Group #28503393 — subgroups

Subgroups — #28503393

Order 15104.

290 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 83 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	#21426320	C₂	1	4	yes	maximal, complemented
2	#21423711	C₂	1	1	yes	maximal, characteristic
3	#21423706	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₉×#1311	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837807	C₂²	1	2	yes	—
6	#16837779	—	2	2	no	—
7	#16837779	C₂²	1	2	yes	—
8	#16837753	—	2	2	no	—
9	#16837204	—	2	8	no	—
10	C₄×C₁₆×C₅₉	C₂²	1	2	yes	—
11	C₈²×C₅₉	C₂²	1	1	yes	characteristic
12	C₂×D₄₇₂	—	2	4	no	—
13	C₂×D₄₇₂	—	4	4	no	—
14	#14273727	—	2	2	no	—
15	#14273724	—	2	2	no	—
16	#14273661	—	2	4	no	—
17	#14273652	—	4	4	no	—
18	C₂×C₁₆×C₅₉	D₄	1	4	yes	—
19	C₄×C₈×C₅₉	D₄	1	2	yes	—
20	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
21	C₂×D₂₃₆	—	4	2	no	—
22	C₂×D₂₃₆	—	4	4	no	—
23	C₂×Dic₁₁₈	—	4	2	no	—
24	C₂×C₈×C₅₉	D₈	1	4	yes	—
25	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
26	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
27	C₄⋊Dic₅₉	—	4	2	no	—
28	C₄⋊Dic₅₉	—	4	2	no	—
29	D₄₇₂	—	4	8	no	—
30	D₄₇₂	—	8	8	no	—
31	C₁₆×C₅₉	—	2	4	no	—
32	C₂²×D₅₉	—	8	2	no	—
33	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
34	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
35	C₂×Dic₅₉	—	8	2	no	—
36	D₂₃₆	—	8	4	no	—
37	D₂₃₆	—	8	8	no	—
38	Dic₁₁₈	—	8	4	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	#495	1	2	yes	—
46	C₄×C₅₉	#509	1	4	yes	—
47	Dic₅₉	—	16	2	no	—
48	#2240	—	59	4	no	—
49	#1712	—	59	1	no	—
50	#1710	—	59	1	no	—
51	#1311	D₅₉	1	1	yes	characteristic, complemented
52	C₂×C₅₉	#1709	1	1	yes	characteristic
53	C₂×C₅₉	#2250	1	2	yes	—
54	D₅₉	—	32	2	no	—
55	#500	—	59	2	no	—
56	#493	—	59	2	no	—
57	#493	—	118	2	no	—
58	#487	—	118	2	no	—
59	#357	—	118	8	no	—
60	C₄×C₁₆	C₂×D₅₉	1	2	yes	—
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	—	118	2	no	—
66	#127	—	118	2	no	—
67	C₂×C₁₆	D₂₃₆	1	4	yes	—
68	#107	—	118	4	no	—
69	#103	—	236	4	no	—
70	C₄×C₈	D₂₃₆	1	2	yes	—
71	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
72	C₂×Q₈	—	236	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	D₄₇₂	1	4	yes	—
78	C₂×C₈	C₂×D₂₃₆	1	2	yes	—
79	C₄⋊C₄	—	236	2	no	—
80	C₄⋊C₄	—	236	2	no	—
81	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
82	C₁₆	—	2	4	no	—
83	C₂³	—	472	2	no	—
84	Q₈	—	472	4	no	—
85	D₄	—	472	4	no	—
86	D₄	—	472	8	no	—
87	C₂×C₄	—	472	2	no	—
88	C₂×C₄	#14273724	1	1	yes	characteristic
89	C₂×C₄	C₂×D₄₇₂	1	2	yes	—
90	C₈	—	2	4	no	—
91	C₈	#14273749	1	4	yes	—
92	C₂²	—	944	2	no	—
93	C₂²	—	944	4	no	—
94	C₂²	#16837779	1	1	yes	characteristic
95	C₄	—	944	2	no	—
96	C₄	#16837786	1	2	yes	—
97	C₄	#16837840	1	4	yes	—
98	C₂	—	1888	2	no	—
99	C₂	#21423702	1	1	yes	characteristic
100	C₂	#21426362	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21426320

C₂

yes

maximal, complemented

#21423711

C₂

yes

maximal, characteristic

#21423706

C₂

yes

maximal, characteristic, complemented

C₅₉×#1311

C₂

yes

maximal, characteristic, complemented

#16837807

C₂²

yes

—

#16837779

—

#16837779

C₂²

yes

—

#16837753

—

#16837204

—

C₄×C₁₆×C₅₉

C₂²

yes

—

C₈²×C₅₉

C₂²

yes

characteristic

C₂×D₄₇₂

—

C₂×D₄₇₂

—

—

—

—

—

C₂×C₁₆×C₅₉

D₄

yes

—

C₄×C₈×C₅₉

D₄

yes

—

C₄×C₈×C₅₉

C₂³

yes

characteristic

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

D₈

yes

—

C₂×C₈×C₅₉

C₂×D₄

yes

—

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

D₄₇₂

—

D₄₇₂

—

C₁₆×C₅₉

—

C₂²×D₅₉

—

C₂×C₄×C₅₉

#127

yes

characteristic

C₂×C₄×C₅₉

C₂×D₈

yes

—

C₂×Dic₅₉

—

D₂₃₆

—

D₂₃₆

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

#135

yes

—

—

C₂²×C₅₉

yes

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

#495

yes

—

C₄×C₅₉

#509

yes

—

Dic₅₉

—

—

—

—

D₅₉

yes

characteristic, complemented

C₂×C₅₉

#1709

yes

characteristic

C₂×C₅₉

#2250

yes

—

D₅₉

—

#500

—

#493

—

#493

—

118

—

#487

—

118

—

#357

—

118

—

C₄×C₁₆

C₂×D₅₉

yes

—

C₈²

C₂×D₅₉

yes

characteristic

C₅₉

#12513

yes

characteristic, complemented, Sylow

C₂×D₈

—

118

—

C₂×D₈

—

236

—

C₄⋊Q₈

—

118

—

#127

—

118

—

C₂×C₁₆

D₂₃₆

yes

—

#107

—

118

—

#103

—

236

—

C₄×C₈

D₂₃₆

yes

—

C₄×C₈

C₂²×D₅₉

yes

characteristic

C₂×Q₈

—

236

—

C₂×D₄

—

236

—

C₂×D₄

—

236

—

D₈

—

472

—

D₈

—

236

—

C₂×C₈

D₄₇₂

yes

—

C₂×C₈

C₂×D₂₃₆

yes

—

C₄⋊C₄

—

236

—

C₄⋊C₄

—

236

—

C₄²

C₂×D₂₃₆

yes

characteristic

C₁₆

—

C₂³

—

472

—

Q₈

—

472

—

D₄

—

472

—

D₄

—

472

—

C₂×C₄

—

472

—

C₂×C₄

#14273724

yes

characteristic

C₂×C₄

C₂×D₄₇₂

yes

—

C₈

—

C₈

#14273749

yes

—

C₂²

—

944

—

C₂²

—

944

—

C₂²

#16837779

yes

characteristic

C₄

—

944

—

C₄

#16837786

yes

—

C₄

#16837840

yes

—

C₂

—

1888

—

C₂

#21423702

yes

characteristic

100

C₂

#21426362

yes

—

	subgroup	G/H
center	C₂²	#16837779
commutator, cosocle	C₄×C₈×C₅₉	C₂³
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₅₉×#1311	C₂
socle	C₂²×C₅₉	#493
prime radical	#21423711	C₂
hypercenter	#1311	D₅₉
hypocenter	C₅₉	#12513
LCS	G ▷ C₄×C₈×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

subgroup

G/H

center

C₂²

#16837779

commutator, cosocle

C₄×C₈×C₅₉

C₂³

Frattini

C₄×C₈

C₂²×D₅₉

Fitting

C₅₉×#1311

C₂

C₂²×C₅₉

C₂

D₅₉

C₅₉

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

N	H ≅ G/N	type
1. #21426320	C₂	semidirect
3. #21423706	C₂	semidirect
4. C₅₉×#1311	C₂	semidirect
51. #1311	D₅₉	semidirect
62. C₅₉	#12513	semidirect

H ≅ G/N

type

1. #21426320

C₂

semidirect

3. #21423706

C₂

semidirect

4. C₅₉×#1311

C₂

semidirect

51. #1311

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
#21426320	2
#21423711	2
#21423706	2
C₅₉×#1311	2
#12513	59

subgroup

index

C₅₉×#1311

Subgroups — #28503393

Conjugacy classes of subgroups

Complements

Maximal subgroups