Group #36475041 — subgroups

Subgroups — #36475041

Order 22784.

184 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 69 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	#26022133	C₂	1	2	yes	maximal, complemented
2	#26018456	C₂	1	1	yes	maximal, characteristic, complemented
3	#26018436	C₂	1	2	yes	maximal, complemented
4	#26018385	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#2262	C₂	1	1	yes	maximal, characteristic
6	#19151715	—	2	2	no	—
7	C₂×#15667918	C₂²	1	1	yes	characteristic
8	#19151428	C₂²	1	2	yes	—
9	#19151405	—	2	2	no	—
10	C₁₆×Dic₈₉	C₂²	1	1	yes	characteristic
11	#508×C₈₉	C₂²	1	2	yes	—
12	Q₆₄×C₈₉	C₄	1	1	yes	characteristic, complemented
13	SD₆₄×C₈₉	C₄	1	2	yes	complemented
14	D₃₂×C₈₉	C₄	1	1	yes	characteristic, complemented
15	C₂×C₃₂×C₈₉	C₂²	1	1	yes	characteristic
16	#15668019	—	4	2	no	—
17	C₂×#13083549	D₄	1	1	yes	characteristic
18	#15667948	—	2	2	no	—
19	#15667939	—	4	2	no	—
20	C₈×Dic₈₉	D₄	1	1	yes	characteristic
21	#15667918	—	2	1	no	—
22	#135×C₈₉	—	2	2	no	—
23	Q₃₂×C₈₉	C₂×C₄	1	2	yes	—
24	SD₃₂×C₈₉	—	2	2	no	—
25	D₁₆×C₈₉	C₂×C₄	1	2	yes	—
26	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
27	C₃₂×C₈₉	C₂×C₄	1	1	yes	characteristic
28	C₃₂×C₈₉	C₂×C₄	1	1	yes	characteristic
29	#13083558	—	4	2	no	—
30	#13083558	—	8	2	no	—
31	C₂×#10623494	—	8	2	no	—
32	C₂×#10623494	D₈	1	1	yes	characteristic
33	C₄×Dic₈₉	D₈	1	1	yes	characteristic
34	#13083549	—	2	1	no	—
35	#41×C₈₉	—	4	2	no	—
36	Q₁₆×C₈₉	—	2	2	no	—
37	SD₁₆×C₈₉	—	4	2	no	—
38	D₈×C₈₉	—	2	2	no	—
39	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
40	C₁₆×C₈₉	C₂²×C₄	1	1	yes	characteristic
41	C₁₆×C₈₉	#41	1	1	yes	characteristic
42	Q₈×C₈₉	—	4	2	no	—
43	D₄×C₈₉	—	4	2	no	—
44	D₄×C₈₉	—	8	2	no	—
45	C₂×C₄×C₈₉	—	8	2	no	—
46	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
47	C₂×Dic₈₉	—	2	1	no	—
48	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
49	C₈×C₈₉	#135	1	1	yes	characteristic
50	#10623494	—	8	2	no	—
51	#10623494	D₁₆	1	2	yes	—
52	C₂²×C₈₉	—	8	2	no	—
53	C₂²×C₈₉	#508	1	1	yes	characteristic
54	C₄×C₈₉	—	8	2	no	—
55	C₄×C₈₉	C₄×D₈	1	1	yes	characteristic
56	C₄×C₈₉	C₂×D₁₆	1	1	yes	characteristic
57	Dic₈₉	—	4	1	no	—
58	#13656	—	89	1	no	maximal, Sylow
59	C₂×C₈₉	—	2	1	no	—
60	C₂×C₈₉	—	16	2	no	—
61	C₂×C₈₉	C₄×D₁₆	1	1	yes	characteristic
62	#2262	D₈₉	1	1	yes	characteristic
63	#2178	—	89	2	no	—
64	#1425	—	89	1	no	—
65	#1417	—	89	2	no	—
66	#1397	—	89	1	no	—
67	C₈₉	#13656	1	1	yes	characteristic, complemented, Sylow
68	#508	C₂×D₈₉	1	2	yes	—
69	#443	—	178	2	no	—
70	Q₆₄	Dic₈₉	1	1	yes	characteristic, complemented
71	SD₆₄	Dic₈₉	1	2	yes	complemented
72	D₃₂	Dic₈₉	1	1	yes	characteristic, complemented
73	C₂×C₃₂	—	89	1	no	—
74	C₂×C₃₂	C₂×D₈₉	1	1	yes	characteristic
75	#368	—	89	2	no	—
76	#359	—	178	2	no	—
77	C₄×C₁₆	—	89	1	no	—
78	#135	—	2	2	no	—
79	#131	—	356	2	no	—
80	Q₃₂	C₂×Dic₈₉	1	2	yes	—
81	SD₃₂	—	2	2	no	—
82	D₁₆	C₂×Dic₈₉	1	2	yes	—
83	C₂×C₁₆	—	89	1	no	—
84	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
85	#108	—	178	2	no	—
86	#104	—	356	2	no	—
87	C₄×C₈	—	89	1	no	—
88	C₃₂	—	178	1	no	—
89	C₃₂	C₂×Dic₈₉	1	1	yes	characteristic
90	C₃₂	C₂×Dic₈₉	1	1	yes	characteristic
91	#41	—	4	2	no	—
92	Q₁₆	—	2	2	no	—
93	SD₁₆	—	4	2	no	—
94	D₈	—	2	2	no	—
95	M₁₆	—	356	2	no	—
96	M₁₆	—	712	2	no	—
97	C₂×C₈	—	89	1	no	—
98	C₂×C₈	—	712	2	no	—
99	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
100	C₄²	—	89	1	no	—
101	C₁₆	—	178	1	no	—
102	C₁₆	C₂²×Dic₈₉	1	1	yes	characteristic
103	C₁₆	#13083573	1	1	yes	characteristic
104	Q₈	—	4	2	no	—
105	D₄	—	4	2	no	—
106	D₄	—	8	2	no	—
107	C₂×C₄	—	8	2	no	—
108	C₂×C₄	—	178	1	no	—
109	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
110	C₈	—	89	2	no	—
111	C₈	—	712	2	no	—
112	C₈	D₄×Dic₈₉	1	1	yes	characteristic
113	C₈	#15668031	1	1	yes	characteristic
114	C₂²	—	8	2	no	—
115	C₂²	#19152027	1	1	yes	characteristic
116	C₄	—	8	2	no	—
117	C₄	—	356	1	no	—
118	C₄	D₈×Dic₈₉	1	1	yes	characteristic
119	C₄	D₁₆×D₈₉	1	1	yes	characteristic
120	C₂	—	2	1	no	—
121	C₂	—	16	2	no	—
122	C₂	D₁₆×Dic₈₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26022133

C₂

yes

maximal, complemented

#26018456

C₂

yes

maximal, characteristic, complemented

#26018436

C₂

yes

maximal, complemented

#26018385

C₂

yes

maximal, characteristic, complemented

C₈₉×#2262

C₂

yes

maximal, characteristic

#19151715

—

C₂×#15667918

C₂²

yes

characteristic

#19151428

C₂²

yes

—

#19151405

—

C₁₆×Dic₈₉

C₂²

yes

characteristic

#508×C₈₉

C₂²

yes

—

Q₆₄×C₈₉

C₄

yes

characteristic, complemented

SD₆₄×C₈₉

C₄

yes

complemented

D₃₂×C₈₉

C₄

yes

characteristic, complemented

C₂×C₃₂×C₈₉

C₂²

yes

characteristic

#15668019

—

C₂×#13083549

D₄

yes

characteristic

#15667948

—

#15667939

—

C₈×Dic₈₉

D₄

yes

characteristic

#15667918

—

#135×C₈₉

—

Q₃₂×C₈₉

C₂×C₄

yes

—

SD₃₂×C₈₉

—

D₁₆×C₈₉

C₂×C₄

yes

—

C₂×C₁₆×C₈₉

C₂³

yes

characteristic

C₃₂×C₈₉

C₂×C₄

yes

characteristic

C₃₂×C₈₉

C₂×C₄

yes

—

—

—

D₈

yes

characteristic

C₄×Dic₈₉

D₈

yes

characteristic

#13083549

—

#41×C₈₉

—

Q₁₆×C₈₉

—

SD₁₆×C₈₉

—

D₈×C₈₉

—

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₁₆×C₈₉

C₂²×C₄

yes

characteristic

C₁₆×C₈₉

#41

yes

characteristic

Q₈×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×Dic₈₉

—

C₈×C₈₉

C₄×D₄

yes

C₈×C₈₉

yes

—

D₁₆

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

#508

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

C₄×D₈

yes

characteristic

C₄×C₈₉

C₂×D₁₆

yes

characteristic

Dic₈₉

—

#13656

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

C₄×D₁₆

yes

characteristic

#2262

D₈₉

yes

—

—

—

—

C₈₉

#13656

yes

characteristic, complemented, Sylow

#508

C₂×D₈₉

yes

—

#443

—

178

—

Q₆₄

Dic₈₉

yes

characteristic, complemented

SD₆₄

Dic₈₉

yes

complemented

D₃₂

Dic₈₉

yes

characteristic, complemented

C₂×C₃₂

—

C₂×C₃₂

C₂×D₈₉

yes

characteristic

#368

—

#359

—

178

—

C₄×C₁₆

—

#135

—

#131

—

356

—

Q₃₂

C₂×Dic₈₉

yes

—

SD₃₂

—

D₁₆

C₂×Dic₈₉

yes

—

C₂×C₁₆

—

C₂×C₁₆

C₂²×D₈₉

yes

characteristic

#108

—

178

—

#104

—

356

—

C₄×C₈

—

C₃₂

—

178

—

C₃₂

C₂×Dic₈₉

yes

characteristic

C₃₂

C₂×Dic₈₉

yes

characteristic

#41

—

Q₁₆

—

SD₁₆

—

D₈

—

M₁₆

—

356

—

M₁₆

—

712

—

C₂×C₈

—

C₂×C₈

—

712

—

C₂×C₈

D₄×D₈₉

yes

characteristic

100

C₄²

—

101

C₁₆

—

178

—

102

C₁₆

C₂²×Dic₈₉

yes

characteristic

103

C₁₆

#13083573

yes

characteristic

104

Q₈

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

178

—

109

C₂×C₄

D₈×D₈₉

yes

characteristic

110

C₈

—

111

C₈

—

712

—

112

C₈

D₄×Dic₈₉

yes

characteristic

113

C₈

#15668031

yes

characteristic

114

C₂²

—

115

C₂²

#19152027

yes

characteristic

116

C₄

—

117

C₄

—

356

—

118

C₄

D₈×Dic₈₉

yes

characteristic

119

C₄

D₁₆×D₈₉

yes

characteristic

120

C₂

—

121

C₂

—

122

C₂

D₁₆×Dic₈₉

yes

characteristic

	subgroup	G/H
center	C₄	D₁₆×D₈₉
commutator	C₁₆×C₈₉	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₈₉
Fitting, prime radical	C₈₉×#2262	C₂
socle	C₂×C₈₉	C₄×D₁₆
cosocle	C₂×C₁₆×C₈₉	C₂³
hypercenter	#2262	D₈₉
hypocenter	C₈₉	#13656
LCS	G ▷ C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #2262

subgroup

G/H

center

C₄

D₁₆×D₈₉

commutator

C₁₆×C₈₉

C₂²×C₄

Frattini

C₂×C₁₆

C₂²×D₈₉

Fitting, prime radical

C₈₉×#2262

C₂

socle

C₂×C₈₉

C₄×D₁₆

cosocle

C₂×C₁₆×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #2262

N	H ≅ G/N	type
1. #26022133	C₂	semidirect
2. #26018456	C₂	semidirect
3. #26018436	C₂	semidirect
4. #26018385	C₂	semidirect
12. Q₆₄×C₈₉	C₄	semidirect
13. SD₆₄×C₈₉	C₄	semidirect
14. D₃₂×C₈₉	C₄	semidirect
67. C₈₉	#13656	semidirect
70. Q₆₄	Dic₈₉	semidirect
71. SD₆₄	Dic₈₉	semidirect
72. D₃₂	Dic₈₉	semidirect

H ≅ G/N

type

1. #26022133

C₂

semidirect

2. #26018456

C₂

semidirect

3. #26018436

C₂

semidirect

4. #26018385

C₂

semidirect

12. Q₆₄×C₈₉

C₄

semidirect

13. SD₆₄×C₈₉

C₄

semidirect

14. D₃₂×C₈₉

C₄

semidirect

67. C₈₉

#13656

semidirect

70. Q₆₄

Dic₈₉

semidirect

71. SD₆₄

Dic₈₉

semidirect

72. D₃₂

Dic₈₉

semidirect

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

subgroup	index
#26022133	2
#26018456	2
#26018436	2
#26018385	2
C₈₉×#2262	2
#13656	89

subgroup

index

C₈₉×#2262

Subgroups — #36475041

Conjugacy classes of subgroups

Complements

Maximal subgroups