Group #36475035 — subgroups

Subgroups — #36475035

Order 22784.

186 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 61 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	#26022561	C₂	1	1	yes	maximal, characteristic
2	#26022132	C₂	1	2	yes	maximal, complemented
3	#26018435	C₂	1	2	yes	maximal, complemented
4	C₃₂×Dic₈₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#1425	C₂	1	1	yes	maximal, characteristic, complemented
6	#19152030	C₂²	1	2	yes	—
7	#19151714	—	2	2	no	—
8	Dic₁₄₂₄	C₄	1	1	yes	characteristic, complemented
9	#19151437	C₄	1	2	yes	complemented
10	D₂₈₄₈	C₄	1	1	yes	characteristic, complemented
11	C₂×#15667918	C₂²	1	1	yes	characteristic
12	#19151404	—	2	2	no	—
13	C₁₆×Dic₈₉	C₂²	1	1	yes	characteristic
14	C₂×C₃₂×C₈₉	C₂²	1	1	yes	characteristic
15	#368×C₈₉	C₂²	1	2	yes	—
16	#15668034	—	2	2	no	—
17	#15668020	—	4	2	no	—
18	Dic₇₁₂	C₂×C₄	1	2	yes	—
19	#15667957	—	2	2	no	—
20	D₁₄₂₄	C₂×C₄	1	2	yes	—
21	C₂×#13083549	D₄	1	1	yes	characteristic
22	#15667938	—	4	2	no	—
23	C₈×Dic₈₉	D₄	1	1	yes	characteristic
24	#15667918	—	2	1	no	—
25	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
26	#108×C₈₉	—	2	2	no	—
27	C₃₂×C₈₉	C₂×C₄	1	1	yes	characteristic
28	C₃₂×C₈₉	C₂×C₄	1	1	yes	characteristic
29	#13083574	—	4	2	no	—
30	Dic₃₅₆	—	2	2	no	—
31	#13083563	—	4	2	no	—
32	D₇₁₂	—	2	2	no	—
33	#13083557	—	8	2	no	—
34	C₂×#10623494	D₈	1	1	yes	characteristic
35	C₈×D₈₉	—	8	2	no	—
36	C₄×Dic₈₉	D₈	1	1	yes	characteristic
37	#13083549	—	2	1	no	—
38	M₁₆×C₈₉	—	4	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	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
43	C₂²⋊D₈₉	—	8	2	no	—
44	C₂×Dic₈₉	—	2	1	no	—
45	D₃₅₆	—	4	2	no	—
46	C₄×D₈₉	—	8	2	no	—
47	Dic₁₇₈	—	4	2	no	—
48	C₈×C₈₉	—	8	2	no	—
49	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
50	C₈×C₈₉	#135	1	1	yes	characteristic
51	#10623494	D₁₆	1	1	yes	characteristic
52	#10623494	D₁₆	1	1	yes	characteristic
53	C₂²×C₈₉	#508	1	1	yes	characteristic
54	C₂×D₈₉	—	8	2	no	—
55	C₄×C₈₉	C₄×D₈	1	1	yes	characteristic
56	C₄×C₈₉	C₂×D₁₆	1	1	yes	characteristic
57	Dic₈₉	—	2	2	no	—
58	Dic₈₉	—	8	2	no	—
59	#13655	—	89	1	no	maximal, Sylow
60	C₂×C₈₉	—	2	1	no	—
61	C₂×C₈₉	C₄×D₁₆	1	1	yes	characteristic
62	D₈₉	—	16	2	no	—
63	#2262	—	89	1	no	—
64	#2178	—	89	2	no	—
65	#1425	D₈₉	1	1	yes	characteristic, complemented
66	#1417	—	89	2	no	—
67	C₄×C₃₂	—	89	1	no	—
68	C₈₉	#13655	1	1	yes	characteristic, complemented, Sylow
69	#508	—	89	2	no	—
70	#443	—	178	2	no	—
71	Q₆₄	—	89	1	no	—
72	SD₆₄	—	89	2	no	—
73	D₃₂	—	89	1	no	—
74	C₂×C₃₂	—	89	1	no	—
75	C₂×C₃₂	C₂×D₈₉	1	1	yes	characteristic
76	#368	C₂×D₈₉	1	2	yes	—
77	#359	—	178	2	no	—
78	C₄×C₁₆	—	89	1	no	—
79	#135	—	178	2	no	—
80	#131	—	356	2	no	—
81	Q₃₂	—	89	2	no	—
82	SD₃₂	—	178	2	no	—
83	D₁₆	—	89	2	no	—
84	C₂×C₁₆	—	89	1	no	—
85	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
86	#108	—	2	2	no	—
87	#104	—	356	2	no	—
88	C₄×C₈	—	89	1	no	—
89	C₃₂	—	178	1	no	—
90	C₃₂	C₄×D₈₉	1	1	yes	characteristic
91	C₃₂	C₄×D₈₉	1	1	yes	characteristic
92	#41	—	356	2	no	—
93	Q₁₆	—	178	2	no	—
94	SD₁₆	—	356	2	no	—
95	D₈	—	178	2	no	—
96	M₁₆	—	4	2	no	—
97	M₁₆	—	712	2	no	—
98	C₂×C₈	—	89	1	no	—
99	C₂×C₈	—	712	2	no	—
100	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
101	C₄²	—	89	1	no	—
102	C₁₆	—	178	1	no	—
103	C₁₆	C₂×C₄×D₈₉	1	1	yes	characteristic
104	C₁₆	#13083575	1	1	yes	characteristic
105	Q₈	—	356	2	no	—
106	D₄	—	356	2	no	—
107	D₄	—	712	2	no	—
108	C₂×C₄	—	178	1	no	—
109	C₂×C₄	—	712	2	no	—
110	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
111	C₈	—	8	2	no	—
112	C₈	—	89	1	no	—
113	C₈	—	89	1	no	—
114	C₈	#15667974	1	1	yes	characteristic
115	C₈	#15668035	1	1	yes	characteristic
116	C₂²	—	712	2	no	—
117	C₂²	#19152031	1	1	yes	characteristic
118	C₄	—	178	2	no	—
119	C₄	—	712	2	no	—
120	C₄	#19151680	1	1	yes	characteristic
121	C₄	D₁₆×D₈₉	1	1	yes	characteristic
122	C₂	—	2	1	no	—
123	C₂	—	1424	2	no	—
124	C₂	#26022098	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26022561

C₂

yes

maximal, characteristic

#26022132

C₂

yes

maximal, complemented

#26018435

C₂

yes

maximal, complemented

C₃₂×Dic₈₉

C₂

yes

maximal, characteristic, complemented

C₈₉×#1425

C₂

yes

maximal, characteristic, complemented

#19152030

C₂²

yes

—

#19151714

—

Dic₁₄₂₄

C₄

yes

characteristic, complemented

#19151437

C₄

yes

complemented

D₂₈₄₈

C₄

yes

characteristic, complemented

C₂×#15667918

C₂²

yes

characteristic

#19151404

—

C₁₆×Dic₈₉

C₂²

yes

characteristic

C₂×C₃₂×C₈₉

C₂²

yes

characteristic

#368×C₈₉

C₂²

yes

—

#15668034

—

#15668020

—

Dic₇₁₂

C₂×C₄

yes

—

#15667957

—

D₁₄₂₄

C₂×C₄

yes

—

C₂×#13083549

D₄

yes

characteristic

#15667938

—

C₈×Dic₈₉

D₄

yes

characteristic

#15667918

—

C₂×C₁₆×C₈₉

C₂³

yes

characteristic

#108×C₈₉

—

C₃₂×C₈₉

C₂×C₄

yes

characteristic

C₃₂×C₈₉

C₂×C₄

yes

characteristic

#13083574

—

Dic₃₅₆

—

#13083563

—

D₇₁₂

—

#13083557

—

C₂×#10623494

D₈

yes

characteristic

C₈×D₈₉

—

C₄×Dic₈₉

D₈

yes

characteristic

#13083549

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₁₆×C₈₉

C₂²×C₄

yes

characteristic

C₁₆×C₈₉

#41

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂²⋊D₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

C₄×D₈₉

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

C₄×D₄

yes

C₈×C₈₉

yes

D₁₆

yes

characteristic

#10623494

D₁₆

yes

characteristic

C₂²×C₈₉

#508

yes

characteristic

C₂×D₈₉

—

C₄×C₈₉

C₄×D₈

yes

characteristic

C₄×C₈₉

C₂×D₁₆

yes

characteristic

Dic₈₉

—

Dic₈₉

—

#13655

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

C₄×D₁₆

yes

characteristic

D₈₉

—

#2262

—

#2178

—

#1425

D₈₉

yes

characteristic, complemented

#1417

—

C₄×C₃₂

—

C₈₉

#13655

yes

characteristic, complemented, Sylow

#508

—

#443

—

178

—

Q₆₄

—

SD₆₄

—

D₃₂

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₈₉

yes

characteristic

#368

C₂×D₈₉

yes

—

#359

—

178

—

C₄×C₁₆

—

#135

—

178

—

#131

—

356

—

Q₃₂

—

SD₃₂

—

178

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²×D₈₉

yes

characteristic

#108

—

#104

—

356

—

C₄×C₈

—

C₃₂

—

178

—

C₃₂

C₄×D₈₉

yes

characteristic

C₃₂

C₄×D₈₉

yes

characteristic

#41

—

356

—

Q₁₆

—

178

—

SD₁₆

—

356

—

D₈

—

178

—

M₁₆

—

M₁₆

—

712

—

C₂×C₈

—

C₂×C₈

—

712

—

100

C₂×C₈

D₄×D₈₉

yes

characteristic

101

C₄²

—

102

C₁₆

—

178

—

103

C₁₆

C₂×C₄×D₈₉

yes

characteristic

104

C₁₆

#13083575

yes

characteristic

105

Q₈

—

356

—

106

D₄

—

356

—

107

D₄

—

712

—

108

C₂×C₄

—

178

—

109

C₂×C₄

—

712

—

110

C₂×C₄

D₈×D₈₉

yes

characteristic

111

C₈

—

112

C₈

—

113

C₈

—

114

C₈

#15667974

yes

characteristic

115

C₈

#15668035

yes

characteristic

116

C₂²

—

712

—

117

C₂²

#19152031

yes

characteristic

118

C₄

—

178

—

119

C₄

—

712

—

120

C₄

#19151680

yes

characteristic

121

C₄

D₁₆×D₈₉

yes

characteristic

122

C₂

—

123

C₂

—

1424

—

124

C₂

#26022098

yes

characteristic

	subgroup	G/H
center	C₄	D₁₆×D₈₉
commutator	C₁₆×C₈₉	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₈₉
Fitting	C₈₉×#1425	C₂
socle	C₂×C₈₉	C₄×D₁₆
prime radical	#26022561	C₂
cosocle	C₂×C₁₆×C₈₉	C₂³
hypercenter	#1425	D₈₉
hypocenter	C₈₉	#13655
LCS	G ▷ C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1425

subgroup

G/H

center

C₄

D₁₆×D₈₉

commutator

C₁₆×C₈₉

C₂²×C₄

Frattini

C₂×C₁₆

C₂²×D₈₉

Fitting

C₈₉×#1425

C₂

socle

C₂×C₈₉

C₄×D₁₆

prime radical

#26022561

C₂

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 ◁ #1425

N	H ≅ G/N	type
2. #26022132	C₂	semidirect
3. #26018435	C₂	semidirect
4. C₃₂×Dic₈₉	C₂	semidirect
5. C₈₉×#1425	C₂	semidirect
8. Dic₁₄₂₄	C₄	semidirect
9. #19151437	C₄	semidirect
10. D₂₈₄₈	C₄	semidirect
65. #1425	D₈₉	semidirect
68. C₈₉	#13655	semidirect

H ≅ G/N

type

2. #26022132

C₂

semidirect

3. #26018435

C₂

semidirect

4. C₃₂×Dic₈₉

C₂

semidirect

5. C₈₉×#1425

C₂

semidirect

8. Dic₁₄₂₄

C₄

semidirect

9. #19151437

C₄

semidirect

10. D₂₈₄₈

C₄

semidirect

65. #1425

D₈₉

semidirect

68. C₈₉

#13655

semidirect

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

subgroup	index
#26022561	2
#26022132	2
#26018435	2
C₃₂×Dic₈₉	2
C₈₉×#1425	2
#13655	89

subgroup

index

#26022561

#26022132

#26018435

C₃₂×Dic₈₉

C₈₉×#1425

#13655

Subgroups — #36475035

Conjugacy classes of subgroups

Complements

Maximal subgroups