Group #28482386 — subgroups

Subgroups — #28482386

Order 15104.

270 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 107 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	#21424268	C₂	1	1	yes	maximal, characteristic
2	#21422388	C₂	1	1	yes	maximal, characteristic
3	#21421954	C₂	1	4	yes	maximal
4	C₅₉×#1835	C₂	1	1	yes	maximal, characteristic
5	#16837455	C₂²	1	2	yes	—
6	C₂×#14273663	—	2	2	no	—
7	#16837426	C₂²	1	2	yes	—
8	#16837124	C₄	1	4	yes	complemented
9	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
10	C₅₉×#382	C₂²	1	2	yes	—
11	C₅₉×#340	—	2	2	no	—
12	C₂×#11745664	—	2	2	no	—
13	#14273663	—	4	4	no	—
14	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
15	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
16	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
17	C₄×C₅₉⋊C₈	C₂×C₄	1	4	yes	—
18	C₂²×C₈×C₅₉	D₄	1	2	yes	—
19	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
20	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
21	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₅₉	Q₈	1	2	yes	—
23	#95×C₅₉	—	2	4	no	—
24	C₂²×C₄×C₅₉	—	4	2	no	—
25	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
26	C₂²×C₄×C₅₉	#41	1	2	yes	—
27	C₂×C₈×C₅₉	—	2	2	no	—
28	C₂×C₈×C₅₉	—	2	2	no	—
29	C₂×C₈×C₅₉	C₄⋊C₄	1	4	yes	—
30	C₄⋊C₄×C₅₉	—	4	2	no	—
31	C₄⋊C₄×C₅₉	—	4	2	no	—
32	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
33	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
34	C₄²×C₅₉	#41	1	2	yes	—
35	#11745664	—	4	4	no	—
36	C₂×C₅₉⋊C₈	—	2	4	no	—
37	C₂×C₅₉⋊C₈	—	4	2	no	—
38	C₂³×C₅₉	#124	1	1	yes	characteristic
39	C₂×C₄×C₅₉	—	4	2	no	—
40	C₂×C₄×C₅₉	—	8	4	no	—
41	C₂×C₄×C₅₉	—	8	2	no	—
42	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₅₉	#117	1	2	yes	—
44	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
45	C₂×C₄×C₅₉	#125	1	2	yes	—
46	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
47	C₂×C₄×C₅₉	#137	1	2	yes	—
48	C₈×C₅₉	—	2	4	no	—
49	C₅₉⋊C₈	—	4	4	no	—
50	C₅₉⋊C₈	—	8	2	no	—
51	#8794	—	59	1	no	maximal, Sylow
52	C₂²×C₅₉	—	2	2	no	—
53	C₂²×C₅₉	#382	1	1	yes	characteristic
54	C₂²×C₅₉	#489	1	2	yes	—
55	C₄×C₅₉	—	2	2	no	—
56	C₄×C₅₉	—	2	2	no	—
57	C₄×C₅₉	—	16	2	no	—
58	C₄×C₅₉	#428	1	2	yes	—
59	C₄×C₅₉	#490	1	1	yes	characteristic
60	C₄×C₅₉	#491	1	1	yes	characteristic
61	#1848	—	59	1	no	—
62	#1835	D₅₉	1	1	yes	characteristic
63	#1452	—	59	1	no	—
64	#1286	—	59	4	no	—
65	C₂×C₅₉	—	2	2	no	—
66	C₂×C₅₉	#1833	1	2	yes	—
67	C₂×C₅₉	#1836	1	1	yes	characteristic
68	#431	—	59	2	no	—
69	C₂×#108	—	118	2	no	—
70	#423	—	59	2	no	—
71	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
72	#382	C₂×D₅₉	1	2	yes	—
73	#340	—	2	2	no	—
74	#322	—	59	4	no	—
75	C₅₉	#8794	1	1	yes	characteristic, complemented, Sylow
76	C₂×M₁₆	—	118	2	no	—
77	C₂²×C₈	C₂²⋊D₅₉	1	2	yes	—
78	C₂×C₄⋊C₄	—	2	2	no	—
79	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
80	#108	—	236	4	no	—
81	C₄⋊C₈	—	118	2	no	—
82	C₄⋊C₈	—	118	4	no	—
83	C₂²⋊C₈	—	118	2	no	—
84	C₄×C₈	—	59	4	no	—
85	C₄×C₈	Dic₁₁₈	1	2	yes	—
86	C₄×C₈	C₄×D₅₉	1	2	yes	—
87	#95	—	2	4	no	—
88	C₂²×C₄	—	4	2	no	—
89	C₂²×C₄	#11745687	1	2	yes	—
90	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
91	M₁₆	—	236	4	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	118	4	no	—
95	C₂×C₈	—	236	2	no	—
96	C₂×C₈	Dic₅₉⋊C₄	1	4	yes	—
97	C₄⋊C₄	—	4	2	no	—
98	C₄⋊C₄	—	4	2	no	—
99	C₄²	C₂×Dic₁₁₈	1	1	yes	characteristic
100	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
101	C₄²	#11745689	1	2	yes	—
102	C₂³	#14273714	1	1	yes	characteristic
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	—	8	4	no	—
105	C₂×C₄	—	8	2	no	—
106	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
107	C₂×C₄	#14273685	1	2	yes	—
108	C₂×C₄	#14273716	1	1	yes	characteristic
109	C₂×C₄	#14273718	1	2	yes	—
110	C₂×C₄	#14273752	1	2	yes	—
111	C₂×C₄	#14273757	1	2	yes	—
112	C₈	—	2	4	no	—
113	C₈	—	472	2	no	—
114	C₈	—	236	4	no	—
115	C₂²	—	2	2	no	—
116	C₂²	#16837267	1	1	yes	characteristic
117	C₂²	#16837761	1	2	yes	—
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	16	2	no	—
121	C₄	#16837445	1	2	yes	—
122	C₄	#16837768	1	1	yes	characteristic
123	C₄	#16837772	1	1	yes	characteristic
124	C₂	—	2	2	no	—
125	C₂	#21424219	1	2	yes	—
126	C₂	#21424229	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21424268

C₂

yes

maximal, characteristic

#21422388

C₂

yes

maximal, characteristic

#21421954

C₂

yes

maximal

C₅₉×#1835

C₂

yes

maximal, characteristic

#16837455

C₂²

yes

—

C₂×#14273663

—

#16837426

C₂²

yes

—

#16837124

C₄

yes

complemented

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₅₉×#382

C₂²

yes

—

C₅₉×#340

—

C₂×#11745664

—

#14273663

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

C₂²⋊(C₅₉⋊C₈)

—

C₄×C₅₉⋊C₈

C₂×C₄

yes

—

C₂²×C₈×C₅₉

D₄

yes

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₄×C₈×C₅₉

Q₈

yes

—

#95×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₉

#41

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₄⋊C₄

yes

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

C₂²×C₄

yes

characteristic

C₄²×C₅₉

C₂×Q₈

yes

characteristic

C₄²×C₅₉

#41

yes

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

#124

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

#117

yes

—

C₂×C₄×C₅₉

#124

yes

characteristic

C₂×C₄×C₅₉

#125

yes

—

C₂×C₄×C₅₉

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₅₉

#137

yes

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

#8794

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

#382

yes

characteristic

C₂²×C₅₉

#489

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#428

yes

—

C₄×C₅₉

yes

C₄×C₅₉

yes

—

D₅₉

yes

characteristic

#1452

—

#1286

—

C₂×C₅₉

—

C₂×C₅₉

#1833

yes

—

C₂×C₅₉

yes

—

—

118

—

#423

—

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

#382

C₂×D₅₉

yes

—

#340

—

#322

—

C₅₉

#8794

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

118

—

C₂²×C₈

C₂²⋊D₅₉

yes

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₅₉

yes

characteristic

#108

—

236

—

C₄⋊C₈

—

118

—

C₄⋊C₈

—

118

—

C₂²⋊C₈

—

118

—

C₄×C₈

—

C₄×C₈

Dic₁₁₈

yes

—

C₄×C₈

C₄×D₅₉

yes

—

#95

—

C₂²×C₄

—

C₂²×C₄

#11745687

yes

—

C₂²×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

M₁₆

—

236

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

118

—

C₂×C₈

—

236

—

C₂×C₈

Dic₅₉⋊C₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×Dic₁₁₈

yes

characteristic

100

C₄²

C₂×C₄×D₅₉

yes

characteristic

101

C₄²

#11745689

yes

—

102

C₂³

#14273714

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

C₂×Dic₅₉⋊C₄

yes

characteristic

107

C₂×C₄

#14273685

yes

—

108

C₂×C₄

#14273716

yes

characteristic

109

C₂×C₄

#14273718

yes

—

110

C₂×C₄

#14273752

yes

—

111

C₂×C₄

#14273757

yes

—

112

C₈

—

113

C₈

—

472

—

114

C₈

—

236

—

115

C₂²

—

116

C₂²

#16837267

yes

characteristic

117

C₂²

#16837761

yes

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

#16837445

yes

—

122

C₄

#16837768

yes

characteristic

123

C₄

#16837772

yes

characteristic

124

C₂

—

125

C₂

#21424219

yes

—

126

C₂

#21424229

yes

characteristic

	subgroup	G/H
center	C₂²	#16837267
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1835	C₂
socle	C₂²×C₅₉	#382
prime radical	C₂³×C₅₉	#124
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1835	D₅₉
hypocenter	C₅₉	#8794
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1835

subgroup

G/H

center

C₂²

#16837267

commutator

C₄²×C₅₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₅₉×#1835

C₂

C₂²×C₅₉

C₂³×C₅₉

C₂×C₄²×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1835

N	H ≅ G/N	type
8. #16837124	C₄	semidirect
75. C₅₉	#8794	semidirect

H ≅ G/N

type

8. #16837124

C₄

semidirect

75. C₅₉

#8794

semidirect

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

subgroup	index
#21424268	2
#21422388	2
#21421954	2
C₅₉×#1835	2
#8794	59

subgroup

index

C₅₉×#1835

Subgroups — #28482386

Conjugacy classes of subgroups

Complements

Maximal subgroups