Group #36474991 — subgroups

Subgroups — #36474991

Order 22784.

186 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 81 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	#26022547	C₂	1	2	yes	maximal, complemented
2	#26022090	C₂	1	1	yes	maximal, characteristic
3	#26018395	C₂	1	2	yes	maximal, complemented
4	C₄×#15667918	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#1422	C₂	1	1	yes	maximal, characteristic, complemented
6	#19152014	C₂²	1	2	yes	—
7	#19151716	C₄	1	1	yes	characteristic
8	#19151431	—	2	2	no	—
9	C₂×#15667918	C₂²	1	1	yes	characteristic
10	C₂×#15667918	C₂²	1	1	yes	characteristic
11	C₃₂×D₈₉	—	2	2	no	—
12	#19151419	C₄	1	1	yes	characteristic
13	#19151381	C₄	1	2	yes	—
14	M₆₄×C₈₉	C₂²	1	2	yes	—
15	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
16	#15668034	C₈	1	1	yes	characteristic
17	#15668018	C₂×C₄	1	2	yes	—
18	#15667952	C₂×C₄	1	2	yes	—
19	#15667951	—	2	2	no	—
20	C₁₆×D₈₉	—	2	2	no	—
21	#15667948	C₈	1	1	yes	characteristic
22	#15667940	C₈	1	2	yes	—
23	#15667918	—	2	1	no	—
24	#15667918	D₄	1	1	yes	characteristic
25	#15667918	D₄	1	1	yes	characteristic
26	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
28	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
29	C₃₂×C₈₉	—	2	2	no	—
30	#13083574	C₂×C₈	1	2	yes	—
31	Dic₃₅₆	C₁₆	1	1	yes	characteristic
32	#13083563	C₁₆	1	2	yes	—
33	D₇₁₂	C₁₆	1	1	yes	characteristic
34	#13083558	C₂×C₈	1	2	yes	—
35	#13083557	—	2	2	no	—
36	C₈×D₈₉	—	2	2	no	—
37	#13083549	—	2	2	no	—
38	C₂×C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
39	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
40	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
41	C₁₆×C₈₉	—	2	1	no	—
42	C₁₆×C₈₉	C₂×D₄	1	1	yes	characteristic
43	C₁₆×C₈₉	#41	1	1	yes	characteristic
44	C₂×C₄×C₈₉	—	2	1	no	—
45	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
46	C₂²⋊D₈₉	—	2	2	no	—
47	D₃₅₆	C₂×C₁₆	1	2	yes	—
48	C₄×D₈₉	—	2	2	no	—
49	Dic₁₇₈	C₂×C₁₆	1	2	yes	—
50	C₈×C₈₉	C₂×C₁₆	1	1	yes	characteristic
51	C₈×C₈₉	C₂×C₁₆	1	1	yes	characteristic
52	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
53	C₈×C₈₉	#131	1	1	yes	characteristic
54	#10623494	—	2	2	no	—
55	C₂²×C₈₉	#504	1	1	yes	characteristic
56	C₂×D₈₉	—	2	2	no	—
57	C₄×C₈₉	—	2	2	no	—
58	C₄×C₈₉	C₈×D₄	1	1	yes	characteristic
59	C₄×C₈₉	C₂²×C₁₆	1	1	yes	characteristic
60	Dic₈₉	—	2	2	no	—
61	#13647	—	89	1	no	maximal, Sylow
62	C₂×C₈₉	—	2	1	no	—
63	C₂×C₈₉	D₄×C₁₆	1	1	yes	characteristic
64	D₈₉	—	4	2	no	—
65	#2258	—	89	2	no	—
66	#2170	—	89	1	no	—
67	#1422	D₈₉	1	1	yes	characteristic, complemented
68	#1401	—	89	2	no	—
69	C₄×C₃₂	—	89	1	no	—
70	C₈₉	#13647	1	1	yes	characteristic, complemented, Sylow
71	#504	—	89	2	no	—
72	#443	—	89	1	no	—
73	M₆₄	—	178	2	no	—
74	M₆₄	C₂×D₈₉	1	2	yes	—
75	C₂×C₃₂	—	89	1	no	—
76	C₂×C₃₂	—	89	1	no	—
77	C₂×C₃₂	—	178	2	no	—
78	#364	—	89	1	no	—
79	#350	—	89	2	no	—
80	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
81	#135	—	89	1	no	—
82	#131	—	89	2	no	—
83	M₃₂	—	89	2	no	—
84	M₃₂	—	178	2	no	—
85	C₂×C₁₆	—	178	2	no	—
86	C₂×C₁₆	C₄×D₈₉	1	1	yes	characteristic
87	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
88	#108	—	89	1	no	—
89	#104	—	89	2	no	—
90	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
91	C₃₂	—	2	2	no	—
92	C₃₂	—	89	1	no	—
93	C₃₂	—	89	1	no	—
94	C₃₂	—	178	1	no	—
95	#41	—	89	2	no	—
96	Q₁₆	—	89	1	no	—
97	SD₁₆	—	89	2	no	—
98	D₈	—	89	1	no	—
99	M₁₆	—	89	2	no	—
100	M₁₆	—	178	2	no	—
101	C₂×C₈	—	178	2	no	—
102	C₂×C₈	C₈×D₈₉	1	1	yes	characteristic
103	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
104	C₄²	C₈×D₈₉	1	1	yes	characteristic
105	C₁₆	—	2	1	no	—
106	C₁₆	—	178	2	no	—
107	C₁₆	D₄×D₈₉	1	1	yes	characteristic
108	C₁₆	#13083575	1	1	yes	characteristic
109	Q₈	—	89	2	no	—
110	D₄	—	89	2	no	—
111	D₄	—	178	2	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	178	2	no	—
114	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
115	C₈	—	178	2	no	—
116	C₈	C₁₆×D₈₉	1	1	yes	characteristic
117	C₈	C₁₆×D₈₉	1	1	yes	characteristic
118	C₈	#15667974	1	1	yes	characteristic
119	C₈	#15668020	1	1	yes	characteristic
120	C₂²	—	178	2	no	—
121	C₂²	#19152016	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	178	2	no	—
124	C₄	#19151663	1	1	yes	characteristic
125	C₄	C₂×C₁₆×D₈₉	1	1	yes	characteristic
126	C₂	—	2	1	no	—
127	C₂	—	356	2	no	—
128	C₂	#26022071	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26022547

C₂

yes

maximal, complemented

#26022090

C₂

yes

maximal, characteristic

#26018395

C₂

yes

maximal, complemented

C₄×#15667918

C₂

yes

maximal, characteristic, complemented

C₈₉×#1422

C₂

yes

maximal, characteristic, complemented

#19152014

C₂²

yes

—

#19151716

C₄

yes

characteristic

#19151431

—

C₂×#15667918

C₂²

yes

characteristic

C₂×#15667918

C₂²

yes

characteristic

C₃₂×D₈₉

—

#19151419

C₄

yes

characteristic

#19151381

C₄

yes

—

M₆₄×C₈₉

C₂²

yes

—

C₄×C₁₆×C₈₉

C₂²

yes

characteristic

#15668034

C₈

yes

characteristic

#15668018

C₂×C₄

yes

—

#15667952

C₂×C₄

yes

—

#15667951

—

C₁₆×D₈₉

—

#15667948

C₈

yes

characteristic

#15667940

C₈

yes

—

#15667918

—

#15667918

D₄

yes

characteristic

#15667918

D₄

yes

characteristic

C₂×C₁₆×C₈₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₈₉

C₂³

yes

characteristic

C₄×C₈×C₈₉

C₂×C₄

yes

characteristic

C₃₂×C₈₉

—

#13083574

C₂×C₈

yes

—

Dic₃₅₆

C₁₆

yes

characteristic

#13083563

C₁₆

yes

—

D₇₁₂

C₁₆

yes

characteristic

#13083558

C₂×C₈

yes

—

#13083557

—

C₈×D₈₉

—

#13083549

—

C₂×C₈×C₈₉

C₂×C₈

yes

characteristic

C₂×C₈×C₈₉

C₂²×C₄

yes

characteristic

C₄²×C₈₉

C₂×C₈

yes

characteristic

C₁₆×C₈₉

—

C₁₆×C₈₉

C₂×D₄

yes

characteristic

C₁₆×C₈₉

#41

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²×C₈

yes

characteristic

C₂²⋊D₈₉

—

D₃₅₆

C₂×C₁₆

yes

—

C₄×D₈₉

—

Dic₁₇₈

C₂×C₁₆

yes

—

C₈×C₈₉

C₂×C₁₆

yes

characteristic

C₈×C₈₉

C₂×C₁₆

yes

characteristic

C₈×C₈₉

C₄×D₄

yes

C₈×C₈₉

yes

—

C₂²×C₈₉

#504

yes

characteristic

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₈×D₄

yes

characteristic

C₄×C₈₉

C₂²×C₁₆

yes

characteristic

Dic₈₉

—

#13647

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

D₄×C₁₆

yes

characteristic

D₈₉

—

#2258

—

#2170

—

#1422

D₈₉

yes

characteristic, complemented

#1401

—

C₄×C₃₂

—

C₈₉

#13647

yes

characteristic, complemented, Sylow

#504

—

#443

—

M₆₄

—

178

—

M₆₄

C₂×D₈₉

yes

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

178

—

#364

—

#350

—

C₄×C₁₆

C₂×D₈₉

yes

characteristic

#135

—

#131

—

M₃₂

—

M₃₂

—

178

—

C₂×C₁₆

—

178

—

C₂×C₁₆

C₄×D₈₉

yes

characteristic

C₂×C₁₆

C₂²×D₈₉

yes

characteristic

#108

—

#104

—

C₄×C₈

C₄×D₈₉

yes

characteristic

C₃₂

—

C₃₂

—

C₃₂

—

C₃₂

—

178

—

#41

—

Q₁₆

—

SD₁₆

—

D₈

—

M₁₆

—

100

M₁₆

—

178

—

101

C₂×C₈

—

178

—

102

C₂×C₈

C₈×D₈₉

yes

characteristic

103

C₂×C₈

C₂×C₄×D₈₉

yes

characteristic

104

C₄²

C₈×D₈₉

yes

characteristic

105

C₁₆

—

106

C₁₆

—

178

—

107

C₁₆

D₄×D₈₉

yes

characteristic

108

C₁₆

#13083575

yes

characteristic

109

Q₈

—

110

D₄

—

111

D₄

—

178

—

112

C₂×C₄

—

113

C₂×C₄

—

178

—

114

C₂×C₄

C₂×C₈×D₈₉

yes

characteristic

115

C₈

—

178

—

116

C₈

C₁₆×D₈₉

yes

characteristic

117

C₈

C₁₆×D₈₉

yes

characteristic

118

C₈

#15667974

yes

characteristic

119

C₈

#15668020

yes

characteristic

120

C₂²

—

178

—

121

C₂²

#19152016

yes

characteristic

122

C₄

—

123

C₄

—

178

—

124

C₄

#19151663

yes

characteristic

125

C₄

C₂×C₁₆×D₈₉

yes

characteristic

126

C₂

—

127

C₂

—

356

—

128

C₂

#26022071

yes

characteristic

	subgroup	G/H
center	C₁₆	D₄×D₈₉
commutator	C₄×C₈₉	C₂²×C₁₆
Frattini	C₂×C₁₆	C₂²×D₈₉
Fitting	C₈₉×#1422	C₂
socle	C₂×C₈₉	D₄×C₁₆
prime radical	#15668034	C₈
cosocle	C₂×C₁₆×C₈₉	C₂³
hypercenter	#1422	D₈₉
hypocenter	C₈₉	#13647
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ #1422

subgroup

G/H

center

C₁₆

D₄×D₈₉

commutator

C₄×C₈₉

C₂²×C₁₆

Frattini

C₂×C₁₆

C₂²×D₈₉

Fitting

C₈₉×#1422

C₂

socle

C₂×C₈₉

D₄×C₁₆

prime radical

#15668034

C₈

cosocle

C₂×C₁₆×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ #1422

N	H ≅ G/N	type
1. #26022547	C₂	semidirect
3. #26018395	C₂	semidirect
4. C₄×#15667918	C₂	semidirect
5. C₈₉×#1422	C₂	semidirect
67. #1422	D₈₉	semidirect
70. C₈₉	#13647	semidirect

H ≅ G/N

type

1. #26022547

C₂

semidirect

3. #26018395

C₂

semidirect

4. C₄×#15667918

C₂

semidirect

5. C₈₉×#1422

C₂

semidirect

67. #1422

D₈₉

semidirect

70. C₈₉

#13647

semidirect

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

subgroup	index
#26022547	2
#26022090	2
#26018395	2
C₄×#15667918	2
C₈₉×#1422	2
#13647	89

subgroup

index

C₈₉×#1422

Subgroups — #36474991

Conjugacy classes of subgroups

Complements

Maximal subgroups