Group #28482402 — subgroups

Subgroups — #28482402

Order 15104.

220 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 99 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	#21422859	C₂	1	2	yes	maximal
2	#21421958	C₂	1	4	yes	maximal
3	C₅₉×#1452	C₂	1	1	yes	maximal, characteristic
4	#16837459	C₂²	1	4	yes	—
5	#16837155	—	2	2	no	—
6	#16837153	—	2	2	no	—
7	C₅₉×#431	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
9	C₅₉×#322	C₄	1	4	yes	—
10	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
11	C₂²⋊(C₅₉⋊C₈)	—	2	4	no	—
12	#14273638	—	2	4	no	—
13	C₂²×C₈×C₅₉	Q₈	1	2	yes	—
14	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
15	C₄⋊C₈×C₅₉	—	2	2	no	—
16	C₂²⋊C₈×C₅₉	—	2	2	no	—
17	C₄×C₈×C₅₉	C₂×C₄	1	4	yes	—
18	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
20	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
22	C₂²×C₄×C₅₉	#41	1	2	yes	—
23	C₂×C₈×C₅₉	—	2	4	no	—
24	C₂×C₈×C₅₉	—	4	2	no	—
25	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
26	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
27	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
28	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
29	C₄²×C₅₉	#41	1	2	yes	—
30	C₂×C₅₉⋊C₈	—	4	4	no	—
31	C₂³×C₅₉	#125	1	1	yes	characteristic
32	C₂×C₄×C₅₉	—	2	2	no	—
33	C₂×C₄×C₅₉	—	4	2	no	—
34	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₅₉	#117	1	2	yes	—
36	C₂×C₄×C₅₉	#124	1	2	yes	—
37	C₂×C₄×C₅₉	#125	1	1	yes	characteristic
38	C₈×C₅₉	—	2	2	no	—
39	C₈×C₅₉	—	2	2	no	—
40	C₈×C₅₉	—	4	4	no	—
41	C₅₉⋊C₈	—	8	4	no	—
42	#8798	—	59	1	no	maximal, Sylow
43	C₂²×C₅₉	—	2	2	no	—
44	C₂²×C₅₉	—	2	1	no	—
45	C₂²×C₅₉	#382	1	1	yes	characteristic
46	C₄×C₅₉	—	2	2	no	—
47	C₄×C₅₉	—	2	2	no	—
48	C₄×C₅₉	—	2	1	no	—
49	C₄×C₅₉	#430	1	2	yes	—
50	#1568	—	59	2	no	—
51	#1452	D₅₉	1	1	yes	characteristic
52	#1287	—	59	4	no	—
53	C₂×C₅₉	—	4	1	no	—
54	C₂×C₅₉	#1833	1	2	yes	—
55	C₂×C₅₉	#1838	1	1	yes	characteristic
56	#432	—	59	4	no	—
57	#431	C₂×D₅₉	1	2	yes	—
58	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
59	#335	—	118	2	no	—
60	#334	—	118	2	no	—
61	#322	Dic₅₉	1	4	yes	—
62	C₅₉	#8798	1	1	yes	characteristic, complemented, Sylow
63	C₂²×C₈	Dic₁₁₈	1	2	yes	—
64	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
65	C₄⋊C₈	—	2	2	no	—
66	C₄⋊C₈	—	118	4	no	—
67	C₂²⋊C₈	—	2	2	no	—
68	C₂²⋊C₈	—	118	4	no	—
69	#97	—	118	4	no	—
70	C₄×C₈	D₂₃₆	1	1	yes	characteristic
71	C₄×C₈	D₂₃₆	1	1	yes	characteristic
72	C₄×C₈	C₂×Dic₅₉	1	4	yes	—
73	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
74	C₂²×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
75	C₂²×C₄	#11745685	1	2	yes	—
76	C₂×C₈	—	2	4	no	—
77	C₂×C₈	—	4	2	no	—
78	C₂×C₈	—	236	4	no	—
79	C₂×C₈	C₄⋊Dic₅₉	1	2	yes	—
80	C₂×C₈	C₄⋊Dic₅₉	1	2	yes	—
81	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
82	C₄²	#11745685	1	2	yes	—
83	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
84	C₂³	#14273719	1	1	yes	characteristic
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	4	2	no	—
87	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
88	C₂×C₄	#14273686	1	2	yes	—
89	C₂×C₄	#14273715	1	2	yes	—
90	C₂×C₄	#14273719	1	1	yes	characteristic
91	C₈	—	2	2	no	—
92	C₈	—	2	2	no	—
93	C₈	—	4	4	no	—
94	C₈	—	472	4	no	—
95	C₂²	—	2	2	no	—
96	C₂²	—	2	1	no	—
97	C₂²	#16837268	1	1	yes	characteristic
98	C₄	—	2	2	no	—
99	C₄	—	2	2	no	—
100	C₄	—	2	1	no	—
101	C₄	#16837451	1	2	yes	—
102	C₂	—	4	1	no	—
103	C₂	#21424220	1	2	yes	—
104	C₂	#21424237	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21422859

C₂

yes

maximal

#21421958

C₂

yes

maximal

C₅₉×#1452

C₂

yes

maximal, characteristic

#16837459

C₂²

yes

—

#16837155

—

#16837153

—

C₅₉×#431

C₂²

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₅₉×#322

C₄

yes

—

C₄⋊(C₅₉⋊C₈)

—

C₂²⋊(C₅₉⋊C₈)

—

#14273638

—

C₂²×C₈×C₅₉

Q₈

yes

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

C₄⋊C₈×C₅₉

—

C₂²⋊C₈×C₅₉

—

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₄×C₈×C₅₉

D₄

yes

characteristic

C₄×C₈×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×Q₈

yes

characteristic

C₂²×C₄×C₅₉

#41

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₄⋊C₄

yes

—

C₂×C₈×C₅₉

C₄⋊C₄

yes

—

C₄²×C₅₉

C₂²×C₄

yes

characteristic

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄²×C₅₉

#41

yes

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

#125

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

#117

yes

—

C₂×C₄×C₅₉

#124

yes

—

C₂×C₄×C₅₉

#125

yes

characteristic

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

#8798

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#382

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#430

yes

—

#1568

—

#1452

D₅₉

yes

characteristic

#1287

—

C₂×C₅₉

—

C₂×C₅₉

#1833

yes

—

C₂×C₅₉

yes

—

C₂×D₅₉

yes

—

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

#335

—

118

—

#334

—

118

—

#322

Dic₅₉

yes

—

C₅₉

#8798

yes

characteristic, complemented, Sylow

C₂²×C₈

Dic₁₁₈

yes

—

C₂×C₄²

C₂²×D₅₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

118

—

C₂²⋊C₈

—

C₂²⋊C₈

—

118

—

#97

—

118

—

C₄×C₈

D₂₃₆

yes

characteristic

C₄×C₈

D₂₃₆

yes

characteristic

C₄×C₈

C₂×Dic₅₉

yes

—

C₄×C₈

C₂×Dic₅₉

yes

—

C₂²×C₄

C₂×Dic₁₁₈

yes

characteristic

C₂²×C₄

#11745685

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

236

—

C₂×C₈

C₄⋊Dic₅₉

yes

—

C₂×C₈

C₄⋊Dic₅₉

yes

—

C₄²

C₂×D₂₃₆

yes

characteristic

C₄²

#11745685

yes

—

C₄²

C₂²×Dic₅₉

yes

characteristic

C₂³

#14273719

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄⋊Dic₅₉

yes

characteristic

C₂×C₄

#14273686

yes

—

C₂×C₄

#14273715

yes

—

C₂×C₄

#14273719

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

472

—

C₂²

—

C₂²

—

C₂²

#16837268

yes

characteristic

C₄

—

C₄

—

100

C₄

—

101

C₄

#16837451

yes

—

102

C₂

—

103

C₂

#21424220

yes

—

104

C₂

#21424237

yes

characteristic

	subgroup	G/H
center	C₂²	#16837268
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1452	C₂
socle	C₂²×C₅₉	#382
prime radical	C₂³×C₅₉	#125
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1452	D₅₉
hypocenter	C₅₉	#8798
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452

subgroup

G/H

center

C₂²

#16837268

commutator

C₄²×C₅₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₅₉×#1452

C₂

C₂²×C₅₉

C₂³×C₅₉

C₂×C₄²×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452

N	H ≅ G/N	type
62. C₅₉	#8798	semidirect

H ≅ G/N

type

62. C₅₉

#8798

semidirect

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

subgroup	index
#21422859	2
#21421958	2
C₅₉×#1452	2
#8798	59

subgroup

index

#21422859

#21421958

C₅₉×#1452

#8798

Subgroups — #28482402

Conjugacy classes of subgroups

Complements

Maximal subgroups