Group #36441811 — subgroups

Subgroups — #36441811

Order 22784.

268 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 47 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	#26018198	C₂	1	2	yes	maximal, complemented
2	C₈₉×#1957	C₂	1	1	yes	maximal, characteristic
3	#19151324	—	2	2	no	—
4	C₂×#127×C₈₉	C₄	1	1	yes	characteristic
5	C₂×#102×C₈₉	—	2	2	no	—
6	C₂×C₄×C₈×C₈₉	C₄	1	1	yes	characteristic
7	#384×C₈₉	C₂²	1	1	yes	characteristic
8	C₂²⋊#10623494	—	4	2	no	—
9	C₂²×D₄×C₈₉	—	2	1	no	—
10	C₂²×D₄×C₈₉	—	4	1	no	—
11	C₂²×C₈×C₈₉	—	2	1	no	—
12	#127×C₈₉	—	2	1	no	—
13	#127×C₈₉	—	4	1	no	—
14	#127×C₈₉	C₈	1	2	yes	complemented
15	C₂×C₄⋊C₄×C₈₉	—	4	1	no	—
16	C₂×C₄⋊C₄×C₈₉	D₄	1	2	yes	—
17	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
18	#102×C₈₉	—	4	4	no	—
19	C₄×C₈×C₈₉	—	4	1	no	—
20	C₂×#10623494	—	8	2	no	—
21	C₂⁴×C₈₉	—	4	1	no	—
22	C₂×D₄×C₈₉	—	2	2	no	—
23	C₂×D₄×C₈₉	—	4	2	no	—
24	C₂×D₄×C₈₉	—	4	2	no	—
25	C₂×D₄×C₈₉	—	4	2	no	—
26	C₂×D₄×C₈₉	—	4	2	no	—
27	C₂×D₄×C₈₉	—	4	1	no	—
28	C₂×D₄×C₈₉	—	8	2	no	—
29	C₂×D₄×C₈₉	—	8	2	no	—
30	C₂²×C₄×C₈₉	—	2	1	no	—
31	C₂²×C₄×C₈₉	—	2	2	no	—
32	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₈×C₈₉	—	2	2	no	—
34	C₂×C₈×C₈₉	—	4	1	no	—
35	C₂×C₈×C₈₉	—	4	1	no	—
36	C₄⋊C₄×C₈₉	—	4	2	no	—
37	C₄⋊C₄×C₈₉	—	8	2	no	—
38	C₄²×C₈₉	—	2	1	no	—
39	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
40	C₄²×C₈₉	M₁₆	1	1	yes	characteristic
41	C₂³×C₈₉	—	4	2	no	—
42	C₂³×C₈₉	—	8	2	no	—
43	C₂³×C₈₉	—	8	2	no	—
44	C₂³×C₈₉	—	8	1	no	—
45	C₂³×C₈₉	—	8	1	no	—
46	C₂³×C₈₉	#99	1	1	yes	characteristic
47	D₄×C₈₉	—	4	4	no	—
48	D₄×C₈₉	—	8	2	no	—
49	D₄×C₈₉	—	8	4	no	—
50	D₄×C₈₉	—	8	4	no	—
51	C₂×C₄×C₈₉	—	2	2	no	—
52	C₂×C₄×C₈₉	—	2	2	no	—
53	C₂×C₄×C₈₉	—	2	1	no	—
54	C₂×C₄×C₈₉	—	2	1	no	—
55	C₂×C₄×C₈₉	—	2	1	no	—
56	C₂×C₄×C₈₉	—	2	1	no	—
57	C₂×C₄×C₈₉	—	4	2	no	—
58	C₂×C₄×C₈₉	—	8	2	no	—
59	C₂×C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
60	C₂×C₄×C₈₉	#100	1	1	yes	characteristic
61	C₈×C₈₉	—	4	2	no	—
62	#10623494	—	16	2	no	—
63	C₂²×C₈₉	—	2	1	no	—
64	C₂²×C₈₉	—	2	1	no	—
65	C₂²×C₈₉	—	8	4	no	—
66	C₂²×C₈₉	—	8	2	no	—
67	C₂²×C₈₉	—	16	2	no	—
68	C₂²×C₈₉	—	16	1	no	—
69	C₂²×C₈₉	—	16	1	no	—
70	C₂²×C₈₉	#323	1	1	yes	characteristic
71	C₂²×C₈₉	#353	1	1	yes	characteristic
72	C₂²×C₈₉	#355	1	1	yes	characteristic
73	C₄×C₈₉	—	2	2	no	—
74	C₄×C₈₉	—	2	2	no	—
75	C₄×C₈₉	—	2	1	no	—
76	C₄×C₈₉	—	2	1	no	—
77	C₄×C₈₉	—	8	2	no	—
78	#7354	—	89	1	no	maximal, Sylow
79	C₂×C₈₉	—	2	1	no	—
80	C₂×C₈₉	—	2	1	no	—
81	C₂×C₈₉	—	16	2	no	—
82	C₂×C₈₉	#1318	1	1	yes	characteristic
83	C₂×C₈₉	#1408	1	1	yes	characteristic
84	C₂×C₈₉	#1410	1	1	yes	characteristic
85	#1957	D₈₉	1	1	yes	characteristic
86	#1325	—	89	2	no	—
87	C₈₉	#7354	1	1	yes	characteristic, complemented, Sylow
88	C₂×#127	Dic₈₉	1	1	yes	characteristic
89	C₂×#102	—	2	2	no	—
90	C₂×C₄×C₈	Dic₈₉	1	1	yes	characteristic
91	#384	C₂×D₈₉	1	1	yes	characteristic
92	#324	—	178	2	no	—
93	C₂²×D₄	—	2	1	no	—
94	C₂²×D₄	—	4	1	no	—
95	C₂²×C₈	—	2	1	no	—
96	#127	—	2	1	no	—
97	#127	—	4	1	no	—
98	#127	#10623494	1	2	yes	complemented
99	C₂×C₄⋊C₄	—	4	1	no	—
100	C₂×C₄⋊C₄	C₂²⋊D₈₉	1	2	yes	—
101	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
102	#102	—	4	4	no	—
103	C₂²⋊C₈	—	356	2	no	—
104	C₄×C₈	—	4	1	no	—
105	C₂⁴	—	4	1	no	—
106	C₂×D₄	—	2	2	no	—
107	C₂×D₄	—	4	2	no	—
108	C₂×D₄	—	4	2	no	—
109	C₂×D₄	—	4	2	no	—
110	C₂×D₄	—	4	2	no	—
111	C₂×D₄	—	4	1	no	—
112	C₂×D₄	—	8	2	no	—
113	C₂×D₄	—	8	2	no	—
114	C₂²×C₄	—	2	1	no	—
115	C₂²×C₄	—	2	2	no	—
116	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	4	1	no	—
119	C₂×C₈	—	4	1	no	—
120	C₂×C₈	—	712	2	no	—
121	C₄⋊C₄	—	4	2	no	—
122	C₄⋊C₄	—	8	2	no	—
123	C₄²	—	2	1	no	—
124	C₄²	C₂×#10623494	1	1	yes	characteristic
125	C₄²	#13083558	1	1	yes	characteristic
126	C₂³	—	4	2	no	—
127	C₂³	—	8	2	no	—
128	C₂³	—	8	2	no	—
129	C₂³	—	8	1	no	—
130	C₂³	—	8	1	no	—
131	C₂³	#15667927	1	1	yes	characteristic
132	D₄	—	4	4	no	—
133	D₄	—	8	2	no	—
134	D₄	—	8	4	no	—
135	D₄	—	8	4	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	8	2	no	—
144	C₂×C₄	C₂²⋊#10623494	1	1	yes	characteristic
145	C₂×C₄	#15667929	1	1	yes	characteristic
146	C₈	—	4	2	no	—
147	C₈	—	1424	2	no	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	—	8	4	no	—
151	C₂²	—	8	2	no	—
152	C₂²	—	16	2	no	—
153	C₂²	—	16	1	no	—
154	C₂²	—	16	1	no	—
155	C₂²	#19151322	1	1	yes	characteristic
156	C₂²	#19151391	1	1	yes	characteristic
157	C₂²	#19151395	1	1	yes	characteristic
158	C₄	—	2	2	no	—
159	C₄	—	2	2	no	—
160	C₄	—	2	1	no	—
161	C₄	—	2	1	no	—
162	C₄	—	8	2	no	—
163	C₂	—	2	1	no	—
164	C₂	—	2	1	no	—
165	C₂	—	16	2	no	—
166	C₂	#26018184	1	1	yes	characteristic
167	C₂	#26018415	1	1	yes	characteristic
168	C₂	#26018419	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26018198

C₂

yes

maximal, complemented

C₈₉×#1957

C₂

yes

maximal, characteristic

#19151324

—

C₂×#127×C₈₉

C₄

yes

characteristic

C₂×#102×C₈₉

—

C₂×C₄×C₈×C₈₉

C₄

yes

characteristic

#384×C₈₉

C₂²

yes

characteristic

C₂²⋊#10623494

—

C₂²×D₄×C₈₉

—

C₂²×D₄×C₈₉

—

C₂²×C₈×C₈₉

—

#127×C₈₉

—

#127×C₈₉

—

#127×C₈₉

C₈

yes

complemented

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

C₂×C₄

yes

characteristic

#102×C₈₉

—

C₄×C₈×C₈₉

—

C₂×#10623494

—

C₂⁴×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

C₂×C₈

yes

characteristic

C₄²×C₈₉

M₁₆

yes

characteristic

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

#99

yes

characteristic

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×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

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₈₉

#100

yes

characteristic

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

#7354

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

D₈₉

yes

characteristic

#1325

—

C₈₉

#7354

yes

characteristic, complemented, Sylow

C₂×#127

Dic₈₉

yes

characteristic

C₂×#102

—

C₂×C₄×C₈

Dic₈₉

yes

characteristic

#384

C₂×D₈₉

yes

characteristic

#324

—

178

—

C₂²×D₄

—

C₂²×D₄

—

C₂²×C₈

—

—

—

yes

C₂×C₄⋊C₄

—

100

C₂×C₄⋊C₄

C₂²⋊D₈₉

yes

—

101

C₂×C₄²

C₂×Dic₈₉

yes

characteristic

102

#102

—

103

C₂²⋊C₈

—

356

—

104

C₄×C₈

—

105

C₂⁴

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

C₂²⋊Dic₈₉

yes

characteristic

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

712

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄²

—

124

C₄²

C₂×#10623494

yes

characteristic

125

C₄²

#13083558

yes

characteristic

126

C₂³

—

127

C₂³

—

128

C₂³

—

129

C₂³

—

130

C₂³

—

131

C₂³

#15667927

yes

characteristic

132

D₄

—

133

D₄

—

134

D₄

—

135

D₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

C₂²⋊#10623494

yes

characteristic

145

C₂×C₄

#15667929

yes

characteristic

146

C₈

—

147

C₈

—

1424

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

#19151322

yes

characteristic

156

C₂²

#19151391

yes

characteristic

157

C₂²

#19151395

yes

characteristic

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₂

—

164

C₂

—

165

C₂

—

166

C₂

#26018184

yes

characteristic

167

C₂

#26018415

yes

characteristic

168

C₂

#26018419

yes

characteristic

	subgroup	G/H
center	C₂²	#19151391
commutator	C₄²×C₈₉	C₂×C₈
Frattini	#384	C₂×D₈₉
Fitting	C₈₉×#1957	C₂
socle	C₂²×C₈₉	#353
prime radical	C₂×#127×C₈₉	C₄
cosocle	#384×C₈₉	C₂²
hypercenter	#1957	D₈₉
hypocenter	C₈₉	#7354
LCS	G ▷ C₄²×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ #384 ◁ #1957

subgroup

G/H

center

C₂²

#19151391

commutator

C₄²×C₈₉

C₂×C₈

Frattini

#384

C₂×D₈₉

Fitting

C₈₉×#1957

C₂

socle

C₂²×C₈₉

#353

prime radical

C₂×#127×C₈₉

C₄

cosocle

#384×C₈₉

C₂²

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ #384 ◁ #1957

N	H ≅ G/N	type
1. #26018198	C₂	semidirect
14. #127×C₈₉	C₈	semidirect
87. C₈₉	#7354	semidirect
98. #127	#10623494	semidirect

H ≅ G/N

type

1. #26018198

C₂

semidirect

14. #127×C₈₉

C₈

semidirect

87. C₈₉

#7354

semidirect

98. #127

#10623494

semidirect

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

subgroup	index
#26018198	2
C₈₉×#1957	2
#7354	89

subgroup

index

#26018198

C₈₉×#1957

#7354

Subgroups — #36441811

Conjugacy classes of subgroups

Complements

Maximal subgroups