Group #28500680 — subgroups

Subgroups — #28500680

Order 15104.

210 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 85 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	#21422993	C₂	1	2	yes	maximal
2	#21422168	C₂	1	1	yes	maximal, characteristic
3	#21422060	C₂	1	2	yes	maximal
4	C₁₆×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₄×Q₃₂×C₅₉	C₂	1	1	yes	maximal, characteristic
6	#16837533	—	2	2	no	—
7	C₄×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
8	#16837155	C₂²	1	2	yes	—
9	Q₈⋊(C₅₉⋊C₈)	—	2	2	no	—
10	C₈×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×Q₃₂×C₅₉	C₄	1	1	yes	characteristic
12	C₄×Q₁₆×C₅₉	C₂²	1	2	yes	—
13	C₅₉×#366	C₄	1	1	yes	characteristic
14	C₅₉×#358	C₄	1	2	yes	—
15	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
16	C₂×C₅₉⋊C₁₆	—	2	1	no	—
17	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
18	C₄⋊(C₅₉⋊C₈)	—	4	2	no	—
19	#14273638	—	4	2	no	—
20	C₄×C₅₉⋊C₈	D₄	1	1	yes	characteristic
21	C₄×C₅₉⋊C₈	D₄	1	1	yes	characteristic
22	C₂×Q₁₆×C₅₉	C₂×C₄	1	2	yes	—
23	C₄×Q₈×C₅₉	—	2	2	no	—
24	Q₃₂×C₅₉	—	2	2	no	—
25	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
27	#107×C₅₉	C₂×C₄	1	2	yes	—
28	#103×C₅₉	—	2	2	no	—
29	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
30	C₂×Q₈×C₅₉	—	2	2	no	—
31	Q₁₆×C₅₉	—	2	2	no	—
32	Q₁₆×C₅₉	—	2	2	no	—
33	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
35	C₄⋊C₄×C₅₉	—	2	2	no	—
36	C₄⋊C₄×C₅₉	—	4	2	no	—
37	C₄²×C₅₉	—	4	2	no	—
38	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₅₉⋊C₈	—	2	1	no	—
40	C₂×C₅₉⋊C₈	—	4	2	no	—
41	C₂×C₅₉⋊C₈	D₈	1	1	yes	characteristic
42	C₂×C₅₉⋊C₈	D₈	1	1	yes	characteristic
43	C₁₆×C₅₉	—	2	1	no	—
44	C₁₆×C₅₉	M₁₆	1	2	yes	—
45	C₅₉⋊C₁₆	—	2	2	no	—
46	Q₈×C₅₉	—	4	2	no	—
47	Q₈×C₅₉	—	4	2	no	—
48	C₂×C₄×C₅₉	—	4	2	no	—
49	C₂×C₄×C₅₉	—	4	2	no	—
50	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
51	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
52	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
53	C₈×C₅₉	—	2	1	no	—
54	C₈×C₅₉	C₂×M₁₆	1	1	yes	characteristic
55	C₈×C₅₉	#131	1	1	yes	characteristic
56	C₅₉⋊C₈	—	2	2	no	—
57	C₅₉⋊C₈	—	2	1	no	—
58	C₅₉⋊C₈	—	8	2	no	—
59	C₅₉⋊C₈	Q₃₂	1	2	yes	complemented
60	#12083	—	59	1	no	maximal, Sylow
61	C₂²×C₅₉	C₄×D₈	1	1	yes	characteristic
62	C₄×C₅₉	—	2	1	no	—
63	C₄×C₅₉	—	8	2	no	—
64	C₄×C₅₉	—	8	2	no	—
65	C₄×C₅₉	#436	1	1	yes	characteristic
66	C₄×C₅₉	#443	1	1	yes	characteristic
67	C₄×C₅₉	C₂×Q₃₂	1	1	yes	characteristic
68	C₄×C₅₉	#508	1	1	yes	characteristic
69	C₄×Q₃₂	D₅₉	1	1	yes	characteristic
70	#1591	—	59	2	no	—
71	#1371	—	59	1	no	—
72	#1332	—	59	2	no	—
73	C₈×C₁₆	—	59	1	no	—
74	C₂×C₅₉	#1589	1	1	yes	characteristic
75	C₂×C₅₉	C₄×Q₃₂	1	1	yes	characteristic
76	C₂×C₅₉	#2178	1	1	yes	characteristic
77	C₂×Q₃₂	Dic₅₉	1	1	yes	characteristic
78	#446	—	118	2	no	—
79	C₄×Q₁₆	C₂×D₅₉	1	2	yes	—
80	#366	Dic₅₉	1	1	yes	characteristic
81	#358	Dic₅₉	1	2	yes	—
82	C₄×C₁₆	—	59	1	no	—
83	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
84	#335	—	59	2	no	—
85	Q₈⋊C₈	—	118	2	no	—
86	C₈²	—	59	1	no	—
87	C₅₉	#12083	1	1	yes	characteristic, complemented, Sylow
88	C₂×Q₁₆	C₂×Dic₅₉	1	2	yes	—
89	C₄×Q₈	—	2	2	no	—
90	Q₃₂	—	2	2	no	—
91	C₂×C₁₆	—	118	1	no	—
92	C₂×C₁₆	C₂×Dic₅₉	1	1	yes	characteristic
93	C₂×C₁₆	C₂×Dic₅₉	1	1	yes	characteristic
94	#107	C₂×Dic₅₉	1	2	yes	—
95	C₄⋊C₈	—	118	2	no	—
96	C₄⋊C₈	—	236	2	no	—
97	#103	—	2	2	no	—
98	#97	—	236	2	no	—
99	C₄×C₈	—	59	1	no	—
100	C₄×C₈	—	59	1	no	—
101	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
102	C₂×Q₈	—	2	2	no	—
103	Q₁₆	—	2	2	no	—
104	Q₁₆	—	2	2	no	—
105	C₂×C₈	—	59	1	no	—
106	C₂×C₈	—	59	1	no	—
107	C₂×C₈	—	118	1	no	—
108	C₂×C₈	—	236	2	no	—
109	C₂×C₈	#11745687	1	1	yes	characteristic
110	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	4	2	no	—
113	C₄²	—	4	2	no	—
114	C₄²	D₄×D₅₉	1	1	yes	characteristic
115	C₁₆	—	2	1	no	—
116	C₁₆	—	118	2	no	—
117	C₁₆	#11745664	1	2	yes	—
118	Q₈	—	4	2	no	—
119	Q₈	—	4	2	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	D₄×Dic₅₉	1	1	yes	characteristic
123	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
124	C₂×C₄	#14273746	1	1	yes	characteristic
125	C₈	—	2	1	no	—
126	C₈	—	59	2	no	—
127	C₈	—	118	2	no	—
128	C₈	—	118	1	no	—
129	C₈	—	472	2	no	—
130	C₈	C₂×#11745664	1	1	yes	characteristic
131	C₈	#14273734	1	1	yes	characteristic
132	C₂²	D₈×Dic₅₉	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	8	2	no	—
135	C₄	—	8	2	no	—
136	C₄	#16837480	1	1	yes	characteristic
137	C₄	#16837519	1	1	yes	characteristic
138	C₄	Q₃₂×D₅₉	1	1	yes	characteristic
139	C₄	#16837835	1	1	yes	characteristic
140	C₂	#21422981	1	1	yes	characteristic
141	C₂	Q₃₂×Dic₅₉	1	1	yes	characteristic
142	C₂	#21425977	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21422993

C₂

yes

maximal

#21422168

C₂

yes

maximal, characteristic

#21422060

C₂

yes

maximal

C₁₆×C₅₉⋊C₈

C₂

yes

maximal, characteristic

C₄×Q₃₂×C₅₉

C₂

yes

maximal, characteristic

#16837533

—

C₄×C₅₉⋊C₁₆

C₂²

yes

characteristic

#16837155

C₂²

yes

—

Q₈⋊(C₅₉⋊C₈)

—

C₈×C₅₉⋊C₈

C₂²

yes

characteristic

C₂×Q₃₂×C₅₉

C₄

yes

characteristic

C₄×Q₁₆×C₅₉

C₂²

yes

—

C₅₉×#366

C₄

yes

characteristic

C₅₉×#358

C₄

yes

—

C₄×C₁₆×C₅₉

C₂²

yes

characteristic

C₂×C₅₉⋊C₁₆

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

#14273638

—

C₄×C₅₉⋊C₈

D₄

yes

characteristic

C₄×C₅₉⋊C₈

D₄

yes

characteristic

C₂×Q₁₆×C₅₉

C₂×C₄

yes

—

C₄×Q₈×C₅₉

—

Q₃₂×C₅₉

—

C₂×C₁₆×C₅₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₉

C₂×C₄

yes

characteristic

#107×C₅₉

C₂×C₄

yes

—

#103×C₅₉

—

C₄×C₈×C₅₉

C₂³

yes

characteristic

C₂×Q₈×C₅₉

—

Q₁₆×C₅₉

—

Q₁₆×C₅₉

—

C₂×C₈×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₉

#41

yes

characteristic

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

D₈

yes

characteristic

C₂×C₅₉⋊C₈

D₈

yes

characteristic

C₁₆×C₅₉

—

C₁₆×C₅₉

M₁₆

yes

—

C₅₉⋊C₁₆

—

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₄×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₈

yes

characteristic

C₂×C₄×C₅₉

#135

yes

characteristic

C₈×C₅₉

—

C₈×C₅₉

C₂×M₁₆

yes

characteristic

C₈×C₅₉

#131

yes

characteristic

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

Q₃₂

yes

complemented

#12083

—

maximal, Sylow

C₂²×C₅₉

C₄×D₈

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

C₂×Q₃₂

yes

characteristic

C₄×C₅₉

#508

yes

characteristic

C₄×Q₃₂

D₅₉

yes

—

—

—

C₈×C₁₆

—

C₂×C₅₉

#1589

yes

characteristic

C₂×C₅₉

C₄×Q₃₂

yes

characteristic

C₂×C₅₉

#2178

yes

characteristic

C₂×Q₃₂

Dic₅₉

yes

characteristic

#446

—

118

—

C₄×Q₁₆

C₂×D₅₉

yes

—

#366

Dic₅₉

yes

characteristic

#358

Dic₅₉

yes

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₅₉

yes

characteristic

#335

—

Q₈⋊C₈

—

118

—

C₈²

—

C₅₉

#12083

yes

characteristic, complemented, Sylow

C₂×Q₁₆

C₂×Dic₅₉

yes

—

C₄×Q₈

—

Q₃₂

—

C₂×C₁₆

—

118

—

C₂×C₁₆

C₂×Dic₅₉

yes

characteristic

C₂×C₁₆

C₂×Dic₅₉

yes

characteristic

#107

C₂×Dic₅₉

yes

—

C₄⋊C₈

—

118

—

C₄⋊C₈

—

236

—

#103

—

#97

—

236

—

C₄×C₈

—

100

C₄×C₈

—

101

C₄×C₈

C₂²×D₅₉

yes

characteristic

102

C₂×Q₈

—

103

Q₁₆

—

104

Q₁₆

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

118

—

108

C₂×C₈

—

236

—

109

C₂×C₈

#11745687

yes

characteristic

110

C₂×C₈

C₂²×Dic₅₉

yes

characteristic

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄²

—

114

C₄²

D₄×D₅₉

yes

characteristic

115

C₁₆

—

116

C₁₆

—

118

—

117

C₁₆

#11745664

yes

—

118

Q₈

—

119

Q₈

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

D₄×Dic₅₉

yes

characteristic

123

C₂×C₄

D₈×D₅₉

yes

characteristic

124

C₂×C₄

#14273746

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

118

—

128

C₈

—

118

—

129

C₈

—

472

—

130

C₈

C₂×#11745664

yes

characteristic

131

C₈

#14273734

yes

characteristic

132

C₂²

D₈×Dic₅₉

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

#16837480

yes

characteristic

137

C₄

#16837519

yes

characteristic

138

C₄

Q₃₂×D₅₉

yes

characteristic

139

C₄

#16837835

yes

characteristic

140

C₂

#21422981

yes

characteristic

141

C₂

Q₃₂×Dic₅₉

yes

characteristic

142

C₂

#21425977

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₅₉
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₄×Q₃₂×C₅₉	C₂
socle, prime radical	C₂²×C₅₉	C₄×D₈
cosocle	C₄×C₈×C₅₉	C₂³
hypercenter	C₄×Q₃₂	D₅₉
hypocenter	C₅₉	#12083
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

subgroup

G/H

center

C₂×C₄

D₈×D₅₉

commutator

C₂×C₈×C₅₉

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₅₉

Fitting

C₄×Q₃₂×C₅₉

C₂

socle, prime radical

C₂²×C₅₉

C₄×D₈

cosocle

C₄×C₈×C₅₉

C₂³

hypercenter

C₄×Q₃₂

D₅₉

hypocenter

C₅₉

#12083

LCS

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

N	H ≅ G/N	type
59. C₅₉⋊C₈	Q₃₂	semidirect
87. C₅₉	#12083	semidirect

H ≅ G/N

type

59. C₅₉⋊C₈

Q₃₂

semidirect

87. C₅₉

#12083

semidirect

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

subgroup	index
#21422993	2
#21422168	2
#21422060	2
C₁₆×C₅₉⋊C₈	2
C₄×Q₃₂×C₅₉	2
#12083	59

subgroup

index

#21422993

#21422168

#21422060

C₁₆×C₅₉⋊C₈

C₄×Q₃₂×C₅₉

#12083

Subgroups — #28500680

Conjugacy classes of subgroups

Complements

Maximal subgroups