Group #36441689 — subgroups

Subgroups — #36441689

Order 22784.

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

normal subgroups only

This group has 212 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	#26022055	C₂	1	1	yes	maximal, characteristic
2	#26019008	C₂	1	1	yes	maximal, characteristic
3	C₄×#108×C₈₉	C₂	1	1	yes	maximal, characteristic
4	#19151622	—	2	1	no	—
5	C₄⋊#13083549	C₄	1	2	yes	—
6	C₂²⋊#13083549	—	2	1	no	—
7	#19151374	—	2	1	no	—
8	#19151349	—	2	1	no	—
9	#19151349	C₄	1	2	yes	—
10	C₂×#108×C₈₉	—	2	1	no	—
11	C₄×M₁₆×C₈₉	—	2	1	no	—
12	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
13	#341×C₈₉	C₄	1	2	yes	—
14	C₂×#13083558	—	4	1	no	—
15	C₂×#13083549	—	2	2	no	—
16	C₄⋊#10623494	—	2	2	no	—
17	C₄⋊#10623494	—	4	1	no	—
18	C₂×M₁₆×C₈₉	—	2	2	no	—
19	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
21	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
22	#108×C₈₉	—	2	4	no	—
23	#97×C₈₉	—	4	1	no	—
24	C₄×C₈×C₈₉	—	4	1	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	—	8	2	no	—
30	C₂×#10623494	—	4	2	no	—
31	#13083549	—	4	2	no	—
32	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
34	C₂²×C₄×C₈₉	Q₁₆	1	1	yes	characteristic
35	M₁₆×C₈₉	—	2	4	no	—
36	M₁₆×C₈₉	—	4	2	no	—
37	C₂×C₈×C₈₉	—	2	2	no	—
38	C₂×C₈×C₈₉	—	4	1	no	—
39	C₂×C₈×C₈₉	—	4	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	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
45	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
46	C₄²×C₈₉	D₈	1	1	yes	characteristic
47	C₄²×C₈₉	SD₁₆	1	1	yes	characteristic
48	C₂³×C₈₉	#103	1	1	yes	characteristic
49	C₂×C₄×C₈₉	—	2	1	no	—
50	C₂×C₄×C₈₉	—	2	1	no	—
51	C₂×C₄×C₈₉	—	2	2	no	—
52	C₂×C₄×C₈₉	—	4	1	no	—
53	C₂×C₄×C₈₉	#95	1	1	yes	characteristic
54	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
55	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
56	C₂×C₄×C₈₉	#103	1	1	yes	characteristic
57	C₂×C₄×C₈₉	#106	1	1	yes	characteristic
58	C₂×C₄×C₈₉	#107	1	1	yes	characteristic
59	C₈×C₈₉	—	2	1	no	—
60	C₈×C₈₉	—	2	1	no	—
61	C₈×C₈₉	—	2	1	no	—
62	C₈×C₈₉	—	4	4	no	—
63	C₈×C₈₉	#104	1	2	yes	—
64	#10623494	—	8	2	no	—
65	C₂²×C₈₉	—	2	1	no	—
66	C₂²×C₈₉	—	2	1	no	—
67	C₂²×C₈₉	—	2	1	no	—
68	C₂²×C₈₉	#340	1	1	yes	characteristic
69	C₄×C₈₉	—	2	1	no	—
70	C₄×C₈₉	—	2	1	no	—
71	C₄×C₈₉	—	2	1	no	—
72	C₄×C₈₉	—	2	1	no	—
73	C₄×C₈₉	#339	1	1	yes	characteristic
74	C₄×C₈₉	#344	1	1	yes	characteristic
75	C₄×C₈₉	#365	1	2	yes	—
76	#7300	—	89	1	no	maximal, Sylow
77	C₂×C₈₉	—	4	1	no	—
78	C₂×C₈₉	#1303	1	1	yes	characteristic
79	C₂×C₈₉	#1383	1	1	yes	characteristic
80	C₂×C₈₉	#1388	1	1	yes	characteristic
81	#2163	—	89	1	no	—
82	C₄×#108	D₈₉	1	1	yes	characteristic
83	#1566	—	89	1	no	—
84	C₈₉	#7300	1	1	yes	characteristic, complemented, Sylow
85	C₂×#108	—	2	1	no	—
86	#423	—	178	1	no	—
87	C₄×M₁₆	—	2	1	no	—
88	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
89	C₄⋊C₁₆	—	89	2	no	—
90	C₂²⋊C₁₆	—	178	1	no	—
91	#346	—	178	1	no	—
92	#341	Dic₈₉	1	2	yes	—
93	#334	—	89	2	no	—
94	#334	—	178	1	no	—
95	C₂×M₁₆	—	2	2	no	—
96	C₂×M₁₆	—	356	1	no	—
97	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
98	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
99	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
100	C₂×C₁₆	—	178	2	no	—
101	#108	—	2	4	no	—
102	C₄⋊C₈	—	356	1	no	—
103	C₄⋊C₈	—	178	2	no	—
104	#97	—	4	1	no	—
105	C₄×C₈	—	4	1	no	—
106	C₄×C₈	Dic₁₇₈	1	1	yes	characteristic
107	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
108	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
109	C₄×C₈	D₃₅₆	1	1	yes	characteristic
110	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
111	C₂²×C₄	#13083561	1	1	yes	characteristic
112	C₂²×C₄	#13083564	1	1	yes	characteristic
113	M₁₆	—	2	4	no	—
114	M₁₆	—	4	2	no	—
115	M₁₆	—	712	2	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	4	1	no	—
118	C₂×C₈	—	4	2	no	—
119	C₂×C₈	—	356	2	no	—
120	C₂×C₈	C₄×Dic₈₉	1	1	yes	characteristic
121	C₂×C₈	#13083551	1	1	yes	characteristic
122	C₂×C₈	#13083553	1	1	yes	characteristic
123	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
124	C₄²	#13083551	1	1	yes	characteristic
125	C₄²	#13083553	1	1	yes	characteristic
126	C₄²	#13083559	1	1	yes	characteristic
127	C₄²	#13083562	1	1	yes	characteristic
128	C₁₆	—	356	2	no	—
129	C₂³	#15667936	1	1	yes	characteristic
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	#15667919	1	1	yes	characteristic
135	C₂×C₄	#15667932	1	1	yes	characteristic
136	C₂×C₄	#15667933	1	1	yes	characteristic
137	C₂×C₄	#15667935	1	1	yes	characteristic
138	C₂×C₄	#15667943	1	1	yes	characteristic
139	C₂×C₄	#15667945	1	1	yes	characteristic
140	C₈	—	2	1	no	—
141	C₈	—	2	1	no	—
142	C₈	—	2	1	no	—
143	C₈	—	4	4	no	—
144	C₈	—	712	2	no	—
145	C₈	#15667940	1	2	yes	—
146	C₂²	—	2	1	no	—
147	C₂²	—	2	1	no	—
148	C₂²	—	2	1	no	—
149	C₂²	#19151360	1	1	yes	characteristic
150	C₄	—	2	1	no	—
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	#19151358	1	1	yes	characteristic
155	C₄	#19151368	1	1	yes	characteristic
156	C₄	#19151422	1	2	yes	—
157	C₂	—	4	1	no	—
158	C₂	#26018155	1	1	yes	characteristic
159	C₂	#26018352	1	1	yes	characteristic
160	C₂	#26018363	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26022055

C₂

yes

maximal, characteristic

#26019008

C₂

yes

maximal, characteristic

C₄×#108×C₈₉

C₂

yes

maximal, characteristic

#19151622

—

C₄⋊#13083549

C₄

yes

—

—

—

—

C₄

yes

—

C₂×#108×C₈₉

—

C₄×M₁₆×C₈₉

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

#341×C₈₉

C₄

yes

—

—

—

—

—

C₂×M₁₆×C₈₉

—

C₂²×C₈×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₈₉

D₄

yes

characteristic

C₂×C₄²×C₈₉

D₄

yes

characteristic

#108×C₈₉

—

#97×C₈₉

—

C₄×C₈×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

M₁₆×C₈₉

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

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

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

—

C₈×C₈₉

#104

yes

—

#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

—

#7300

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

D₈₉

yes

characteristic

#1566

—

C₈₉

#7300

yes

characteristic, complemented, Sylow

C₂×#108

—

#423

—

178

—

C₄×M₁₆

—

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

178

—

#346

—

178

—

#341

Dic₈₉

yes

—

#334

—

#334

—

178

—

C₂×M₁₆

—

C₂×M₁₆

—

356

—

C₂²×C₈

C₂×Dic₈₉

yes

characteristic

C₂²×C₈

C₂²⋊D₈₉

yes

characteristic

C₂×C₄²

C₂²⋊D₈₉

yes

characteristic

100

C₂×C₁₆

—

178

—

101

#108

—

102

C₄⋊C₈

—

356

—

103

C₄⋊C₈

—

178

—

104

#97

—

105

C₄×C₈

—

106

C₄×C₈

Dic₁₇₈

yes

characteristic

107

C₄×C₈

C₄×D₈₉

yes

characteristic

108

C₄×C₈

C₄×D₈₉

yes

characteristic

109

C₄×C₈

D₃₅₆

yes

characteristic

110

C₂²×C₄

C₂²⋊Dic₈₉

yes

characteristic

111

C₂²×C₄

#13083561

yes

characteristic

112

C₂²×C₄

#13083564

yes

characteristic

113

M₁₆

—

114

M₁₆

—

115

M₁₆

—

712

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

356

—

120

C₂×C₈

C₄×Dic₈₉

yes

characteristic

121

C₂×C₈

#13083551

yes

characteristic

122

C₂×C₈

#13083553

yes

characteristic

123

C₂×C₈

C₄⋊Dic₈₉

yes

characteristic

124

C₄²

#13083551

yes

characteristic

125

C₄²

#13083553

yes

characteristic

126

C₄²

#13083559

yes

characteristic

127

C₄²

#13083562

yes

characteristic

128

C₁₆

—

356

—

129

C₂³

#15667936

yes

characteristic

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

#15667919

yes

characteristic

135

C₂×C₄

#15667932

yes

characteristic

136

C₂×C₄

#15667933

yes

characteristic

137

C₂×C₄

#15667935

yes

characteristic

138

C₂×C₄

#15667943

yes

characteristic

139

C₂×C₄

#15667945

yes

characteristic

140

C₈

—

141

C₈

—

142

C₈

—

143

C₈

—

144

C₈

—

712

—

145

C₈

#15667940

yes

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

#19151360

yes

characteristic

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

#19151358

yes

characteristic

155

C₄

#19151368

yes

characteristic

156

C₄

#19151422

yes

—

157

C₂

—

158

C₂

#26018155

yes

characteristic

159

C₂

#26018352

yes

characteristic

160

C₂

#26018363

yes

characteristic

	subgroup	G/H
center	C₂²	#19151360
commutator	C₂×C₈×C₈₉	C₄²
Frattini	C₂×C₄×C₈	C₂×D₈₉
Fitting	C₄×#108×C₈₉	C₂
socle	C₂²×C₈₉	#340
prime radical	C₂³×C₈₉	#103
cosocle	C₂×C₄×C₈×C₈₉	C₂²
hypercenter	C₄×#108	D₈₉
hypocenter	C₈₉	#7300
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#108

subgroup

G/H

center

C₂²

#19151360

commutator

C₂×C₈×C₈₉

C₄²

Frattini

C₂×C₄×C₈

C₂×D₈₉

Fitting

C₄×#108×C₈₉

C₂

C₂²×C₈₉

C₂³×C₈₉

C₂×C₄×C₈×C₈₉

C₂²

D₈₉

C₈₉

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

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#108

N	H ≅ G/N	type
84. C₈₉	#7300	semidirect

H ≅ G/N

type

84. C₈₉

#7300

semidirect

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

subgroup	index
#26022055	2
#26019008	2
C₄×#108×C₈₉	2
#7300	89

subgroup

index

#26022055

#26019008

C₄×#108×C₈₉

#7300

Subgroups — #36441689

Conjugacy classes of subgroups

Complements

Maximal subgroups