Group #28509724 — subgroups

Subgroups — D₅₉×M₁₂₈ #28509724

Order 15104.

134 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 81 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	C₂×C₃₂×D₅₉	C₂	1	1	yes	maximal, characteristic
2	#21422307	C₂	1	1	yes	maximal, characteristic, complemented
3	#21422306	C₂	1	2	yes	maximal, complemented
4	C₆₄×D₅₉	C₂	1	2	yes	maximal, complemented
5	C₅₉×M₁₂₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₆×D₅₉	C₄	1	1	yes	characteristic
7	C₂×C₅₉⋊C₃₂	C₄	1	1	yes	characteristic
8	C₃₂×D₅₉	C₄	1	1	yes	characteristic
9	C₃₂×D₅₉	C₄	1	1	yes	characteristic
10	C₃₂×D₅₉	C₂²	1	1	yes	characteristic
11	C₃₂×D₅₉	C₂²	1	1	yes	characteristic
12	C₅₉⋊C₆₄	C₂²	1	2	yes	complemented
13	C₂×C₃₂×C₅₉	C₂²	1	1	yes	characteristic
14	C₅₉×C₆₄	C₂²	1	2	yes	complemented
15	C₂×C₈×D₅₉	C₈	1	1	yes	characteristic
16	C₂×C₅₉⋊C₁₆	C₈	1	1	yes	characteristic
17	C₁₆×D₅₉	C₈	1	1	yes	characteristic
18	C₁₆×D₅₉	C₈	1	1	yes	characteristic
19	C₁₆×D₅₉	C₂×C₄	1	1	yes	characteristic
20	C₁₆×D₅₉	C₂×C₄	1	1	yes	characteristic
21	C₅₉⋊C₃₂	C₂×C₄	1	1	yes	characteristic
22	C₅₉⋊C₃₂	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
24	C₃₂×C₅₉	C₂×C₄	1	1	yes	characteristic
25	C₃₂×C₅₉	C₂³	1	1	yes	characteristic
26	C₂×C₄×D₅₉	C₁₆	1	1	yes	characteristic
27	C₂×C₈×C₅₉	C₂×C₈	1	1	yes	characteristic
28	C₂×C₅₉⋊C₈	C₁₆	1	1	yes	characteristic
29	C₈×D₅₉	C₁₆	1	1	yes	characteristic
30	C₈×D₅₉	C₁₆	1	1	yes	characteristic
31	C₈×D₅₉	C₂×C₈	1	1	yes	characteristic
32	C₈×D₅₉	C₂×C₈	1	1	yes	characteristic
33	C₁₆×C₅₉	C₂×C₈	1	1	yes	characteristic
34	C₁₆×C₅₉	C₂²×C₄	1	1	yes	characteristic
35	C₅₉⋊C₁₆	C₂×C₈	1	1	yes	characteristic
36	C₅₉⋊C₁₆	C₂×C₈	1	1	yes	characteristic
37	C₂²×D₅₉	C₃₂	1	1	yes	characteristic
38	C₂×C₄×C₅₉	C₂×C₁₆	1	1	yes	characteristic
39	C₂×Dic₅₉	C₃₂	1	1	yes	characteristic
40	C₄×D₅₉	C₃₂	1	1	yes	characteristic
41	C₄×D₅₉	C₃₂	1	1	yes	characteristic
42	C₄×D₅₉	C₂×C₁₆	1	1	yes	characteristic
43	C₄×D₅₉	C₂×C₁₆	1	1	yes	characteristic
44	C₈×C₅₉	C₂×C₁₆	1	1	yes	characteristic
45	C₈×C₅₉	C₂²×C₈	1	1	yes	characteristic
46	C₅₉⋊C₈	C₂×C₁₆	1	1	yes	characteristic
47	C₅₉⋊C₈	C₂×C₁₆	1	1	yes	characteristic
48	C₂×M₁₂₈	—	59	1	no	maximal, Sylow
49	C₂²×C₅₉	C₂×C₃₂	1	1	yes	characteristic
50	C₂×D₅₉	—	2	2	no	—
51	C₂×D₅₉	C₂×C₃₂	1	1	yes	characteristic
52	C₂×D₅₉	C₂×C₃₂	1	1	yes	characteristic
53	C₄×C₅₉	C₂×C₃₂	1	1	yes	characteristic
54	C₄×C₅₉	C₂²×C₁₆	1	1	yes	characteristic
55	Dic₅₉	C₂×C₃₂	1	1	yes	characteristic
56	Dic₅₉	C₂×C₃₂	1	1	yes	characteristic
57	C₂²×C₃₂	—	59	1	no	—
58	M₁₂₈	—	59	1	no	—
59	M₁₂₈	—	59	2	no	—
60	M₁₂₈	D₅₉	1	1	yes	characteristic, direct product factor
61	C₂×C₆₄	—	59	2	no	—
62	C₂×C₅₉	—	2	1	no	—
63	C₂×C₅₉	C₂²×C₃₂	1	1	yes	characteristic
64	D₅₉	—	2	1	no	—
65	D₅₉	M₁₂₈	1	2	yes	direct product factor
66	C₂²×C₁₆	—	59	1	no	—
67	C₂×C₃₂	—	59	1	no	—
68	C₂×C₃₂	—	59	1	no	—
69	C₂×C₃₂	—	59	1	no	—
70	C₂×C₃₂	—	59	1	no	—
71	C₂×C₃₂	—	59	1	no	—
72	C₂×C₃₂	C₂×D₅₉	1	1	yes	characteristic
73	C₆₄	—	59	2	no	—
74	C₆₄	C₂×D₅₉	1	2	yes	complemented
75	C₅₉	C₂×M₁₂₈	1	1	yes	characteristic, complemented, Sylow
76	C₂²×C₈	—	59	1	no	—
77	C₂×C₁₆	—	59	1	no	—
78	C₂×C₁₆	—	59	1	no	—
79	C₂×C₁₆	—	59	1	no	—
80	C₂×C₁₆	—	59	1	no	—
81	C₂×C₁₆	—	59	1	no	—
82	C₂×C₁₆	C₄×D₅₉	1	1	yes	characteristic
83	C₃₂	—	59	1	no	—
84	C₃₂	—	59	1	no	—
85	C₃₂	C₄×D₅₉	1	1	yes	characteristic
86	C₃₂	C₂²×D₅₉	1	1	yes	characteristic
87	C₂²×C₄	—	59	1	no	—
88	C₂×C₈	—	59	1	no	—
89	C₂×C₈	—	59	1	no	—
90	C₂×C₈	—	59	1	no	—
91	C₂×C₈	—	59	1	no	—
92	C₂×C₈	—	59	1	no	—
93	C₂×C₈	C₈×D₅₉	1	1	yes	characteristic
94	C₁₆	—	59	1	no	—
95	C₁₆	—	59	1	no	—
96	C₁₆	C₈×D₅₉	1	1	yes	characteristic
97	C₁₆	C₂×C₄×D₅₉	1	1	yes	characteristic
98	C₂³	—	59	1	no	—
99	C₂×C₄	—	59	1	no	—
100	C₂×C₄	—	59	1	no	—
101	C₂×C₄	—	59	1	no	—
102	C₂×C₄	—	59	1	no	—
103	C₂×C₄	—	59	1	no	—
104	C₂×C₄	C₁₆×D₅₉	1	1	yes	characteristic
105	C₈	—	59	1	no	—
106	C₈	—	59	1	no	—
107	C₈	C₁₆×D₅₉	1	1	yes	characteristic
108	C₈	C₂×C₈×D₅₉	1	1	yes	characteristic
109	C₂²	—	59	1	no	—
110	C₂²	—	59	1	no	—
111	C₂²	—	118	2	no	—
112	C₂²	C₃₂×D₅₉	1	1	yes	characteristic
113	C₄	—	59	1	no	—
114	C₄	—	59	1	no	—
115	C₄	C₃₂×D₅₉	1	1	yes	characteristic
116	C₄	C₂×C₁₆×D₅₉	1	1	yes	characteristic
117	C₂	—	2	1	no	—
118	C₂	—	59	2	no	—
119	C₂	—	118	1	no	—
120	C₂	C₂×C₃₂×D₅₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃₂×D₅₉

C₂

yes

maximal, characteristic

#21422307

C₂

yes

maximal, characteristic, complemented

#21422306

C₂

yes

maximal, complemented

C₆₄×D₅₉

C₂

yes

maximal, complemented

C₅₉×M₁₂₈

C₂

yes

maximal, characteristic, complemented

C₂×C₁₆×D₅₉

C₄

yes

characteristic

C₂×C₅₉⋊C₃₂

C₄

yes

characteristic

C₃₂×D₅₉

C₄

yes

characteristic

C₃₂×D₅₉

C₄

yes

characteristic

C₃₂×D₅₉

C₂²

yes

characteristic

C₃₂×D₅₉

C₂²

yes

characteristic

C₅₉⋊C₆₄

C₂²

yes

complemented

C₂×C₃₂×C₅₉

C₂²

yes

characteristic

C₅₉×C₆₄

C₂²

yes

complemented

C₂×C₈×D₅₉

C₈

yes

characteristic

C₂×C₅₉⋊C₁₆

C₈

yes

characteristic

C₁₆×D₅₉

C₈

yes

characteristic

C₁₆×D₅₉

C₈

yes

characteristic

C₁₆×D₅₉

C₂×C₄

yes

characteristic

C₁₆×D₅₉

C₂×C₄

yes

characteristic

C₅₉⋊C₃₂

C₂×C₄

yes

characteristic

C₅₉⋊C₃₂

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₉

C₂×C₄

yes

characteristic

C₃₂×C₅₉

C₂×C₄

yes

characteristic

C₃₂×C₅₉

C₂³

yes

characteristic

C₂×C₄×D₅₉

C₁₆

yes

characteristic

C₂×C₈×C₅₉

C₂×C₈

yes

characteristic

C₂×C₅₉⋊C₈

C₁₆

yes

characteristic

C₈×D₅₉

C₁₆

yes

characteristic

C₈×D₅₉

C₁₆

yes

characteristic

C₈×D₅₉

C₂×C₈

yes

characteristic

C₈×D₅₉

C₂×C₈

yes

characteristic

C₁₆×C₅₉

C₂×C₈

yes

characteristic

C₁₆×C₅₉

C₂²×C₄

yes

characteristic

C₅₉⋊C₁₆

C₂×C₈

yes

characteristic

C₅₉⋊C₁₆

C₂×C₈

yes

characteristic

C₂²×D₅₉

C₃₂

yes

characteristic

C₂×C₄×C₅₉

C₂×C₁₆

yes

characteristic

C₂×Dic₅₉

C₃₂

yes

characteristic

C₄×D₅₉

C₃₂

yes

characteristic

C₄×D₅₉

C₃₂

yes

characteristic

C₄×D₅₉

C₂×C₁₆

yes

characteristic

C₄×D₅₉

C₂×C₁₆

yes

characteristic

C₈×C₅₉

C₂×C₁₆

yes

characteristic

C₈×C₅₉

C₂²×C₈

yes

characteristic

C₅₉⋊C₈

C₂×C₁₆

yes

characteristic

C₅₉⋊C₈

C₂×C₁₆

yes

characteristic

C₂×M₁₂₈

—

maximal, Sylow

C₂²×C₅₉

C₂×C₃₂

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

C₂×C₃₂

yes

characteristic

C₂×D₅₉

C₂×C₃₂

yes

characteristic

C₄×C₅₉

C₂×C₃₂

yes

characteristic

C₄×C₅₉

C₂²×C₁₆

yes

characteristic

Dic₅₉

C₂×C₃₂

yes

characteristic

Dic₅₉

C₂×C₃₂

yes

characteristic

C₂²×C₃₂

—

M₁₂₈

—

M₁₂₈

—

M₁₂₈

D₅₉

yes

characteristic, direct product factor

C₂×C₆₄

—

C₂×C₅₉

—

C₂×C₅₉

C₂²×C₃₂

yes

characteristic

D₅₉

—

D₅₉

M₁₂₈

yes

direct product factor

C₂²×C₁₆

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₅₉

yes

characteristic

C₆₄

—

C₆₄

C₂×D₅₉

yes

complemented

C₅₉

C₂×M₁₂₈

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₅₉

yes

characteristic

C₃₂

—

C₃₂

—

C₃₂

C₄×D₅₉

yes

characteristic

C₃₂

C₂²×D₅₉

yes

characteristic

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₈×D₅₉

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

C₈×D₅₉

yes

characteristic

C₁₆

C₂×C₄×D₅₉

yes

characteristic

C₂³

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₁₆×D₅₉

yes

characteristic

105

C₈

—

106

C₈

—

107

C₈

C₁₆×D₅₉

yes

characteristic

108

C₈

C₂×C₈×D₅₉

yes

characteristic

109

C₂²

—

110

C₂²

—

111

C₂²

—

118

—

112

C₂²

C₃₂×D₅₉

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

C₃₂×D₅₉

yes

characteristic

116

C₄

C₂×C₁₆×D₅₉

yes

characteristic

117

C₂

—

118

C₂

—

119

C₂

—

118

—

120

C₂

C₂×C₃₂×D₅₉

yes

characteristic

	subgroup	G/H
center, Frattini	C₃₂	C₂²×D₅₉
commutator, socle	C₂×C₅₉	C₂²×C₃₂
Fitting	C₅₉×M₁₂₈	C₂
prime radical	C₂²×D₅₉	C₃₂
cosocle	C₃₂×C₅₉	C₂³
hypercenter	M₁₂₈	D₅₉
hypocenter	C₅₉	C₂×M₁₂₈
LCS	G ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₃₂ ◁ M₁₂₈

subgroup

G/H

center, Frattini

C₃₂

C₂²×D₅₉

commutator, socle

C₂×C₅₉

C₂²×C₃₂

Fitting

C₅₉×M₁₂₈

C₂

prime radical

C₂²×D₅₉

C₃₂

cosocle

C₃₂×C₅₉

C₂³

hypercenter

M₁₂₈

D₅₉

hypocenter

C₅₉

C₂×M₁₂₈

LCS

G ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₃₂ ◁ M₁₂₈

N	H ≅ G/N	type
2. #21422307	C₂	semidirect
3. #21422306	C₂	semidirect
4. C₆₄×D₅₉	C₂	semidirect
5. C₅₉×M₁₂₈	C₂	semidirect
12. C₅₉⋊C₆₄	C₂²	semidirect
14. C₅₉×C₆₄	C₂²	semidirect
60. M₁₂₈	D₅₉	direct (N × H)
65. D₅₉	M₁₂₈	direct (N × H)
74. C₆₄	C₂×D₅₉	semidirect
75. C₅₉	C₂×M₁₂₈	semidirect

H ≅ G/N

type

2. #21422307

C₂

semidirect

3. #21422306

C₂

semidirect

4. C₆₄×D₅₉

C₂

semidirect

5. C₅₉×M₁₂₈

C₂

semidirect

12. C₅₉⋊C₆₄

C₂²

semidirect

14. C₅₉×C₆₄

C₂²

semidirect

60. M₁₂₈

D₅₉

direct (N × H)

65. D₅₉

M₁₂₈

direct (N × H)

74. C₆₄

C₂×D₅₉

semidirect

75. C₅₉

C₂×M₁₂₈

semidirect

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

subgroup	index
C₂×C₃₂×D₅₉	2
#21422307	2
#21422306	2
C₆₄×D₅₉	2
C₅₉×M₁₂₈	2
C₂×M₁₂₈	59

subgroup

index

C₂×C₃₂×D₅₉

#21422307

#21422306

C₆₄×D₅₉

C₅₉×M₁₂₈

C₂×M₁₂₈

Subgroups — D₅₉×M₁₂₈ #28509724

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D59×M128 #28509724

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₉×M₁₂₈ #28509724