristic
117	C₂²×C₄	#13083563	1	1	yes	characteristic
118	C₂²×C₄	Dic₃₅₆	1	1	yes	characteristic
119	M₁₆	—	356	2	no	—
120	M₁₆	—	178	4	no	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	356	2	no	—
123	C₂×C₈	—	356	2	no	—
124	C₂×C₈	—	712	2	no	—
125	C₂×C₈	C₄×Dic₈₉	1	1	yes	characteristic
126	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
127	C₂×C₈	#13083553	1	1	yes	characteristic
128	C₂×C₈	#13083553	1	1	yes	characteristic
129	C₂×C₈	C₈×D₈₉	1	2	yes	—
130	C₂×C₈	#13083557	1	2	yes	—
131	C₄²	#13083551	1	1	yes	characteristic
132	C₄²	C₄⋊Dic₈₉	1	1	yes	characteristic
133	C₄²	D₇₁₂	1	1	yes	characteristic
134	C₄²	#13083563	1	1	yes	characteristic
135	C₁₆	—	2	4	no	—
136	C₂³	#15667937	1	1	yes	characteristic
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	4	1	no	—
139	C₂×C₄	#15667919	1	1	yes	characteristic
140	C₂×C₄	#15667924	1	2	yes	—
141	C₂×C₄	#15667934	1	1	yes	characteristic
142	C₂×C₄	#15667934	1	1	yes	characteristic
143	C₂×C₄	#15667937	1	1	yes	characteristic
144	C₂×C₄	#15667940	1	2	yes	—
145	C₂×C₄	#15667944	1	1	yes	characteristic
146	C₂×C₄	#15667946	1	1	yes	characteristic
147	C₈	—	2	1	no	—
148	C₈	—	2	1	no	—
149	C₈	—	356	4	no	—
150	C₈	—	712	2	no	—
151	C₈	C₈×Dic₈₉	1	1	yes	characteristic
152	C₈	#15667922	1	1	yes	characteristic
153	C₈	#15667941	1	2	yes	—
154	C₂²	—	2	1	no	—
155	C₂²	—	2	1	no	—
156	C₂²	#19151330	1	2	yes	—
157	C₂²	#19151361	1	1	yes	characteristic
158	C₄	—	2	1	no	—
159	C₄	—	2	1	no	—
160	C₄	—	2	1	no	—
161	C₄	#19151327	1	2	yes	—
162	C₄	#19151352	1	1	yes	characteristic
163	C₄	#19151358	1	1	yes	characteristic
164	C₄	#19151421	1	2	yes	—
165	C₂	—	2	2	no	—
166	C₂	#26018092	1	1	yes	characteristic
167	C₂	#26018351	1	1	yes	characteristic
168	C₂	#26018356	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#15667948

C₂

yes

maximal, characteristic

C₂×#19151349

C₂

yes

maximal, characteristic

C₄×M₃₂×C₈₉

C₂

yes

maximal, characteristic

C₂×#15667948

C₄

yes

—

—

—

—

—

C₄

yes

—

C₂×M₃₂×C₈₉

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

#346×C₈₉

—

C₄×C₁₆×C₈₉

C₄

yes

—

—

—

—

C₈

yes

—

—

—

—

C₂²×C₈×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₈₉

D₄

yes

characteristic

C₂×C₄²×C₈₉

D₄

yes

characteristic

M₃₂×C₈₉

—

C₂×C₁₆×C₈₉

—

C₄×C₈×C₈₉

C₂×C₄

yes

characteristic

C₄×C₈×C₈₉

C₂×C₄

yes

characteristic

C₄×C₈×C₈₉

D₄

yes

characteristic

C₄×C₈×C₈₉

Q₈

yes

—

—

—

—

—

C₂²×C₄×C₈₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₈₉

SD₁₆

yes

characteristic

C₂²×C₄×C₈₉

Q₁₆

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₄²

yes

characteristic

C₂×C₈×C₈₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₈₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₈₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₈₉

C₂×C₈

yes

—

C₂×C₈×C₈₉

M₁₆

yes

—

C₄²×C₈₉

C₂²⋊C₄

yes

characteristic

C₄²×C₈₉

C₄⋊C₄

yes

characteristic

C₄²×C₈₉

D₈

yes

characteristic

C₄²×C₈₉

SD₁₆

yes

characteristic

C₁₆×C₈₉

—

C₂³×C₈₉

#103

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#95

yes

characteristic

C₂×C₄×C₈₉

C₂²⋊C₈

yes

—

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

—

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

C₄×C₈

yes

characteristic

C₈×C₈₉

#97

yes

characteristic

C₈×C₈₉

C₄⋊C₈

yes

—

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

Q₈⋊C₈

yes

—

C₂²×C₈₉

#340

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

D₄⋊C₈

yes

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

#7173

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₄×M₃₂

D₈₉

yes

characteristic

C₄×#108

—

C₂×#334

—

C₈₉

#7173

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×#108

—

C₂×C₄⋊C₈

—

178

—

C₄×M₁₆

—

178

—

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

#346

—

C₄×C₁₆

Dic₈₉

yes

—

#341

—

178

—

#334

—

#334

—

178

—

C₂×M₁₆

—

178

—

100

C₂²×C₈

—

356

—

101

C₂²×C₈

C₄×D₈₉

yes

characteristic

102

C₂²×C₈

C₂²⋊D₈₉

yes

characteristic

103

C₂×C₄²

D₃₅₆

yes

characteristic

104

M₃₂

—

105

C₂×C₁₆

—

106

#108

—

107

#108

—

178

—

108

C₄⋊C₈

—

356

—

109

C₄⋊C₈

—

178

—

110

#97

—

356

—

111

C₄×C₈

—

356

—

112

C₄×C₈

Dic₁₇₈

yes

characteristic

113

C₄×C₈

C₄×D₈₉

yes

characteristic

114

C₄×C₈

C₂×Dic₈₉

yes

characteristic

115

C₄×C₈

C₂²⋊D₈₉

yes

characteristic

116

C₂²×C₄

#13083551

yes

characteristic

117

C₂²×C₄

#13083563

yes

characteristic

118

C₂²×C₄

Dic₃₅₆

yes

characteristic

119

M₁₆

—

356

—

120

M₁₆

—

178

—

121

C₂×C₈

—

122

C₂×C₈

—

356

—

123

C₂×C₈

—

356

—

124

C₂×C₈

—

712

—

125

C₂×C₈

C₄×Dic₈₉

yes

characteristic

126

C₂×C₈

C₂²⋊Dic₈₉

yes

characteristic

127

C₂×C₈

#13083553

yes

characteristic

128

C₂×C₈

#13083553

yes

characteristic

129

C₂×C₈

C₈×D₈₉

yes

—

130

C₂×C₈

#13083557

yes

—

131

C₄²

#13083551

yes

characteristic

132

C₄²

C₄⋊Dic₈₉

yes

characteristic

133

C₄²

D₇₁₂

yes

characteristic

134

C₄²

#13083563

yes

characteristic

135

C₁₆

—

136

C₂³

#15667937

yes

characteristic

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

#15667919

yes

characteristic

140

C₂×C₄

#15667924

yes

—

141

C₂×C₄

#15667934

yes

characteristic

142

C₂×C₄

#15667934

yes

characteristic

143

C₂×C₄

#15667937

yes

characteristic

144

C₂×C₄

#15667940

yes

—

145

C₂×C₄

#15667944

yes

characteristic

146

C₂×C₄

#15667946

yes

characteristic

147

C₈

—

148

C₈

—

149

C₈

—

356

—

150

C₈

—

712

—

151

C₈

C₈×Dic₈₉

yes

characteristic

152

C₈

#15667922

yes

characteristic

153

C₈

#15667941

yes

—

154

C₂²

—

155

C₂²

—

156

C₂²

#19151330

yes

—

157

C₂²

#19151361

yes

characteristic

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

#19151327

yes

—

162

C₄

#19151352

yes

characteristic

163

C₄

#19151358

yes

characteristic

164

C₄

#19151421

yes

—

165

C₂

—

166

C₂

#26018092

yes

characteristic

167

C₂

#26018351

yes

characteristic

168

C₂

#26018356

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#15667944
commutator	C₈×C₈₉	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₈₉
Fitting	C₄×M₃₂×C₈₉	C₂
socle	C₂²×C₈₉	#340
prime radical	C₂³×C₈₉	#103
cosocle	C₂×C₄×C₈×C₈₉	C₂²
hypercenter	C₄×M₃₂	D₈₉
hypocenter	C₈₉	#7173
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

subgroup

G/H

center

C₂×C₄

#15667944

commutator

C₈×C₈₉

C₄×C₈

Frattini

C₂×C₄×C₈

C₂×D₈₉

Fitting

C₄×M₃₂×C₈₉

C₂

C₂²×C₈₉

C₂³×C₈₉

C₂×C₄×C₈×C₈₉

C₂²

hypercenter

C₄×M₃₂

D₈₉

hypocenter

C₈₉

#7173

LCS

G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

N	H ≅ G/N	type
17. #15667948	C₈	semidirect
88. C₈₉	#7173	semidirect

H ≅ G/N

type

17. #15667948

C₈

semidirect

88. C₈₉

#7173

semidirect

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

subgroup	index
C₄×#15667948	2
C₂×#19151349	2
C₄×M₃₂×C₈₉	2
#7173	89

subgroup

index

C₄×#15667948

C₂×#19151349

C₄×M₃₂×C₈₉

#7173

Subgroups — #36441349

Conjugacy classes of subgroups

Complements

Maximal subgroups