Group #36441039 — subgroups

Subgroups — #36441039

Order 22784.

304 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 115 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	#26020105	C₂	1	2	yes	maximal
2	C₈₉×#1831	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊#10623494	—	2	2	no	—
4	C₂×#15667923	—	2	2	no	—
5	#19151353	C₄	1	4	yes	—
6	C₂×C₄⋊C₈×C₈₉	C₄	1	2	yes	—
7	C₄³×C₈₉	C₂²	1	1	yes	characteristic
8	#336×C₈₉	—	2	2	no	—
9	C₂²×#10623494	—	2	4	no	—
10	C₄⋊#10623494	—	2	8	no	—
11	#15667923	—	4	2	no	—
12	#15667923	—	4	2	no	—
13	C₂²×C₈×C₈₉	—	2	2	no	—
14	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
15	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
17	C₂×C₄²×C₈₉	D₄	1	2	yes	—
18	C₂×C₄²×C₈₉	Q₈	1	1	yes	characteristic
19	C₄⋊C₈×C₈₉	—	4	2	no	—
20	C₂×#10623494	—	2	8	no	—
21	C₂×#10623494	—	4	4	no	—
22	C₂×#10623494	—	4	4	no	—
23	C₂²×C₄×C₈₉	C₄²	1	1	yes	characteristic
24	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
25	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
26	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
27	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₈×C₈₉	—	4	2	no	—
29	C₂×C₈×C₈₉	—	4	4	no	—
30	C₄²×C₈₉	—	2	1	no	—
31	C₄²×C₈₉	—	2	1	no	—
32	C₄²×C₈₉	—	2	4	no	—
33	C₄²×C₈₉	—	4	2	no	—
34	C₄²×C₈₉	—	4	1	no	—
35	C₄²×C₈₉	D₈	1	1	yes	characteristic
36	C₄²×C₈₉	SD₁₆	1	2	yes	—
37	C₄²×C₈₉	Q₁₆	1	1	yes	characteristic
38	C₂³×C₈₉	#95	1	1	yes	characteristic
39	C₂×C₄×C₈₉	—	2	1	no	—
40	C₂×C₄×C₈₉	—	2	2	no	—
41	C₂×C₄×C₈₉	—	2	1	no	—
42	C₂×C₄×C₈₉	—	2	2	no	—
43	C₂×C₄×C₈₉	—	2	2	no	—
44	C₂×C₄×C₈₉	—	2	2	no	—
45	C₂×C₄×C₈₉	—	2	1	no	—
46	C₂×C₄×C₈₉	—	2	2	no	—
47	C₂×C₄×C₈₉	—	2	2	no	—
48	C₂×C₄×C₈₉	#100	1	1	yes	characteristic
49	C₂×C₄×C₈₉	#101	1	1	yes	characteristic
50	C₂×C₄×C₈₉	#102	1	2	yes	—
51	C₂×C₄×C₈₉	#103	1	2	yes	—
52	C₂×C₄×C₈₉	#104	1	4	yes	—
53	C₂×C₄×C₈₉	#106	1	1	yes	characteristic
54	C₂×C₄×C₈₉	#107	1	1	yes	characteristic
55	C₈×C₈₉	—	8	2	no	—
56	#10623494	—	4	8	no	—
57	C₂²×C₈₉	#338	1	1	yes	characteristic
58	C₂²×C₈₉	#339	1	2	yes	—
59	C₂²×C₈₉	#340	1	1	yes	characteristic
60	C₂²×C₈₉	#343	1	1	yes	characteristic
61	C₂²×C₈₉	#344	1	2	yes	—
62	C₄×C₈₉	—	2	1	no	—
63	C₄×C₈₉	—	2	1	no	—
64	C₄×C₈₉	—	2	2	no	—
65	C₄×C₈₉	—	2	2	no	—
66	C₄×C₈₉	—	4	2	no	—
67	C₄×C₈₉	—	4	1	no	—
68	C₄×C₈₉	#329	1	2	yes	—
69	C₄×C₈₉	#330	1	2	yes	—
70	#7049	—	89	1	no	maximal, Sylow
71	C₂×C₈₉	#1291	1	2	yes	—
72	C₂×C₈₉	#1292	1	1	yes	characteristic
73	C₂×C₈₉	#1295	1	1	yes	characteristic
74	C₂×C₈₉	#1303	1	2	yes	—
75	C₂×C₈₉	#1307	1	1	yes	characteristic
76	#1831	D₈₉	1	1	yes	characteristic
77	#1767	—	89	2	no	—
78	C₈₉	#7049	1	1	yes	characteristic, complemented, Sylow
79	C₂×C₄⋊C₈	—	178	2	no	—
80	C₂×C₄⋊C₈	Dic₈₉	1	2	yes	—
81	C₂×#97	—	178	2	no	—
82	C₄³	C₂×D₈₉	1	1	yes	characteristic
83	#336	—	2	2	no	—
84	#336	—	89	4	no	—
85	C₂²×C₈	—	2	2	no	—
86	C₂²×C₈	—	178	4	no	—
87	C₂×C₄²	Dic₁₇₈	1	1	yes	characteristic
88	C₂×C₄²	C₄×D₈₉	1	2	yes	—
89	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
90	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
91	C₂×C₄²	C₂²⋊D₈₉	1	2	yes	—
92	C₄⋊C₈	—	4	2	no	—
93	C₄⋊C₈	—	178	8	no	—
94	#97	—	356	2	no	—
95	#97	—	356	2	no	—
96	C₂²×C₄	C₄×Dic₈₉	1	1	yes	characteristic
97	C₂²×C₄	#13083551	1	2	yes	—
98	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
99	C₂²×C₄	#13083553	1	2	yes	—
100	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
101	C₂×C₈	—	4	2	no	—
102	C₂×C₈	—	4	4	no	—
103	C₂×C₈	—	356	4	no	—
104	C₂×C₈	—	356	4	no	—
105	C₂×C₈	—	178	8	no	—
106	C₄²	—	2	1	no	—
107	C₄²	—	2	1	no	—
108	C₄²	—	2	4	no	—
109	C₄²	—	4	1	no	—
110	C₄²	—	4	2	no	—
111	C₄²	D₇₁₂	1	1	yes	characteristic
112	C₄²	#13083563	1	2	yes	—
113	C₄²	Dic₃₅₆	1	1	yes	characteristic
114	C₂³	#15667919	1	1	yes	characteristic
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	#15667929	1	1	yes	characteristic
125	C₂×C₄	#15667931	1	1	yes	characteristic
126	C₂×C₄	#15667934	1	2	yes	—
127	C₂×C₄	#15667937	1	2	yes	—
128	C₂×C₄	#15667939	1	4	yes	—
129	C₂×C₄	#15667944	1	1	yes	characteristic
130	C₂×C₄	#15667946	1	1	yes	characteristic
131	C₈	—	8	2	no	—
132	C₈	—	356	8	no	—
133	C₂²	#19151356	1	1	yes	characteristic
134	C₂²	#19151357	1	2	yes	—
135	C₂²	#19151361	1	1	yes	characteristic
136	C₂²	#19151367	1	1	yes	characteristic
137	C₂²	#19151370	1	2	yes	—
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	—	2	2	no	—
141	C₄	—	2	2	no	—
142	C₄	—	4	2	no	—
143	C₄	—	4	1	no	—
144	C₄	#19151338	1	2	yes	—
145	C₄	#19151340	1	2	yes	—
146	C₂	#26018123	1	2	yes	—
147	C₂	#26018127	1	1	yes	characteristic
148	C₂	#26018133	1	1	yes	characteristic
149	C₂	#26018154	1	2	yes	—
150	C₂	#26018163	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26020105

C₂

yes

maximal

C₈₉×#1831

C₂

yes

maximal, characteristic

C₂×C₄⋊#10623494

—

C₂×#15667923

—

#19151353

C₄

yes

—

C₂×C₄⋊C₈×C₈₉

C₄

yes

—

C₄³×C₈₉

C₂²

yes

characteristic

#336×C₈₉

—

—

—

—

—

C₂²×C₈×C₈₉

—

C₂×C₄²×C₈₉

C₂×C₄

yes

—

C₂×C₄²×C₈₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₈₉

D₄

yes

characteristic

C₂×C₄²×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

Q₈

yes

characteristic

C₄⋊C₈×C₈₉

—

C₂×#10623494

—

C₂×#10623494

—

C₂×#10623494

—

C₂²×C₄×C₈₉

C₄²

yes

characteristic

C₂²×C₄×C₈₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₈₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₈₉

C₄⋊C₄

yes

—

C₂²×C₄×C₈₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

D₈

yes

characteristic

C₄²×C₈₉

SD₁₆

yes

—

C₄²×C₈₉

Q₁₆

yes

characteristic

C₂³×C₈₉

#95

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

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

#103

yes

—

C₂×C₄×C₈₉

#104

yes

—

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

#338

yes

characteristic

C₂²×C₈₉

#339

yes

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#329

yes

—

C₄×C₈₉

#330

yes

—

#7049

—

maximal, Sylow

C₂×C₈₉

#1291

yes

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₂×C₈₉

#1307

yes

characteristic

#1831

D₈₉

yes

characteristic

#1767

—

C₈₉

#7049

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

178

—

C₂×C₄⋊C₈

Dic₈₉

yes

—

C₂×#97

—

178

—

C₄³

C₂×D₈₉

yes

characteristic

#336

—

#336

—

C₂²×C₈

—

C₂²×C₈

—

178

—

C₂×C₄²

Dic₁₇₈

yes

characteristic

C₂×C₄²

C₄×D₈₉

yes

—

C₂×C₄²

D₃₅₆

yes

characteristic

C₂×C₄²

C₂×Dic₈₉

yes

characteristic

C₂×C₄²

C₂²⋊D₈₉

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

178

—

#97

—

356

—

#97

—

356

—

C₂²×C₄

C₄×Dic₈₉

yes

characteristic

C₂²×C₄

#13083551

yes

—

C₂²×C₄

C₂²⋊Dic₈₉

yes

characteristic

C₂²×C₄

#13083553

yes

—

100

C₂²×C₄

C₄⋊Dic₈₉

yes

characteristic

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

356

—

104

C₂×C₈

—

356

—

105

C₂×C₈

—

178

—

106

C₄²

—

107

C₄²

—

108

C₄²

—

109

C₄²

—

110

C₄²

—

111

C₄²

D₇₁₂

yes

characteristic

112

C₄²

#13083563

yes

—

113

C₄²

Dic₃₅₆

yes

characteristic

114

C₂³

#15667919

yes

characteristic

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

#15667929

yes

characteristic

125

C₂×C₄

#15667931

yes

characteristic

126

C₂×C₄

#15667934

yes

—

127

C₂×C₄

#15667937

yes

—

128

C₂×C₄

#15667939

yes

—

129

C₂×C₄

#15667944

yes

characteristic

130

C₂×C₄

#15667946

yes

characteristic

131

C₈

—

132

C₈

—

356

—

133

C₂²

#19151356

yes

characteristic

134

C₂²

#19151357

yes

—

135

C₂²

#19151361

yes

characteristic

136

C₂²

#19151367

yes

characteristic

137

C₂²

#19151370

yes

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

#19151338

yes

—

145

C₄

#19151340

yes

—

146

C₂

#26018123

yes

—

147

C₂

#26018127

yes

characteristic

148

C₂

#26018133

yes

characteristic

149

C₂

#26018154

yes

—

150

C₂

#26018163

yes

characteristic

	subgroup	G/H
center	C₂³	#15667919
commutator	C₂²×C₄×C₈₉	C₄²
Frattini	C₄³	C₂×D₈₉
Fitting	C₈₉×#1831	C₂
socle, prime radical	C₂³×C₈₉	#95
cosocle	C₄³×C₈₉	C₂²
hypercenter	#1831	D₈₉
hypocenter	C₈₉	#7049
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

subgroup

G/H

center

C₂³

#15667919

commutator

C₂²×C₄×C₈₉

C₄²

Frattini

C₄³

C₂×D₈₉

Fitting

C₈₉×#1831

C₂

socle, prime radical

C₂³×C₈₉

#95

cosocle

C₄³×C₈₉

C₂²

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

N	H ≅ G/N	type
78. C₈₉	#7049	semidirect

H ≅ G/N

type

78. C₈₉

#7049

semidirect

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

subgroup	index
#26020105	2
C₈₉×#1831	2
#7049	89

subgroup

index

#26020105

C₈₉×#1831

#7049

Subgroups — #36441039

Conjugacy classes of subgroups

Complements

Maximal subgroups