Group #28509287 — subgroups

Subgroups — #28509287

Order 15104.

188 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 73 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	Q₃₂×Dic₅₉	C₂	1	2	yes	maximal
2	#21422298	C₂	1	1	yes	maximal, characteristic
3	#21422277	C₂	1	2	yes	maximal
4	#21422229	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅₉×Q₆₄	C₂	1	1	yes	maximal, characteristic
6	Q₁₆×Dic₅₉	—	2	2	no	—
7	C₂×C₅₉⋊C₃₂	C₂²	1	1	yes	characteristic
8	#16837228	C₂²	1	2	yes	—
9	#16837206	—	2	2	no	—
10	C₁₆×Dic₅₉	C₂²	1	1	yes	characteristic
11	C₂×Q₃₂×C₅₉	C₂²	1	2	yes	—
12	C₅₉×Q₆₄	C₄	1	4	yes	complemented
13	C₂×C₃₂×C₅₉	C₂²	1	1	yes	characteristic
14	Q₈×Dic₅₉	—	4	2	no	—
15	C₂×C₅₉⋊C₁₆	D₄	1	1	yes	characteristic
16	#14273661	—	2	2	no	—
17	#14273651	—	4	2	no	—
18	C₈×Dic₅₉	D₄	1	1	yes	characteristic
19	C₅₉⋊C₃₂	—	2	1	no	—
20	C₂×Q₁₆×C₅₉	—	2	2	no	—
21	Q₃₂×C₅₉	—	2	2	no	—
22	Q₃₂×C₅₉	C₂×C₄	1	4	yes	—
23	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
24	C₃₂×C₅₉	C₂×C₄	1	2	yes	—
25	C₂×Q₈×C₅₉	—	4	2	no	—
26	Q₁₆×C₅₉	—	2	4	no	—
27	Q₁₆×C₅₉	—	4	2	no	—
28	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
29	C₄⋊Dic₅₉	—	4	2	no	—
30	C₄⋊Dic₅₉	—	8	2	no	—
31	C₄×Dic₅₉	—	8	2	no	—
32	C₄×Dic₅₉	D₈	1	1	yes	characteristic
33	C₂×C₅₉⋊C₈	D₈	1	1	yes	characteristic
34	C₁₆×C₅₉	C₂²×C₄	1	1	yes	characteristic
35	C₁₆×C₅₉	#41	1	1	yes	characteristic
36	C₅₉⋊C₁₆	—	2	1	no	—
37	Q₈×C₅₉	—	4	4	no	—
38	Q₈×C₅₉	—	8	2	no	—
39	C₂×C₄×C₅₉	—	8	2	no	—
40	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
41	C₂×Dic₅₉	—	8	2	no	—
42	C₂×Dic₅₉	D₁₆	1	2	yes	—
43	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
44	C₈×C₅₉	#135	1	1	yes	characteristic
45	C₅₉⋊C₈	—	2	1	no	—
46	#13653	—	59	1	no	maximal, Sylow
47	C₂²×C₅₉	C₂×D₁₆	1	1	yes	characteristic
48	C₄×C₅₉	—	8	4	no	—
49	C₄×C₅₉	C₄×D₈	1	1	yes	characteristic
50	C₄×C₅₉	#508	1	1	yes	characteristic
51	Dic₅₉	—	2	2	no	—
52	Dic₅₉	—	16	2	no	—
53	C₂×Q₆₄	D₅₉	1	1	yes	characteristic
54	C₄×Q₃₂	—	59	2	no	—
55	#1424	—	59	1	no	—
56	#1416	—	59	2	no	—
57	#1397	—	59	1	no	—
58	C₂×C₅₉	C₄×D₁₆	1	1	yes	characteristic
59	C₂×C₅₉	#2264	1	2	yes	—
60	C₂×Q₃₂	C₂×D₅₉	1	2	yes	—
61	C₄×Q₁₆	—	118	2	no	—
62	Q₆₄	Dic₅₉	1	4	yes	complemented
63	C₂×C₃₂	—	59	1	no	—
64	C₂×C₃₂	C₂×D₅₉	1	1	yes	characteristic
65	#366	—	59	2	no	—
66	#358	—	118	2	no	—
67	C₄×C₁₆	—	59	1	no	—
68	C₅₉	#13653	1	1	yes	characteristic, complemented, Sylow
69	C₂×Q₁₆	—	2	2	no	—
70	C₄×Q₈	—	236	2	no	—
71	Q₃₂	—	2	2	no	—
72	Q₃₂	C₂×Dic₅₉	1	4	yes	—
73	C₂×C₁₆	—	59	1	no	—
74	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
75	#107	—	118	2	no	—
76	#103	—	236	2	no	—
77	C₄×C₈	—	59	1	no	—
78	C₃₂	—	118	1	no	—
79	C₃₂	C₂×Dic₅₉	1	2	yes	—
80	C₂×Q₈	—	4	2	no	—
81	Q₁₆	—	2	4	no	—
82	Q₁₆	—	4	2	no	—
83	C₂×C₈	—	59	1	no	—
84	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
85	C₄⋊C₄	—	472	2	no	—
86	C₄⋊C₄	—	236	2	no	—
87	C₄²	—	59	1	no	—
88	C₄²	—	472	2	no	—
89	C₁₆	—	118	1	no	—
90	C₁₆	#11745687	1	1	yes	characteristic
91	C₁₆	C₂²×Dic₅₉	1	1	yes	characteristic
92	Q₈	—	4	4	no	—
93	Q₈	—	8	2	no	—
94	C₂×C₄	—	8	2	no	—
95	C₂×C₄	—	59	2	no	—
96	C₂×C₄	—	472	2	no	—
97	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
98	C₈	—	118	1	no	—
99	C₈	D₄×Dic₅₉	1	1	yes	characteristic
100	C₈	#14273746	1	1	yes	characteristic
101	C₂²	D₁₆×D₅₉	1	1	yes	characteristic
102	C₄	—	8	4	no	—
103	C₄	—	118	2	no	—
104	C₄	—	944	2	no	—
105	C₄	D₈×Dic₅₉	1	1	yes	characteristic
106	C₄	#16837831	1	1	yes	characteristic
107	C₂	D₁₆×Dic₅₉	1	1	yes	characteristic
108	C₂	#21426415	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

Q₃₂×Dic₅₉

C₂

yes

maximal

#21422298

C₂

yes

maximal, characteristic

#21422277

C₂

yes

maximal

#21422229

C₂

yes

maximal, characteristic

C₂×C₅₉×Q₆₄

C₂

yes

maximal, characteristic

Q₁₆×Dic₅₉

—

C₂×C₅₉⋊C₃₂

C₂²

yes

characteristic

#16837228

C₂²

yes

—

#16837206

—

C₁₆×Dic₅₉

C₂²

yes

characteristic

C₂×Q₃₂×C₅₉

C₂²

yes

—

C₅₉×Q₆₄

C₄

yes

complemented

C₂×C₃₂×C₅₉

C₂²

yes

characteristic

Q₈×Dic₅₉

—

C₂×C₅₉⋊C₁₆

D₄

yes

characteristic

#14273661

—

#14273651

—

C₈×Dic₅₉

D₄

yes

characteristic

C₅₉⋊C₃₂

—

C₂×Q₁₆×C₅₉

—

Q₃₂×C₅₉

—

Q₃₂×C₅₉

C₂×C₄

yes

—

C₂×C₁₆×C₅₉

C₂³

yes

characteristic

C₃₂×C₅₉

C₂×C₄

yes

—

C₂×Q₈×C₅₉

—

Q₁₆×C₅₉

—

Q₁₆×C₅₉

—

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄×Dic₅₉

—

C₄×Dic₅₉

D₈

yes

characteristic

C₂×C₅₉⋊C₈

D₈

yes

characteristic

C₁₆×C₅₉

C₂²×C₄

yes

characteristic

C₁₆×C₅₉

#41

yes

characteristic

C₅₉⋊C₁₆

—

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₈

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

D₁₆

yes

—

C₈×C₅₉

C₄×D₄

yes

characteristic

C₈×C₅₉

#135

yes

characteristic

C₅₉⋊C₈

—

#13653

—

maximal, Sylow

C₂²×C₅₉

C₂×D₁₆

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

C₄×D₈

yes

characteristic

C₄×C₅₉

#508

yes

characteristic

Dic₅₉

—

Dic₅₉

—

C₂×Q₆₄

D₅₉

yes

characteristic

C₄×Q₃₂

—

#1424

—

#1416

—

#1397

—

C₂×C₅₉

C₄×D₁₆

yes

characteristic

C₂×C₅₉

#2264

yes

—

C₂×Q₃₂

C₂×D₅₉

yes

—

C₄×Q₁₆

—

118

—

Q₆₄

Dic₅₉

yes

complemented

C₂×C₃₂

—

C₂×C₃₂

C₂×D₅₉

yes

characteristic

#366

—

#358

—

118

—

C₄×C₁₆

—

C₅₉

#13653

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄×Q₈

—

236

—

Q₃₂

—

Q₃₂

C₂×Dic₅₉

yes

—

C₂×C₁₆

—

C₂×C₁₆

C₂²×D₅₉

yes

characteristic

#107

—

118

—

#103

—

236

—

C₄×C₈

—

C₃₂

—

118

—

C₃₂

C₂×Dic₅₉

yes

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

D₄×D₅₉

yes

characteristic

C₄⋊C₄

—

472

—

C₄⋊C₄

—

236

—

C₄²

—

C₄²

—

472

—

C₁₆

—

118

—

C₁₆

#11745687

yes

characteristic

C₁₆

C₂²×Dic₅₉

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

472

—

C₂×C₄

D₈×D₅₉

yes

characteristic

C₈

—

118

—

C₈

D₄×Dic₅₉

yes

characteristic

100

C₈

#14273746

yes

characteristic

101

C₂²

D₁₆×D₅₉

yes

characteristic

102

C₄

—

103

C₄

—

118

—

104

C₄

—

944

—

105

C₄

D₈×Dic₅₉

yes

characteristic

106

C₄

#16837831

yes

characteristic

107

C₂

D₁₆×Dic₅₉

yes

characteristic

108

C₂

#21426415

yes

—

	subgroup	G/H
center	C₂²	D₁₆×D₅₉
commutator	C₁₆×C₅₉	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₅₉
Fitting	C₂×C₅₉×Q₆₄	C₂
socle, prime radical	C₂²×C₅₉	C₂×D₁₆
cosocle	C₂×C₁₆×C₅₉	C₂³
hypercenter	C₂×Q₆₄	D₅₉
hypocenter	C₅₉	#13653
LCS	G ▷ C₁₆×C₅₉ ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×Q₆₄

subgroup

G/H

center

C₂²

D₁₆×D₅₉

commutator

C₁₆×C₅₉

C₂²×C₄

Frattini

C₂×C₁₆

C₂²×D₅₉

Fitting

C₂×C₅₉×Q₆₄

C₂

socle, prime radical

C₂²×C₅₉

C₂×D₁₆

cosocle

C₂×C₁₆×C₅₉

C₂³

hypercenter

C₂×Q₆₄

D₅₉

hypocenter

C₅₉

#13653

LCS

G ▷ C₁₆×C₅₉ ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×Q₆₄

N	H ≅ G/N	type
12. C₅₉×Q₆₄	C₄	semidirect
62. Q₆₄	Dic₅₉	semidirect
68. C₅₉	#13653	semidirect

H ≅ G/N

type

12. C₅₉×Q₆₄

C₄

semidirect

62. Q₆₄

Dic₅₉

semidirect

68. C₅₉

#13653

semidirect

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

subgroup	index
Q₃₂×Dic₅₉	2
#21422298	2
#21422277	2
#21422229	2
C₂×C₅₉×Q₆₄	2
#13653	59

subgroup

index

Q₃₂×Dic₅₉

C₂×C₅₉×Q₆₄

Subgroups — #28509287

Conjugacy classes of subgroups

Complements

Maximal subgroups