Group #28475790 — subgroups

Subgroups — #28475790

Order 15104.

190 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 87 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	#21422859	C₂	1	2	yes	maximal
2	C₂×C₅₉×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#16837459	—	2	2	no	—
4	#16837155	—	2	2	no	—
5	#16837153	C₄	1	4	yes	—
6	C₂²×C₁₆×C₅₉	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
8	C₅₉×C₄⋊C₁₆	—	2	2	no	—
9	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
10	C₂²⋊(C₅₉⋊C₈)	—	4	2	no	—
11	#14273638	—	4	2	no	—
12	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₅₉	Q₈	1	1	yes	characteristic
14	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
15	C₂×C₁₆×C₅₉	—	2	2	no	—
16	C₂×C₁₆×C₅₉	—	4	2	no	—
17	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
18	C₄×C₈×C₅₉	D₄	1	2	yes	—
19	C₂²×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
20	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
21	C₂²×C₄×C₅₉	Q₁₆	1	1	yes	characteristic
22	C₂×C₈×C₅₉	—	2	1	no	—
23	C₂×C₈×C₅₉	—	2	1	no	—
24	C₂×C₈×C₅₉	—	4	1	no	—
25	C₂×C₈×C₅₉	C₄²	1	1	yes	characteristic
26	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
28	C₄²×C₅₉	C₂²⋊C₄	1	2	yes	—
29	C₄²×C₅₉	SD₁₆	1	2	yes	—
30	C₂×C₅₉⋊C₈	—	4	4	no	—
31	C₁₆×C₅₉	—	4	2	no	—
32	C₂³×C₅₉	#107	1	1	yes	characteristic
33	C₂×C₄×C₅₉	—	2	2	no	—
34	C₂×C₄×C₅₉	—	2	1	no	—
35	C₂×C₄×C₅₉	—	2	1	no	—
36	C₂×C₄×C₅₉	—	4	1	no	—
37	C₂×C₄×C₅₉	#95	1	1	yes	characteristic
38	C₂×C₄×C₅₉	#102	1	2	yes	—
39	C₂×C₄×C₅₉	#103	1	2	yes	—
40	C₂×C₄×C₅₉	#106	1	1	yes	characteristic
41	C₈×C₅₉	—	2	2	no	—
42	C₈×C₅₉	—	2	1	no	—
43	C₈×C₅₉	#100	1	1	yes	characteristic
44	C₈×C₅₉	#101	1	1	yes	characteristic
45	C₅₉⋊C₈	—	8	4	no	—
46	#7242	—	59	1	no	maximal, Sylow
47	C₂²×C₅₉	—	2	1	no	—
48	C₂²×C₅₉	—	2	1	no	—
49	C₂²×C₅₉	#340	1	1	yes	characteristic
50	C₂²×C₅₉	#368	1	2	yes	—
51	C₄×C₅₉	—	2	1	no	—
52	C₄×C₅₉	—	2	2	no	—
53	C₄×C₅₉	—	2	2	no	—
54	C₄×C₅₉	#338	1	1	yes	characteristic
55	C₄×C₅₉	#343	1	1	yes	characteristic
56	C₂×C₄⋊C₁₆	D₅₉	1	1	yes	characteristic
57	#1568	—	59	2	no	—
58	C₂×C₅₉	—	2	2	no	—
59	C₂×C₅₉	#1292	1	1	yes	characteristic
60	C₂×C₅₉	#1381	1	1	yes	characteristic
61	C₂×C₅₉	#1384	1	1	yes	characteristic
62	C₂²×C₁₆	Dic₅₉	1	2	yes	—
63	#432	—	118	2	no	—
64	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
65	C₄⋊C₁₆	—	2	2	no	—
66	#335	—	118	2	no	—
67	#334	—	59	4	no	—
68	C₅₉	#7242	1	1	yes	characteristic, complemented, Sylow
69	C₂²×C₈	Dic₁₁₈	1	1	yes	characteristic
70	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
71	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
72	C₂×C₁₆	—	2	2	no	—
73	C₂×C₁₆	—	4	2	no	—
74	C₄⋊C₈	—	118	4	no	—
75	C₂²⋊C₈	—	236	2	no	—
76	#97	—	236	2	no	—
77	C₄×C₈	C₄×D₅₉	1	2	yes	—
78	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
79	C₂²×C₄	D₄₇₂	1	1	yes	characteristic
80	C₂²×C₄	Dic₂₃₆	1	1	yes	characteristic
81	C₂²×C₄	C₄⋊Dic₅₉	1	1	yes	characteristic
82	C₂×C₈	—	2	1	no	—
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	4	1	no	—
85	C₂×C₈	—	236	4	no	—
86	C₂×C₈	C₄×Dic₅₉	1	1	yes	characteristic
87	C₂×C₈	Dic₅₉⋊C₄	1	2	yes	—
88	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
89	C₄²	#11745660	1	2	yes	—
90	C₄²	(C₂×D₅₉)⋊C₄	1	2	yes	—
91	C₁₆	—	4	2	no	—
92	C₂³	#14273661	1	1	yes	characteristic
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	2	1	no	—
96	C₂×C₄	—	4	1	no	—
97	C₂×C₄	C₅₉⋊#95	1	1	yes	characteristic
98	C₂×C₄	#14273649	1	2	yes	—
99	C₂×C₄	#14273652	1	2	yes	—
100	C₂×C₄	#14273659	1	1	yes	characteristic
101	C₈	—	2	2	no	—
102	C₈	—	2	1	no	—
103	C₈	—	472	4	no	—
104	C₈	#14273644	1	1	yes	characteristic
105	C₈	#14273646	1	1	yes	characteristic
106	C₂²	—	2	1	no	—
107	C₂²	—	2	1	no	—
108	C₂²	#16837165	1	1	yes	characteristic
109	C₂²	#16837232	1	2	yes	—
110	C₄	—	2	2	no	—
111	C₄	—	2	2	no	—
112	C₄	—	2	1	no	—
113	C₄	C₅₉⋊#338	1	1	yes	characteristic
114	C₄	#16837171	1	1	yes	characteristic
115	C₂	—	2	2	no	—
116	C₂	#21421971	1	1	yes	characteristic
117	C₂	#21422190	1	1	yes	characteristic
118	C₂	#21422198	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21422859

C₂

yes

maximal

C₂×C₅₉×C₄⋊C₁₆

C₂

yes

maximal, characteristic

#16837459

—

#16837155

—

#16837153

C₄

yes

—

C₂²×C₁₆×C₅₉

C₄

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₅₉×C₄⋊C₁₆

—

C₄⋊(C₅₉⋊C₈)

—

C₂²⋊(C₅₉⋊C₈)

—

#14273638

—

C₂²×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₉

Q₈

yes

characteristic

C₂×C₄²×C₅₉

D₄

yes

characteristic

C₂×C₁₆×C₅₉

—

C₂×C₁₆×C₅₉

—

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₄×C₈×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₅₉

D₈

yes

characteristic

C₂²×C₄×C₅₉

Q₁₆

yes

characteristic

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₄²

yes

characteristic

C₂×C₈×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₅₉

C₄⋊C₄

yes

—

C₄²×C₅₉

C₂²⋊C₄

yes

—

C₄²×C₅₉

SD₁₆

yes

—

C₂×C₅₉⋊C₈

—

C₁₆×C₅₉

—

C₂³×C₅₉

#107

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#95

yes

characteristic

C₂×C₄×C₅₉

#102

yes

—

C₂×C₄×C₅₉

#103

yes

—

C₂×C₄×C₅₉

#106

yes

characteristic

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

yes

C₈×C₅₉

yes

C₅₉⋊C₈

—

#7242

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#340

yes

characteristic

C₂²×C₅₉

#368

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₂×C₄⋊C₁₆

D₅₉

yes

characteristic

#1568

—

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂²×C₁₆

Dic₅₉

yes

—

#432

—

118

—

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

C₄⋊C₁₆

—

#335

—

118

—

#334

—

C₅₉

#7242

yes

characteristic, complemented, Sylow

C₂²×C₈

Dic₁₁₈

yes

characteristic

C₂²×C₈

C₂×Dic₅₉

yes

characteristic

C₂×C₄²

D₂₃₆

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

118

—

C₂²⋊C₈

—

236

—

#97

—

236

—

C₄×C₈

C₄×D₅₉

yes

—

C₄×C₈

C₂²⋊D₅₉

yes

—

C₂²×C₄

D₄₇₂

yes

characteristic

C₂²×C₄

Dic₂₃₆

yes

characteristic

C₂²×C₄

C₄⋊Dic₅₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

236

—

C₂×C₈

C₄×Dic₅₉

yes

characteristic

C₂×C₈

Dic₅₉⋊C₄

yes

—

C₂×C₈

C₂²⋊Dic₅₉

yes

characteristic

C₄²

#11745660

yes

—

C₄²

(C₂×D₅₉)⋊C₄

yes

—

C₁₆

—

C₂³

#14273661

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₅₉⋊#95

yes

characteristic

C₂×C₄

#14273649

yes

—

C₂×C₄

#14273652

yes

—

100

C₂×C₄

#14273659

yes

characteristic

101

C₈

—

102

C₈

—

103

C₈

—

472

—

104

C₈

#14273644

yes

characteristic

105

C₈

#14273646

yes

characteristic

106

C₂²

—

107

C₂²

—

108

C₂²

#16837165

yes

characteristic

109

C₂²

#16837232

yes

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

C₅₉⋊#338

yes

characteristic

114

C₄

#16837171

yes

characteristic

115

C₂

—

116

C₂

#21421971

yes

characteristic

117

C₂

#21422190

yes

characteristic

118

C₂

#21422198

yes

characteristic

	subgroup	G/H
center	C₂²	#16837165
commutator	C₂×C₈×C₅₉	C₄²
Frattini	C₂×C₄×C₈	C₂×D₅₉
Fitting	C₂×C₅₉×C₄⋊C₁₆	C₂
socle	C₂²×C₅₉	#340
prime radical	C₂³×C₅₉	#107
cosocle	C₂×C₄×C₈×C₅₉	C₂²
hypercenter	C₂×C₄⋊C₁₆	D₅₉
hypocenter	C₅₉	#7242
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

subgroup

G/H

center

C₂²

#16837165

commutator

C₂×C₈×C₅₉

C₄²

Frattini

C₂×C₄×C₈

C₂×D₅₉

Fitting

C₂×C₅₉×C₄⋊C₁₆

C₂

C₂²×C₅₉

C₂³×C₅₉

C₂×C₄×C₈×C₅₉

C₂²

hypercenter

C₂×C₄⋊C₁₆

D₅₉

hypocenter

C₅₉

#7242

LCS

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

N	H ≅ G/N	type
68. C₅₉	#7242	semidirect

H ≅ G/N

type

68. C₅₉

#7242

semidirect

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

subgroup	index
#21422859	2
C₂×C₅₉×C₄⋊C₁₆	2
#7242	59

subgroup

index

#21422859

C₂×C₅₉×C₄⋊C₁₆

#7242

Subgroups — #28475790

Conjugacy classes of subgroups

Complements

Maximal subgroups