Group #28493198 — subgroups

Subgroups — #28493198

Order 15104.

350 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 97 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	#21424229	C₂	1	3	yes	maximal
2	#21421977	C₂	1	3	yes	maximal
3	C₅₉×#2845	C₂	1	1	yes	maximal, characteristic
4	C₂×#14273638	C₂²	1	3	yes	—
5	#16837272	C₂²	1	1	yes	characteristic
6	#16837164	—	2	6	no	—
7	C₅₉×#398	—	2	6	no	—
8	C₅₉×#382	C₂²	1	3	yes	—
9	C₂²×C₅₉⋊C₈	—	2	3	no	—
10	C₂×C₄⋊Dic₅₉	—	2	3	no	—
11	#14273638	—	2	3	no	—
12	#14273638	Q₈	1	6	yes	—
13	C₂×C₄⋊C₄×C₅₉	—	2	6	no	—
14	C₂×C₄⋊C₄×C₅₉	D₄	1	6	yes	—
15	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
16	#95×C₅₉	—	2	2	no	—
17	#95×C₅₉	—	2	6	no	—
18	C₂²×C₄×C₅₉	—	2	6	no	—
19	C₂²×C₄×C₅₉	C₂×D₄	1	3	yes	—
20	C₂²×Dic₅₉	—	4	1	no	—
21	C₄⋊C₄×C₅₉	—	4	6	no	—
22	C₄⋊C₄×C₅₉	—	8	6	no	—
23	C₄²×C₅₉	C₂×Q₈	1	3	yes	—
24	C₄²×C₅₉	#41	1	1	yes	characteristic
25	C₄⋊Dic₅₉	—	4	6	no	—
26	C₂×C₅₉⋊C₈	—	2	6	no	—
27	C₂×C₅₉⋊C₈	—	4	3	no	—
28	C₂×C₅₉⋊C₈	—	4	3	no	—
29	C₂³×C₅₉	#120	1	1	yes	characteristic
30	C₂×C₄×C₅₉	—	2	3	no	—
31	C₂×C₄×C₅₉	—	2	3	no	—
32	C₂×C₄×C₅₉	—	4	6	no	—
33	C₂×C₄×C₅₉	—	4	6	no	—
34	C₂×C₄×C₅₉	—	4	6	no	—
35	C₂×C₄×C₅₉	C₂²⋊Q₈	1	3	yes	—
36	C₂×C₄×C₅₉	C₄⋊Q₈	1	3	yes	—
37	C₂×Dic₅₉	—	8	3	no	—
38	C₅₉⋊C₈	—	4	6	no	—
39	#10586	—	59	1	no	maximal, Sylow
40	C₂²×C₅₉	#395	1	1	yes	characteristic
41	C₂²×C₅₉	#454	1	3	yes	—
42	C₂²×C₅₉	#455	1	3	yes	—
43	C₄×C₅₉	—	2	3	no	—
44	C₄×C₅₉	—	2	3	no	—
45	C₄×C₅₉	—	8	6	no	—
46	Dic₅₉	—	16	1	no	—
47	#2845	D₅₉	1	1	yes	characteristic
48	#1836	—	59	3	no	—
49	#1295	—	59	3	no	—
50	C₂×C₅₉	#1658	1	1	yes	characteristic
51	C₂×C₅₉	#1662	1	3	yes	—
52	C₂×C₅₉	#2061	1	3	yes	—
53	C₂×#97	—	59	3	no	—
54	#398	—	2	6	no	—
55	#384	—	59	1	no	—
56	#382	C₂×D₅₉	1	3	yes	—
57	#340	—	118	6	no	—
58	C₅₉	#10586	1	1	yes	characteristic, complemented, Sylow
59	C₂²×C₈	—	118	3	no	—
60	C₂×C₄⋊C₄	—	2	6	no	—
61	C₂×C₄⋊C₄	—	118	3	no	—
62	C₂×C₄⋊C₄	C₂²⋊D₅₉	1	6	yes	—
63	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
64	#97	—	59	6	no	—
65	#97	—	118	3	no	—
66	#95	—	2	2	no	—
67	#95	—	2	6	no	—
68	C₂²×C₄	—	2	6	no	—
69	C₂²×C₄	—	236	1	no	—
70	C₂²×C₄	C₂×C₂²⋊D₅₉	1	3	yes	—
71	C₂×C₈	—	118	6	no	—
72	C₂×C₈	—	236	3	no	—
73	C₂×C₈	—	236	3	no	—
74	C₄⋊C₄	—	4	6	no	—
75	C₄⋊C₄	—	8	6	no	—
76	C₄⋊C₄	—	236	6	no	—
77	C₄²	Q₈×D₅₉	1	3	yes	—
78	C₄²	#11745689	1	1	yes	characteristic
79	C₂³	#14273696	1	1	yes	characteristic
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	—	2	3	no	—
82	C₂×C₄	—	4	6	no	—
83	C₂×C₄	—	4	6	no	—
84	C₂×C₄	—	4	6	no	—
85	C₂×C₄	—	472	3	no	—
86	C₂×C₄	#14273703	1	3	yes	—
87	C₂×C₄	#14273725	1	3	yes	—
88	C₈	—	236	6	no	—
89	C₂²	#16837321	1	1	yes	characteristic
90	C₂²	#16837575	1	3	yes	—
91	C₂²	#16837580	1	3	yes	—
92	C₄	—	2	3	no	—
93	C₄	—	2	3	no	—
94	C₄	—	8	6	no	—
95	C₄	—	944	1	no	—
96	C₂	#21423398	1	1	yes	characteristic
97	C₂	#21423419	1	3	yes	—
98	C₂	#21425443	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21424229

C₂

yes

maximal

#21421977

C₂

yes

maximal

C₅₉×#2845

C₂

yes

maximal, characteristic

C₂×#14273638

C₂²

yes

—

#16837272

C₂²

yes

characteristic

#16837164

—

C₅₉×#398

—

C₅₉×#382

C₂²

yes

—

C₂²×C₅₉⋊C₈

—

C₂×C₄⋊Dic₅₉

—

#14273638

—

#14273638

Q₈

yes

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄⋊C₄×C₅₉

D₄

yes

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

#95×C₅₉

—

#95×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂×D₄

yes

—

C₂²×Dic₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

C₂×Q₈

yes

—

C₄²×C₅₉

#41

yes

characteristic

C₄⋊Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

#120

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₉

C₄⋊Q₈

yes

—

C₂×Dic₅₉

—

C₅₉⋊C₈

—

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₂²×C₅₉

#455

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

Dic₅₉

—

#2845

D₅₉

yes

characteristic

#1836

—

#1295

—

C₂×C₅₉

#1658

yes

characteristic

C₂×C₅₉

#1662

yes

—

C₂×C₅₉

#2061

yes

—

C₂×#97

—

#398

—

#384

—

#382

C₂×D₅₉

yes

—

#340

—

118

—

C₅₉

#10586

yes

characteristic, complemented, Sylow

C₂²×C₈

—

118

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

118

—

C₂×C₄⋊C₄

C₂²⋊D₅₉

yes

—

C₂×C₄²

C₂²×D₅₉

yes

characteristic

#97

—

#97

—

118

—

#95

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

236

—

C₂²×C₄

C₂×C₂²⋊D₅₉

yes

—

C₂×C₈

—

118

—

C₂×C₈

—

236

—

C₂×C₈

—

236

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

236

—

C₄²

Q₈×D₅₉

yes

—

C₄²

yes

C₂³

yes

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

472

—

C₂×C₄

#14273703

yes

—

C₂×C₄

#14273725

yes

—

C₈

—

236

—

C₂²

#16837321

yes

characteristic

C₂²

#16837575

yes

—

C₂²

#16837580

yes

—

C₄

—

C₄

—

C₄

—

C₄

—

944

—

C₂

#21423398

yes

characteristic

C₂

#21423419

yes

—

C₂

#21425443

yes

—

	subgroup	G/H
center	C₂³	#14273696
commutator, cosocle	C₂×C₄²×C₅₉	C₂³
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#2845	C₂
socle, prime radical	C₂³×C₅₉	#120
hypercenter	#2845	D₅₉
hypocenter	C₅₉	#10586
LCS	G ▷ C₂×C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2845

subgroup

G/H

center

C₂³

#14273696

commutator, cosocle

C₂×C₄²×C₅₉

C₂³

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₅₉×#2845

C₂

C₂³×C₅₉

D₅₉

C₅₉

G ▷ C₂×C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2845

N	H ≅ G/N	type
58. C₅₉	#10586	semidirect

H ≅ G/N

type

58. C₅₉

#10586

semidirect

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

subgroup	index
#21424229	2
#21421977	2
C₅₉×#2845	2
#10586	59

subgroup

index

#21424229

#21421977

C₅₉×#2845

#10586

Subgroups — #28493198

Conjugacy classes of subgroups

Complements

Maximal subgroups