Group #28500325 — subgroups

Subgroups — #28500325

Order 15104.

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

normal subgroups only

This group has 232 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	#21422852	C₂	1	2	yes	maximal
2	#21422162	C₂	1	4	yes	maximal, complemented
3	C₄×M₃₂×C₅₉	C₂	1	1	yes	maximal, characteristic
4	#16837426	—	2	2	no	—
5	#16837153	—	2	2	no	—
6	#16837153	C₂²	1	4	yes	—
7	C₂×M₃₂×C₅₉	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
9	C₅₉×#346	C₂²	1	2	yes	—
10	C₄×C₁₆×C₅₉	C₄	1	1	yes	characteristic
11	C₄×C₁₆×C₅₉	C₄	1	1	yes	characteristic
12	C₂×#11745664	—	4	2	no	—
13	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
14	C₄⋊(C₅₉⋊C₈)	—	4	2	no	—
15	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₅₉	Q₈	1	1	yes	characteristic
17	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
18	M₃₂×C₅₉	—	2	4	no	—
19	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
20	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
21	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
23	C₄×C₈×C₅₉	Q₈	1	1	yes	characteristic
24	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
26	C₂²×C₄×C₅₉	SD₁₆	1	2	yes	—
27	C₂×C₈×C₅₉	—	2	2	no	—
28	C₂×C₈×C₅₉	—	2	2	no	—
29	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
33	C₄²×C₅₉	C₄⋊C₄	1	1	yes	characteristic
34	C₄²×C₅₉	SD₁₆	1	2	yes	—
35	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
36	#11745664	—	8	4	no	—
37	C₂×C₅₉⋊C₈	—	4	4	no	—
38	C₁₆×C₅₉	—	2	2	no	—
39	C₁₆×C₅₉	M₁₆	1	4	yes	complemented
40	C₂³×C₅₉	#106	1	1	yes	characteristic
41	C₂×C₄×C₅₉	—	2	4	no	—
42	C₂×C₄×C₅₉	—	2	1	no	—
43	C₂×C₄×C₅₉	—	2	1	no	—
44	C₂×C₄×C₅₉	#106	1	1	yes	characteristic
45	C₂×C₄×C₅₉	#106	1	2	yes	—
46	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
48	C₈×C₅₉	—	2	1	no	—
49	C₈×C₅₉	—	2	1	no	—
50	C₈×C₅₉	#108	1	2	yes	—
51	C₈×C₅₉	C₂×M₁₆	1	2	yes	—
52	C₅₉⋊C₈	—	8	4	no	—
53	#12020	—	59	1	no	maximal, Sylow
54	C₂²×C₅₉	—	2	1	no	—
55	C₂²×C₅₉	—	2	2	no	—
56	C₂²×C₅₉	C₂×#106	1	1	yes	characteristic
57	C₄×C₅₉	—	2	1	no	—
58	C₄×C₅₉	—	2	2	no	—
59	C₄×C₅₉	#365	1	4	yes	—
60	C₄×C₅₉	#423	1	1	yes	characteristic
61	C₄×C₅₉	C₂×#108	1	1	yes	characteristic
62	C₄×M₃₂	D₅₉	1	1	yes	characteristic
63	#1566	—	59	2	no	—
64	#1368	—	59	4	no	—
65	C₂×C₅₉	—	4	1	no	—
66	C₂×C₅₉	#1566	1	1	yes	characteristic
67	C₂×C₅₉	C₂×#365	1	2	yes	—
68	C₂×M₃₂	Dic₅₉	1	2	yes	—
69	#423	—	118	2	no	—
70	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
71	#346	C₂×D₅₉	1	2	yes	—
72	C₄×C₁₆	Dic₅₉	1	1	yes	characteristic
73	C₄×C₁₆	Dic₅₉	1	1	yes	characteristic
74	#334	—	59	4	no	—
75	#334	—	118	2	no	—
76	C₅₉	#12020	1	1	yes	characteristic, complemented, Sylow
77	C₂×M₁₆	—	236	2	no	—
78	C₂²×C₈	Dic₁₁₈	1	1	yes	characteristic
79	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
80	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
81	M₃₂	—	2	4	no	—
82	C₂×C₁₆	C₂×Dic₅₉	1	2	yes	—
83	C₂×C₁₆	C₂×Dic₅₉	1	2	yes	—
84	C₄⋊C₈	—	118	4	no	—
85	C₄⋊C₈	—	236	2	no	—
86	C₄×C₈	Dic₁₁₈	1	1	yes	characteristic
87	C₄×C₈	D₂₃₆	1	1	yes	characteristic
88	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
89	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
90	C₂²×C₄	#11745660	1	2	yes	—
91	C₂²×C₄	C₄⋊Dic₅₉	1	1	yes	characteristic
92	M₁₆	—	472	4	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	236	4	no	—
96	C₂×C₈	C₄⋊Dic₅₉	1	1	yes	characteristic
97	C₂×C₈	C₄⋊Dic₅₉	1	1	yes	characteristic
98	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
99	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
100	C₄²	#11745660	1	2	yes	—
101	C₄²	C₄⋊Dic₅₉	1	1	yes	characteristic
102	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
103	C₁₆	—	2	2	no	—
104	C₁₆	#11745664	1	4	yes	complemented
105	C₂³	#14273659	1	1	yes	characteristic
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	#14273659	1	1	yes	characteristic
110	C₂×C₄	#14273659	1	2	yes	—
111	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
112	C₂×C₄	C₂×#11745660	1	2	yes	—
113	C₈	—	2	1	no	—
114	C₈	—	2	1	no	—
115	C₈	—	472	4	no	—
116	C₈	#14273663	1	2	yes	—
117	C₈	C₂×#11745664	1	2	yes	—
118	C₂²	—	2	1	no	—
119	C₂²	—	2	2	no	—
120	C₂²	C₂×#14273659	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	2	2	no	—
123	C₄	#16837225	1	4	yes	—
124	C₄	#16837426	1	1	yes	characteristic
125	C₄	C₂×#14273663	1	1	yes	characteristic
126	C₂	—	4	1	no	—
127	C₂	#21422852	1	1	yes	characteristic
128	C₂	C₂×#16837225	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21422852

C₂

yes

maximal

#21422162

C₂

yes

maximal, complemented

C₄×M₃₂×C₅₉

C₂

yes

maximal, characteristic

#16837426

—

#16837153

—

#16837153

C₂²

yes

—

C₂×M₃₂×C₅₉

C₄

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₅₉×#346

C₂²

yes

—

C₄×C₁₆×C₅₉

C₄

yes

characteristic

C₄×C₁₆×C₅₉

C₄

yes

characteristic

C₂×#11745664

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

C₂²×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₉

Q₈

yes

characteristic

C₂×C₄²×C₅₉

D₄

yes

characteristic

M₃₂×C₅₉

—

C₂×C₁₆×C₅₉

C₂×C₄

yes

—

C₂×C₁₆×C₅₉

C₂×C₄

yes

—

C₄×C₈×C₅₉

C₂×C₄

yes

characteristic

C₄×C₈×C₅₉

D₄

yes

characteristic

C₄×C₈×C₅₉

Q₈

yes

characteristic

C₄×C₈×C₅₉

C₂³

yes

characteristic

C₂²×C₄×C₅₉

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₅₉

SD₁₆

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂×Q₈

yes

characteristic

C₄²×C₅₉

C₄⋊C₄

yes

characteristic

C₄²×C₅₉

SD₁₆

yes

—

C₄²×C₅₉

C₂×D₄

yes

characteristic

#11745664

—

C₂×C₅₉⋊C₈

—

C₁₆×C₅₉

—

C₁₆×C₅₉

M₁₆

yes

complemented

C₂³×C₅₉

#106

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#106

yes

characteristic

C₂×C₄×C₅₉

#106

yes

—

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×SD₁₆

yes

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

#108

yes

—

C₈×C₅₉

C₂×M₁₆

yes

—

C₅₉⋊C₈

—

#12020

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×#106

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#365

yes

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×M₃₂

D₅₉

yes

characteristic

#1566

—

#1368

—

C₂×C₅₉

—

C₂×C₅₉

#1566

yes

characteristic

C₂×C₅₉

C₂×#365

yes

—

C₂×M₃₂

Dic₅₉

yes

—

#423

—

118

—

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

#346

C₂×D₅₉

yes

—

C₄×C₁₆

Dic₅₉

yes

characteristic

C₄×C₁₆

Dic₅₉

yes

characteristic

#334

—

#334

—

118

—

C₅₉

#12020

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

236

—

C₂²×C₈

Dic₁₁₈

yes

characteristic

C₂²×C₈

C₂×Dic₅₉

yes

characteristic

C₂×C₄²

D₂₃₆

yes

characteristic

M₃₂

—

C₂×C₁₆

C₂×Dic₅₉

yes

—

C₂×C₁₆

C₂×Dic₅₉

yes

—

C₄⋊C₈

—

118

—

C₄⋊C₈

—

236

—

C₄×C₈

Dic₁₁₈

yes

characteristic

C₄×C₈

D₂₃₆

yes

characteristic

C₄×C₈

C₂×Dic₅₉

yes

characteristic

C₄×C₈

C₂²×D₅₉

yes

characteristic

C₂²×C₄

#11745660

yes

—

C₂²×C₄

C₄⋊Dic₅₉

yes

characteristic

M₁₆

—

472

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

236

—

C₂×C₈

C₄⋊Dic₅₉

yes

characteristic

C₂×C₈

C₄⋊Dic₅₉

yes

characteristic

C₂×C₈

C₂×Dic₁₁₈

yes

characteristic

C₂×C₈

C₂²×Dic₅₉

yes

characteristic

100

C₄²

#11745660

yes

—

101

C₄²

C₄⋊Dic₅₉

yes

characteristic

102

C₄²

C₂×D₂₃₆

yes

characteristic

103

C₁₆

—

104

C₁₆

#11745664

yes

complemented

105

C₂³

#14273659

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

#14273659

yes

characteristic

110

C₂×C₄

#14273659

yes

—

111

C₂×C₄

C₂×C₄⋊Dic₅₉

yes

characteristic

112

C₂×C₄

C₂×#11745660

yes

—

113

C₈

—

114

C₈

—

115

C₈

—

472

—

116

C₈

#14273663

yes

—

117

C₈

C₂×#11745664

yes

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂×#14273659

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

#16837225

yes

—

124

C₄

#16837426

yes

characteristic

125

C₄

C₂×#14273663

yes

characteristic

126

C₂

—

127

C₂

#21422852

yes

characteristic

128

C₂

C₂×#16837225

yes

—

	subgroup	G/H
center	C₂²	C₂×#14273659
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₄×M₃₂×C₅₉	C₂
socle	C₂²×C₅₉	C₂×#106
prime radical	C₂³×C₅₉	#106
cosocle	C₄×C₈×C₅₉	C₂³
hypercenter	C₄×M₃₂	D₅₉
hypocenter	C₅₉	#12020
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

subgroup

G/H

center

C₂²

C₂×#14273659

commutator

C₂×C₈×C₅₉

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₅₉

Fitting

C₄×M₃₂×C₅₉

C₂

C₂²×C₅₉

C₂³×C₅₉

C₄×C₈×C₅₉

C₂³

hypercenter

C₄×M₃₂

D₅₉

hypocenter

C₅₉

#12020

LCS

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

N	H ≅ G/N	type
2. #21422162	C₂	semidirect
39. C₁₆×C₅₉	M₁₆	semidirect
76. C₅₉	#12020	semidirect
104. C₁₆	#11745664	semidirect

H ≅ G/N

type

2. #21422162

C₂

semidirect

39. C₁₆×C₅₉

M₁₆

semidirect

76. C₅₉

#12020

semidirect

104. C₁₆

#11745664

semidirect

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

subgroup	index
#21422852	2
#21422162	2
C₄×M₃₂×C₅₉	2
#12020	59

subgroup

index

#21422852

#21422162

C₄×M₃₂×C₅₉

#12020

Subgroups — #28500325

Conjugacy classes of subgroups

Complements

Maximal subgroups