Group #28515923 — subgroups

Subgroups — #28515923

Order 15104.

1118 conjugacy classes of subgroups; only the 411 normal subgroups are shown.

normal subgroups only

This group has 1118 conjugacy classes of subgroups — too many to draw. Only its 411 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	#21422358	C₂	1	8	yes	maximal
2	#21422326	C₂	1	6	yes	maximal
3	C₂×C₅₉×#376	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄⋊Dic₅₉	C₂²	1	4	yes	—
5	C₂×C₅₉⋊#95	C₂²	1	24	yes	—
6	C₂²×C₄²×C₅₉	C₂²	1	3	yes	—
7	C₅₉×#376	C₄	1	8	yes	complemented
8	C₂×#95×C₅₉	C₂²	1	4	yes	—
9	C₂³×Dic₅₉	C₂³	1	8	yes	—
10	C₂×C₄⋊Dic₅₉	—	2	24	no	—
11	C₅₉⋊#95	—	4	24	no	—
12	C₂³×C₄×C₅₉	C₂³	1	6	yes	—
13	C₂³×C₄×C₅₉	C₂³	1	1	yes	characteristic
14	C₂×C₄²×C₅₉	—	2	12	no	—
15	C₂×C₄²×C₅₉	C₂×C₄	1	12	yes	—
16	#95×C₅₉	C₂×C₄	1	16	yes	—
17	C₂⁴×C₅₉	C₂⁴	1	1	yes	characteristic
18	C₂²×C₄×C₅₉	—	2	6	no	—
19	C₂²×C₄×C₅₉	—	2	6	no	—
20	C₂²×C₄×C₅₉	—	2	12	no	—
21	C₂²×C₄×C₅₉	—	2	12	no	—
22	C₂²×C₄×C₅₉	C₂²×C₄	1	12	yes	—
23	C₂²×C₄×C₅₉	C₂²×C₄	1	2	yes	—
24	C₂²×C₄×C₅₉	#41	1	12	yes	—
25	C₂²×Dic₅₉	—	2	24	no	—
26	C₂²×Dic₅₉	—	2	24	no	—
27	C₂²×Dic₅₉	—	2	8	no	—
28	C₄²×C₅₉	—	2	24	no	—
29	C₄⋊Dic₅₉	—	4	16	no	—
30	C₂³×C₅₉	C₂³×C₄	1	1	yes	characteristic
31	C₂³×C₅₉	C₂×#41	1	6	yes	—
32	C₂³×C₅₉	H(5,𝔽₂)	1	4	yes	—
33	C₂³×C₅₉	2₋¹⁺⁴	1	4	yes	—
34	C₂×C₄×C₅₉	—	2	12	no	—
35	C₂×C₄×C₅₉	—	2	24	no	—
36	C₂×C₄×C₅₉	—	4	6	no	—
37	C₂×C₄×C₅₉	—	4	6	no	—
38	C₂×C₄×C₅₉	—	4	12	no	—
39	C₂×C₄×C₅₉	#117	1	12	yes	—
40	C₂×C₄×C₅₉	#126	1	16	yes	—
41	C₂×Dic₅₉	—	4	8	no	—
42	C₂×Dic₅₉	—	4	24	no	—
43	C₂×Dic₅₉	—	4	24	no	—
44	#14350	—	59	1	no	maximal, Sylow
45	C₂²×C₅₉	C₂×#117	1	3	yes	—
46	C₂²×C₅₉	#520	1	4	yes	—
47	C₂²×C₅₉	C₂×#126	1	4	yes	—
48	C₂²×C₅₉	#540	1	6	yes	—
49	C₂²×C₅₉	#541	1	6	yes	—
50	C₂²×C₅₉	#542	1	12	yes	—
51	C₄×C₅₉	—	4	12	no	—
52	C₄×C₅₉	#383	1	8	yes	—
53	Dic₅₉	—	8	8	no	—
54	C₂×#376	D₅₉	1	1	yes	characteristic
55	#1443	—	59	8	no	—
56	C₄⋊#95	—	59	6	no	—
57	C₂×C₅₉	C₂×#383	1	1	yes	characteristic
58	C₂×C₅₉	#2314	1	6	yes	—
59	C₂×C₅₉	#2644	1	4	yes	—
60	C₂×C₅₉	#2645	1	4	yes	—
61	C₂²×C₄⋊C₄	—	59	4	no	—
62	C₂²×C₄²	C₂×D₅₉	1	3	yes	—
63	#376	Dic₅₉	1	8	yes	complemented
64	C₂×#95	—	59	24	no	—
65	C₂×#95	C₂×D₅₉	1	4	yes	—
66	C₅₉	#14350	1	1	yes	characteristic, complemented, Sylow
67	C₂³×C₄	—	59	8	no	—
68	C₂³×C₄	C₂²×D₅₉	1	6	yes	—
69	C₂³×C₄	C₂²×D₅₉	1	1	yes	characteristic
70	C₂×C₄⋊C₄	—	118	24	no	—
71	C₂×C₄²	—	2	12	no	—
72	C₂×C₄²	C₂×Dic₅₉	1	12	yes	—
73	#95	—	236	24	no	—
74	#95	C₂×Dic₅₉	1	16	yes	—
75	C₂⁴	C₂³×D₅₉	1	1	yes	characteristic
76	C₂²×C₄	—	2	6	no	—
77	C₂²×C₄	—	2	6	no	—
78	C₂²×C₄	—	2	12	no	—
79	C₂²×C₄	—	2	12	no	—
80	C₂²×C₄	—	118	24	no	—
81	C₂²×C₄	—	118	24	no	—
82	C₂²×C₄	—	118	8	no	—
83	C₂²×C₄	#11745685	1	12	yes	—
84	C₂²×C₄	C₂²×Dic₅₉	1	12	yes	—
85	C₂²×C₄	C₂²×Dic₅₉	1	2	yes	—
86	C₄⋊C₄	—	236	16	no	—
87	C₄²	—	2	24	no	—
88	C₂³	C₂³×Dic₅₉	1	1	yes	characteristic
89	C₂³	C₂×#11745685	1	6	yes	—
90	C₂³	#14273773	1	4	yes	—
91	C₂³	#14273774	1	4	yes	—
92	C₂×C₄	—	2	12	no	—
93	C₂×C₄	—	2	24	no	—
94	C₂×C₄	—	4	6	no	—
95	C₂×C₄	—	4	6	no	—
96	C₂×C₄	—	4	12	no	—
97	C₂×C₄	—	236	8	no	—
98	C₂×C₄	—	236	24	no	—
99	C₂×C₄	—	236	24	no	—
100	C₂×C₄	#14273686	1	12	yes	—
101	C₂×C₄	#14273721	1	16	yes	—
102	C₂²	C₂×#14273686	1	3	yes	—
103	C₂²	#16837885	1	4	yes	—
104	C₂²	C₂×#14273721	1	4	yes	—
105	C₂²	#16838007	1	6	yes	—
106	C₂²	#16838014	1	6	yes	—
107	C₂²	#16838023	1	12	yes	—
108	C₄	—	4	12	no	—
109	C₄	—	472	8	no	—
110	C₄	#16837270	1	8	yes	—
111	C₂	C₂×#16837270	1	1	yes	characteristic
112	C₂	#21426705	1	6	yes	—
113	C₂	#21430056	1	4	yes	—
114	C₂	#21430061	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21422358

C₂

yes

maximal

#21422326

C₂

yes

maximal

C₂×C₅₉×#376

C₂

yes

maximal, characteristic

C₂²×C₄⋊Dic₅₉

C₂²

yes

—

C₂×C₅₉⋊#95

C₂²

yes

—

C₂²×C₄²×C₅₉

C₂²

yes

—

C₅₉×#376

C₄

yes

complemented

C₂×#95×C₅₉

C₂²

yes

—

C₂³×Dic₅₉

C₂³

yes

—

C₂×C₄⋊Dic₅₉

—

C₅₉⋊#95

—

C₂³×C₄×C₅₉

C₂³

yes

—

C₂³×C₄×C₅₉

C₂³

yes

characteristic

C₂×C₄²×C₅₉

—

C₂×C₄²×C₅₉

C₂×C₄

yes

—

#95×C₅₉

C₂×C₄

yes

—

C₂⁴×C₅₉

C₂⁴

yes

characteristic

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂²×C₄

yes

—

C₂²×C₄×C₅₉

C₂²×C₄

yes

—

C₂²×C₄×C₅₉

#41

yes

—

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₄²×C₅₉

—

C₄⋊Dic₅₉

—

C₂³×C₅₉

C₂³×C₄

yes

characteristic

C₂³×C₅₉

C₂×#41

yes

—

C₂³×C₅₉

H(5,𝔽₂)

yes

—

C₂³×C₅₉

2₋¹⁺⁴

yes

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#117

yes

—

C₂×C₄×C₅₉

#126

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

#14350

—

maximal, Sylow

C₂²×C₅₉

C₂×#117

yes

—

C₂²×C₅₉

#520

yes

—

C₂²×C₅₉

C₂×#126

yes

—

C₂²×C₅₉

#540

yes

—

C₂²×C₅₉

#541

yes

—

C₂²×C₅₉

#542

yes

—

C₄×C₅₉

—

C₄×C₅₉

#383

yes

—

Dic₅₉

—

C₂×#376

D₅₉

yes

characteristic

#1443

—

C₄⋊#95

—

C₂×C₅₉

C₂×#383

yes

characteristic

C₂×C₅₉

#2314

yes

—

C₂×C₅₉

#2644

yes

—

C₂×C₅₉

#2645

yes

—

C₂²×C₄⋊C₄

—

C₂²×C₄²

C₂×D₅₉

yes

—

#376

Dic₅₉

yes

complemented

C₂×#95

—

C₂×#95

C₂×D₅₉

yes

—

C₅₉

#14350

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂³×C₄

C₂²×D₅₉

yes

—

C₂³×C₄

C₂²×D₅₉

yes

characteristic

C₂×C₄⋊C₄

—

118

—

C₂×C₄²

—

C₂×C₄²

C₂×Dic₅₉

yes

—

#95

—

236

—

#95

C₂×Dic₅₉

yes

—

C₂⁴

C₂³×D₅₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

118

—

C₂²×C₄

—

118

—

C₂²×C₄

—

118

—

C₂²×C₄

#11745685

yes

—

C₂²×C₄

C₂²×Dic₅₉

yes

—

C₂²×C₄

C₂²×Dic₅₉

yes

—

C₄⋊C₄

—

236

—

C₄²

—

C₂³

C₂³×Dic₅₉

yes

characteristic

C₂³

C₂×#11745685

yes

—

C₂³

#14273773

yes

—

C₂³

#14273774

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

236

—

C₂×C₄

—

236

—

C₂×C₄

—

236

—

100

C₂×C₄

#14273686

yes

—

101

C₂×C₄

#14273721

yes

—

102

C₂²

C₂×#14273686

yes

—

103

C₂²

#16837885

yes

—

104

C₂²

C₂×#14273721

yes

—

105

C₂²

#16838007

yes

—

106

C₂²

#16838014

yes

—

107

C₂²

#16838023

yes

—

108

C₄

—

109

C₄

—

472

—

110

C₄

#16837270

yes

—

111

C₂

C₂×#16837270

yes

characteristic

112

C₂

#21426705

yes

—

113

C₂

#21430056

yes

—

114

C₂

#21430061

yes

—

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₅₉
commutator	C₂³×C₅₉	C₂³×C₄
Fitting	C₂×C₅₉×#376	C₂
socle, prime radical, cosocle	C₂⁴×C₅₉	C₂⁴
hypercenter	C₂×#376	D₅₉
hypocenter	C₅₉	#14350
LCS	G ▷ C₂³×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376

subgroup

G/H

center, Frattini

C₂⁴

C₂³×D₅₉

commutator

C₂³×C₅₉

C₂³×C₄

Fitting

C₂×C₅₉×#376

C₂

socle, prime radical, cosocle

C₂⁴×C₅₉

C₂⁴

D₅₉

C₅₉

G ▷ C₂³×C₅₉ ▷ C₅₉

C₁ ◁ C₂⁴ ◁ C₂×#376

N	H ≅ G/N	type
7. C₅₉×#376	C₄	semidirect
63. #376	Dic₅₉	semidirect
66. C₅₉	#14350	semidirect

H ≅ G/N

type

7. C₅₉×#376

C₄

semidirect

63. #376

Dic₅₉

semidirect

66. C₅₉

#14350

semidirect

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

subgroup	index
#21422358	2
#21422326	2
C₂×C₅₉×#376	2
#14350	59

subgroup

index

#21422358

#21422326

C₂×C₅₉×#376

#14350

Subgroups — #28515923

Conjugacy classes of subgroups

Complements

Maximal subgroups