Group #28509708 — subgroups

Subgroups — #28509708

Order 15104.

158 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 67 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	#21426370	C₂	1	2	yes	maximal
2	#21422296	C₂	1	1	yes	maximal, characteristic
3	#21422295	C₂	1	2	yes	maximal
4	C₃₂×Dic₅₉	C₂	1	1	yes	maximal, characteristic
5	C₅₉×#1423	C₂	1	1	yes	maximal, characteristic
6	#16837806	—	2	2	no	—
7	C₂×C₅₉⋊C₃₂	C₂²	1	1	yes	characteristic
8	#16837228	C₂²	1	2	yes	—
9	#16837227	—	2	2	no	—
10	C₁₆×Dic₅₉	C₂²	1	1	yes	characteristic
11	C₂×C₃₂×C₅₉	C₂²	1	1	yes	characteristic
12	C₅₉×#366	C₂²	1	2	yes	—
13	#14273726	—	4	2	no	—
14	C₂×C₅₉⋊C₁₆	D₄	1	1	yes	characteristic
15	#14273661	—	2	2	no	—
16	#14273660	—	4	2	no	—
17	C₈×Dic₅₉	D₄	1	1	yes	characteristic
18	C₅₉⋊C₃₂	Q₈	1	2	yes	complemented
19	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
20	#107×C₅₉	—	2	2	no	—
21	C₃₂×C₅₉	Q₈	1	2	yes	complemented
22	C₂×Dic₁₁₈	—	8	2	no	—
23	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
24	C₄⋊C₄×C₅₉	—	4	2	no	—
25	C₄⋊Dic₅₉	—	4	2	no	—
26	Dic₅₉⋊C₄	—	8	2	no	—
27	C₄×Dic₅₉	D₈	1	1	yes	characteristic
28	C₂×C₅₉⋊C₈	D₈	1	1	yes	characteristic
29	C₁₆×C₅₉	C₂×Q₈	1	1	yes	characteristic
30	C₁₆×C₅₉	C₂×Q₈	1	1	yes	characteristic
31	C₅₉⋊C₁₆	Q₁₆	1	2	yes	—
32	C₂×C₄×C₅₉	—	8	2	no	—
33	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
34	C₂×Dic₅₉	—	8	2	no	—
35	C₂×Dic₅₉	D₁₆	1	1	yes	characteristic
36	C₂×Dic₅₉	D₁₆	1	1	yes	characteristic
37	Dic₁₁₈	—	16	4	no	—
38	C₈×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
39	C₈×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
40	C₅₉⋊C₈	Q₃₂	1	2	yes	—
41	#13731	—	59	1	no	maximal, Sylow
42	C₂²×C₅₉	C₂×D₁₆	1	1	yes	characteristic
43	C₄×C₅₉	—	16	2	no	—
44	C₄×C₅₉	#500	1	1	yes	characteristic
45	C₄×C₅₉	C₂×Q₃₂	1	1	yes	characteristic
46	Dic₅₉	—	16	2	no	—
47	Dic₅₉	D₃₂	1	2	yes	—
48	Dic₅₉	Q₆₄	1	2	yes	—
49	#2252	—	59	2	no	—
50	#1423	—	59	1	no	—
51	#1423	—	59	2	no	—
52	#1423	D₅₉	1	1	yes	characteristic
53	C₄×C₃₂	—	59	1	no	—
54	C₂×C₅₉	#2252	1	1	yes	characteristic
55	C₂×C₅₉	C₂×D₃₂	1	1	yes	characteristic
56	C₂×C₅₉	C₂×Q₆₄	1	1	yes	characteristic
57	#500	—	118	2	no	—
58	C₂×C₃₂	—	59	1	no	—
59	C₂×C₃₂	C₂×D₅₉	1	1	yes	characteristic
60	#366	—	59	2	no	—
61	#366	—	118	2	no	—
62	#366	C₂×D₅₉	1	2	yes	—
63	C₄×C₁₆	—	59	1	no	—
64	C₅₉	#13731	1	1	yes	characteristic, complemented, Sylow
65	C₄⋊Q₈	—	236	2	no	—
66	C₂×C₁₆	—	59	1	no	—
67	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
68	#107	—	2	2	no	—
69	#107	—	118	2	no	—
70	#107	—	236	2	no	—
71	C₄×C₈	—	59	1	no	—
72	C₃₂	—	59	2	no	—
73	C₃₂	Dic₁₁₈	1	2	yes	complemented
74	C₂×Q₈	—	472	2	no	—
75	C₂×C₈	—	59	1	no	—
76	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
77	C₄⋊C₄	—	4	2	no	—
78	C₄⋊C₄	—	472	2	no	—
79	C₄⋊C₄	—	236	2	no	—
80	C₄²	—	59	1	no	—
81	C₁₆	—	59	2	no	—
82	C₁₆	C₂×Dic₁₁₈	1	1	yes	characteristic
83	C₁₆	Q₈×D₅₉	1	1	yes	characteristic
84	Q₈	—	944	4	no	—
85	C₂×C₄	—	8	2	no	—
86	C₂×C₄	—	59	1	no	—
87	C₂×C₄	—	59	1	no	—
88	C₂×C₄	—	472	2	no	—
89	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
90	C₈	—	59	2	no	—
91	C₈	#14273726	1	1	yes	characteristic
92	C₈	Q₁₆×D₅₉	1	1	yes	characteristic
93	C₂²	D₁₆×D₅₉	1	1	yes	characteristic
94	C₄	—	16	2	no	—
95	C₄	—	59	2	no	—
96	C₄	—	59	2	no	—
97	C₄	—	944	2	no	—
98	C₄	#16837806	1	1	yes	characteristic
99	C₄	Q₃₂×D₅₉	1	1	yes	characteristic
100	C₂	#21426370	1	1	yes	characteristic
101	C₂	D₃₂×D₅₉	1	1	yes	characteristic
102	C₂	Q₆₄×D₅₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21426370

C₂

yes

maximal

#21422296

C₂

yes

maximal, characteristic

#21422295

C₂

yes

maximal

C₃₂×Dic₅₉

C₂

yes

maximal, characteristic

C₅₉×#1423

C₂

yes

maximal, characteristic

#16837806

—

C₂×C₅₉⋊C₃₂

C₂²

yes

characteristic

#16837228

C₂²

yes

—

#16837227

—

C₁₆×Dic₅₉

C₂²

yes

characteristic

C₂×C₃₂×C₅₉

C₂²

yes

characteristic

C₅₉×#366

C₂²

yes

—

#14273726

—

C₂×C₅₉⋊C₁₆

D₄

yes

characteristic

#14273661

—

#14273660

—

C₈×Dic₅₉

D₄

yes

characteristic

C₅₉⋊C₃₂

Q₈

yes

complemented

C₂×C₁₆×C₅₉

C₂³

yes

characteristic

#107×C₅₉

—

C₃₂×C₅₉

Q₈

yes

complemented

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₄⋊C₄×C₅₉

—

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

D₈

yes

characteristic

C₂×C₅₉⋊C₈

D₈

yes

characteristic

C₁₆×C₅₉

C₂×Q₈

yes

characteristic

C₁₆×C₅₉

C₂×Q₈

yes

characteristic

C₅₉⋊C₁₆

Q₁₆

yes

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₈

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

D₁₆

yes

characteristic

C₂×Dic₅₉

D₁₆

yes

characteristic

Dic₁₁₈

—

C₈×C₅₉

C₄⋊Q₈

yes

characteristic

C₈×C₅₉

C₂×Q₁₆

yes

characteristic

C₅₉⋊C₈

Q₃₂

yes

—

#13731

—

maximal, Sylow

C₂²×C₅₉

C₂×D₁₆

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

#500

yes

characteristic

C₄×C₅₉

C₂×Q₃₂

yes

characteristic

Dic₅₉

—

Dic₅₉

D₃₂

yes

—

Dic₅₉

Q₆₄

yes

—

—

—

—

D₅₉

yes

characteristic

C₄×C₃₂

—

C₂×C₅₉

#2252

yes

characteristic

C₂×C₅₉

C₂×D₃₂

yes

characteristic

C₂×C₅₉

C₂×Q₆₄

yes

characteristic

#500

—

118

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₅₉

yes

characteristic

#366

—

#366

—

118

—

#366

C₂×D₅₉

yes

—

C₄×C₁₆

—

C₅₉

#13731

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

236

—

C₂×C₁₆

—

C₂×C₁₆

C₂²×D₅₉

yes

characteristic

#107

—

#107

—

118

—

#107

—

236

—

C₄×C₈

—

C₃₂

—

C₃₂

Dic₁₁₈

yes

complemented

C₂×Q₈

—

472

—

C₂×C₈

—

C₂×C₈

D₄×D₅₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

472

—

C₄⋊C₄

—

236

—

C₄²

—

C₁₆

—

C₁₆

C₂×Dic₁₁₈

yes

characteristic

C₁₆

Q₈×D₅₉

yes

characteristic

Q₈

—

944

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

472

—

C₂×C₄

D₈×D₅₉

yes

characteristic

C₈

—

C₈

#14273726

yes

characteristic

C₈

Q₁₆×D₅₉

yes

characteristic

C₂²

D₁₆×D₅₉

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

944

—

C₄

#16837806

yes

characteristic

C₄

Q₃₂×D₅₉

yes

characteristic

100

C₂

#21426370

yes

characteristic

101

C₂

D₃₂×D₅₉

yes

characteristic

102

C₂

Q₆₄×D₅₉

yes

characteristic

	subgroup	G/H
center	C₂²	D₁₆×D₅₉
commutator, cosocle	C₂×C₁₆×C₅₉	C₂³
Frattini	C₂×C₁₆	C₂²×D₅₉
Fitting	C₅₉×#1423	C₂
socle, prime radical	C₂²×C₅₉	C₂×D₁₆
hypercenter	#1423	D₅₉
hypocenter	C₅₉	#13731
LCS	G ▷ C₂×C₁₆×C₅₉ ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1423

subgroup

G/H

center

C₂²

D₁₆×D₅₉

commutator, cosocle

C₂×C₁₆×C₅₉

C₂³

Frattini

C₂×C₁₆

C₂²×D₅₉

Fitting

C₅₉×#1423

C₂

socle, prime radical

C₂²×C₅₉

C₂×D₁₆

D₅₉

C₅₉

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

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1423

N	H ≅ G/N	type
18. C₅₉⋊C₃₂	Q₈	semidirect
21. C₃₂×C₅₉	Q₈	semidirect
64. C₅₉	#13731	semidirect
73. C₃₂	Dic₁₁₈	semidirect

H ≅ G/N

type

18. C₅₉⋊C₃₂

Q₈

semidirect

21. C₃₂×C₅₉

Q₈

semidirect

64. C₅₉

#13731

semidirect

73. C₃₂

Dic₁₁₈

semidirect

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

subgroup	index
#21426370	2
#21422296	2
#21422295	2
C₃₂×Dic₅₉	2
C₅₉×#1423	2
#13731	59

subgroup

index

#21426370

#21422296

#21422295

C₃₂×Dic₅₉

C₅₉×#1423

#13731

Subgroups — #28509708

Conjugacy classes of subgroups

Complements

Maximal subgroups