Group #36441672 — subgroups

Subgroups — #36441672

Order 22784.

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

normal subgroups only

This group has 188 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	#26019000	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#1566	C₂	1	1	yes	maximal, characteristic
4	#19151651	—	2	1	no	—
5	C₄⋊#13083549	—	2	1	no	—
6	C₂²⋊#13083549	C₄	1	2	yes	—
7	#19151374	—	2	1	no	—
8	#19151351	C₄	1	2	yes	—
9	#19151349	—	2	1	no	—
10	#423×C₈₉	—	2	1	no	—
11	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
12	#334×C₈₉	—	2	1	no	—
13	#334×C₈₉	C₄	1	2	yes	—
14	C₂×#13083549	—	2	2	no	—
15	C₄⋊#10623494	—	2	2	no	—
16	C₂²⋊#10623494	—	4	1	no	—
17	C₄×#10623494	—	4	1	no	—
18	C₂×M₁₆×C₈₉	—	4	1	no	—
19	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₈₉	Q₈	1	1	yes	characteristic
21	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
22	C₄⋊C₈×C₈₉	—	2	2	no	—
23	C₄⋊C₈×C₈₉	—	4	1	no	—
24	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
27	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
28	C₂×#10623494	—	4	2	no	—
29	#13083549	—	4	2	no	—
30	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
31	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
32	C₂²×C₄×C₈₉	Q₁₆	1	1	yes	characteristic
33	M₁₆×C₈₉	—	8	2	no	—
34	C₂×C₈×C₈₉	—	2	2	no	—
35	C₂×C₈×C₈₉	—	4	1	no	—
36	C₂×C₈×C₈₉	—	4	2	no	—
37	C₂×C₈×C₈₉	C₄²	1	1	yes	characteristic
38	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
41	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
42	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
43	C₄²×C₈₉	SD₁₆	1	1	yes	characteristic
44	C₄²×C₈₉	SD₁₆	1	1	yes	characteristic
45	C₂³×C₈₉	#107	1	1	yes	characteristic
46	C₂×C₄×C₈₉	—	2	1	no	—
47	C₂×C₄×C₈₉	—	2	1	no	—
48	C₂×C₄×C₈₉	—	2	2	no	—
49	C₂×C₄×C₈₉	—	4	1	no	—
50	C₂×C₄×C₈₉	#95	1	1	yes	characteristic
51	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
52	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
53	C₂×C₄×C₈₉	#103	1	1	yes	characteristic
54	C₂×C₄×C₈₉	#103	1	1	yes	characteristic
55	C₂×C₄×C₈₉	#106	1	1	yes	characteristic
56	C₈×C₈₉	—	2	1	no	—
57	C₈×C₈₉	—	2	1	no	—
58	C₈×C₈₉	—	2	1	no	—
59	C₈×C₈₉	—	8	2	no	—
60	C₈×C₈₉	#108	1	2	yes	—
61	#10623494	—	8	2	no	—
62	C₂²×C₈₉	—	2	1	no	—
63	C₂²×C₈₉	—	2	1	no	—
64	C₂²×C₈₉	—	2	1	no	—
65	C₂²×C₈₉	#340	1	1	yes	characteristic
66	C₄×C₈₉	—	2	1	no	—
67	C₄×C₈₉	—	2	1	no	—
68	C₄×C₈₉	—	2	1	no	—
69	C₄×C₈₉	—	2	1	no	—
70	C₄×C₈₉	#341	1	1	yes	characteristic
71	C₄×C₈₉	#344	1	1	yes	characteristic
72	C₄×C₈₉	#360	1	2	yes	—
73	#7295	—	89	1	no	maximal, Sylow
74	C₂×C₈₉	—	4	1	no	—
75	C₂×C₈₉	#1301	1	1	yes	characteristic
76	C₂×C₈₉	#1385	1	1	yes	characteristic
77	C₂×C₈₉	#1386	1	1	yes	characteristic
78	#2163	—	89	1	no	—
79	#1566	D₈₉	1	1	yes	characteristic
80	#1564	—	89	1	no	—
81	C₈₉	#7295	1	1	yes	characteristic, complemented, Sylow
82	#431	—	178	1	no	—
83	#423	—	2	1	no	—
84	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
85	C₄⋊C₁₆	—	178	1	no	—
86	C₂²⋊C₁₆	—	89	2	no	—
87	#346	—	178	1	no	—
88	#335	—	89	2	no	—
89	#334	—	2	1	no	—
90	#334	—	178	1	no	—
91	#334	Dic₈₉	1	2	yes	—
92	C₂×M₁₆	—	4	1	no	—
93	C₂²×C₈	Dic₁₇₈	1	1	yes	characteristic
94	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
95	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
96	C₂×C₁₆	—	178	2	no	—
97	C₄⋊C₈	—	2	2	no	—
98	C₄⋊C₈	—	4	1	no	—
99	C₄⋊C₈	—	178	2	no	—
100	C₂²⋊C₈	—	356	1	no	—
101	C₄×C₈	—	356	1	no	—
102	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
103	C₄×C₈	D₃₅₆	1	1	yes	characteristic
104	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
105	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
106	C₂²×C₄	#13083553	1	1	yes	characteristic
107	C₂²×C₄	#13083559	1	1	yes	characteristic
108	C₂²×C₄	#13083564	1	1	yes	characteristic
109	M₁₆	—	8	2	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	4	1	no	—
112	C₂×C₈	—	4	2	no	—
113	C₂×C₈	—	356	2	no	—
114	C₂×C₈	C₄×Dic₈₉	1	1	yes	characteristic
115	C₂×C₈	#13083551	1	1	yes	characteristic
116	C₂×C₈	#13083553	1	1	yes	characteristic
117	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
118	C₄²	#13083551	1	1	yes	characteristic
119	C₄²	C₂²⋊Dic₈₉	1	1	yes	characteristic
120	C₄²	#13083561	1	1	yes	characteristic
121	C₄²	#13083562	1	1	yes	characteristic
122	C₁₆	—	356	2	no	—
123	C₂³	#15667945	1	1	yes	characteristic
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	#15667919	1	1	yes	characteristic
129	C₂×C₄	#15667932	1	1	yes	characteristic
130	C₂×C₄	#15667933	1	1	yes	characteristic
131	C₂×C₄	#15667935	1	1	yes	characteristic
132	C₂×C₄	#15667936	1	1	yes	characteristic
133	C₂×C₄	#15667943	1	1	yes	characteristic
134	C₈	—	2	1	no	—
135	C₈	—	2	1	no	—
136	C₈	—	2	1	no	—
137	C₈	—	8	2	no	—
138	C₈	—	712	2	no	—
139	C₈	#15667947	1	2	yes	—
140	C₂²	—	2	1	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	#19151360	1	1	yes	characteristic
144	C₄	—	2	1	no	—
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	#19151362	1	1	yes	characteristic
149	C₄	#19151369	1	1	yes	characteristic
150	C₄	#19151407	1	2	yes	—
151	C₂	—	4	1	no	—
152	C₂	#26018147	1	1	yes	characteristic
153	C₂	#26018355	1	1	yes	characteristic
154	C₂	#26018359	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26022055

C₂

yes

maximal, characteristic

#26019000

C₂

yes

maximal, characteristic

C₈₉×#1566

C₂

yes

maximal, characteristic

#19151651

—

C₄⋊#13083549

—

C₂²⋊#13083549

C₄

yes

—

#19151374

—

#19151351

C₄

yes

—

#19151349

—

#423×C₈₉

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

#334×C₈₉

—

#334×C₈₉

C₄

yes

—

—

—

—

—

C₂×M₁₆×C₈₉

—

C₂²×C₈×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₈₉

Q₈

yes

characteristic

C₂×C₄²×C₈₉

D₄

yes

characteristic

C₄⋊C₈×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

D₄

yes

characteristic

C₂×#10623494

—

#13083549

—

C₂²×C₄×C₈₉

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₈₉

D₈

yes

characteristic

C₂²×C₄×C₈₉

Q₁₆

yes

characteristic

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

C₂²⋊C₄

yes

characteristic

C₄²×C₈₉

SD₁₆

yes

characteristic

C₄²×C₈₉

SD₁₆

yes

characteristic

C₂³×C₈₉

#107

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

#108

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

—

#7295

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

D₈₉

yes

characteristic

#1564

—

C₈₉

#7295

yes

characteristic, complemented, Sylow

#431

—

178

—

#423

—

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

C₄⋊C₁₆

—

178

—

C₂²⋊C₁₆

—

#346

—

178

—

#335

—

#334

—

#334

—

178

—

#334

Dic₈₉

yes

—

C₂×M₁₆

—

C₂²×C₈

Dic₁₇₈

yes

characteristic

C₂²×C₈

C₄×D₈₉

yes

characteristic

C₂×C₄²

C₂²⋊D₈₉

yes

characteristic

C₂×C₁₆

—

178

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

178

—

100

C₂²⋊C₈

—

356

—

101

C₄×C₈

—

356

—

102

C₄×C₈

C₄×D₈₉

yes

characteristic

103

C₄×C₈

D₃₅₆

yes

characteristic

104

C₄×C₈

C₂×Dic₈₉

yes

characteristic

105

C₄×C₈

C₂²⋊D₈₉

yes

characteristic

106

C₂²×C₄

#13083553

yes

characteristic

107

C₂²×C₄

#13083559

yes

characteristic

108

C₂²×C₄

#13083564

yes

characteristic

109

M₁₆

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

356

—

114

C₂×C₈

C₄×Dic₈₉

yes

characteristic

115

C₂×C₈

#13083551

yes

characteristic

116

C₂×C₈

#13083553

yes

characteristic

117

C₂×C₈

C₄⋊Dic₈₉

yes

characteristic

118

C₄²

#13083551

yes

characteristic

119

C₄²

C₂²⋊Dic₈₉

yes

characteristic

120

C₄²

#13083561

yes

characteristic

121

C₄²

#13083562

yes

characteristic

122

C₁₆

—

356

—

123

C₂³

#15667945

yes

characteristic

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

#15667919

yes

characteristic

129

C₂×C₄

#15667932

yes

characteristic

130

C₂×C₄

#15667933

yes

characteristic

131

C₂×C₄

#15667935

yes

characteristic

132

C₂×C₄

#15667936

yes

characteristic

133

C₂×C₄

#15667943

yes

characteristic

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

—

712

—

139

C₈

#15667947

yes

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

#19151360

yes

characteristic

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

#19151362

yes

characteristic

149

C₄

#19151369

yes

characteristic

150

C₄

#19151407

yes

—

151

C₂

—

152

C₂

#26018147

yes

characteristic

153

C₂

#26018355

yes

characteristic

154

C₂

#26018359

yes

characteristic

	subgroup	G/H
center	C₂²	#19151360
commutator	C₂×C₈×C₈₉	C₄²
Frattini	C₂×C₄×C₈	C₂×D₈₉
Fitting	C₈₉×#1566	C₂
socle	C₂²×C₈₉	#340
prime radical	C₂³×C₈₉	#107
cosocle	C₂×C₄×C₈×C₈₉	C₂²
hypercenter	#1566	D₈₉
hypocenter	C₈₉	#7295
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1566

subgroup

G/H

center

C₂²

#19151360

commutator

C₂×C₈×C₈₉

C₄²

Frattini

C₂×C₄×C₈

C₂×D₈₉

Fitting

C₈₉×#1566

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 ◁ #1566

N	H ≅ G/N	type
81. C₈₉	#7295	semidirect

H ≅ G/N

type

81. C₈₉

#7295

semidirect

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

subgroup	index
#26022055	2
#26019000	2
C₈₉×#1566	2
#7295	89

subgroup

index

#26022055

#26019000

C₈₉×#1566

#7295

Subgroups — #36441672

Conjugacy classes of subgroups

Complements

Maximal subgroups