Group #36441679 — subgroups

Subgroups — #36441679

Order 22784.

208 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 87 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	#26019015	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#2162	C₂	1	1	yes	maximal, characteristic
4	#19151655	—	2	1	no	—
5	C₂×#15667948	—	2	1	no	—
6	C₄×#13083558	—	2	1	no	—
7	#19151363	C₄	1	2	yes	—
8	#19151351	C₄	1	2	yes	—
9	#19151349	—	2	1	no	—
10	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
11	C₄⋊C₁₆×C₈₉	C₄	1	2	yes	—
12	C₂²⋊C₁₆×C₈₉	—	2	1	no	—
13	C₄×C₁₆×C₈₉	—	2	1	no	—
14	C₂×#13083558	—	2	2	no	—
15	#15667948	—	2	4	no	—
16	C₄⋊#10623494	—	2	2	no	—
17	C₂²⋊#10623494	—	4	1	no	—
18	#15667923	—	4	1	no	—
19	#15667923	—	4	1	no	—
20	C₄×#10623494	—	4	1	no	—
21	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
23	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
24	C₂×C₁₆×C₈₉	—	2	2	no	—
25	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
28	C₄×C₈×C₈₉	Q₈	1	1	yes	characteristic
29	#13083558	—	2	4	no	—
30	#13083558	—	4	2	no	—
31	C₂×#10623494	—	4	2	no	—
32	C₂×#10623494	—	4	2	no	—
33	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
35	C₂²×C₄×C₈₉	Q₁₆	1	1	yes	characteristic
36	C₂×C₈×C₈₉	—	2	2	no	—
37	C₂×C₈×C₈₉	—	4	1	no	—
38	C₂×C₈×C₈₉	C₄²	1	1	yes	characteristic
39	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
42	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
43	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
44	C₄²×C₈₉	D₈	1	1	yes	characteristic
45	C₄²×C₈₉	SD₁₆	1	1	yes	characteristic
46	C₁₆×C₈₉	—	4	2	no	—
47	C₂³×C₈₉	#103	1	1	yes	characteristic
48	C₂×C₄×C₈₉	—	2	1	no	—
49	C₂×C₄×C₈₉	—	2	1	no	—
50	C₂×C₄×C₈₉	—	2	2	no	—
51	C₂×C₄×C₈₉	—	4	1	no	—
52	C₂×C₄×C₈₉	#95	1	1	yes	characteristic
53	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
54	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
55	C₂×C₄×C₈₉	#103	1	1	yes	characteristic
56	C₂×C₄×C₈₉	#106	1	1	yes	characteristic
57	C₂×C₄×C₈₉	#107	1	1	yes	characteristic
58	C₈×C₈₉	—	2	1	no	—
59	C₈×C₈₉	—	2	1	no	—
60	C₈×C₈₉	—	2	1	no	—
61	C₈×C₈₉	#104	1	2	yes	—
62	#10623494	—	4	4	no	—
63	#10623494	—	8	2	no	—
64	C₂²×C₈₉	—	2	1	no	—
65	C₂²×C₈₉	—	2	1	no	—
66	C₂²×C₈₉	—	2	1	no	—
67	C₂²×C₈₉	#340	1	1	yes	characteristic
68	C₄×C₈₉	—	2	1	no	—
69	C₄×C₈₉	—	2	1	no	—
70	C₄×C₈₉	—	2	1	no	—
71	C₄×C₈₉	—	2	1	no	—
72	C₄×C₈₉	#339	1	1	yes	characteristic
73	C₄×C₈₉	#344	1	1	yes	characteristic
74	C₄×C₈₉	#359	1	2	yes	—
75	#7297	—	89	1	no	maximal, Sylow
76	C₂×C₈₉	—	4	1	no	—
77	C₂×C₈₉	#1303	1	1	yes	characteristic
78	C₂×C₈₉	#1382	1	1	yes	characteristic
79	C₂×C₈₉	#1388	1	1	yes	characteristic
80	#2162	D₈₉	1	1	yes	characteristic
81	C₄×#108	—	89	1	no	—
82	#1568	—	89	1	no	—
83	C₈₉	#7297	1	1	yes	characteristic, complemented, Sylow
84	#432	—	178	1	no	—
85	C₂×#108	—	178	1	no	—
86	C₄×M₁₆	—	178	1	no	—
87	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
88	C₄⋊C₁₆	Dic₈₉	1	2	yes	—
89	C₂²⋊C₁₆	—	2	1	no	—
90	C₄×C₁₆	—	2	1	no	—
91	#341	—	89	2	no	—
92	#335	—	89	2	no	—
93	#334	—	178	1	no	—
94	C₂×M₁₆	—	178	2	no	—
95	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
96	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
97	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
98	C₂×C₁₆	—	2	2	no	—
99	#108	—	178	4	no	—
100	C₄⋊C₈	—	178	2	no	—
101	C₂²⋊C₈	—	356	1	no	—
102	#97	—	356	1	no	—
103	#97	—	356	1	no	—
104	C₄×C₈	—	356	1	no	—
105	C₄×C₈	Dic₁₇₈	1	1	yes	characteristic
106	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
107	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
108	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
109	C₂²×C₄	#13083551	1	1	yes	characteristic
110	C₂²×C₄	#13083563	1	1	yes	characteristic
111	C₂²×C₄	Dic₃₅₆	1	1	yes	characteristic
112	M₁₆	—	356	2	no	—
113	M₁₆	—	178	4	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	4	1	no	—
116	C₂×C₈	—	356	2	no	—
117	C₂×C₈	—	356	2	no	—
118	C₂×C₈	C₄×Dic₈₉	1	1	yes	characteristic
119	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
120	C₂×C₈	#13083553	1	1	yes	characteristic
121	C₂×C₈	#13083553	1	1	yes	characteristic
122	C₄²	#13083551	1	1	yes	characteristic
123	C₄²	C₄⋊Dic₈₉	1	1	yes	characteristic
124	C₄²	D₇₁₂	1	1	yes	characteristic
125	C₄²	#13083563	1	1	yes	characteristic
126	C₁₆	—	4	2	no	—
127	C₂³	#15667937	1	1	yes	characteristic
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	#15667919	1	1	yes	characteristic
133	C₂×C₄	#15667934	1	1	yes	characteristic
134	C₂×C₄	#15667934	1	1	yes	characteristic
135	C₂×C₄	#15667937	1	1	yes	characteristic
136	C₂×C₄	#15667944	1	1	yes	characteristic
137	C₂×C₄	#15667946	1	1	yes	characteristic
138	C₈	—	2	1	no	—
139	C₈	—	2	1	no	—
140	C₈	—	2	1	no	—
141	C₈	—	356	4	no	—
142	C₈	—	712	2	no	—
143	C₈	#15667939	1	2	yes	—
144	C₂²	—	2	1	no	—
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	#19151361	1	1	yes	characteristic
148	C₄	—	2	1	no	—
149	C₄	—	2	1	no	—
150	C₄	—	2	1	no	—
151	C₄	—	2	1	no	—
152	C₄	#19151357	1	1	yes	characteristic
153	C₄	#19151370	1	1	yes	characteristic
154	C₄	#19151406	1	2	yes	—
155	C₂	—	4	1	no	—
156	C₂	#26018154	1	1	yes	characteristic
157	C₂	#26018348	1	1	yes	characteristic
158	C₂	#26018364	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#15667948

C₂

yes

maximal, characteristic

#26019015

C₂

yes

maximal, characteristic

C₈₉×#2162

C₂

yes

maximal, characteristic

—

—

—

C₄

yes

—

#19151351

C₄

yes

—

#19151349

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

C₄⋊C₁₆×C₈₉

C₄

yes

—

C₂²⋊C₁₆×C₈₉

—

C₄×C₁₆×C₈₉

—

—

—

—

—

—

—

—

C₂²×C₈×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₈₉

D₄

yes

characteristic

C₂×C₄²×C₈₉

D₄

yes

characteristic

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

—

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

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

—

C₂×C₄×C₈₉

—

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

—

C₈×C₈₉

#104

yes

—

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#340

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

#7297

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

D₈₉

yes

characteristic

C₄×#108

—

#1568

—

C₈₉

#7297

yes

characteristic, complemented, Sylow

#432

—

178

—

C₂×#108

—

178

—

C₄×M₁₆

—

178

—

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

C₄⋊C₁₆

Dic₈₉

yes

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

#341

—

#335

—

#334

—

178

—

C₂×M₁₆

—

178

—

C₂²×C₈

C₄×D₈₉

yes

characteristic

C₂²×C₈

C₂²⋊D₈₉

yes

characteristic

C₂×C₄²

D₃₅₆

yes

characteristic

C₂×C₁₆

—

#108

—

178

—

100

C₄⋊C₈

—

178

—

101

C₂²⋊C₈

—

356

—

102

#97

—

356

—

103

#97

—

356

—

104

C₄×C₈

—

356

—

105

C₄×C₈

Dic₁₇₈

yes

characteristic

106

C₄×C₈

C₄×D₈₉

yes

characteristic

107

C₄×C₈

C₂×Dic₈₉

yes

characteristic

108

C₄×C₈

C₂²⋊D₈₉

yes

characteristic

109

C₂²×C₄

#13083551

yes

characteristic

110

C₂²×C₄

#13083563

yes

characteristic

111

C₂²×C₄

Dic₃₅₆

yes

characteristic

112

M₁₆

—

356

—

113

M₁₆

—

178

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

356

—

117

C₂×C₈

—

356

—

118

C₂×C₈

C₄×Dic₈₉

yes

characteristic

119

C₂×C₈

C₂²⋊Dic₈₉

yes

characteristic

120

C₂×C₈

#13083553

yes

characteristic

121

C₂×C₈

#13083553

yes

characteristic

122

C₄²

#13083551

yes

characteristic

123

C₄²

C₄⋊Dic₈₉

yes

characteristic

124

C₄²

D₇₁₂

yes

characteristic

125

C₄²

#13083563

yes

characteristic

126

C₁₆

—

127

C₂³

#15667937

yes

characteristic

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

#15667919

yes

characteristic

133

C₂×C₄

#15667934

yes

characteristic

134

C₂×C₄

#15667934

yes

characteristic

135

C₂×C₄

#15667937

yes

characteristic

136

C₂×C₄

#15667944

yes

characteristic

137

C₂×C₄

#15667946

yes

characteristic

138

C₈

—

139

C₈

—

140

C₈

—

141

C₈

—

356

—

142

C₈

—

712

—

143

C₈

#15667939

yes

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

#19151361

yes

characteristic

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

#19151357

yes

characteristic

153

C₄

#19151370

yes

characteristic

154

C₄

#19151406

yes

—

155

C₂

—

156

C₂

#26018154

yes

characteristic

157

C₂

#26018348

yes

characteristic

158

C₂

#26018364

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#15667934
commutator	C₂×C₈×C₈₉	C₄²
Frattini	C₂×C₄×C₈	C₂×D₈₉
Fitting	C₈₉×#2162	C₂
socle	C₂²×C₈₉	#340
prime radical	C₂³×C₈₉	#103
cosocle	C₂×C₄×C₈×C₈₉	C₂²
hypercenter	#2162	D₈₉
hypocenter	C₈₉	#7297
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2162

subgroup

G/H

center

C₂×C₄

#15667934

commutator

C₂×C₈×C₈₉

C₄²

Frattini

C₂×C₄×C₈

C₂×D₈₉

Fitting

C₈₉×#2162

C₂

C₂²×C₈₉

C₂³×C₈₉

C₂×C₄×C₈×C₈₉

C₂²

D₈₉

C₈₉

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

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2162

N	H ≅ G/N	type
83. C₈₉	#7297	semidirect

H ≅ G/N

type

83. C₈₉

#7297

semidirect

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

subgroup	index
C₄×#15667948	2
#26019015	2
C₈₉×#2162	2
#7297	89

subgroup

index

C₄×#15667948

#26019015

C₈₉×#2162

#7297

Subgroups — #36441679

Conjugacy classes of subgroups

Complements

Maximal subgroups