Group #28482363 — subgroups

Subgroups — #28482363

Order 15104.

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

normal subgroups only

This group has 322 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	#21424213	C₂	1	2	yes	maximal
2	#21421931	C₂	1	4	yes	maximal
3	C₅₉×#1744	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	4	yes	—
5	#16837157	—	2	4	no	—
6	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
7	C₄³×C₅₉	C₂²	1	1	yes	characteristic
8	C₅₉×#336	C₄	1	4	yes	—
9	C₂²×C₅₉⋊C₈	—	2	4	no	—
10	C₄⋊(C₅₉⋊C₈)	—	2	8	no	—
11	C₂²×C₈×C₅₉	C₂×C₄	1	4	yes	—
12	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
13	C₂×C₄²×C₅₉	D₄	1	2	yes	—
14	C₂×C₄²×C₅₉	Q₈	1	2	yes	—
15	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
16	C₄×C₈×C₅₉	—	2	4	no	—
17	C₂²×C₄×C₅₉	C₄⋊C₄	1	4	yes	—
18	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
20	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
21	C₂×C₈×C₅₉	—	2	4	no	—
22	C₂×C₈×C₅₉	—	2	8	no	—
23	C₄²×C₅₉	—	2	4	no	—
24	C₄²×C₅₉	—	2	4	no	—
25	C₄²×C₅₉	—	4	2	no	—
26	C₄²×C₅₉	#41	1	4	yes	—
27	C₂×C₅₉⋊C₈	—	4	4	no	—
28	C₂×C₅₉⋊C₈	—	4	8	no	—
29	C₂³×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₅₉	—	2	2	no	—
31	C₂×C₄×C₅₉	—	2	4	no	—
32	C₂×C₄×C₅₉	—	2	4	no	—
33	C₂×C₄×C₅₉	—	2	4	no	—
34	C₂×C₄×C₅₉	#117	1	2	yes	—
35	C₂×C₄×C₅₉	#124	1	2	yes	—
36	C₂×C₄×C₅₉	#125	1	2	yes	—
37	C₂×C₄×C₅₉	#131	1	4	yes	—
38	C₂×C₄×C₅₉	#135	1	4	yes	—
39	C₈×C₅₉	—	4	4	no	—
40	C₅₉⋊C₈	—	8	4	no	—
41	#8788	—	59	1	no	maximal, Sylow
42	C₂²×C₅₉	#382	1	1	yes	characteristic
43	C₂²×C₅₉	#424	1	2	yes	—
44	C₂²×C₅₉	#427	1	2	yes	—
45	C₂²×C₅₉	#430	1	2	yes	—
46	C₄×C₅₉	—	2	2	no	—
47	C₄×C₅₉	—	4	4	no	—
48	C₄×C₅₉	#433	1	4	yes	—
49	C₄×C₅₉	#488	1	4	yes	—
50	#1831	—	59	2	no	—
51	#1744	D₅₉	1	1	yes	characteristic
52	#1277	—	59	4	no	—
53	C₂×C₅₉	#1572	1	4	yes	—
54	C₂×C₅₉	#1832	1	1	yes	characteristic
55	C₂×C₅₉	#1835	1	1	yes	characteristic
56	C₂×C₅₉	#1838	1	1	yes	characteristic
57	C₂×C₄⋊C₈	—	59	4	no	—
58	C₂×C₄×C₈	C₂×D₅₉	1	2	yes	—
59	C₄³	C₂×D₅₉	1	1	yes	characteristic
60	#336	—	118	4	no	—
61	#336	Dic₅₉	1	4	yes	—
62	C₅₉	#8788	1	1	yes	characteristic, complemented, Sylow
63	C₂²×C₈	—	118	4	no	—
64	C₂²×C₈	C₂×Dic₅₉	1	4	yes	—
65	C₂×C₄²	Dic₁₁₈	1	2	yes	—
66	C₂×C₄²	D₂₃₆	1	2	yes	—
67	C₂×C₄²	C₂×Dic₅₉	1	2	yes	—
68	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
69	C₄⋊C₈	—	118	8	no	—
70	C₄×C₈	—	2	4	no	—
71	C₂²×C₄	C₄⋊Dic₅₉	1	4	yes	—
72	C₂²×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
73	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
74	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
75	C₂×C₈	—	2	4	no	—
76	C₂×C₈	—	2	8	no	—
77	C₂×C₈	—	236	4	no	—
78	C₂×C₈	—	236	8	no	—
79	C₄²	—	2	4	no	—
80	C₄²	—	2	4	no	—
81	C₄²	—	4	2	no	—
82	C₄²	#11745685	1	4	yes	—
83	C₂³	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	4	no	—
86	C₂×C₄	—	2	4	no	—
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	#14273686	1	2	yes	—
89	C₂×C₄	#14273715	1	2	yes	—
90	C₂×C₄	#14273719	1	2	yes	—
91	C₂×C₄	#14273734	1	4	yes	—
92	C₂×C₄	#14273749	1	4	yes	—
93	C₈	—	4	4	no	—
94	C₈	—	472	4	no	—
95	C₂²	#16837268	1	1	yes	characteristic
96	C₂²	#16837429	1	2	yes	—
97	C₂²	#16837440	1	2	yes	—
98	C₂²	#16837451	1	2	yes	—
99	C₄	—	2	2	no	—
100	C₄	—	4	4	no	—
101	C₄	#16837463	1	4	yes	—
102	C₄	#16837757	1	4	yes	—
103	C₂	#21422876	1	4	yes	—
104	C₂	#21424216	1	1	yes	characteristic
105	C₂	#21424227	1	1	yes	characteristic
106	C₂	#21424237	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21424213

C₂

yes

maximal

#21421931

C₂

yes

maximal

C₅₉×#1744

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

#16837157

—

C₂×C₄×C₈×C₅₉

C₂²

yes

—

C₄³×C₅₉

C₂²

yes

characteristic

C₅₉×#336

C₄

yes

—

C₂²×C₅₉⋊C₈

—

C₄⋊(C₅₉⋊C₈)

—

C₂²×C₈×C₅₉

C₂×C₄

yes

—

C₂×C₄²×C₅₉

C₂×C₄

yes

—

C₂×C₄²×C₅₉

D₄

yes

—

C₂×C₄²×C₅₉

Q₈

yes

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

C₄×C₈×C₅₉

—

C₂²×C₄×C₅₉

C₄⋊C₄

yes

—

C₂²×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

#41

yes

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#117

yes

—

C₂×C₄×C₅₉

#124

yes

—

C₂×C₄×C₅₉

#125

yes

—

C₂×C₄×C₅₉

#131

yes

—

C₂×C₄×C₅₉

#135

yes

—

C₈×C₅₉

—

C₅₉⋊C₈

—

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₂²×C₅₉

#427

yes

—

C₂²×C₅₉

#430

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#433

yes

—

C₄×C₅₉

#488

yes

—

#1831

—

#1744

D₅₉

yes

characteristic

#1277

—

C₂×C₅₉

#1572

yes

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₄⋊C₈

—

C₂×C₄×C₈

C₂×D₅₉

yes

—

C₄³

C₂×D₅₉

yes

characteristic

#336

—

118

—

#336

Dic₅₉

yes

—

C₅₉

#8788

yes

characteristic, complemented, Sylow

C₂²×C₈

—

118

—

C₂²×C₈

C₂×Dic₅₉

yes

—

C₂×C₄²

Dic₁₁₈

yes

—

C₂×C₄²

D₂₃₆

yes

—

C₂×C₄²

C₂×Dic₅₉

yes

—

C₂×C₄²

C₂²×D₅₉

yes

characteristic

C₄⋊C₈

—

118

—

C₄×C₈

—

C₂²×C₄

C₄⋊Dic₅₉

yes

—

C₂²×C₄

C₂×Dic₁₁₈

yes

characteristic

C₂²×C₄

C₂×D₂₃₆

yes

characteristic

C₂²×C₄

C₂²×Dic₅₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

236

—

C₂×C₈

—

236

—

C₄²

—

C₄²

—

C₄²

—

C₄²

#11745685

yes

—

C₂³

C₂×C₄⋊Dic₅₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#14273686

yes

—

C₂×C₄

#14273715

yes

—

C₂×C₄

#14273719

yes

—

C₂×C₄

#14273734

yes

—

C₂×C₄

#14273749

yes

—

C₈

—

C₈

—

472

—

C₂²

#16837268

yes

characteristic

C₂²

#16837429

yes

—

C₂²

#16837440

yes

—

C₂²

#16837451

yes

—

C₄

—

100

C₄

—

101

C₄

#16837463

yes

—

102

C₄

#16837757

yes

—

103

C₂

#21422876

yes

—

104

C₂

#21424216

yes

characteristic

105

C₂

#21424227

yes

characteristic

106

C₂

#21424237

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₃₆
commutator	C₂²×C₄×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1744	C₂
socle, prime radical	C₂³×C₅₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1744	D₅₉
hypocenter	C₅₉	#8788
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

subgroup

G/H

center

C₂²×C₄

C₂×D₂₃₆

commutator

C₂²×C₄×C₅₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₅₉×#1744

C₂

socle, prime radical

C₂³×C₅₉

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

N	H ≅ G/N	type
62. C₅₉	#8788	semidirect

H ≅ G/N

type

62. C₅₉

#8788

semidirect

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

subgroup	index
#21424213	2
#21421931	2
C₅₉×#1744	2
#8788	59

subgroup

index

#21424213

#21421931

C₅₉×#1744

#8788

Subgroups — #28482363

Conjugacy classes of subgroups

Complements

Maximal subgroups