Group #28482491 — subgroups

Subgroups — #28482491

Order 15104.

354 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 171 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	#21422854	C₂	1	4	yes	maximal
3	C₂×C₈×C₅₉⋊C₈	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	#16837154	—	2	8	no	—
9	C₈×C₅₉⋊C₈	C₄	1	1	yes	characteristic
10	C₈×C₅₉⋊C₈	C₄	1	2	yes	—
11	C₈×C₅₉⋊C₈	C₄	1	1	yes	characteristic
12	C₂×#108×C₅₉	—	2	2	no	—
13	C₅₉×#423	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₂×#11745664	—	2	2	no	—
16	C₂²×C₅₉⋊C₈	Q₈	1	4	yes	—
17	#14273663	—	4	4	no	—
18	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
19	C₄×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
20	C₄×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
21	C₄×C₅₉⋊C₈	D₄	1	4	yes	—
22	C₂×M₁₆×C₅₉	—	2	2	no	—
23	C₂²×C₈×C₅₉	D₄	1	2	yes	—
24	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
25	#108×C₅₉	—	4	4	no	—
26	C₄⋊C₈×C₅₉	—	2	4	no	—
27	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₅₉	Q₈	1	2	yes	—
30	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
31	C₂²×C₄×C₅₉	C₂×Q₈	1	2	yes	—
32	M₁₆×C₅₉	—	4	4	no	—
33	C₂×C₈×C₅₉	—	4	2	no	—
34	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
35	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
36	C₂×C₈×C₅₉	D₈	1	4	yes	—
37	C₂×C₈×C₅₉	Q₁₆	1	4	yes	—
38	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
39	C₄²×C₅₉	C₂×D₄	1	2	yes	—
40	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
41	#11745664	—	4	4	no	—
42	C₂×C₅₉⋊C₈	—	2	8	no	—
43	C₂×C₅₉⋊C₈	—	4	2	no	—
44	C₂×C₅₉⋊C₈	C₄⋊C₄	1	4	yes	—
45	C₂×C₅₉⋊C₈	C₄⋊C₄	1	4	yes	—
46	C₂³×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
47	C₂×C₄×C₅₉	—	2	4	no	—
48	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
49	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
51	C₂×C₄×C₅₉	C₄⋊Q₈	1	2	yes	—
52	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
53	C₂×C₄×C₅₉	C₂×Q₁₆	1	2	yes	—
54	C₈×C₅₉	—	8	2	no	—
55	C₈×C₅₉	#107	1	4	yes	—
56	C₈×C₅₉	#107	1	4	yes	—
57	C₅₉⋊C₈	—	2	4	no	—
58	C₅₉⋊C₈	—	8	2	no	—
59	C₅₉⋊C₈	#108	1	8	yes	complemented
60	#8823	—	59	1	no	maximal, Sylow
61	C₂²×C₅₉	—	2	2	no	—
62	C₂²×C₅₉	#384	1	1	yes	characteristic
63	C₂²×C₅₉	#500	1	2	yes	—
64	C₄×C₅₉	—	2	2	no	—
65	C₄×C₅₉	C₂×#107	1	2	yes	—
66	C₄×C₅₉	C₂×#108	1	4	yes	—
67	C₄×C₅₉	#493	1	1	yes	characteristic
68	C₄×C₅₉	#494	1	1	yes	characteristic
69	#1848	—	59	1	no	—
70	#1848	D₅₉	1	1	yes	characteristic
71	#1567	—	59	4	no	—
72	C₂×C₈²	—	59	1	no	—
73	C₂×C₅₉	—	2	2	no	—
74	C₂×C₅₉	#1845	1	1	yes	characteristic
75	C₂×C₅₉	#1848	1	2	yes	—
76	C₂×#108	—	2	2	no	—
77	C₂×#108	—	118	2	no	—
78	#423	—	59	2	no	—
79	#423	C₂×D₅₉	1	2	yes	—
80	C₂×C₄×C₈	—	59	2	no	—
81	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
82	#335	—	118	8	no	—
83	C₈²	—	59	1	no	—
84	C₈²	—	59	2	no	—
85	C₈²	—	59	1	no	—
86	C₅₉	#8823	1	1	yes	characteristic, complemented, Sylow
87	C₂×M₁₆	—	2	2	no	—
88	C₂×M₁₆	—	118	2	no	—
89	C₂²×C₈	—	59	4	no	—
90	C₂²×C₈	C₂²⋊D₅₉	1	2	yes	—
91	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
92	#108	—	4	4	no	—
93	#108	—	236	4	no	—
94	C₄⋊C₈	—	2	4	no	—
95	C₄⋊C₈	—	118	4	no	—
96	C₄×C₈	—	59	2	no	—
97	C₄×C₈	—	59	2	no	—
98	C₄×C₈	—	59	4	no	—
99	C₄×C₈	Dic₁₁₈	1	2	yes	—
100	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
101	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
102	C₂²×C₄	Q₈×D₅₉	1	2	yes	—
103	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
104	M₁₆	—	4	4	no	—
105	M₁₆	—	236	4	no	—
106	C₂×C₈	—	4	2	no	—
107	C₂×C₈	—	59	4	no	—
108	C₂×C₈	—	59	4	no	—
109	C₂×C₈	—	118	8	no	—
110	C₂×C₈	—	236	2	no	—
111	C₂×C₈	Dic₅₉⋊C₄	1	2	yes	—
112	C₂×C₈	Dic₅₉⋊C₄	1	2	yes	—
113	C₂×C₈	#11745669	1	4	yes	—
114	C₂×C₈	#11745672	1	4	yes	—
115	C₄²	C₂×Dic₁₁₈	1	1	yes	characteristic
116	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
117	C₄²	D₄×D₅₉	1	2	yes	—
118	C₂³	#14273725	1	1	yes	characteristic
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	C₄⋊C₄×D₅₉	1	2	yes	—
121	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
122	C₂×C₄	#14273723	1	1	yes	characteristic
123	C₂×C₄	#14273726	1	2	yes	—
124	C₂×C₄	C₂×#11745669	1	2	yes	—
125	C₂×C₄	C₂×#11745672	1	2	yes	—
126	C₈	—	8	2	no	—
127	C₈	—	59	8	no	—
128	C₈	—	118	4	no	—
129	C₈	—	472	2	no	—
130	C₈	#14273660	1	4	yes	—
131	C₈	#14273660	1	4	yes	—
132	C₂²	—	2	2	no	—
133	C₂²	#16837271	1	1	yes	characteristic
134	C₂²	#16837805	1	2	yes	—
135	C₄	—	2	2	no	—
136	C₄	C₂×#14273660	1	2	yes	—
137	C₄	#108×D₅₉	1	4	yes	—
138	C₄	#16837778	1	1	yes	characteristic
139	C₄	#16837781	1	1	yes	characteristic
140	C₂	—	2	2	no	—
141	C₂	#21424255	1	1	yes	characteristic
142	C₂	#21424266	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21424268

C₂

yes

maximal, characteristic

#21422854

C₂

yes

maximal

C₂×C₈×C₅₉⋊C₈

C₂

yes

maximal, characteristic

C₅₉×#1848

C₂

yes

maximal, characteristic

C₂×#14273663

—

#16837426

C₂²

yes

—

C₂×C₄×C₅₉⋊C₈

C₂²

yes

—

#16837154

—

C₈×C₅₉⋊C₈

C₄

yes

characteristic

C₈×C₅₉⋊C₈

C₄

yes

—

C₈×C₅₉⋊C₈

C₄

yes

characteristic

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

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

characteristic

C₄×C₈×C₅₉

C₂×C₄

yes

characteristic

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

C₄⋊C₄

yes

—

C₂×C₈×C₅₉

D₈

yes

—

C₂×C₈×C₅₉

Q₁₆

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

—

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×D₈

yes

—

C₂×C₄×C₅₉

C₂×Q₁₆

yes

—

C₈×C₅₉

—

C₈×C₅₉

#107

yes

—

C₈×C₅₉

#107

yes

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

yes

—

C₂²×C₅₉

—

C₂²×C₅₉

#384

yes

characteristic

C₂²×C₅₉

#500

yes

—

C₄×C₅₉

—

C₄×C₅₉

C₂×#107

yes

—

C₄×C₅₉

C₂×#108

yes

—

C₄×C₅₉

yes

C₄×C₅₉

yes

—

D₅₉

yes

characteristic

#1567

—

C₂×C₈²

—

C₂×C₅₉

—

C₂×C₅₉

#1845

yes

characteristic

C₂×C₅₉

#1848

yes

—

C₂×#108

—

C₂×#108

—

118

—

#423

—

#423

C₂×D₅₉

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

#335

—

118

—

C₈²

—

C₈²

—

C₈²

—

C₅₉

#8823

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

—

C₄×C₈

Dic₁₁₈

yes

—

100

C₄×C₈

C₄×D₅₉

yes

characteristic

101

C₄×C₈

C₄×D₅₉

yes

characteristic

102

C₂²×C₄

Q₈×D₅₉

yes

—

103

C₂²×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

104

M₁₆

—

105

M₁₆

—

236

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

118

—

110

C₂×C₈

—

236

—

111

C₂×C₈

Dic₅₉⋊C₄

yes

—

112

C₂×C₈

Dic₅₉⋊C₄

yes

—

113

C₂×C₈

#11745669

yes

—

114

C₂×C₈

#11745672

yes

—

115

C₄²

C₂×Dic₁₁₈

yes

characteristic

116

C₄²

C₂×C₄×D₅₉

yes

characteristic

117

C₄²

D₄×D₅₉

yes

—

118

C₂³

#14273725

yes

characteristic

119

C₂×C₄

—

120

C₂×C₄

C₄⋊C₄×D₅₉

yes

—

121

C₂×C₄

C₂×Dic₅₉⋊C₄

yes

characteristic

122

C₂×C₄

#14273723

yes

characteristic

123

C₂×C₄

#14273726

yes

—

124

C₂×C₄

C₂×#11745669

yes

—

125

C₂×C₄

C₂×#11745672

yes

—

126

C₈

—

127

C₈

—

128

C₈

—

118

—

129

C₈

—

472

—

130

C₈

#14273660

yes

—

131

C₈

#14273660

yes

—

132

C₂²

—

133

C₂²

#16837271

yes

characteristic

134

C₂²

#16837805

yes

—

135

C₄

—

136

C₄

C₂×#14273660

yes

—

137

C₄

#108×D₅₉

yes

—

138

C₄

#16837778

yes

characteristic

139

C₄

#16837781

yes

characteristic

140

C₂

—

141

C₂

#21424255

yes

characteristic

142

C₂

#21424266

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	#8823
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
59. C₅₉⋊C₈	#108	semidirect
86. C₅₉	#8823	semidirect

H ≅ G/N

type

59. C₅₉⋊C₈

#108

semidirect

86. C₅₉

#8823

semidirect

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

subgroup	index
#21424268	2
#21422854	2
C₂×C₈×C₅₉⋊C₈	2
C₅₉×#1848	2
#8823	59

subgroup

index

#21424268

#21422854

C₂×C₈×C₅₉⋊C₈

C₅₉×#1848

#8823

Subgroups — #28482491

Conjugacy classes of subgroups

Complements

Maximal subgroups