Group #36470821 — subgroups

Subgroups — #36470821

Order 22784.

274 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 147 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	#26020424	C₂	1	2	yes	maximal
2	#26018346	C₂	1	4	yes	maximal
3	C₂×#346×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#15667948	—	2	2	no	—
5	C₂×#15667948	C₂²	1	4	yes	—
6	#19151622	—	2	2	no	—
7	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
9	#346×C₈₉	C₄	1	4	yes	—
10	C₂×#13083558	—	2	4	no	—
11	#15667948	—	2	8	no	—
12	#15667948	—	4	4	no	—
13	C₄⋊#10623494	—	4	4	no	—
14	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
15	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
16	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
17	C₂×C₁₆×C₈₉	—	2	4	no	—
18	C₂×C₁₆×C₈₉	C₂×C₄	1	4	yes	—
19	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
20	C₄×C₈×C₈₉	Q₈	1	2	yes	—
21	#13083558	—	4	4	no	—
22	#13083558	—	4	4	no	—
23	C₂×#10623494	—	4	4	no	—
24	C₂²×C₄×C₈₉	Q₁₆	1	2	yes	—
25	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
26	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
27	C₂×C₈×C₈₉	SD₁₆	1	4	yes	—
28	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₈×C₈₉	#41	1	4	yes	—
31	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
32	C₄²×C₈₉	D₈	1	2	yes	—
33	C₁₆×C₈₉	—	2	4	no	—
34	C₂³×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
35	C₂×C₄×C₈₉	—	4	2	no	—
36	C₂×C₄×C₈₉	#107	1	4	yes	—
37	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₈₉	#124	1	2	yes	—
39	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
40	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
41	C₈×C₈₉	#106	1	4	yes	—
42	C₈×C₈₉	#117	1	2	yes	—
43	C₈×C₈₉	#125	1	2	yes	—
44	#10623494	—	8	4	no	—
45	C₂²×C₈₉	—	2	2	no	—
46	C₂²×C₈₉	C₂×#107	1	1	yes	characteristic
47	C₂²×C₈₉	#487	1	2	yes	—
48	C₄×C₈₉	—	2	4	no	—
49	C₄×C₈₉	#382	1	1	yes	characteristic
50	C₄×C₈₉	C₂×#106	1	1	yes	characteristic
51	C₄×C₈₉	#486	1	2	yes	—
52	#12892	—	89	1	no	maximal, Sylow
53	C₂×C₈₉	—	2	2	no	—
54	C₂×C₈₉	#1834	1	1	yes	characteristic
55	C₂×C₈₉	#2161	1	2	yes	—
56	C₂×#346	D₈₉	1	1	yes	characteristic
57	#1848	—	89	2	no	—
58	#1381	—	89	4	no	—
59	C₈₉	#12892	1	1	yes	characteristic, complemented, Sylow
60	C₂²×C₁₆	C₂×D₈₉	1	2	yes	—
61	C₂×#108	—	89	4	no	—
62	C₂×#108	—	178	2	no	—
63	#423	—	178	2	no	—
64	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
65	#346	Dic₈₉	1	4	yes	—
66	C₂×M₁₆	—	178	4	no	—
67	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
68	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
69	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
70	C₂×C₁₆	—	2	4	no	—
71	C₂×C₁₆	C₂×Dic₈₉	1	4	yes	—
72	#108	—	356	4	no	—
73	#108	—	178	8	no	—
74	C₄⋊C₈	—	356	4	no	—
75	C₄×C₈	Dic₁₇₈	1	2	yes	—
76	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
77	C₂²×C₄	Dic₃₅₆	1	2	yes	—
78	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
79	M₁₆	—	356	4	no	—
80	M₁₆	—	356	4	no	—
81	C₂×C₈	—	356	4	no	—
82	C₂×C₈	C₄⋊Dic₈₉	1	2	yes	—
83	C₂×C₈	#13083563	1	4	yes	—
84	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
85	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
86	C₂×C₈	#13083574	1	4	yes	—
87	C₄²	C₄⋊Dic₈₉	1	2	yes	—
88	C₄²	D₇₁₂	1	2	yes	—
89	C₁₆	—	2	4	no	—
90	C₂³	C₂×Dic₃₅₆	1	1	yes	characteristic
91	C₂×C₄	—	4	2	no	—
92	C₂×C₄	#15667946	1	4	yes	—
93	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
94	C₂×C₄	#15668000	1	2	yes	—
95	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
96	C₂×C₄	C₂×#13083563	1	2	yes	—
97	C₈	—	712	4	no	—
98	C₈	#15667944	1	4	yes	—
99	C₈	#15667971	1	2	yes	—
100	C₈	#15668004	1	2	yes	—
101	C₂²	—	2	2	no	—
102	C₂²	C₂×#15667946	1	1	yes	characteristic
103	C₂²	#19151949	1	2	yes	—
104	C₄	—	2	4	no	—
105	C₄	#19151464	1	1	yes	characteristic
106	C₄	C₂×#15667944	1	1	yes	characteristic
107	C₄	#19151945	1	2	yes	—
108	C₂	—	2	2	no	—
109	C₂	#26020380	1	1	yes	characteristic
110	C₂	#26022047	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26020424

C₂

yes

maximal

#26018346

C₂

yes

maximal

C₂×#346×C₈₉

C₂

yes

maximal, characteristic

C₂×#15667948

—

C₂×#15667948

C₂²

yes

—

#19151622

—

C₂²×C₁₆×C₈₉

C₂²

yes

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

#346×C₈₉

C₄

yes

—

—

—

—

—

C₂²×C₈×C₈₉

D₄

yes

characteristic

C₂²×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₄²×C₈₉

D₄

yes

characteristic

C₂×C₁₆×C₈₉

—

C₂×C₁₆×C₈₉

C₂×C₄

yes

—

C₄×C₈×C₈₉

C₂×C₄

yes

—

C₄×C₈×C₈₉

Q₈

yes

—

#13083558

—

#13083558

—

C₂×#10623494

—

C₂²×C₄×C₈₉

Q₁₆

yes

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

C₄⋊C₄

yes

—

C₂×C₈×C₈₉

SD₁₆

yes

—

C₂×C₈×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₈₉

#41

yes

—

C₄²×C₈₉

C₄⋊C₄

yes

—

C₄²×C₈₉

D₈

yes

—

C₁₆×C₈₉

—

C₂³×C₈₉

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#107

yes

—

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

#124

yes

—

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

C₂×SD₁₆

yes

—

C₈×C₈₉

#106

yes

—

C₈×C₈₉

#117

yes

—

C₈×C₈₉

#125

yes

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×#107

yes

characteristic

C₂²×C₈₉

#487

yes

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

#12892

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

#1834

yes

characteristic

C₂×C₈₉

#2161

yes

—

C₂×#346

D₈₉

yes

characteristic

#1848

—

#1381

—

C₈₉

#12892

yes

characteristic, complemented, Sylow

C₂²×C₁₆

C₂×D₈₉

yes

—

C₂×#108

—

C₂×#108

—

178

—

#423

—

178

—

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

#346

Dic₈₉

yes

—

C₂×M₁₆

—

178

—

C₂²×C₈

D₃₅₆

yes

characteristic

C₂²×C₈

C₂²×D₈₉

yes

characteristic

C₂×C₄²

D₃₅₆

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂×Dic₈₉

yes

—

#108

—

356

—

#108

—

178

—

C₄⋊C₈

—

356

—

C₄×C₈

Dic₁₇₈

yes

—

C₄×C₈

C₂×Dic₈₉

yes

—

C₂²×C₄

Dic₃₅₆

yes

—

C₂²×C₄

C₂×D₃₅₆

yes

characteristic

M₁₆

—

356

—

M₁₆

—

356

—

C₂×C₈

—

356

—

C₂×C₈

C₄⋊Dic₈₉

yes

—

C₂×C₈

#13083563

yes

—

C₂×C₈

C₂²×Dic₈₉

yes

characteristic

C₂×C₈

C₂×Dic₁₇₈

yes

characteristic

C₂×C₈

#13083574

yes

—

C₄²

C₄⋊Dic₈₉

yes

—

C₄²

D₇₁₂

yes

—

C₁₆

—

C₂³

C₂×Dic₃₅₆

yes

characteristic

C₂×C₄

—

C₂×C₄

#15667946

yes

—

C₂×C₄

C₂×C₄⋊Dic₈₉

yes

characteristic

C₂×C₄

#15668000

yes

—

C₂×C₄

C₂×D₇₁₂

yes

characteristic

C₂×C₄

C₂×#13083563

yes

—

C₈

—

712

—

C₈

#15667944

yes

—

C₈

#15667971

yes

—

100

C₈

#15668004

yes

—

101

C₂²

—

102

C₂²

C₂×#15667946

yes

characteristic

103

C₂²

#19151949

yes

—

104

C₄

—

105

C₄

#19151464

yes

characteristic

106

C₄

C₂×#15667944

yes

characteristic

107

C₄

#19151945

yes

—

108

C₂

—

109

C₂

#26020380

yes

characteristic

110

C₂

#26022047

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₇₁₂
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂×#346×C₈₉	C₂
socle	C₂²×C₈₉	C₂×#107
prime radical	C₂³×C₈₉	C₂×Q₁₆
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×#346	D₈₉
hypocenter	C₈₉	#12892
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂×C₄

C₂×D₇₁₂

commutator

C₂×C₈×C₈₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₈₉

Fitting

C₂×#346×C₈₉

C₂

socle

C₂²×C₈₉

C₂×#107

prime radical

C₂³×C₈₉

C₂×Q₁₆

cosocle

C₂²×C₈×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
59. C₈₉	#12892	semidirect

H ≅ G/N

type

59. C₈₉

#12892

semidirect

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

subgroup	index
#26020424	2
#26018346	2
C₂×#346×C₈₉	2
#12892	89

subgroup

index

#26020424

#26018346

C₂×#346×C₈₉

#12892

Subgroups — #36470821

Conjugacy classes of subgroups

Complements

Maximal subgroups