Group #28500482 — subgroups

Subgroups — #28500482

Order 15104.

218 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 95 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	#21425998	C₂	1	2	yes	maximal
2	C₈×Dic₂₃₆	C₂	1	1	yes	maximal, characteristic
3	#21422074	C₂	1	2	yes	maximal
4	C₈×C₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₅₉×#1370	C₂	1	1	yes	maximal, characteristic
6	C₈×Dic₁₁₈	C₂²	1	2	yes	—
7	C₄×Dic₂₃₆	C₄	1	1	yes	characteristic
8	Dic₅₉⋊C₁₆	—	2	2	no	—
9	C₄×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₄×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₁₆×Dic₅₉	—	2	2	no	—
12	#16837155	C₄	1	1	yes	characteristic
13	#16837134	C₄	1	2	yes	—
14	C₅₉×C₄⋊C₁₆	C₂²	1	2	yes	—
15	C₈²×C₅₉	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₃₆	C₈	1	1	yes	characteristic
17	C₄×Dic₁₁₈	C₂×C₄	1	2	yes	—
18	C₂×C₅₉⋊C₁₆	—	2	1	no	—
19	C₂×C₅₉⋊C₁₆	D₄	1	1	yes	characteristic
20	C₂×C₅₉⋊C₁₆	D₄	1	1	yes	characteristic
21	#14273661	C₈	1	1	yes	characteristic
22	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	2	yes	—
23	Dic₅₉⋊C₈	—	2	2	no	—
24	#14273652	C₈	1	2	yes	—
25	C₈×Dic₅₉	—	2	2	no	—
26	C₂×C₁₆×C₅₉	—	2	2	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	C₂³	1	1	yes	characteristic
30	C₂×Dic₁₁₈	C₂×C₈	1	2	yes	—
31	C₂×C₈×C₅₉	—	2	1	no	—
32	C₂×C₈×C₅₉	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈×C₅₉	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
35	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
36	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
37	C₄⋊Dic₅₉	C₂×C₈	1	2	yes	—
38	Dic₅₉⋊C₄	—	2	2	no	—
39	C₄×Dic₅₉	—	2	2	no	—
40	C₂×C₅₉⋊C₈	—	2	2	no	—
41	Dic₂₃₆	C₁₆	1	4	yes	complemented
42	C₁₆×C₅₉	—	4	2	no	—
43	C₅₉⋊C₁₆	—	2	2	no	—
44	C₅₉⋊C₁₆	—	2	1	no	—
45	C₅₉⋊C₁₆	Q₁₆	1	2	yes	complemented
46	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
47	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
48	C₂×C₄×C₅₉	#131	1	1	yes	characteristic
49	C₂×Dic₅₉	—	2	2	no	—
50	C₂×Dic₅₉	—	2	2	no	—
51	Dic₁₁₈	—	2	2	no	—
52	Dic₁₁₈	C₂×C₁₆	1	4	yes	—
53	C₈×C₅₉	—	2	2	no	—
54	C₈×C₅₉	—	2	1	no	—
55	C₈×C₅₉	—	2	1	no	—
56	C₈×C₅₉	C₂×C₁₆	1	2	yes	—
57	C₈×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
58	C₈×C₅₉	#135	1	1	yes	characteristic
59	C₅₉⋊C₈	—	4	2	no	—
60	C₁₆×Q₁₆	—	59	1	no	maximal, Sylow
61	C₂²×C₅₉	C₈×D₄	1	1	yes	characteristic
62	C₄×C₅₉	—	2	1	no	—
63	C₄×C₅₉	C₄×Q₁₆	1	1	yes	characteristic
64	C₄×C₅₉	#443	1	1	yes	characteristic
65	C₄×C₅₉	C₂²×C₁₆	1	1	yes	characteristic
66	C₄×C₅₉	#504	1	1	yes	characteristic
67	Dic₅₉	—	2	4	no	—
68	Dic₅₉	—	4	2	no	—
69	Q₈×C₁₆	—	59	2	no	—
70	C₈×Q₁₆	—	59	1	no	—
71	#1370	D₅₉	1	1	yes	characteristic
72	Q₈⋊C₁₆	—	59	2	no	—
73	C₈×C₁₆	—	59	1	no	—
74	C₂×C₅₉	C₈×Q₁₆	1	1	yes	characteristic
75	C₂×C₅₉	D₄×C₁₆	1	1	yes	characteristic
76	C₂×C₅₉	#2170	1	1	yes	characteristic
77	C₈×Q₈	—	59	2	no	—
78	C₄×Q₁₆	—	59	1	no	—
79	C₄⋊C₁₆	—	118	2	no	—
80	C₄⋊C₁₆	C₂×D₅₉	1	2	yes	—
81	C₄×C₁₆	—	59	1	no	—
82	C₄×C₁₆	—	59	1	no	—
83	C₄×C₁₆	—	118	2	no	—
84	#335	—	59	1	no	—
85	Q₈⋊C₈	—	59	2	no	—
86	C₈²	C₂×D₅₉	1	1	yes	characteristic
87	C₅₉	C₁₆×Q₁₆	1	1	yes	characteristic, complemented, Sylow
88	C₂×Q₁₆	—	59	1	no	—
89	C₄×Q₈	—	59	2	no	—
90	C₂×C₁₆	—	2	2	no	—
91	C₂×C₁₆	—	59	1	no	—
92	C₂×C₁₆	—	59	1	no	—
93	C₂×C₁₆	—	118	1	no	—
94	#107	—	59	1	no	—
95	C₄⋊C₈	—	59	2	no	—
96	C₄⋊C₈	—	118	2	no	—
97	#103	—	59	2	no	—
98	C₄×C₈	—	118	2	no	—
99	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
100	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
101	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
102	C₂×Q₈	—	59	2	no	—
103	Q₁₆	—	59	4	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	118	2	no	—
106	C₂×C₈	C₈×D₅₉	1	1	yes	characteristic
107	C₂×C₈	C₈×D₅₉	1	1	yes	characteristic
108	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
109	C₂×C₈	#11745689	1	1	yes	characteristic
110	C₄⋊C₄	—	59	2	no	—
111	C₄⋊C₄	—	118	2	no	—
112	C₄²	—	118	2	no	—
113	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
114	C₁₆	—	4	2	no	—
115	C₁₆	—	59	2	no	—
116	C₁₆	—	118	2	no	—
117	C₁₆	—	118	1	no	—
118	Q₈	—	59	4	no	—
119	Q₈	—	118	2	no	—
120	C₂×C₄	—	118	2	no	—
121	C₂×C₄	—	118	2	no	—
122	C₂×C₄	#14273689	1	1	yes	characteristic
123	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
124	C₂×C₄	#14273735	1	1	yes	characteristic
125	C₈	—	2	2	no	—
126	C₈	—	2	1	no	—
127	C₈	—	2	1	no	—
128	C₈	—	236	2	no	—
129	C₈	C₁₆×D₅₉	1	2	yes	—
130	C₈	Q₁₆×D₅₉	1	1	yes	characteristic
131	C₈	#14273746	1	1	yes	characteristic
132	C₂²	#16837467	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	118	4	no	—
135	C₄	—	236	2	no	—
136	C₄	#16837496	1	1	yes	characteristic
137	C₄	#16837518	1	1	yes	characteristic
138	C₄	C₂×C₁₆×D₅₉	1	1	yes	characteristic
139	C₄	#16837820	1	1	yes	characteristic
140	C₂	#21422902	1	1	yes	characteristic
141	C₂	#21425915	1	1	yes	characteristic
142	C₂	#21425932	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21425998

C₂

yes

maximal

C₈×Dic₂₃₆

C₂

yes

maximal, characteristic

#21422074

C₂

yes

maximal

C₈×C₅₉⋊C₁₆

C₂

yes

maximal, characteristic

C₅₉×#1370

C₂

yes

maximal, characteristic

C₈×Dic₁₁₈

C₂²

yes

—

C₄×Dic₂₃₆

C₄

yes

characteristic

Dic₅₉⋊C₁₆

—

C₄×C₅₉⋊C₁₆

C₂²

yes

characteristic

C₄×C₅₉⋊C₁₆

C₂²

yes

characteristic

C₁₆×Dic₅₉

—

#16837155

C₄

yes

characteristic

#16837134

C₄

yes

—

C₅₉×C₄⋊C₁₆

C₂²

yes

—

C₈²×C₅₉

C₂²

yes

characteristic

C₂×Dic₂₃₆

C₈

yes

characteristic

C₄×Dic₁₁₈

C₂×C₄

yes

—

C₂×C₅₉⋊C₁₆

—

C₂×C₅₉⋊C₁₆

D₄

yes

characteristic

C₂×C₅₉⋊C₁₆

D₄

yes

characteristic

#14273661

C₈

yes

characteristic

C₄⋊(C₅₉⋊C₈)

C₂×C₄

yes

—

Dic₅₉⋊C₈

—

#14273652

C₈

yes

—

C₈×Dic₅₉

—

C₂×C₁₆×C₅₉

—

C₄×C₈×C₅₉

C₂×C₄

yes

characteristic

C₄×C₈×C₅₉

C₂×C₄

yes

characteristic

C₄×C₈×C₅₉

C₂³

yes

characteristic

C₂×Dic₁₁₈

C₂×C₈

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂×C₈

yes

characteristic

C₂×C₈×C₅₉

C₂×C₈

yes

characteristic

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

#41

yes

characteristic

C₄²×C₅₉

C₂²×C₄

yes

characteristic

C₄⋊Dic₅₉

C₂×C₈

yes

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

Dic₂₃₆

C₁₆

yes

complemented

C₁₆×C₅₉

—

C₅₉⋊C₁₆

—

C₅₉⋊C₁₆

—

C₅₉⋊C₁₆

Q₁₆

yes

complemented

C₂×C₄×C₅₉

C₄×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₉

#131

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

C₂×C₁₆

yes

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

C₂×C₁₆

yes

—

C₈×C₅₉

C₂×Q₁₆

yes

characteristic

C₈×C₅₉

#135

yes

characteristic

C₅₉⋊C₈

—

C₁₆×Q₁₆

—

maximal, Sylow

C₂²×C₅₉

C₈×D₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

C₄×Q₁₆

yes

characteristic

C₄×C₅₉

#443

yes

characteristic

C₄×C₅₉

C₂²×C₁₆

yes

characteristic

C₄×C₅₉

#504

yes

characteristic

Dic₅₉

—

Dic₅₉

—

Q₈×C₁₆

—

C₈×Q₁₆

—

#1370

D₅₉

yes

characteristic

Q₈⋊C₁₆

—

C₈×C₁₆

—

C₂×C₅₉

C₈×Q₁₆

yes

characteristic

C₂×C₅₉

D₄×C₁₆

yes

characteristic

C₂×C₅₉

#2170

yes

characteristic

C₈×Q₈

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

118

—

C₄⋊C₁₆

C₂×D₅₉

yes

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

118

—

#335

—

Q₈⋊C₈

—

C₈²

C₂×D₅₉

yes

characteristic

C₅₉

C₁₆×Q₁₆

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

118

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

118

—

#103

—

C₄×C₈

—

118

—

C₄×C₈

C₄×D₅₉

yes

characteristic

100

C₄×C₈

C₄×D₅₉

yes

characteristic

101

C₄×C₈

C₂²×D₅₉

yes

characteristic

102

C₂×Q₈

—

103

Q₁₆

—

104

C₂×C₈

—

105

C₂×C₈

—

118

—

106

C₂×C₈

C₈×D₅₉

yes

characteristic

107

C₂×C₈

C₈×D₅₉

yes

characteristic

108

C₂×C₈

D₄×D₅₉

yes

characteristic

109

C₂×C₈

#11745689

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

—

118

—

112

C₄²

—

118

—

113

C₄²

C₂×C₄×D₅₉

yes

characteristic

114

C₁₆

—

115

C₁₆

—

116

C₁₆

—

118

—

117

C₁₆

—

118

—

118

Q₈

—

119

Q₈

—

118

—

120

C₂×C₄

—

118

—

121

C₂×C₄

—

118

—

122

C₂×C₄

#14273689

yes

characteristic

123

C₂×C₄

C₂×C₈×D₅₉

yes

characteristic

124

C₂×C₄

#14273735

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

236

—

129

C₈

C₁₆×D₅₉

yes

—

130

C₈

Q₁₆×D₅₉

yes

characteristic

131

C₈

#14273746

yes

characteristic

132

C₂²

#16837467

yes

characteristic

133

C₄

—

134

C₄

—

118

—

135

C₄

—

236

—

136

C₄

#16837496

yes

characteristic

137

C₄

#16837518

yes

characteristic

138

C₄

C₂×C₁₆×D₅₉

yes

characteristic

139

C₄

#16837820

yes

characteristic

140

C₂

#21422902

yes

characteristic

141

C₂

#21425915

yes

characteristic

142

C₂

#21425932

yes

characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₅₉
commutator	C₄×C₅₉	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₅₉×#1370	C₂
socle, prime radical	C₂²×C₅₉	C₈×D₄
cosocle	C₄×C₈×C₅₉	C₂³
hypercenter	#1370	D₅₉
hypocenter	C₅₉	C₁₆×Q₁₆
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1370

subgroup

G/H

center

C₂×C₈

D₄×D₅₉

commutator

C₄×C₅₉

C₂²×C₁₆

Frattini

C₄×C₈

C₂²×D₅₉

Fitting

C₅₉×#1370

C₂

socle, prime radical

C₂²×C₅₉

C₈×D₄

cosocle

C₄×C₈×C₅₉

C₂³

hypercenter

#1370

D₅₉

hypocenter

C₅₉

C₁₆×Q₁₆

LCS

G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1370

N	H ≅ G/N	type
41. Dic₂₃₆	C₁₆	semidirect
45. C₅₉⋊C₁₆	Q₁₆	semidirect
87. C₅₉	C₁₆×Q₁₆	semidirect

H ≅ G/N

type

41. Dic₂₃₆

C₁₆

semidirect

45. C₅₉⋊C₁₆

Q₁₆

semidirect

87. C₅₉

C₁₆×Q₁₆

semidirect

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

subgroup	index
#21425998	2
C₈×Dic₂₃₆	2
#21422074	2
C₈×C₅₉⋊C₁₆	2
C₅₉×#1370	2
C₁₆×Q₁₆	59

subgroup

index

#21425998

C₈×Dic₂₃₆

#21422074

C₈×C₅₉⋊C₁₆

C₅₉×#1370

C₁₆×Q₁₆

Subgroups — #28500482

Conjugacy classes of subgroups

Complements

Maximal subgroups