Group #28482498 — subgroups

Subgroups — #28482498

Order 15104.

338 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 163 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	#21422851	C₂	1	4	yes	maximal
3	C₂×#16837123	C₂	1	1	yes	maximal, characteristic
4	C₅₉×#1848	C₂	1	1	yes	maximal, characteristic
5	C₂×#14273663	—	2	2	no	—
6	#16837426	C₂²	1	2	yes	—
7	C₂×C₄×C₅₉⋊C₈	C₂²	1	2	yes	—
8	#16837152	—	2	8	no	—
9	#16837123	C₄	1	4	yes	—
10	C₂×#108×C₅₉	—	2	2	no	—
11	C₅₉×#423	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
13	C₂×#11745664	—	2	2	no	—
14	C₂²×C₅₉⋊C₈	Q₈	1	4	yes	—
15	#14273663	—	4	4	no	—
16	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
17	C₄×C₅₉⋊C₈	C₂×C₄	1	4	yes	—
18	C₄×C₅₉⋊C₈	D₄	1	4	yes	—
19	C₂×M₁₆×C₅₉	—	2	2	no	—
20	C₂²×C₈×C₅₉	D₄	1	2	yes	—
21	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
22	#108×C₅₉	—	4	4	no	—
23	C₄⋊C₈×C₅₉	—	2	4	no	—
24	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₅₉	Q₈	1	2	yes	—
26	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
27	C₂²×C₄×C₅₉	C₂×Q₈	1	2	yes	—
28	M₁₆×C₅₉	—	4	4	no	—
29	C₂×C₈×C₅₉	—	4	2	no	—
30	C₂×C₈×C₅₉	C₄⋊C₄	1	4	yes	—
31	C₂×C₈×C₅₉	SD₁₆	1	4	yes	—
32	C₂×C₈×C₅₉	SD₁₆	1	4	yes	—
33	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
34	C₄²×C₅₉	C₂×D₄	1	2	yes	—
35	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
36	#11745664	—	4	4	no	—
37	C₂×C₅₉⋊C₈	—	4	4	no	—
38	C₂×C₅₉⋊C₈	—	4	2	no	—
39	C₂×C₅₉⋊C₈	C₄⋊C₄	1	8	yes	—
40	C₂³×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
41	C₂×C₄×C₅₉	—	2	4	no	—
42	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
43	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
45	C₂×C₄×C₅₉	C₄⋊Q₈	1	2	yes	—
46	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
47	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
48	C₈×C₅₉	—	8	2	no	—
49	C₈×C₅₉	#106	1	8	yes	—
50	C₅₉⋊C₈	—	2	8	no	—
51	C₅₉⋊C₈	—	8	2	no	—
52	#8825	—	59	1	no	maximal, Sylow
53	C₂²×C₅₉	—	2	2	no	—
54	C₂²×C₅₉	#384	1	1	yes	characteristic
55	C₂²×C₅₉	#498	1	2	yes	—
56	C₄×C₅₉	—	2	2	no	—
57	C₄×C₅₉	C₂×#106	1	2	yes	—
58	C₄×C₅₉	#430	1	4	yes	—
59	C₄×C₅₉	#492	1	1	yes	characteristic
60	C₄×C₅₉	#492	1	1	yes	characteristic
61	#1848	—	59	1	no	—
62	#1848	D₅₉	1	1	yes	characteristic
63	#1566	—	59	4	no	—
64	C₂×#322	—	59	1	no	—
65	C₂×C₅₉	—	2	2	no	—
66	C₂×C₅₉	#1844	1	1	yes	characteristic
67	C₂×C₅₉	#1849	1	2	yes	—
68	C₂×#108	—	2	2	no	—
69	C₂×#108	—	118	2	no	—
70	#423	—	59	2	no	—
71	#423	C₂×D₅₉	1	2	yes	—
72	C₂×C₄×C₈	—	59	2	no	—
73	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
74	#334	—	118	8	no	—
75	#322	—	59	4	no	—
76	C₅₉	#8825	1	1	yes	characteristic, complemented, Sylow
77	C₂×M₁₆	—	2	2	no	—
78	C₂×M₁₆	—	118	2	no	—
79	C₂²×C₈	—	59	4	no	—
80	C₂²×C₈	C₂²⋊D₅₉	1	2	yes	—
81	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
82	#108	—	4	4	no	—
83	#108	—	236	4	no	—
84	C₄⋊C₈	—	2	4	no	—
85	C₄⋊C₈	—	118	4	no	—
86	C₄×C₈	—	59	4	no	—
87	C₄×C₈	—	59	4	no	—
88	C₄×C₈	Dic₁₁₈	1	2	yes	—
89	C₄×C₈	C₄×D₅₉	1	2	yes	—
90	C₂²×C₄	Q₈×D₅₉	1	2	yes	—
91	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
92	M₁₆	—	4	4	no	—
93	M₁₆	—	236	4	no	—
94	C₂×C₈	—	4	2	no	—
95	C₂×C₈	—	59	8	no	—
96	C₂×C₈	—	236	4	no	—
97	C₂×C₈	—	236	2	no	—
98	C₂×C₈	Dic₅₉⋊C₄	1	4	yes	—
99	C₂×C₈	#11745670	1	4	yes	—
100	C₂×C₈	#11745671	1	4	yes	—
101	C₄²	C₂×Dic₁₁₈	1	1	yes	characteristic
102	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
103	C₄²	D₄×D₅₉	1	2	yes	—
104	C₂³	#14273725	1	1	yes	characteristic
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	C₄⋊C₄×D₅₉	1	2	yes	—
107	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
108	C₂×C₄	#14273723	1	1	yes	characteristic
109	C₂×C₄	#14273726	1	2	yes	—
110	C₂×C₄	C₂×#11745671	1	2	yes	—
111	C₂×C₄	C₂×#11745670	1	2	yes	—
112	C₈	—	8	2	no	—
113	C₈	—	118	8	no	—
114	C₈	—	472	2	no	—
115	C₈	#14273658	1	8	yes	—
116	C₂²	—	2	2	no	—
117	C₂²	#16837271	1	1	yes	characteristic
118	C₂²	#16837798	1	2	yes	—
119	C₄	—	2	2	no	—
120	C₄	C₂×#14273658	1	2	yes	—
121	C₄	#16837449	1	4	yes	—
122	C₄	#16837774	1	1	yes	characteristic
123	C₄	#16837776	1	1	yes	characteristic
124	C₂	—	2	2	no	—
125	C₂	#21424252	1	1	yes	characteristic
126	C₂	#21424269	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21424268

C₂

yes

maximal, characteristic

#21422851

C₂

yes

maximal

C₂×#16837123

C₂

yes

maximal, characteristic

C₅₉×#1848

C₂

yes

maximal, characteristic

C₂×#14273663

—

#16837426

C₂²

yes

—

C₂×C₄×C₅₉⋊C₈

C₂²

yes

—

#16837152

—

#16837123

C₄

yes

—

C₂×#108×C₅₉

—

C₅₉×#423

C₂²

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×#11745664

—

C₂²×C₅₉⋊C₈

Q₈

yes

—

#14273663

—

C₄⋊(C₅₉⋊C₈)

—

C₄×C₅₉⋊C₈

C₂×C₄

yes

—

C₄×C₅₉⋊C₈

D₄

yes

—

C₂×M₁₆×C₅₉

—

C₂²×C₈×C₅₉

D₄

yes

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

#108×C₅₉

—

C₄⋊C₈×C₅₉

—

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₄×C₈×C₅₉

Q₈

yes

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×Q₈

yes

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₄⋊C₄

yes

—

C₂×C₈×C₅₉

SD₁₆

yes

—

C₂×C₈×C₅₉

SD₁₆

yes

—

C₄²×C₅₉

C₂²×C₄

yes

characteristic

C₄²×C₅₉

C₂×D₄

yes

—

C₄²×C₅₉

C₂×Q₈

yes

characteristic

#11745664

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₄⋊C₄

yes

—

C₂³×C₅₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

#127

yes

characteristic

C₂×C₄×C₅₉

C₄⋊Q₈

yes

—

C₂×C₄×C₅₉

C₂×SD₁₆

yes

—

C₂×C₄×C₅₉

C₂×SD₁₆

yes

—

C₈×C₅₉

—

C₈×C₅₉

#106

yes

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

#8825

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

#384

yes

characteristic

C₂²×C₅₉

#498

yes

—

C₄×C₅₉

—

C₄×C₅₉

C₂×#106

yes

—

C₄×C₅₉

#430

yes

—

C₄×C₅₉

yes

C₄×C₅₉

yes

—

D₅₉

yes

characteristic

#1566

—

C₂×#322

—

C₂×C₅₉

—

C₂×C₅₉

#1844

yes

characteristic

C₂×C₅₉

#1849

yes

—

C₂×#108

—

C₂×#108

—

118

—

#423

—

#423

C₂×D₅₉

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

#334

—

118

—

#322

—

C₅₉

#8825

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

118

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₅₉

yes

—

C₂×C₄²

C₂²×D₅₉

yes

characteristic

#108

—

#108

—

236

—

C₄⋊C₈

—

C₄⋊C₈

—

118

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Dic₁₁₈

yes

—

C₄×C₈

C₄×D₅₉

yes

—

C₂²×C₄

Q₈×D₅₉

yes

—

C₂²×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

M₁₆

—

M₁₆

—

236

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

236

—

C₂×C₈

—

236

—

C₂×C₈

Dic₅₉⋊C₄

yes

—

C₂×C₈

#11745670

yes

—

100

C₂×C₈

#11745671

yes

—

101

C₄²

C₂×Dic₁₁₈

yes

characteristic

102

C₄²

C₂×C₄×D₅₉

yes

characteristic

103

C₄²

D₄×D₅₉

yes

—

104

C₂³

#14273725

yes

characteristic

105

C₂×C₄

—

106

C₂×C₄

C₄⋊C₄×D₅₉

yes

—

107

C₂×C₄

C₂×Dic₅₉⋊C₄

yes

characteristic

108

C₂×C₄

#14273723

yes

characteristic

109

C₂×C₄

#14273726

yes

—

110

C₂×C₄

C₂×#11745671

yes

—

111

C₂×C₄

C₂×#11745670

yes

—

112

C₈

—

113

C₈

—

118

—

114

C₈

—

472

—

115

C₈

#14273658

yes

—

116

C₂²

—

117

C₂²

#16837271

yes

characteristic

118

C₂²

#16837798

yes

—

119

C₄

—

120

C₄

C₂×#14273658

yes

—

121

C₄

#16837449

yes

—

122

C₄

#16837774

yes

characteristic

123

C₄

#16837776

yes

characteristic

124

C₂

—

125

C₂

#21424252

yes

characteristic

126

C₂

#21424269

yes

—

	subgroup	G/H
center	C₂×C₄	#14273723
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1848	C₂
socle	C₂²×C₅₉	#384
prime radical	C₂³×C₅₉	C₄⋊Q₈
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1848	D₅₉
hypocenter	C₅₉	#8825
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

subgroup

G/H

center

C₂×C₄

#14273723

commutator

C₄²×C₅₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₅₉×#1848

C₂

socle

C₂²×C₅₉

#384

prime radical

C₂³×C₅₉

C₄⋊Q₈

cosocle

C₂×C₄²×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

N	H ≅ G/N	type
76. C₅₉	#8825	semidirect

H ≅ G/N

type

76. C₅₉

#8825

semidirect

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

subgroup	index
#21424268	2
#21422851	2
C₂×#16837123	2
C₅₉×#1848	2
#8825	59

subgroup

index

C₅₉×#1848

Subgroups — #28482498

Conjugacy classes of subgroups

Complements

Maximal subgroups