Group #28509661 — subgroups

Subgroups — #28509661

Order 15104.

182 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 75 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	#21426371	C₂	1	1	yes	maximal, characteristic
2	#21422277	C₂	1	4	yes	maximal
3	C₄×C₅₉⋊C₃₂	C₂	1	1	yes	maximal, characteristic
4	C₅₉×#2247	C₂	1	1	yes	maximal, characteristic
5	#16837807	—	2	1	no	—
6	C₂×C₅₉⋊C₃₂	C₂²	1	2	yes	—
7	#16837228	—	2	1	no	—
8	#16837228	C₂²	1	2	yes	—
9	C₂×Q₃₂×C₅₉	—	2	1	no	—
10	C₂×Q₃₂×C₅₉	C₂²	1	2	yes	—
11	C₅₉×#494	—	2	1	no	—
12	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
13	#14273727	—	4	1	no	—
14	#14273661	—	2	2	no	—
15	#14273661	—	4	1	no	—
16	C₅₉⋊C₃₂	—	2	2	no	—
17	C₂×Q₁₆×C₅₉	—	2	2	no	—
18	C₂×Q₁₆×C₅₉	—	4	1	no	—
19	C₄⋊Q₈×C₅₉	—	4	1	no	—
20	Q₃₂×C₅₉	—	2	4	no	—
21	Q₃₂×C₅₉	—	4	2	no	—
22	C₂×C₁₆×C₅₉	D₄	1	1	yes	characteristic
23	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
24	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
25	C₂×Q₈×C₅₉	—	4	2	no	—
26	C₂×Dic₁₁₈	—	8	1	no	—
27	Q₁₆×C₅₉	—	4	2	no	—
28	Q₁₆×C₅₉	—	4	2	no	—
29	Q₁₆×C₅₉	—	8	2	no	—
30	C₂×C₈×C₅₉	D₈	1	1	yes	characteristic
31	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
32	C₄⋊C₄×C₅₉	—	8	1	no	—
33	C₄⋊C₄×C₅₉	—	8	1	no	—
34	C₄²×C₅₉	D₈	1	1	yes	characteristic
35	C₄⋊Dic₅₉	—	4	2	no	—
36	C₄⋊Dic₅₉	—	8	1	no	—
37	C₁₆×C₅₉	SD₁₆	1	2	yes	—
38	C₁₆×C₅₉	#41	1	2	yes	—
39	Q₈×C₅₉	—	8	2	no	—
40	Q₈×C₅₉	—	8	2	no	—
41	C₂×C₄×C₅₉	—	8	2	no	—
42	C₂×C₄×C₅₉	D₁₆	1	1	yes	characteristic
43	C₂×C₄×C₅₉	D₁₆	1	1	yes	characteristic
44	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
45	C₂×Dic₅₉	—	8	2	no	—
46	Dic₁₁₈	—	16	2	no	—
47	C₈×C₅₉	SD₃₂	1	2	yes	—
48	C₈×C₅₉	#124	1	1	yes	characteristic
49	C₈×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
50	#13720	—	59	1	no	maximal, Sylow
51	C₂²×C₅₉	C₂×D₁₆	1	1	yes	characteristic
52	C₄×C₅₉	—	16	2	no	—
53	C₄×C₅₉	SD₆₄	1	2	yes	—
54	C₄×C₅₉	Q₆₄	1	2	yes	—
55	C₄×C₅₉	#486	1	1	yes	characteristic
56	C₄×C₅₉	C₂×SD₃₂	1	1	yes	characteristic
57	Dic₅₉	—	16	2	no	—
58	#2252	—	59	1	no	—
59	#2247	D₅₉	1	1	yes	characteristic
60	#1416	—	59	4	no	—
61	C₄×C₃₂	—	59	1	no	—
62	C₂×C₅₉	#2240	1	1	yes	characteristic
63	C₂×C₅₉	C₂×SD₆₄	1	1	yes	characteristic
64	C₂×C₅₉	C₂×Q₆₄	1	1	yes	characteristic
65	C₂×Q₃₂	—	2	1	no	—
66	C₂×Q₃₂	C₂×D₅₉	1	2	yes	—
67	#500	—	118	1	no	—
68	#494	—	2	1	no	—
69	C₂×C₃₂	—	59	2	no	—
70	#366	—	59	2	no	—
71	#366	—	118	1	no	—
72	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
73	C₅₉	#13720	1	1	yes	characteristic, complemented, Sylow
74	C₂×Q₁₆	—	2	2	no	—
75	C₂×Q₁₆	—	4	1	no	—
76	C₄⋊Q₈	—	4	1	no	—
77	C₄⋊Q₈	—	236	1	no	—
78	Q₃₂	—	2	4	no	—
79	Q₃₂	—	4	2	no	—
80	C₂×C₁₆	C₂²⋊D₅₉	1	1	yes	characteristic
81	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
82	#107	—	118	2	no	—
83	#107	—	236	1	no	—
84	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
85	C₃₂	—	118	2	no	—
86	C₂×Q₈	—	4	2	no	—
87	C₂×Q₈	—	472	1	no	—
88	Q₁₆	—	4	2	no	—
89	Q₁₆	—	4	2	no	—
90	Q₁₆	—	8	2	no	—
91	C₂×C₈	#11745669	1	1	yes	characteristic
92	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
93	C₄⋊C₄	—	8	1	no	—
94	C₄⋊C₄	—	8	1	no	—
95	C₄⋊C₄	—	472	1	no	—
96	C₄⋊C₄	—	236	2	no	—
97	C₄²	#11745669	1	1	yes	characteristic
98	C₁₆	#11745670	1	2	yes	—
99	C₁₆	#11745687	1	2	yes	—
100	Q₈	—	8	2	no	—
101	Q₈	—	8	2	no	—
102	Q₈	—	944	2	no	—
103	C₂×C₄	—	8	2	no	—
104	C₂×C₄	—	472	2	no	—
105	C₂×C₄	#14273668	1	1	yes	characteristic
106	C₂×C₄	#14273668	1	1	yes	characteristic
107	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
108	C₈	#14273671	1	2	yes	—
109	C₈	#14273714	1	1	yes	characteristic
110	C₈	C₂×#11745670	1	1	yes	characteristic
111	C₂²	C₂×#14273668	1	1	yes	characteristic
112	C₄	—	16	2	no	—
113	C₄	—	944	2	no	—
114	C₄	#16837239	1	2	yes	—
115	C₄	#16837242	1	2	yes	—
116	C₄	#16837748	1	1	yes	characteristic
117	C₄	C₂×#14273671	1	1	yes	characteristic
118	C₂	#21426319	1	1	yes	characteristic
119	C₂	C₂×#16837239	1	1	yes	characteristic
120	C₂	C₂×#16837242	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21426371

C₂

yes

maximal, characteristic

#21422277

C₂

yes

maximal

C₄×C₅₉⋊C₃₂

C₂

yes

maximal, characteristic

C₅₉×#2247

C₂

yes

maximal, characteristic

#16837807

—

C₂×C₅₉⋊C₃₂

C₂²

yes

—

#16837228

—

#16837228

C₂²

yes

—

C₂×Q₃₂×C₅₉

—

C₂×Q₃₂×C₅₉

C₂²

yes

—

C₅₉×#494

—

C₄×C₁₆×C₅₉

C₂²

yes

—

—

—

C₅₉⋊C₃₂

—

C₂×Q₁₆×C₅₉

—

C₂×Q₁₆×C₅₉

—

C₄⋊Q₈×C₅₉

—

Q₃₂×C₅₉

—

Q₃₂×C₅₉

—

C₂×C₁₆×C₅₉

D₄

yes

characteristic

C₂×C₁₆×C₅₉

C₂³

yes

characteristic

C₄×C₈×C₅₉

D₄

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×Dic₁₁₈

—

Q₁₆×C₅₉

—

Q₁₆×C₅₉

—

Q₁₆×C₅₉

—

C₂×C₈×C₅₉

D₈

yes

characteristic

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

D₈

yes

characteristic

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₁₆×C₅₉

SD₁₆

yes

—

C₁₆×C₅₉

#41

yes

—

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

D₁₆

yes

characteristic

C₂×C₄×C₅₉

D₁₆

yes

characteristic

C₂×C₄×C₅₉

C₂×D₈

yes

characteristic

C₂×Dic₅₉

—

Dic₁₁₈

—

C₈×C₅₉

SD₃₂

yes

—

C₈×C₅₉

#124

yes

characteristic

C₈×C₅₉

C₂×SD₁₆

yes

characteristic

#13720

—

maximal, Sylow

C₂²×C₅₉

C₂×D₁₆

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

SD₆₄

yes

—

C₄×C₅₉

Q₆₄

yes

—

C₄×C₅₉

#486

yes

characteristic

C₄×C₅₉

C₂×SD₃₂

yes

characteristic

Dic₅₉

—

#2252

—

#2247

D₅₉

yes

characteristic

#1416

—

C₄×C₃₂

—

C₂×C₅₉

#2240

yes

characteristic

C₂×C₅₉

C₂×SD₆₄

yes

characteristic

C₂×C₅₉

C₂×Q₆₄

yes

characteristic

C₂×Q₃₂

—

C₂×Q₃₂

C₂×D₅₉

yes

—

#500

—

118

—

#494

—

C₂×C₃₂

—

#366

—

#366

—

118

—

C₄×C₁₆

C₂×D₅₉

yes

characteristic

C₅₉

#13720

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

236

—

Q₃₂

—

Q₃₂

—

C₂×C₁₆

C₂²⋊D₅₉

yes

characteristic

C₂×C₁₆

C₂²×D₅₉

yes

characteristic

#107

—

118

—

#107

—

236

—

C₄×C₈

C₂²⋊D₅₉

yes

characteristic

C₃₂

—

118

—

C₂×Q₈

—

C₂×Q₈

—

472

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

C₂×C₈

#11745669

yes

characteristic

C₂×C₈

C₂×C₂²⋊D₅₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

472

—

C₄⋊C₄

—

236

—

C₄²

#11745669

yes

characteristic

C₁₆

#11745670

yes

—

C₁₆

#11745687

yes

—

100

Q₈

—

101

Q₈

—

102

Q₈

—

944

—

103

C₂×C₄

—

104

C₂×C₄

—

472

—

105

C₂×C₄

#14273668

yes

characteristic

106

C₂×C₄

#14273668

yes

characteristic

107

C₂×C₄

C₂×#11745669

yes

characteristic

108

C₈

#14273671

yes

—

109

C₈

#14273714

yes

characteristic

110

C₈

C₂×#11745670

yes

characteristic

111

C₂²

C₂×#14273668

yes

characteristic

112

C₄

—

113

C₄

—

944

—

114

C₄

#16837239

yes

—

115

C₄

#16837242

yes

—

116

C₄

#16837748

yes

characteristic

117

C₄

C₂×#14273671

yes

characteristic

118

C₂

#21426319

yes

characteristic

119

C₂

C₂×#16837239

yes

characteristic

120

C₂

C₂×#16837242

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#14273668
commutator, cosocle	C₂×C₁₆×C₅₉	C₂³
Frattini	C₂×C₁₆	C₂²×D₅₉
Fitting	C₅₉×#2247	C₂
socle, prime radical	C₂²×C₅₉	C₂×D₁₆
hypercenter	#2247	D₅₉
hypocenter	C₅₉	#13720
LCS	G ▷ C₂×C₁₆×C₅₉ ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2247

subgroup

G/H

center

C₂²

C₂×#14273668

commutator, cosocle

C₂×C₁₆×C₅₉

C₂³

Frattini

C₂×C₁₆

C₂²×D₅₉

Fitting

C₅₉×#2247

C₂

socle, prime radical

C₂²×C₅₉

C₂×D₁₆

D₅₉

C₅₉

G ▷ C₂×C₁₆×C₅₉ ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2247

N	H ≅ G/N	type
73. C₅₉	#13720	semidirect

H ≅ G/N

type

73. C₅₉

#13720

semidirect

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

subgroup	index
#21426371	2
#21422277	2
C₄×C₅₉⋊C₃₂	2
C₅₉×#2247	2
#13720	59

subgroup

index

#21426371

#21422277

C₄×C₅₉⋊C₃₂

C₅₉×#2247

#13720

Subgroups — #28509661

Conjugacy classes of subgroups

Complements

Maximal subgroups