Group #28509176 — subgroups

Subgroups — #28509176

Order 15104.

206 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 123 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	C₂²×C₅₉⋊C₃₂	C₂	1	1	yes	maximal, characteristic
2	#21425692	C₂	1	1	yes	maximal, characteristic
3	Dic₅₉⋊C₃₂	C₂	1	4	yes	maximal, complemented
4	C₂×C₅₉×M₆₄	C₂	1	1	yes	maximal, characteristic
5	C₂×#14273667	C₄	1	1	yes	characteristic
6	#16837357	C₄	1	1	yes	characteristic
7	C₂×C₅₉⋊C₃₂	—	2	2	no	—
8	C₂×C₅₉⋊C₃₂	C₂²	1	2	yes	—
9	#16837177	C₄	1	2	yes	—
10	C₁₆×Dic₅₉	C₂²	1	2	yes	—
11	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
12	C₅₉×M₆₄	—	2	2	no	—
13	C₂×C₃₂×C₅₉	C₂²	1	2	yes	—
14	C₂×#11745664	C₈	1	1	yes	characteristic
15	#14273686	C₈	1	1	yes	characteristic
16	#14273667	—	2	2	no	—
17	C₂×C₅₉⋊C₁₆	C₂×C₄	1	2	yes	—
18	#14273637	C₈	1	2	yes	—
19	C₈×Dic₅₉	C₂×C₄	1	2	yes	—
20	C₅₉⋊C₃₂	—	2	2	no	—
21	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆×C₅₉	D₄	1	2	yes	—
24	C₂×C₁₆×C₅₉	Q₈	1	2	yes	—
25	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
26	C₃₂×C₅₉	—	2	2	no	—
27	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
28	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
29	C₂×C₈×C₅₉	C₂×C₈	1	1	yes	characteristic
30	C₂×C₈×C₅₉	M₁₆	1	2	yes	—
31	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
32	C₂²⋊Dic₅₉	C₁₆	1	2	yes	—
33	C₄⋊Dic₅₉	C₁₆	1	2	yes	—
34	C₄×Dic₅₉	C₂×C₈	1	2	yes	—
35	#11745664	—	2	2	no	—
36	C₂×C₅₉⋊C₈	C₂×C₈	1	2	yes	—
37	C₁₆×C₅₉	C₄⋊C₄	1	2	yes	—
38	C₁₆×C₅₉	C₂×D₄	1	1	yes	characteristic
39	C₁₆×C₅₉	C₂×Q₈	1	1	yes	characteristic
40	C₅₉⋊C₁₆	—	2	2	no	—
41	C₂³×C₅₉	C₂×C₁₆	1	1	yes	characteristic
42	C₂×C₄×C₅₉	C₄⋊C₈	1	2	yes	—
43	C₂×C₄×C₅₉	C₂×C₁₆	1	1	yes	characteristic
44	C₂×C₄×C₅₉	M₃₂	1	2	yes	—
45	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
46	C₂×Dic₅₉	C₂×C₁₆	1	4	yes	—
47	C₈×C₅₉	C₄⋊C₈	1	2	yes	—
48	C₈×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₈×C₅₉	C₂×M₁₆	1	1	yes	characteristic
50	C₅₉⋊C₈	—	2	2	no	—
51	#13628	—	59	1	no	maximal, Sylow
52	C₂²×C₅₉	—	2	2	no	—
53	C₂²×C₅₉	C₄⋊C₁₆	1	2	yes	—
54	C₂²×C₅₉	C₂²×C₁₆	1	1	yes	characteristic
55	C₄×C₅₉	C₄⋊C₁₆	1	2	yes	—
56	C₄×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
57	C₄×C₅₉	C₂×M₃₂	1	1	yes	characteristic
58	Dic₅₉	—	2	4	no	—
59	C₂×M₆₄	D₅₉	1	1	yes	characteristic
60	C₂²×C₃₂	—	59	1	no	—
61	#2108	—	59	1	no	—
62	C₄⋊C₃₂	—	59	4	no	—
63	C₂×C₅₉	—	2	2	no	—
64	C₂×C₅₉	C₂×C₄⋊C₁₆	1	1	yes	characteristic
65	C₂×C₅₉	#2258	1	2	yes	—
66	C₂×M₃₂	—	59	1	no	—
67	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
68	#405	—	59	1	no	—
69	M₆₄	—	2	2	no	—
70	C₂×C₃₂	—	59	2	no	—
71	C₂×C₃₂	—	118	2	no	—
72	C₂×C₃₂	C₂×D₅₉	1	2	yes	—
73	#346	—	59	2	no	—
74	C₄×C₁₆	—	59	2	no	—
75	C₅₉	#13628	1	1	yes	characteristic, complemented, Sylow
76	C₂×M₁₆	—	59	1	no	—
77	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
78	#117	—	59	1	no	—
79	M₃₂	—	118	2	no	—
80	C₂×C₁₆	—	59	2	no	—
81	C₂×C₁₆	Dic₁₁₈	1	2	yes	—
82	C₂×C₁₆	C₄×D₅₉	1	1	yes	characteristic
83	C₂×C₁₆	C₂²⋊D₅₉	1	2	yes	—
84	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
85	#97	—	59	2	no	—
86	C₄×C₈	—	59	2	no	—
87	C₃₂	—	2	2	no	—
88	C₃₂	—	118	2	no	—
89	C₂²×C₄	C₈×D₅₉	1	1	yes	characteristic
90	M₁₆	—	118	2	no	—
91	C₂×C₈	—	59	2	no	—
92	C₂×C₈	C₈×D₅₉	1	1	yes	characteristic
93	C₂×C₈	#11745659	1	2	yes	—
94	C₂×C₈	Dic₅₉⋊C₄	1	2	yes	—
95	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
96	C₄⋊C₄	—	59	2	no	—
97	C₂²⋊C₄	—	59	2	no	—
98	C₄²	—	59	2	no	—
99	C₁₆	—	118	2	no	—
100	C₁₆	Dic₅₉⋊C₄	1	2	yes	—
101	C₁₆	C₂×Dic₁₁₈	1	1	yes	characteristic
102	C₁₆	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
103	C₂³	C₁₆×D₅₉	1	1	yes	characteristic
104	C₂×C₄	—	59	4	no	—
105	C₂×C₄	Dic₅₉⋊C₈	1	2	yes	—
106	C₂×C₄	C₁₆×D₅₉	1	1	yes	characteristic
107	C₂×C₄	#14273666	1	2	yes	—
108	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
109	C₈	—	118	2	no	—
110	C₈	Dic₅₉⋊C₈	1	2	yes	—
111	C₈	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
112	C₈	C₂×#11745659	1	1	yes	characteristic
113	C₂²	—	2	2	no	—
114	C₂²	Dic₅₉⋊C₁₆	1	2	yes	—
115	C₂²	C₂×C₁₆×D₅₉	1	1	yes	characteristic
116	C₄	—	118	4	no	—
117	C₄	Dic₅₉⋊C₁₆	1	2	yes	—
118	C₄	C₂×Dic₅₉⋊C₈	1	1	yes	characteristic
119	C₄	C₂×#14273666	1	1	yes	characteristic
120	C₂	—	2	2	no	—
121	C₂	C₂×Dic₅₉⋊C₁₆	1	1	yes	characteristic
122	C₂	#21426391	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₅₉⋊C₃₂

C₂

yes

maximal, characteristic

#21425692

C₂

yes

maximal, characteristic

Dic₅₉⋊C₃₂

C₂

yes

maximal, complemented

C₂×C₅₉×M₆₄

C₂

yes

maximal, characteristic

C₂×#14273667

C₄

yes

characteristic

#16837357

C₄

yes

characteristic

C₂×C₅₉⋊C₃₂

—

C₂×C₅₉⋊C₃₂

C₂²

yes

—

#16837177

C₄

yes

—

C₁₆×Dic₅₉

C₂²

yes

—

C₂²×C₁₆×C₅₉

C₂²

yes

characteristic

C₅₉×M₆₄

—

C₂×C₃₂×C₅₉

C₂²

yes

—

C₂×#11745664

C₈

yes

characteristic

#14273686

C₈

yes

characteristic

#14273667

—

C₂×C₅₉⋊C₁₆

C₂×C₄

yes

—

#14273637

C₈

yes

—

C₈×Dic₅₉

C₂×C₄

yes

—

C₅₉⋊C₃₂

—

C₂²×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₉

D₄

yes

—

C₂×C₁₆×C₅₉

Q₈

yes

—

C₂×C₁₆×C₅₉

C₂³

yes

characteristic

C₃₂×C₅₉

—

C₂²×C₄×C₅₉

C₂×C₈

yes

characteristic

C₂×C₈×C₅₉

C₄⋊C₄

yes

—

C₂×C₈×C₅₉

C₂×C₈

yes

characteristic

C₂×C₈×C₅₉

M₁₆

yes

—

C₂×C₈×C₅₉

C₂²×C₄

yes

characteristic

C₂²⋊Dic₅₉

C₁₆

yes

—

C₄⋊Dic₅₉

C₁₆

yes

—

C₄×Dic₅₉

C₂×C₈

yes

—

#11745664

—

C₂×C₅₉⋊C₈

C₂×C₈

yes

—

C₁₆×C₅₉

C₄⋊C₄

yes

—

C₁₆×C₅₉

C₂×D₄

yes

characteristic

C₁₆×C₅₉

C₂×Q₈

yes

characteristic

C₅₉⋊C₁₆

—

C₂³×C₅₉

C₂×C₁₆

yes

characteristic

C₂×C₄×C₅₉

C₄⋊C₈

yes

—

C₂×C₄×C₅₉

C₂×C₁₆

yes

characteristic

C₂×C₄×C₅₉

M₃₂

yes

—

C₂×C₄×C₅₉

C₂²×C₈

yes

characteristic

C₂×Dic₅₉

C₂×C₁₆

yes

—

C₈×C₅₉

C₄⋊C₈

yes

—

C₈×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₈×C₅₉

C₂×M₁₆

yes

characteristic

C₅₉⋊C₈

—

#13628

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

C₄⋊C₁₆

yes

—

C₂²×C₅₉

C₂²×C₁₆

yes

characteristic

C₄×C₅₉

C₄⋊C₁₆

yes

—

C₄×C₅₉

C₂×C₄⋊C₈

yes

characteristic

C₄×C₅₉

C₂×M₃₂

yes

characteristic

Dic₅₉

—

C₂×M₆₄

D₅₉

yes

characteristic

C₂²×C₃₂

—

#2108

—

C₄⋊C₃₂

—

C₂×C₅₉

—

C₂×C₅₉

C₂×C₄⋊C₁₆

yes

characteristic

C₂×C₅₉

#2258

yes

—

C₂×M₃₂

—

C₂²×C₁₆

C₂×D₅₉

yes

characteristic

#405

—

M₆₄

—

C₂×C₃₂

—

C₂×C₃₂

—

118

—

C₂×C₃₂

C₂×D₅₉

yes

—

#346

—

C₄×C₁₆

—

C₅₉

#13628

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

C₄×D₅₉

yes

characteristic

#117

—

M₃₂

—

118

—

C₂×C₁₆

—

C₂×C₁₆

Dic₁₁₈

yes

—

C₂×C₁₆

C₄×D₅₉

yes

characteristic

C₂×C₁₆

C₂²⋊D₅₉

yes

—

C₂×C₁₆

C₂²×D₅₉

yes

characteristic

#97

—

C₄×C₈

—

C₃₂

—

C₃₂

—

118

—

C₂²×C₄

C₈×D₅₉

yes

characteristic

M₁₆

—

118

—

C₂×C₈

—

C₂×C₈

C₈×D₅₉

yes

characteristic

C₂×C₈

#11745659

yes

—

C₂×C₈

Dic₅₉⋊C₄

yes

—

C₂×C₈

C₂×C₄×D₅₉

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

118

—

100

C₁₆

Dic₅₉⋊C₄

yes

—

101

C₁₆

C₂×Dic₁₁₈

yes

characteristic

102

C₁₆

C₂×C₂²⋊D₅₉

yes

characteristic

103

C₂³

C₁₆×D₅₉

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

Dic₅₉⋊C₈

yes

—

106

C₂×C₄

C₁₆×D₅₉

yes

characteristic

107

C₂×C₄

#14273666

yes

—

108

C₂×C₄

C₂×C₈×D₅₉

yes

characteristic

109

C₈

—

118

—

110

C₈

Dic₅₉⋊C₈

yes

—

111

C₈

C₂×Dic₅₉⋊C₄

yes

characteristic

112

C₈

C₂×#11745659

yes

characteristic

113

C₂²

—

114

C₂²

Dic₅₉⋊C₁₆

yes

—

115

C₂²

C₂×C₁₆×D₅₉

yes

characteristic

116

C₄

—

118

—

117

C₄

Dic₅₉⋊C₁₆

yes

—

118

C₄

C₂×Dic₅₉⋊C₈

yes

characteristic

119

C₄

C₂×#14273666

yes

characteristic

120

C₂

—

121

C₂

C₂×Dic₅₉⋊C₁₆

yes

characteristic

122

C₂

#21426391

yes

—

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₅₉
commutator, socle	C₂²×C₅₉	C₂²×C₁₆
Fitting	C₂×C₅₉×M₆₄	C₂
prime radical	C₂³×C₅₉	C₂×C₁₆
cosocle	C₂×C₁₆×C₅₉	C₂³
hypercenter	C₂×M₆₄	D₅₉
hypocenter	C₅₉	#13628
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×M₆₄

subgroup

G/H

center, Frattini

C₂×C₁₆

C₂²×D₅₉

commutator, socle

C₂²×C₅₉

C₂²×C₁₆

Fitting

C₂×C₅₉×M₆₄

C₂

prime radical

C₂³×C₅₉

C₂×C₁₆

cosocle

C₂×C₁₆×C₅₉

C₂³

hypercenter

C₂×M₆₄

D₅₉

C₅₉

G ▷ C₂²×C₅₉ ▷ C₅₉

C₁ ◁ C₂×C₁₆ ◁ C₂×M₆₄

N	H ≅ G/N	type
3. Dic₅₉⋊C₃₂	C₂	semidirect
75. C₅₉	#13628	semidirect

H ≅ G/N

type

3. Dic₅₉⋊C₃₂

C₂

semidirect

75. C₅₉

#13628

semidirect

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

subgroup	index
C₂²×C₅₉⋊C₃₂	2
#21425692	2
Dic₅₉⋊C₃₂	2
C₂×C₅₉×M₆₄	2
#13628	59

subgroup

index

C₂²×C₅₉⋊C₃₂

#21425692

Dic₅₉⋊C₃₂

C₂×C₅₉×M₆₄

#13628

Subgroups — #28509176

Conjugacy classes of subgroups

Complements

Maximal subgroups