Group #28476170 — subgroups

Subgroups — #28476170

Order 15104.

110 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 110 conjugacy classes of subgroups — too many to draw. Only its 53 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	#21426004	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₅₉⋊C₃₂)	C₂	1	1	yes	maximal, characteristic
3	C₅₉×#1397	C₂	1	1	yes	maximal, characteristic
4	C₈×Dic₁₁₈	C₄	1	1	yes	characteristic
5	C₂×C₅₉⋊C₃₂	—	2	1	no	—
6	C₄⋊(C₅₉⋊C₁₆)	C₄	1	1	yes	characteristic
7	Dic₅₉⋊C₁₆	—	2	1	no	—
8	#16837177	—	2	1	no	—
9	C₂×C₃₂×C₅₉	—	2	1	no	—
10	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
11	C₄×Dic₁₁₈	C₈	1	1	yes	characteristic
12	C₂×C₅₉⋊C₁₆	—	2	1	no	—
13	C₄⋊(C₅₉⋊C₈)	C₈	1	1	yes	characteristic
14	Dic₅₉⋊C₈	—	2	1	no	—
15	C₈×Dic₅₉	—	2	1	no	—
16	C₅₉⋊C₃₂	—	4	1	no	—
17	C₂×C₁₆×C₅₉	D₄	1	1	yes	characteristic
18	C₂×C₁₆×C₅₉	D₄	1	1	yes	characteristic
19	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
20	C₃₂×C₅₉	—	2	2	no	—
21	C₂×Dic₁₁₈	C₁₆	1	1	yes	characteristic
22	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×C₈×C₅₉	M₁₆	1	1	yes	characteristic
24	C₄²×C₅₉	C₂×C₈	1	1	yes	characteristic
25	C₄⋊Dic₅₉	C₁₆	1	1	yes	characteristic
26	Dic₅₉⋊C₄	—	2	1	no	—
27	C₄×Dic₅₉	—	2	1	no	—
28	C₂×C₅₉⋊C₈	—	2	1	no	—
29	C₁₆×C₅₉	—	2	1	no	—
30	C₁₆×C₅₉	SD₁₆	1	1	yes	characteristic
31	C₁₆×C₅₉	Q₁₆	1	1	yes	characteristic
32	C₅₉⋊C₁₆	—	4	1	no	—
33	C₂×C₄×C₅₉	C₂²⋊C₈	1	1	yes	characteristic
34	C₂×C₄×C₅₉	C₂×C₁₆	1	1	yes	characteristic
35	C₂×C₄×C₅₉	M₃₂	1	1	yes	characteristic
36	C₂×Dic₅₉	—	2	1	no	—
37	C₂×Dic₅₉	—	2	1	no	—
38	Dic₁₁₈	—	4	1	no	—
39	C₈×C₅₉	—	2	1	no	—
40	C₈×C₅₉	#103	1	1	yes	characteristic
41	C₈×C₅₉	#104	1	1	yes	characteristic
42	C₅₉⋊C₈	—	4	1	no	—
43	#7393	—	59	1	no	maximal, Sylow
44	C₂²×C₅₉	C₂²⋊C₁₆	1	1	yes	characteristic
45	C₄×C₅₉	—	2	1	no	—
46	C₄×C₅₉	—	2	1	no	—
47	C₄×C₅₉	Q₈⋊C₈	1	1	yes	characteristic
48	C₄×C₅₉	#350	1	1	yes	characteristic
49	Dic₅₉	—	4	1	no	—
50	Dic₅₉	—	4	1	no	—
51	#2183	—	59	1	no	—
52	C₄⋊C₃₂	—	59	1	no	—
53	#1397	D₅₉	1	1	yes	characteristic
54	C₂×C₅₉	Q₈⋊C₁₆	1	1	yes	characteristic
55	C₂×C₅₉	#1400	1	1	yes	characteristic
56	C₂×C₅₉	#1401	1	1	yes	characteristic
57	C₈×Q₈	—	59	1	no	—
58	C₂×C₃₂	—	2	1	no	—
59	C₂×C₃₂	—	118	1	no	—
60	C₄⋊C₁₆	—	59	1	no	—
61	C₄⋊C₁₆	—	118	1	no	—
62	#346	—	118	1	no	—
63	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
64	C₅₉	#7393	1	1	yes	characteristic, complemented, Sylow
65	C₄×Q₈	—	59	1	no	—
66	C₂×C₁₆	—	118	1	no	—
67	C₂×C₁₆	D₂₃₆	1	1	yes	characteristic
68	C₂×C₁₆	C₂²⋊D₅₉	1	1	yes	characteristic
69	C₄⋊C₈	—	59	1	no	—
70	C₄⋊C₈	—	118	1	no	—
71	C₄×C₈	—	118	1	no	—
72	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
73	C₃₂	—	2	2	no	—
74	C₃₂	—	236	1	no	—
75	C₂×Q₈	—	59	1	no	—
76	C₂×C₈	—	118	1	no	—
77	C₂×C₈	#11745659	1	1	yes	characteristic
78	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
79	C₄⋊C₄	—	59	1	no	—
80	C₄⋊C₄	—	118	1	no	—
81	C₄²	—	118	1	no	—
82	C₄²	C₈×D₅₉	1	1	yes	characteristic
83	C₁₆	—	2	1	no	—
84	C₁₆	—	236	1	no	—
85	C₁₆	#11745660	1	1	yes	characteristic
86	C₁₆	Dic₂₃₆	1	1	yes	characteristic
87	Q₈	—	236	1	no	—
88	C₂×C₄	—	118	1	no	—
89	C₂×C₄	—	118	1	no	—
90	C₂×C₄	(C₂×D₅₉)⋊C₈	1	1	yes	characteristic
91	C₂×C₄	C₁₆×D₅₉	1	1	yes	characteristic
92	C₂×C₄	#14273666	1	1	yes	characteristic
93	C₈	—	2	1	no	—
94	C₈	—	236	1	no	—
95	C₈	#14273652	1	1	yes	characteristic
96	C₈	#14273655	1	1	yes	characteristic
97	C₂²	(C₂×D₅₉)⋊C₁₆	1	1	yes	characteristic
98	C₄	—	2	1	no	—
99	C₄	—	2	1	no	—
100	C₄	—	236	1	no	—
101	C₄	—	236	1	no	—
102	C₄	#16837134	1	1	yes	characteristic
103	C₄	#16837185	1	1	yes	characteristic
104	C₂	#21422076	1	1	yes	characteristic
105	C₂	#21422235	1	1	yes	characteristic
106	C₂	#21422239	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21426004

C₂

yes

maximal, characteristic

C₄⋊(C₅₉⋊C₃₂)

C₂

yes

maximal, characteristic

C₅₉×#1397

C₂

yes

maximal, characteristic

C₈×Dic₁₁₈

C₄

yes

characteristic

C₂×C₅₉⋊C₃₂

—

C₄⋊(C₅₉⋊C₁₆)

C₄

yes

characteristic

Dic₅₉⋊C₁₆

—

#16837177

—

C₂×C₃₂×C₅₉

—

C₄×C₁₆×C₅₉

C₂²

yes

characteristic

C₄×Dic₁₁₈

C₈

yes

characteristic

C₂×C₅₉⋊C₁₆

—

C₄⋊(C₅₉⋊C₈)

C₈

yes

characteristic

Dic₅₉⋊C₈

—

C₈×Dic₅₉

—

C₅₉⋊C₃₂

—

C₂×C₁₆×C₅₉

D₄

yes

characteristic

C₂×C₁₆×C₅₉

D₄

yes

characteristic

C₄×C₈×C₅₉

C₂×C₄

yes

characteristic

C₃₂×C₅₉

—

C₂×Dic₁₁₈

C₁₆

yes

characteristic

C₂×C₈×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₅₉

M₁₆

yes

characteristic

C₄²×C₅₉

C₂×C₈

yes

characteristic

C₄⋊Dic₅₉

C₁₆

yes

characteristic

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₁₆×C₅₉

—

C₁₆×C₅₉

SD₁₆

yes

characteristic

C₁₆×C₅₉

Q₁₆

yes

characteristic

C₅₉⋊C₁₆

—

C₂×C₄×C₅₉

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₅₉

C₂×C₁₆

yes

characteristic

C₂×C₄×C₅₉

M₃₂

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

yes

C₈×C₅₉

yes

C₅₉⋊C₈

—

#7393

—

maximal, Sylow

C₂²×C₅₉

C₂²⋊C₁₆

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

Q₈⋊C₈

yes

characteristic

C₄×C₅₉

#350

yes

characteristic

Dic₅₉

—

Dic₅₉

—

#2183

—

C₄⋊C₃₂

—

#1397

D₅₉

yes

characteristic

C₂×C₅₉

Q₈⋊C₁₆

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₈×Q₈

—

C₂×C₃₂

—

C₂×C₃₂

—

118

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

118

—

#346

—

118

—

C₄×C₁₆

C₂×D₅₉

yes

characteristic

C₅₉

#7393

yes

characteristic, complemented, Sylow

C₄×Q₈

—

C₂×C₁₆

—

118

—

C₂×C₁₆

D₂₃₆

yes

characteristic

C₂×C₁₆

C₂²⋊D₅₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

118

—

C₄×C₈

—

118

—

C₄×C₈

C₄×D₅₉

yes

characteristic

C₃₂

—

C₃₂

—

236

—

C₂×Q₈

—

C₂×C₈

—

118

—

C₂×C₈

#11745659

yes

characteristic

C₂×C₈

(C₂×D₅₉)⋊C₄

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

118

—

C₄²

—

118

—

C₄²

C₈×D₅₉

yes

characteristic

C₁₆

—

C₁₆

—

236

—

C₁₆

#11745660

yes

characteristic

C₁₆

Dic₂₃₆

yes

characteristic

Q₈

—

236

—

C₂×C₄

—

118

—

C₂×C₄

—

118

—

C₂×C₄

(C₂×D₅₉)⋊C₈

yes

characteristic

C₂×C₄

C₁₆×D₅₉

yes

characteristic

C₂×C₄

#14273666

yes

characteristic

C₈

—

C₈

—

236

—

C₈

yes

C₈

yes

C₂²

(C₂×D₅₉)⋊C₁₆

yes

characteristic

C₄

—

C₄

—

100

C₄

—

236

—

101

C₄

—

236

—

102

C₄

#16837134

yes

characteristic

103

C₄

#16837185

yes

characteristic

104

C₂

#21422076

yes

characteristic

105

C₂

#21422235

yes

characteristic

106

C₂

#21422239

yes

characteristic

	subgroup	G/H
center	C₂×C₈	(C₂×D₅₉)⋊C₄
commutator	C₂×C₄×C₅₉	C₂×C₁₆
Frattini	C₄×C₁₆	C₂×D₅₉
Fitting	C₅₉×#1397	C₂
socle, prime radical	C₂²×C₅₉	C₂²⋊C₁₆
cosocle	C₄×C₁₆×C₅₉	C₂²
hypercenter	#1397	D₅₉
hypocenter	C₅₉	#7393
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₁₆ ◁ #1397

subgroup

G/H

center

C₂×C₈

(C₂×D₅₉)⋊C₄

commutator

C₂×C₄×C₅₉

C₂×C₁₆

Frattini

C₄×C₁₆

C₂×D₅₉

Fitting

C₅₉×#1397

C₂

socle, prime radical

C₂²×C₅₉

C₂²⋊C₁₆

cosocle

C₄×C₁₆×C₅₉

C₂²

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₁₆ ◁ #1397

N	H ≅ G/N	type
64. C₅₉	#7393	semidirect

H ≅ G/N

type

64. C₅₉

#7393

semidirect

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

subgroup	index
#21426004	2
C₄⋊(C₅₉⋊C₃₂)	2
C₅₉×#1397	2
#7393	59

subgroup

index

#21426004

C₄⋊(C₅₉⋊C₃₂)

C₅₉×#1397

#7393

Subgroups — #28476170

Conjugacy classes of subgroups

Complements

Maximal subgroups