Group #28509182 — subgroups

Subgroups — #28509182

Order 15104.

198 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 143 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×#16837236	C₂	1	2	yes	maximal
2	#21422294	C₂	1	4	yes	maximal, complemented
3	C₅₉×#2108	C₂	1	1	yes	maximal, characteristic
4	#16837236	—	2	2	no	—
5	#16837236	C₂²	1	4	yes	—
6	C₂×C₅₉⋊C₃₂	—	2	2	no	—
7	C₂×M₃₂×C₅₉	C₄	1	1	yes	characteristic
8	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
9	C₅₉×#405	C₄	1	1	yes	characteristic
10	C₅₉×#346	C₄	1	2	yes	—
11	C₄×C₁₆×C₅₉	C₂²	1	2	yes	—
12	C₅₉⋊C₃₂	—	2	4	no	—
13	C₂×M₁₆×C₅₉	C₈	1	1	yes	characteristic
14	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
15	#117×C₅₉	C₈	1	1	yes	characteristic
16	M₃₂×C₅₉	—	2	2	no	—
17	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
18	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₆×C₅₉	D₄	1	2	yes	—
20	C₂×C₁₆×C₅₉	Q₈	1	2	yes	—
21	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
22	#97×C₅₉	C₈	1	2	yes	—
23	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
25	M₁₆×C₅₉	C₁₆	1	4	yes	—
26	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
27	C₂×C₈×C₅₉	C₂×C₈	1	1	yes	characteristic
28	C₂×C₈×C₅₉	C₂×C₈	1	2	yes	—
29	C₂×C₈×C₅₉	M₁₆	1	2	yes	—
30	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
31	C₄⋊C₄×C₅₉	—	2	1	no	—
32	C₂²⋊C₄×C₅₉	—	2	1	no	—
33	C₄²×C₅₉	C₂×C₈	1	2	yes	—
34	C₁₆×C₅₉	—	2	2	no	—
35	C₁₆×C₅₉	C₄⋊C₄	1	2	yes	—
36	C₁₆×C₅₉	C₂×D₄	1	1	yes	characteristic
37	C₁₆×C₅₉	C₂×Q₈	1	1	yes	characteristic
38	C₂³×C₅₉	M₃₂	1	1	yes	characteristic
39	C₂×C₄×C₅₉	—	2	2	no	—
40	C₂×C₄×C₅₉	C₄⋊C₈	1	2	yes	—
41	C₂×C₄×C₅₉	C₂×C₁₆	1	2	yes	—
42	C₂×C₄×C₅₉	M₃₂	1	1	yes	characteristic
43	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
44	C₈×C₅₉	C₄⋊C₈	1	2	yes	—
45	C₈×C₅₉	C₂×C₁₆	1	4	yes	—
46	C₈×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₈×C₅₉	C₂×M₁₆	1	1	yes	characteristic
48	#13630	—	59	1	no	maximal, Sylow
49	C₂²×C₅₉	—	4	1	no	—
50	C₂²×C₅₉	C₄⋊C₁₆	1	2	yes	—
51	C₂²×C₅₉	C₂×M₃₂	1	1	yes	characteristic
52	C₄×C₅₉	—	4	2	no	—
53	C₄×C₅₉	C₄⋊C₁₆	1	2	yes	—
54	C₄×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
55	C₄×C₅₉	C₂²×C₁₆	1	1	yes	characteristic
56	C₂×M₆₄	—	59	2	no	—
57	#2108	D₅₉	1	1	yes	characteristic
58	#1422	—	59	4	no	—
59	C₂×C₅₉	—	2	1	no	—
60	C₂×C₅₉	—	2	2	no	—
61	C₂×C₅₉	C₂×C₄⋊C₁₆	1	1	yes	characteristic
62	C₂×M₃₂	Dic₅₉	1	1	yes	characteristic
63	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
64	#405	Dic₅₉	1	1	yes	characteristic
65	M₆₄	—	59	4	no	—
66	M₆₄	—	118	2	no	—
67	C₂×C₃₂	—	118	2	no	—
68	#346	Dic₅₉	1	2	yes	—
69	C₄×C₁₆	C₂×D₅₉	1	2	yes	—
70	C₅₉	#13630	1	1	yes	characteristic, complemented, Sylow
71	C₂×M₁₆	C₅₉⋊C₈	1	1	yes	characteristic
72	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
73	#117	C₅₉⋊C₈	1	1	yes	characteristic
74	M₃₂	—	2	2	no	—
75	C₂×C₁₆	Dic₁₁₈	1	2	yes	—
76	C₂×C₁₆	D₂₃₆	1	2	yes	—
77	C₂×C₁₆	C₂×Dic₅₉	1	2	yes	—
78	C₂×C₁₆	C₂×Dic₅₉	1	1	yes	characteristic
79	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
80	#97	C₅₉⋊C₈	1	2	yes	—
81	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
82	C₃₂	—	118	4	no	—
83	C₂²×C₄	C₂×C₅₉⋊C₈	1	1	yes	characteristic
84	M₁₆	C₅₉⋊C₁₆	1	4	yes	—
85	C₂×C₈	C₂×C₅₉⋊C₈	1	1	yes	characteristic
86	C₂×C₈	C₂×C₅₉⋊C₈	1	2	yes	—
87	C₂×C₈	#11745664	1	2	yes	—
88	C₂×C₈	C₄⋊Dic₅₉	1	2	yes	—
89	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
90	C₄⋊C₄	—	2	1	no	—
91	C₂²⋊C₄	—	2	1	no	—
92	C₄²	C₂×C₅₉⋊C₈	1	2	yes	—
93	C₁₆	—	2	2	no	—
94	C₁₆	C₄⋊Dic₅₉	1	2	yes	—
95	C₁₆	C₂×Dic₁₁₈	1	1	yes	characteristic
96	C₁₆	C₂×D₂₃₆	1	1	yes	characteristic
97	C₂³	#14273667	1	1	yes	characteristic
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	C₄⋊(C₅₉⋊C₈)	1	2	yes	—
100	C₂×C₄	C₂×C₅₉⋊C₁₆	1	2	yes	—
101	C₂×C₄	#14273667	1	1	yes	characteristic
102	C₂×C₄	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
103	C₈	C₄⋊(C₅₉⋊C₈)	1	2	yes	—
104	C₈	C₂×C₅₉⋊C₁₆	1	4	yes	—
105	C₈	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
106	C₈	C₂×#11745664	1	1	yes	characteristic
107	C₂²	—	4	1	no	—
108	C₂²	C₄⋊(C₅₉⋊C₁₆)	1	2	yes	—
109	C₂²	C₂×#14273667	1	1	yes	characteristic
110	C₄	—	4	2	no	—
111	C₄	C₄⋊(C₅₉⋊C₁₆)	1	2	yes	—
112	C₄	C₂×C₄⋊(C₅₉⋊C₈)	1	1	yes	characteristic
113	C₄	C₂²×C₅₉⋊C₁₆	1	1	yes	characteristic
114	C₂	—	2	1	no	—
115	C₂	—	2	2	no	—
116	C₂	C₂×C₄⋊(C₅₉⋊C₁₆)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16837236

C₂

yes

maximal

#21422294

C₂

yes

maximal, complemented

C₅₉×#2108

C₂

yes

maximal, characteristic

#16837236

—

#16837236

C₂²

yes

—

C₂×C₅₉⋊C₃₂

—

C₂×M₃₂×C₅₉

C₄

yes

characteristic

C₂²×C₁₆×C₅₉

C₂²

yes

characteristic

C₅₉×#405

C₄

yes

characteristic

C₅₉×#346

C₄

yes

—

C₄×C₁₆×C₅₉

C₂²

yes

—

C₅₉⋊C₃₂

—

C₂×M₁₆×C₅₉

C₈

yes

characteristic

C₂²×C₈×C₅₉

C₂×C₄

yes

characteristic

#117×C₅₉

C₈

yes

characteristic

M₃₂×C₅₉

—

C₂×C₁₆×C₅₉

C₂×C₄

yes

—

C₂×C₁₆×C₅₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₉

D₄

yes

—

C₂×C₁₆×C₅₉

Q₈

yes

—

C₂×C₁₆×C₅₉

C₂³

yes

characteristic

#97×C₅₉

C₈

yes

—

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₂²×C₄×C₅₉

C₂×C₈

yes

characteristic

M₁₆×C₅₉

C₁₆

yes

—

C₂×C₈×C₅₉

C₄⋊C₄

yes

—

C₂×C₈×C₅₉

C₂×C₈

yes

characteristic

C₂×C₈×C₅₉

C₂×C₈

yes

—

C₂×C₈×C₅₉

M₁₆

yes

—

C₂×C₈×C₅₉

C₂²×C₄

yes

characteristic

C₄⋊C₄×C₅₉

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

C₂×C₈

yes

—

C₁₆×C₅₉

—

C₁₆×C₅₉

C₄⋊C₄

yes

—

C₁₆×C₅₉

C₂×D₄

yes

characteristic

C₁₆×C₅₉

C₂×Q₈

yes

characteristic

C₂³×C₅₉

M₃₂

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₄⋊C₈

yes

—

C₂×C₄×C₅₉

C₂×C₁₆

yes

—

C₂×C₄×C₅₉

M₃₂

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₈

yes

characteristic

C₈×C₅₉

C₄⋊C₈

yes

—

C₈×C₅₉

C₂×C₁₆

yes

—

C₈×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₈×C₅₉

C₂×M₁₆

yes

characteristic

#13630

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

C₄⋊C₁₆

yes

—

C₂²×C₅₉

C₂×M₃₂

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

C₄⋊C₁₆

yes

—

C₄×C₅₉

C₂×C₄⋊C₈

yes

characteristic

C₄×C₅₉

C₂²×C₁₆

yes

characteristic

C₂×M₆₄

—

#2108

D₅₉

yes

characteristic

#1422

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

C₂×C₄⋊C₁₆

yes

characteristic

C₂×M₃₂

Dic₅₉

yes

characteristic

C₂²×C₁₆

C₂×D₅₉

yes

characteristic

#405

Dic₅₉

yes

characteristic

M₆₄

—

M₆₄

—

118

—

C₂×C₃₂

—

118

—

#346

Dic₅₉

yes

—

C₄×C₁₆

C₂×D₅₉

yes

—

C₅₉

#13630

yes

characteristic, complemented, Sylow

C₂×M₁₆

C₅₉⋊C₈

yes

characteristic

C₂²×C₈

C₂×Dic₅₉

yes

characteristic

#117

C₅₉⋊C₈

yes

characteristic

M₃₂

—

C₂×C₁₆

Dic₁₁₈

yes

—

C₂×C₁₆

D₂₃₆

yes

—

C₂×C₁₆

C₂×Dic₅₉

yes

—

C₂×C₁₆

C₂×Dic₅₉

yes

characteristic

C₂×C₁₆

C₂²×D₅₉

yes

characteristic

#97

C₅₉⋊C₈

yes

—

C₄×C₈

C₂×Dic₅₉

yes

—

C₃₂

—

118

—

C₂²×C₄

C₂×C₅₉⋊C₈

yes

characteristic

M₁₆

C₅₉⋊C₁₆

yes

—

C₂×C₈

C₂×C₅₉⋊C₈

yes

characteristic

C₂×C₈

C₂×C₅₉⋊C₈

yes

—

C₂×C₈

#11745664

yes

—

C₂×C₈

C₄⋊Dic₅₉

yes

—

C₂×C₈

C₂²×Dic₅₉

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂×C₅₉⋊C₈

yes

—

C₁₆

—

C₁₆

C₄⋊Dic₅₉

yes

—

C₁₆

C₂×Dic₁₁₈

yes

characteristic

C₁₆

C₂×D₂₃₆

yes

characteristic

C₂³

#14273667

yes

characteristic

C₂×C₄

—

C₂×C₄

C₄⋊(C₅₉⋊C₈)

yes

—

100

C₂×C₄

C₂×C₅₉⋊C₁₆

yes

—

101

C₂×C₄

#14273667

yes

characteristic

102

C₂×C₄

C₂²×C₅₉⋊C₈

yes

characteristic

103

C₈

C₄⋊(C₅₉⋊C₈)

yes

—

104

C₈

C₂×C₅₉⋊C₁₆

yes

—

105

C₈

C₂×C₄⋊Dic₅₉

yes

characteristic

106

C₈

C₂×#11745664

yes

characteristic

107

C₂²

—

108

C₂²

C₄⋊(C₅₉⋊C₁₆)

yes

—

109

C₂²

C₂×#14273667

yes

characteristic

110

C₄

—

111

C₄

C₄⋊(C₅₉⋊C₁₆)

yes

—

112

C₄

C₂×C₄⋊(C₅₉⋊C₈)

yes

characteristic

113

C₄

C₂²×C₅₉⋊C₁₆

yes

characteristic

114

C₂

—

115

C₂

—

116

C₂

C₂×C₄⋊(C₅₉⋊C₁₆)

yes

characteristic

	subgroup	G/H
center	C₁₆	C₂×D₂₃₆
commutator	C₄×C₅₉	C₂²×C₁₆
Frattini	C₂×C₁₆	C₂²×D₅₉
Fitting	C₅₉×#2108	C₂
socle	C₂×C₅₉	C₂×C₄⋊C₁₆
prime radical	C₂³×C₅₉	M₃₂
cosocle	C₂×C₁₆×C₅₉	C₂³
hypercenter	#2108	D₅₉
hypocenter	C₅₉	#13630
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₁₆ ◁ C₂²×C₁₆ ◁ #2108

subgroup

G/H

center

C₁₆

C₂×D₂₃₆

commutator

C₄×C₅₉

C₂²×C₁₆

Frattini

C₂×C₁₆

C₂²×D₅₉

Fitting

C₅₉×#2108

C₂

socle

C₂×C₅₉

C₂×C₄⋊C₁₆

prime radical

C₂³×C₅₉

M₃₂

cosocle

C₂×C₁₆×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₁₆ ◁ C₂²×C₁₆ ◁ #2108

N	H ≅ G/N	type
2. #21422294	C₂	semidirect
70. C₅₉	#13630	semidirect

H ≅ G/N

type

2. #21422294

C₂

semidirect

70. C₅₉

#13630

semidirect

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

subgroup	index
C₂×#16837236	2
#21422294	2
C₅₉×#2108	2
#13630	59

subgroup

index

C₂×#16837236

#21422294

C₅₉×#2108

#13630

Subgroups — #28509182

Conjugacy classes of subgroups

Complements

Maximal subgroups