Group #28482463 — subgroups

Subgroups — #28482463

Order 15104.

422 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 227 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	#21424253	C₂	1	2	yes	maximal
2	#21424224	C₂	1	4	yes	maximal
3	C₂×C₈²×C₅₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#14273659	—	2	4	no	—
5	#16837272	C₂²	1	4	yes	—
6	#16837165	—	2	8	no	—
7	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
9	C₈²×C₅₉	C₄	1	4	yes	complemented
10	C₂×C₄⋊Dic₅₉	—	2	8	no	—
11	C₂×C₄⋊Dic₅₉	—	2	4	no	—
12	#14273659	—	4	8	no	—
13	C₂²×C₈×C₅₉	D₄	1	4	yes	—
14	C₂²×C₈×C₅₉	D₄	1	2	yes	—
15	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
16	C₄×C₈×C₅₉	C₂×C₄	1	4	yes	—
17	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
18	C₄×C₈×C₅₉	Q₈	1	4	yes	—
19	C₄×C₈×C₅₉	Q₈	1	2	yes	—
20	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
22	C₂²×Dic₅₉	—	4	4	no	—
23	C₂×C₈×C₅₉	—	2	8	no	—
24	C₂×C₈×C₅₉	C₄⋊C₄	1	8	yes	—
25	C₂×C₈×C₅₉	C₄⋊C₄	1	4	yes	—
26	C₂×C₈×C₅₉	SD₁₆	1	8	yes	—
27	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
28	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
29	C₄²×C₅₉	C₂×Q₈	1	2	yes	—
30	C₄⋊Dic₅₉	—	4	8	no	—
31	C₄⋊Dic₅₉	—	4	16	no	—
32	C₂³×C₅₉	#127	1	1	yes	characteristic
33	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
35	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
36	C₂×C₄×C₅₉	C₄⋊Q₈	1	2	yes	—
37	C₂×C₄×C₅₉	C₂×SD₁₆	1	4	yes	—
38	C₂×C₄×C₅₉	#135	1	8	yes	—
39	C₂×Dic₅₉	—	8	4	no	—
40	C₂×Dic₅₉	—	8	8	no	—
41	C₈×C₅₉	—	2	8	no	—
42	C₈×C₅₉	#106	1	8	yes	—
43	#8815	—	59	1	no	maximal, Sylow
44	C₂²×C₅₉	#384	1	1	yes	characteristic
45	C₂²×C₅₉	#492	1	2	yes	—
46	C₂²×C₅₉	#495	1	4	yes	—
47	C₄×C₅₉	C₂×#106	1	2	yes	—
48	C₄×C₅₉	#427	1	4	yes	—
49	C₄×C₅₉	#498	1	2	yes	—
50	C₄×C₅₉	#499	1	4	yes	—
51	Dic₅₉	—	16	4	no	—
52	#1844	—	59	2	no	—
53	#1834	—	59	4	no	—
54	C₂×C₈²	D₅₉	1	1	yes	characteristic
55	C₂×C₅₉	#1708	1	4	yes	—
56	C₂×C₅₉	#1844	1	1	yes	characteristic
57	C₂×C₅₉	#1846	1	2	yes	—
58	C₂×#106	—	118	4	no	—
59	C₂×C₄×C₈	C₂×D₅₉	1	2	yes	—
60	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
61	#384	—	59	4	no	—
62	#340	—	118	8	no	—
63	C₈²	Dic₅₉	1	4	yes	complemented
64	C₅₉	#8815	1	1	yes	characteristic, complemented, Sylow
65	C₂²×C₈	D₂₃₆	1	4	yes	—
66	C₂²×C₈	D₂₃₆	1	2	yes	—
67	C₂×C₄⋊C₄	—	118	8	no	—
68	C₂×C₄⋊C₄	—	118	4	no	—
69	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
70	#106	—	236	8	no	—
71	C₄×C₈	Dic₁₁₈	1	4	yes	—
72	C₄×C₈	Dic₁₁₈	1	2	yes	—
73	C₄×C₈	C₂×Dic₅₉	1	4	yes	—
74	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
75	C₂²×C₄	—	236	4	no	—
76	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
77	C₂²×C₄	C₂×D₂₃₆	1	2	yes	—
78	C₂×C₈	—	2	8	no	—
79	C₂×C₈	#11745660	1	8	yes	—
80	C₂×C₈	C₄⋊Dic₅₉	1	8	yes	—
81	C₂×C₈	C₄⋊Dic₅₉	1	4	yes	—
82	C₄⋊C₄	—	236	8	no	—
83	C₄⋊C₄	—	236	16	no	—
84	C₄²	C₂×Dic₁₁₈	1	1	yes	characteristic
85	C₄²	C₂×Dic₁₁₈	1	2	yes	—
86	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
87	C₂³	#14273724	1	1	yes	characteristic
88	C₂×C₄	—	472	4	no	—
89	C₂×C₄	—	472	8	no	—
90	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
91	C₂×C₄	C₂×C₄⋊Dic₅₉	1	2	yes	—
92	C₂×C₄	#14273727	1	1	yes	characteristic
93	C₂×C₄	#14273727	1	2	yes	—
94	C₂×C₄	C₂×#11745660	1	4	yes	—
95	C₂×C₄	#14273749	1	8	yes	—
96	C₈	—	2	8	no	—
97	C₈	#14273659	1	8	yes	—
98	C₂²	#16837272	1	1	yes	characteristic
99	C₂²	#16837775	1	2	yes	—
100	C₂²	#16837786	1	4	yes	—
101	C₄	—	944	4	no	—
102	C₄	C₂×#14273659	1	2	yes	—
103	C₄	#16837440	1	4	yes	—
104	C₄	#16837800	1	2	yes	—
105	C₄	#16837804	1	4	yes	—
106	C₂	#21423697	1	4	yes	—
107	C₂	#21424253	1	1	yes	characteristic
108	C₂	#21424260	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21424253

C₂

yes

maximal

#21424224

C₂

yes

maximal

C₂×C₈²×C₅₉

C₂

yes

maximal, characteristic

C₂×#14273659

—

#16837272

C₂²

yes

—

#16837165

—

C₂×C₄×C₈×C₅₉

C₂²

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₈²×C₅₉

C₄

yes

complemented

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

—

#14273659

—

C₂²×C₈×C₅₉

D₄

yes

—

C₂²×C₈×C₅₉

D₄

yes

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₄×C₈×C₅₉

Q₈

yes

—

C₄×C₈×C₅₉

Q₈

yes

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×D₄

yes

—

C₂²×Dic₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₄⋊C₄

yes

—

C₂×C₈×C₅₉

C₄⋊C₄

yes

—

C₂×C₈×C₅₉

SD₁₆

yes

—

C₄²×C₅₉

C₂²×C₄

yes

characteristic

C₄²×C₅₉

C₂×Q₈

yes

characteristic

C₄²×C₅₉

C₂×Q₈

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₂³×C₅₉

#127

yes

characteristic

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₉

C₄⋊Q₈

yes

—

C₂×C₄×C₅₉

C₂×SD₁₆

yes

—

C₂×C₄×C₅₉

#135

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₈×C₅₉

—

C₈×C₅₉

#106

yes

—

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₂²×C₅₉

#495

yes

—

C₄×C₅₉

C₂×#106

yes

—

C₄×C₅₉

#427

yes

—

C₄×C₅₉

#498

yes

—

C₄×C₅₉

#499

yes

—

Dic₅₉

—

#1844

—

#1834

—

C₂×C₈²

D₅₉

yes

characteristic

C₂×C₅₉

#1708

yes

—

C₂×C₅₉

#1844

yes

characteristic

C₂×C₅₉

#1846

yes

—

C₂×#106

—

118

—

C₂×C₄×C₈

C₂×D₅₉

yes

—

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

#384

—

#340

—

118

—

C₈²

Dic₅₉

yes

complemented

C₅₉

#8815

yes

characteristic, complemented, Sylow

C₂²×C₈

D₂₃₆

yes

—

C₂²×C₈

D₂₃₆

yes

—

C₂×C₄⋊C₄

—

118

—

C₂×C₄⋊C₄

—

118

—

C₂×C₄²

C₂²×D₅₉

yes

characteristic

#106

—

236

—

C₄×C₈

Dic₁₁₈

yes

—

C₄×C₈

Dic₁₁₈

yes

—

C₄×C₈

C₂×Dic₅₉

yes

—

C₄×C₈

C₂×Dic₅₉

yes

—

C₂²×C₄

—

236

—

C₂²×C₄

C₂×D₂₃₆

yes

characteristic

C₂²×C₄

C₂×D₂₃₆

yes

—

C₂×C₈

—

C₂×C₈

#11745660

yes

—

C₂×C₈

C₄⋊Dic₅₉

yes

—

C₂×C₈

C₄⋊Dic₅₉

yes

—

C₄⋊C₄

—

236

—

C₄⋊C₄

—

236

—

C₄²

C₂×Dic₁₁₈

yes

characteristic

C₄²

C₂×Dic₁₁₈

yes

—

C₄²

C₂²×Dic₅₉

yes

characteristic

C₂³

#14273724

yes

characteristic

C₂×C₄

—

472

—

C₂×C₄

—

472

—

C₂×C₄

C₂×C₄⋊Dic₅₉

yes

characteristic

C₂×C₄

C₂×C₄⋊Dic₅₉

yes

—

C₂×C₄

#14273727

yes

characteristic

C₂×C₄

#14273727

yes

—

C₂×C₄

C₂×#11745660

yes

—

C₂×C₄

#14273749

yes

—

C₈

—

C₈

#14273659

yes

—

C₂²

#16837272

yes

characteristic

C₂²

#16837775

yes

—

100

C₂²

#16837786

yes

—

101

C₄

—

944

—

102

C₄

C₂×#14273659

yes

—

103

C₄

#16837440

yes

—

104

C₄

#16837800

yes

—

105

C₄

#16837804

yes

—

106

C₂

#21423697

yes

—

107

C₂

#21424253

yes

characteristic

108

C₂

#21424260

yes

—

	subgroup	G/H
center	C₂³	#14273724
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₂×C₈²×C₅₉	C₂
socle, prime radical	C₂³×C₅₉	#127
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₂×C₈²	D₅₉
hypocenter	C₅₉	#8815
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

subgroup

G/H

center

C₂³

#14273724

commutator

C₄²×C₅₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₂×C₈²×C₅₉

C₂

socle, prime radical

C₂³×C₅₉

#127

cosocle

C₂×C₄²×C₅₉

C₂³

hypercenter

C₂×C₈²

D₅₉

hypocenter

C₅₉

#8815

LCS

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

N	H ≅ G/N	type
9. C₈²×C₅₉	C₄	semidirect
63. C₈²	Dic₅₉	semidirect
64. C₅₉	#8815	semidirect

H ≅ G/N

type

9. C₈²×C₅₉

C₄

semidirect

63. C₈²

Dic₅₉

semidirect

64. C₅₉

#8815

semidirect

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

subgroup	index
#21424253	2
#21424224	2
C₂×C₈²×C₅₉	2
#8815	59

subgroup

index

#21424253

#21424224

C₂×C₈²×C₅₉

#8815

Subgroups — #28482463

Conjugacy classes of subgroups

Complements

Maximal subgroups