Group #28500579 — subgroups

Subgroups — #28500579

Order 15104.

204 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 89 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	#21425938	C₂	1	2	yes	maximal, complemented
2	#21422164	C₂	1	1	yes	maximal, characteristic, complemented
3	#21422072	C₂	1	2	yes	maximal, complemented
4	#21422016	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₉×#2179	C₂	1	1	yes	maximal, characteristic
6	#16837819	—	2	2	no	—
7	#16837223	C₂²	1	2	yes	—
8	#16837186	—	2	2	no	—
9	#16837178	C₂²	1	1	yes	characteristic
10	#16837123	C₂²	1	1	yes	characteristic
11	C₅₉×#508	C₄	1	1	yes	characteristic
12	C₅₉×#443	C₂²	1	2	yes	—
13	C₅₉×#368	C₄	1	1	yes	characteristic
14	C₅₉×#359	C₄	1	2	yes	—
15	C₅₉×#346	C₂²	1	1	yes	characteristic
16	#14273667	—	2	2	no	—
17	#14273667	—	4	2	no	—
18	C₂×C₅₉⋊C₁₆	—	4	2	no	—
19	C₂×C₅₉⋊C₁₆	D₄	1	2	yes	—
20	C₄×C₅₉⋊C₈	—	2	1	no	—
21	#135×C₅₉	C₂×C₄	1	2	yes	—
22	#131×C₅₉	—	2	2	no	—
23	Q₃₂×C₅₉	C₈	1	1	yes	characteristic, complemented
24	SD₃₂×C₅₉	C₈	1	2	yes	complemented
25	D₁₆×C₅₉	C₈	1	1	yes	characteristic, complemented
26	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
28	#108×C₅₉	C₂×C₄	1	2	yes	—
29	#104×C₅₉	—	2	2	no	—
30	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
31	#41×C₅₉	—	2	2	no	—
32	Q₁₆×C₅₉	C₂×C₈	1	2	yes	—
33	SD₁₆×C₅₉	—	2	2	no	—
34	D₈×C₅₉	C₂×C₈	1	2	yes	—
35	M₁₆×C₅₉	—	2	2	no	—
36	M₁₆×C₅₉	—	4	2	no	—
37	C₂×C₈×C₅₉	—	4	2	no	—
38	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
40	C₄²×C₅₉	#41	1	1	yes	characteristic
41	C₂×C₅₉⋊C₈	—	4	1	no	—
42	C₁₆×C₅₉	—	2	1	no	—
43	C₁₆×C₅₉	C₂×C₈	1	1	yes	characteristic
44	C₁₆×C₅₉	C₂×C₈	1	1	yes	characteristic
45	C₅₉⋊C₁₆	—	2	2	no	—
46	C₅₉⋊C₁₆	—	4	2	no	—
47	Q₈×C₅₉	—	2	2	no	—
48	D₄×C₅₉	—	2	2	no	—
49	D₄×C₅₉	—	4	2	no	—
50	C₂×C₄×C₅₉	—	2	1	no	—
51	C₂×C₄×C₅₉	—	4	2	no	—
52	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
53	C₈×C₅₉	—	4	2	no	—
54	C₈×C₅₉	C₂²×C₈	1	1	yes	characteristic
55	C₈×C₅₉	#131	1	1	yes	characteristic
56	C₈×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
57	C₅₉⋊C₈	—	8	1	no	—
58	#12068	—	59	1	no	maximal, Sylow
59	C₂²×C₅₉	—	4	2	no	—
60	C₂²×C₅₉	#444	1	1	yes	characteristic
61	C₄×C₅₉	—	4	1	no	—
62	C₄×C₅₉	—	4	2	no	—
63	C₄×C₅₉	C₈×D₄	1	1	yes	characteristic
64	C₄×C₅₉	#442	1	1	yes	characteristic
65	#2179	D₅₉	1	1	yes	characteristic
66	#2171	—	59	2	no	—
67	#1369	—	59	1	no	—
68	#1336	—	59	2	no	—
69	#1313	—	59	1	no	—
70	C₂×C₅₉	—	2	1	no	—
71	C₂×C₅₉	—	8	2	no	—
72	C₂×C₅₉	#1580	1	1	yes	characteristic
73	#508	Dic₅₉	1	1	yes	characteristic
74	#504	—	118	2	no	—
75	#443	C₂×D₅₉	1	2	yes	—
76	#368	Dic₅₉	1	1	yes	characteristic
77	#364	—	59	2	no	—
78	#359	Dic₅₉	1	2	yes	—
79	#350	—	118	2	no	—
80	#346	—	59	1	no	—
81	#346	C₂×D₅₉	1	1	yes	characteristic
82	#322	—	59	1	no	—
83	C₅₉	#12068	1	1	yes	characteristic, complemented, Sylow
84	#135	C₂×Dic₅₉	1	2	yes	—
85	#131	—	2	2	no	—
86	Q₃₂	C₅₉⋊C₈	1	1	yes	characteristic, complemented
87	SD₃₂	C₅₉⋊C₈	1	2	yes	complemented
88	D₁₆	C₅₉⋊C₈	1	1	yes	characteristic, complemented
89	M₃₂	—	118	2	no	—
90	M₃₂	—	236	2	no	—
91	C₂×C₁₆	—	59	2	no	—
92	C₂×C₁₆	—	236	2	no	—
93	C₂×C₁₆	C₂×Dic₅₉	1	1	yes	characteristic
94	C₂×C₁₆	C₂×Dic₅₉	1	1	yes	characteristic
95	#108	C₂×Dic₅₉	1	2	yes	—
96	#104	—	2	2	no	—
97	C₄×C₈	—	118	1	no	—
98	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
99	#41	—	2	2	no	—
100	Q₁₆	C₂×C₅₉⋊C₈	1	2	yes	—
101	SD₁₆	—	2	2	no	—
102	D₈	C₂×C₅₉⋊C₈	1	2	yes	—
103	M₁₆	—	2	2	no	—
104	M₁₆	—	4	2	no	—
105	C₂×C₈	—	4	2	no	—
106	C₂×C₈	—	236	1	no	—
107	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
108	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
109	C₄²	#11745687	1	1	yes	characteristic
110	C₁₆	—	2	1	no	—
111	C₁₆	—	118	2	no	—
112	C₁₆	—	236	2	no	—
113	C₁₆	C₂×C₅₉⋊C₈	1	1	yes	characteristic
114	C₁₆	C₂×C₅₉⋊C₈	1	1	yes	characteristic
115	Q₈	—	2	2	no	—
116	D₄	—	2	2	no	—
117	D₄	—	4	2	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	D₄×Dic₅₉	1	1	yes	characteristic
121	C₈	—	4	2	no	—
122	C₈	—	472	1	no	—
123	C₈	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
124	C₈	#14273734	1	1	yes	characteristic
125	C₈	#14273751	1	2	yes	—
126	C₂²	—	4	2	no	—
127	C₂²	#16837524	1	1	yes	characteristic
128	C₄	—	4	1	no	—
129	C₄	—	4	2	no	—
130	C₄	D₄×C₅₉⋊C₈	1	1	yes	characteristic
131	C₄	#16837514	1	1	yes	characteristic
132	C₂	—	2	1	no	—
133	C₂	—	8	2	no	—
134	C₂	#21422920	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21425938

C₂

yes

maximal, complemented

#21422164

C₂

yes

maximal, characteristic, complemented

#21422072

C₂

yes

maximal, complemented

#21422016

C₂

yes

maximal, characteristic, complemented

C₅₉×#2179

C₂

yes

maximal, characteristic

#16837819

—

#16837223

C₂²

yes

—

#16837186

—

#16837178

C₂²

yes

characteristic

#16837123

C₂²

yes

characteristic

C₅₉×#508

C₄

yes

characteristic

C₅₉×#443

C₂²

yes

—

C₅₉×#368

C₄

yes

characteristic

C₅₉×#359

C₄

yes

—

C₅₉×#346

C₂²

yes

characteristic

#14273667

—

#14273667

—

C₂×C₅₉⋊C₁₆

—

C₂×C₅₉⋊C₁₆

D₄

yes

—

C₄×C₅₉⋊C₈

—

#135×C₅₉

C₂×C₄

yes

—

#131×C₅₉

—

Q₃₂×C₅₉

C₈

yes

characteristic, complemented

SD₃₂×C₅₉

C₈

yes

complemented

D₁₆×C₅₉

C₈

yes

characteristic, complemented

C₂×C₁₆×C₅₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₉

C₂×C₄

yes

characteristic

#108×C₅₉

C₂×C₄

yes

—

#104×C₅₉

—

C₄×C₈×C₅₉

C₂³

yes

characteristic

#41×C₅₉

—

Q₁₆×C₅₉

C₂×C₈

yes

—

SD₁₆×C₅₉

—

D₈×C₅₉

C₂×C₈

yes

—

M₁₆×C₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₄²×C₅₉

#41

yes

characteristic

C₂×C₅₉⋊C₈

—

C₁₆×C₅₉

—

C₁₆×C₅₉

C₂×C₈

yes

characteristic

C₁₆×C₅₉

C₂×C₈

yes

characteristic

C₅₉⋊C₁₆

—

C₅₉⋊C₁₆

—

Q₈×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₄×D₄

yes

characteristic

C₈×C₅₉

—

C₈×C₅₉

C₂²×C₈

yes

C₈×C₅₉

yes

C₈×C₅₉

yes

—

C₅₉⋊C₈

—

#12068

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

#444

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₈×D₄

yes

C₄×C₅₉

yes

D₅₉

yes

—

—

—

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#1580

yes

characteristic

#508

Dic₅₉

yes

characteristic

#504

—

118

—

#443

C₂×D₅₉

yes

—

#368

Dic₅₉

yes

characteristic

#364

—

#359

Dic₅₉

yes

—

#350

—

118

—

#346

—

#346

C₂×D₅₉

yes

characteristic

#322

—

C₅₉

#12068

yes

characteristic, complemented, Sylow

#135

C₂×Dic₅₉

yes

—

#131

—

Q₃₂

C₅₉⋊C₈

yes

characteristic, complemented

SD₃₂

C₅₉⋊C₈

yes

complemented

D₁₆

C₅₉⋊C₈

yes

characteristic, complemented

M₃₂

—

118

—

M₃₂

—

236

—

C₂×C₁₆

—

C₂×C₁₆

—

236

—

C₂×C₁₆

C₂×Dic₅₉

yes

characteristic

C₂×C₁₆

C₂×Dic₅₉

yes

characteristic

#108

C₂×Dic₅₉

yes

—

#104

—

C₄×C₈

—

118

—

C₄×C₈

C₂²×D₅₉

yes

characteristic

#41

—

100

Q₁₆

C₂×C₅₉⋊C₈

yes

—

101

SD₁₆

—

102

D₈

C₂×C₅₉⋊C₈

yes

—

103

M₁₆

—

104

M₁₆

—

105

C₂×C₈

—

106

C₂×C₈

—

236

—

107

C₂×C₈

D₄×D₅₉

yes

characteristic

108

C₂×C₈

C₂²×Dic₅₉

yes

characteristic

109

C₄²

#11745687

yes

characteristic

110

C₁₆

—

111

C₁₆

—

118

—

112

C₁₆

—

236

—

113

C₁₆

C₂×C₅₉⋊C₈

yes

characteristic

114

C₁₆

C₂×C₅₉⋊C₈

yes

characteristic

115

Q₈

—

116

D₄

—

117

D₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

D₄×Dic₅₉

yes

characteristic

121

C₈

—

122

C₈

—

472

—

123

C₈

C₂²×C₅₉⋊C₈

yes

characteristic

124

C₈

#14273734

yes

characteristic

125

C₈

#14273751

yes

—

126

C₂²

—

127

C₂²

#16837524

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

D₄×C₅₉⋊C₈

yes

characteristic

131

C₄

#16837514

yes

characteristic

132

C₂

—

133

C₂

—

134

C₂

#21422920

yes

characteristic

	subgroup	G/H
center	C₄	#16837514
commutator	C₈×C₅₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₅₉×#2179	C₂
socle	C₂×C₅₉	#1580
prime radical	C₅₉×#508	C₄
cosocle	C₄×C₈×C₅₉	C₂³
hypercenter	#2179	D₅₉
hypocenter	C₅₉	#12068
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #2179

subgroup

G/H

center

C₄

#16837514

commutator

C₈×C₅₉

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₅₉

Fitting

C₅₉×#2179

C₂

socle

C₂×C₅₉

#1580

prime radical

C₅₉×#508

C₄

cosocle

C₄×C₈×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #2179

N	H ≅ G/N	type
1. #21425938	C₂	semidirect
2. #21422164	C₂	semidirect
3. #21422072	C₂	semidirect
4. #21422016	C₂	semidirect
23. Q₃₂×C₅₉	C₈	semidirect
24. SD₃₂×C₅₉	C₈	semidirect
25. D₁₆×C₅₉	C₈	semidirect
83. C₅₉	#12068	semidirect
86. Q₃₂	C₅₉⋊C₈	semidirect
87. SD₃₂	C₅₉⋊C₈	semidirect
88. D₁₆	C₅₉⋊C₈	semidirect

H ≅ G/N

type

1. #21425938

C₂

semidirect

2. #21422164

C₂

semidirect

3. #21422072

C₂

semidirect

4. #21422016

C₂

semidirect

23. Q₃₂×C₅₉

C₈

semidirect

24. SD₃₂×C₅₉

C₈

semidirect

25. D₁₆×C₅₉

C₈

semidirect

83. C₅₉

#12068

semidirect

86. Q₃₂

C₅₉⋊C₈

semidirect

87. SD₃₂

C₅₉⋊C₈

semidirect

88. D₁₆

C₅₉⋊C₈

semidirect

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

subgroup	index
#21425938	2
#21422164	2
#21422072	2
#21422016	2
C₅₉×#2179	2
#12068	59

subgroup

index

C₅₉×#2179

Subgroups — #28500579

Conjugacy classes of subgroups

Complements

Maximal subgroups