Group #28503394 — subgroups

Subgroups — #28503394

Order 15104.

290 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 79 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	#21426321	C₂	1	4	yes	maximal, complemented
2	#21423711	C₂	1	1	yes	maximal, characteristic
3	#21422013	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₉×#1710	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837779	—	2	2	no	—
6	#16837779	C₂²	1	2	yes	—
7	#16837202	—	2	8	no	—
8	C₄×C₅₉⋊C₁₆	C₂²	1	2	yes	—
9	C₅₉×#500	C₂²	1	2	yes	—
10	C₅₉×#487	—	2	2	no	—
11	C₈²×C₅₉	C₂²	1	1	yes	characteristic
12	C₂×D₄₇₂	—	2	4	no	—
13	C₂×D₄₇₂	—	4	4	no	—
14	#14273724	—	2	2	no	—
15	C₂×C₅₉⋊C₁₆	D₄	1	4	yes	—
16	C₄⋊Q₈×C₅₉	—	2	2	no	—
17	#107×C₅₉	—	2	4	no	—
18	#103×C₅₉	—	4	4	no	—
19	C₄×C₈×C₅₉	D₄	1	2	yes	—
20	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
21	C₂×Q₈×C₅₉	—	4	2	no	—
22	C₂×D₂₃₆	—	4	2	no	—
23	C₂×D₂₃₆	—	4	4	no	—
24	C₂×C₈×C₅₉	D₈	1	4	yes	—
25	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
26	C₄⋊C₄×C₅₉	—	4	2	no	—
27	C₄⋊C₄×C₅₉	—	4	2	no	—
28	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
29	D₄₇₂	—	4	8	no	—
30	D₄₇₂	—	8	8	no	—
31	C₅₉⋊C₁₆	—	2	4	no	—
32	C₂²×D₅₉	—	8	2	no	—
33	Q₈×C₅₉	—	8	4	no	—
34	C₂×C₄×C₅₉	—	8	2	no	—
35	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
36	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
37	D₂₃₆	—	8	4	no	—
38	D₂₃₆	—	8	8	no	—
39	C₈×C₅₉	—	2	4	no	—
40	C₈×C₅₉	#135	1	4	yes	—
41	#12513	—	59	1	no	maximal, Sylow
42	C₂²×C₅₉	#493	1	1	yes	characteristic
43	C₂×D₅₉	—	16	2	no	—
44	C₂×D₅₉	—	16	4	no	—
45	C₄×C₅₉	—	16	2	no	—
46	C₄×C₅₉	#495	1	2	yes	—
47	C₄×C₅₉	#509	1	4	yes	—
48	#2240	—	59	4	no	—
49	#1712	—	59	1	no	—
50	#1710	D₅₉	1	1	yes	characteristic, complemented
51	#1311	—	59	1	no	—
52	C₂×C₅₉	#1709	1	1	yes	characteristic
53	C₂×C₅₉	#2250	1	2	yes	—
54	D₅₉	—	32	2	no	—
55	#500	C₂×D₅₉	1	2	yes	—
56	#493	—	59	2	no	—
57	#493	—	118	2	no	—
58	#487	—	2	2	no	—
59	#357	—	118	8	no	—
60	C₄×C₁₆	—	59	2	no	—
61	C₈²	C₂×D₅₉	1	1	yes	characteristic
62	C₅₉	#12513	1	1	yes	characteristic, complemented, Sylow
63	C₂×D₈	—	118	4	no	—
64	C₂×D₈	—	236	4	no	—
65	C₄⋊Q₈	—	2	2	no	—
66	#127	—	118	2	no	—
67	C₂×C₁₆	—	59	4	no	—
68	#107	—	2	4	no	—
69	#103	—	4	4	no	—
70	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
71	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
72	C₂×Q₈	—	4	2	no	—
73	C₂×D₄	—	236	2	no	—
74	C₂×D₄	—	236	4	no	—
75	D₈	—	472	8	no	—
76	D₈	—	236	8	no	—
77	C₂×C₈	#11745669	1	4	yes	—
78	C₂×C₈	D₄×D₅₉	1	2	yes	—
79	C₄⋊C₄	—	4	2	no	—
80	C₄⋊C₄	—	4	2	no	—
81	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
82	C₁₆	—	118	4	no	—
83	C₂³	—	472	2	no	—
84	Q₈	—	8	4	no	—
85	D₄	—	472	4	no	—
86	D₄	—	472	8	no	—
87	C₂×C₄	—	8	2	no	—
88	C₂×C₄	#14273723	1	1	yes	characteristic
89	C₂×C₄	C₂×#11745669	1	2	yes	—
90	C₈	—	2	4	no	—
91	C₈	#14273750	1	4	yes	—
92	C₂²	—	944	2	no	—
93	C₂²	—	944	4	no	—
94	C₂²	#16837778	1	1	yes	characteristic
95	C₄	—	16	2	no	—
96	C₄	#16837787	1	2	yes	—
97	C₄	#16837837	1	4	yes	—
98	C₂	—	1888	2	no	—
99	C₂	#21423703	1	1	yes	characteristic
100	C₂	#21426363	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21426321

C₂

yes

maximal, complemented

#21423711

C₂

yes

maximal, characteristic

#21422013

C₂

yes

maximal, characteristic, complemented

C₅₉×#1710

C₂

yes

maximal, characteristic, complemented

#16837779

—

#16837779

C₂²

yes

—

#16837202

—

C₄×C₅₉⋊C₁₆

C₂²

yes

—

C₅₉×#500

C₂²

yes

—

C₅₉×#487

—

C₈²×C₅₉

C₂²

yes

characteristic

C₂×D₄₇₂

—

C₂×D₄₇₂

—

#14273724

—

C₂×C₅₉⋊C₁₆

D₄

yes

—

C₄⋊Q₈×C₅₉

—

#107×C₅₉

—

#103×C₅₉

—

C₄×C₈×C₅₉

D₄

yes

—

C₄×C₈×C₅₉

C₂³

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×C₈×C₅₉

D₈

yes

—

C₂×C₈×C₅₉

C₂×D₄

yes

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

C₂×D₄

yes

characteristic

D₄₇₂

—

D₄₇₂

—

C₅₉⋊C₁₆

—

C₂²×D₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#127

yes

characteristic

C₂×C₄×C₅₉

C₂×D₈

yes

—

D₂₃₆

—

D₂₃₆

—

C₈×C₅₉

—

C₈×C₅₉

#135

yes

—

—

C₂²×C₅₉

yes

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#495

yes

—

C₄×C₅₉

#509

yes

—

#2240

—

#1712

—

#1710

D₅₉

yes

characteristic, complemented

#1311

—

C₂×C₅₉

#1709

yes

characteristic

C₂×C₅₉

#2250

yes

—

D₅₉

—

#500

C₂×D₅₉

yes

—

#493

—

#493

—

118

—

#487

—

#357

—

118

—

C₄×C₁₆

—

C₈²

C₂×D₅₉

yes

characteristic

C₅₉

#12513

yes

characteristic, complemented, Sylow

C₂×D₈

—

118

—

C₂×D₈

—

236

—

C₄⋊Q₈

—

#127

—

118

—

C₂×C₁₆

—

#107

—

#103

—

C₄×C₈

C₂²⋊D₅₉

yes

—

C₄×C₈

C₂²×D₅₉

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

236

—

C₂×D₄

—

236

—

D₈

—

472

—

D₈

—

236

—

C₂×C₈

#11745669

yes

—

C₂×C₈

D₄×D₅₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×C₂²⋊D₅₉

yes

characteristic

C₁₆

—

118

—

C₂³

—

472

—

Q₈

—

D₄

—

472

—

D₄

—

472

—

C₂×C₄

—

C₂×C₄

#14273723

yes

characteristic

C₂×C₄

C₂×#11745669

yes

—

C₈

—

C₈

#14273750

yes

—

C₂²

—

944

—

C₂²

—

944

—

C₂²

#16837778

yes

characteristic

C₄

—

C₄

#16837787

yes

—

C₄

#16837837

yes

—

C₂

—

1888

—

C₂

#21423703

yes

characteristic

100

C₂

#21426363

yes

—

	subgroup	G/H
center	C₂²	#16837778
commutator, cosocle	C₄×C₈×C₅₉	C₂³
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₅₉×#1710	C₂
socle	C₂²×C₅₉	#493
prime radical	#21423711	C₂
hypercenter	#1710	D₅₉
hypocenter	C₅₉	#12513
LCS	G ▷ C₄×C₈×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

subgroup

G/H

center

C₂²

#16837778

commutator, cosocle

C₄×C₈×C₅₉

C₂³

Frattini

C₄×C₈

C₂²×D₅₉

Fitting

C₅₉×#1710

C₂

C₂²×C₅₉

C₂

D₅₉

C₅₉

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

N	H ≅ G/N	type
1. #21426321	C₂	semidirect
3. #21422013	C₂	semidirect
4. C₅₉×#1710	C₂	semidirect
50. #1710	D₅₉	semidirect
62. C₅₉	#12513	semidirect

H ≅ G/N

type

1. #21426321

C₂

semidirect

3. #21422013

C₂

semidirect

4. C₅₉×#1710

C₂

semidirect

50. #1710

D₅₉

semidirect

62. C₅₉

#12513

semidirect

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

subgroup	index
#21426321	2
#21423711	2
#21422013	2
C₅₉×#1710	2
#12513	59

subgroup

index

C₅₉×#1710

Subgroups — #28503394

Conjugacy classes of subgroups

Complements

Maximal subgroups