Group #28505080 — subgroups

Subgroups — #28505080

Order 15104.

274 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 115 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	C₂×#16837177	C₂	1	1	yes	maximal, characteristic
2	#21424266	C₂	1	2	yes	maximal
3	#21422190	C₂	1	1	yes	maximal, characteristic
4	#21422189	C₂	1	2	yes	maximal
5	C₅₉×#1381	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂×#14273663	C₂²	1	2	yes	—
8	C₂×#14273662	—	2	2	no	—
9	#16837425	—	2	2	no	—
10	C₂×C₈×Dic₅₉	C₂²	1	1	yes	characteristic
11	#16837177	C₄	1	4	yes	—
12	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
13	C₂×#108×C₅₉	C₂²	1	2	yes	—
14	C₂×#11745664	—	2	2	no	—
15	C₂²×C₅₉⋊C₈	D₄	1	1	yes	characteristic
16	C₂×C₄×Dic₅₉	D₄	1	1	yes	characteristic
17	C₂×C₅₉⋊C₁₆	—	2	2	no	—
18	C₂×C₅₉⋊C₁₆	C₂×C₄	1	2	yes	—
19	#14273663	—	2	2	no	—
20	#14273663	—	2	2	no	—
21	#14273662	—	4	4	no	—
22	Dic₅₉⋊C₈	—	4	4	no	—
23	C₈×Dic₅₉	C₂×C₄	1	2	yes	—
24	C₈×Dic₅₉	Q₈	1	2	yes	—
25	C₂×M₁₆×C₅₉	—	2	2	no	—
26	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
27	C₂×C₁₆×C₅₉	—	2	2	no	—
28	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
29	#108×C₅₉	—	2	2	no	—
30	#108×C₅₉	—	2	2	no	—
31	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
32	C₂²×Dic₅₉	Q₁₆	1	2	yes	—
33	M₁₆×C₅₉	—	4	2	no	—
34	M₁₆×C₅₉	—	4	2	no	—
35	C₂×C₈×C₅₉	—	4	2	no	—
36	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₈×C₅₉	#41	1	2	yes	—
39	C₂×C₈×C₅₉	#41	1	2	yes	—
40	C₄×Dic₅₉	C₄⋊C₄	1	2	yes	—
41	C₄×Dic₅₉	D₈	1	2	yes	—
42	#11745664	—	4	2	no	—
43	#11745664	—	4	2	no	—
44	C₂×C₅₉⋊C₈	—	4	2	no	—
45	C₂×C₅₉⋊C₈	C₄⋊C₄	1	2	yes	—
46	C₂×C₅₉⋊C₈	SD₁₆	1	4	yes	—
47	C₁₆×C₅₉	—	2	2	no	—
48	C₅₉⋊C₁₆	—	2	2	no	—
49	C₂³×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
50	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₅₉	#124	1	2	yes	—
52	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
53	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
54	C₂×Dic₅₉	—	4	2	no	—
55	C₂×Dic₅₉	#107	1	4	yes	—
56	C₈×C₅₉	—	8	2	no	—
57	C₈×C₅₉	#117	1	1	yes	characteristic
58	C₈×C₅₉	#117	1	1	yes	characteristic
59	C₈×C₅₉	#125	1	2	yes	—
60	C₅₉⋊C₈	—	8	2	no	—
61	C₅₉⋊C₈	#106	1	4	yes	—
62	#12892	—	59	1	no	maximal, Sylow
63	C₂²×C₅₉	—	2	2	no	—
64	C₂²×C₅₉	C₂×#107	1	1	yes	characteristic
65	C₂²×C₅₉	#487	1	2	yes	—
66	C₄×C₅₉	#382	1	1	yes	characteristic
67	C₄×C₅₉	C₂×#106	1	1	yes	characteristic
68	C₄×C₅₉	#486	1	2	yes	—
69	Dic₅₉	—	2	4	no	—
70	C₂×#346	—	59	1	no	—
71	#1848	—	59	2	no	—
72	#1381	—	59	1	no	—
73	#1381	—	59	2	no	—
74	#1381	D₅₉	1	1	yes	characteristic
75	C₂×C₅₉	—	2	2	no	—
76	C₂×C₅₉	#1834	1	1	yes	characteristic
77	C₂×C₅₉	#2161	1	2	yes	—
78	C₂²×C₁₆	—	59	1	no	—
79	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
80	C₂×#108	—	59	2	no	—
81	C₂×#108	—	118	2	no	—
82	C₂×#108	C₂×D₅₉	1	2	yes	—
83	#423	—	118	2	no	—
84	C₂×C₄×C₈	—	59	1	no	—
85	#346	—	59	4	no	—
86	C₅₉	#12892	1	1	yes	characteristic, complemented, Sylow
87	C₂×M₁₆	—	2	2	no	—
88	C₂×M₁₆	—	118	2	no	—
89	C₂²×C₈	—	59	1	no	—
90	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
91	C₂×C₄²	—	59	1	no	—
92	C₂×C₁₆	—	2	2	no	—
93	C₂×C₁₆	—	59	2	no	—
94	C₂×C₁₆	—	118	2	no	—
95	C₂×C₁₆	C₄×D₅₉	1	2	yes	—
96	#108	—	2	2	no	—
97	#108	—	2	2	no	—
98	#108	—	118	2	no	—
99	#108	—	118	2	no	—
100	#108	—	236	4	no	—
101	C₄⋊C₈	—	236	4	no	—
102	C₄×C₈	—	59	2	no	—
103	C₄×C₈	—	59	2	no	—
104	C₂²×C₄	—	59	2	no	—
105	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
106	M₁₆	—	4	2	no	—
107	M₁₆	—	4	2	no	—
108	M₁₆	—	236	2	no	—
109	M₁₆	—	236	2	no	—
110	C₂×C₈	—	4	2	no	—
111	C₂×C₈	—	59	2	no	—
112	C₂×C₈	—	59	4	no	—
113	C₂×C₈	—	236	2	no	—
114	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
115	C₂×C₈	#11745685	1	2	yes	—
116	C₂×C₈	#11745687	1	2	yes	—
117	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
118	C₄²	—	59	2	no	—
119	C₄²	—	59	2	no	—
120	C₁₆	—	2	2	no	—
121	C₁₆	—	118	2	no	—
122	C₂³	Q₁₆×D₅₉	1	1	yes	characteristic
123	C₂×C₄	—	59	4	no	—
124	C₂×C₄	—	236	2	no	—
125	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
126	C₂×C₄	#14273713	1	2	yes	—
127	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
128	C₂×C₄	SD₁₆×D₅₉	1	2	yes	—
129	C₈	—	8	2	no	—
130	C₈	—	59	4	no	—
131	C₈	—	472	2	no	—
132	C₈	#14273683	1	1	yes	characteristic
133	C₈	#14273684	1	1	yes	characteristic
134	C₈	#14273717	1	2	yes	—
135	C₂²	—	2	2	no	—
136	C₂²	#107×D₅₉	1	1	yes	characteristic
137	C₂²	#16837751	1	2	yes	—
138	C₄	—	118	4	no	—
139	C₄	#16837266	1	1	yes	characteristic
140	C₄	#106×D₅₉	1	1	yes	characteristic
141	C₄	#16837747	1	2	yes	—
142	C₂	—	2	2	no	—
143	C₂	#21424222	1	1	yes	characteristic
144	C₂	#21425889	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16837177

C₂

yes

maximal, characteristic

#21424266

C₂

yes

maximal

#21422190

C₂

yes

maximal, characteristic

#21422189

C₂

yes

maximal

C₅₉×#1381

C₂

yes

maximal, characteristic

C₂²×C₅₉⋊C₁₆

C₂²

yes

characteristic

C₂×#14273663

C₂²

yes

—

C₂×#14273662

—

#16837425

—

C₂×C₈×Dic₅₉

C₂²

yes

characteristic

#16837177

C₄

yes

—

C₂²×C₁₆×C₅₉

C₂²

yes

characteristic

C₂×#108×C₅₉

C₂²

yes

—

C₂×#11745664

—

C₂²×C₅₉⋊C₈

D₄

yes

characteristic

C₂×C₄×Dic₅₉

D₄

yes

characteristic

C₂×C₅₉⋊C₁₆

—

C₂×C₅₉⋊C₁₆

C₂×C₄

yes

—

#14273663

—

#14273663

—

#14273662

—

Dic₅₉⋊C₈

—

C₈×Dic₅₉

C₂×C₄

yes

—

C₈×Dic₅₉

Q₈

yes

—

C₂×M₁₆×C₅₉

—

C₂²×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₁₆×C₅₉

—

C₂×C₁₆×C₅₉

C₂×C₄

yes

—

#108×C₅₉

—

#108×C₅₉

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×Dic₅₉

Q₁₆

yes

—

M₁₆×C₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₉

#41

yes

—

C₂×C₈×C₅₉

#41

yes

—

C₄×Dic₅₉

C₄⋊C₄

yes

—

C₄×Dic₅₉

D₈

yes

—

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₄⋊C₄

yes

—

C₂×C₅₉⋊C₈

SD₁₆

yes

—

C₁₆×C₅₉

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

#124

yes

—

C₂×C₄×C₅₉

C₂×D₈

yes

characteristic

C₂×C₄×C₅₉

C₂×SD₁₆

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

#107

yes

—

C₈×C₅₉

—

C₈×C₅₉

yes

C₈×C₅₉

yes

C₈×C₅₉

yes

—

C₅₉⋊C₈

—

C₅₉⋊C₈

#106

yes

—

#12892

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

C₂×#107

yes

characteristic

C₂²×C₅₉

#487

yes

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

Dic₅₉

—

—

—

—

—

D₅₉

yes

characteristic

C₂×C₅₉

—

C₂×C₅₉

#1834

yes

characteristic

C₂×C₅₉

#2161

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₅₉

yes

characteristic

C₂×#108

—

C₂×#108

—

118

—

C₂×#108

C₂×D₅₉

yes

—

#423

—

118

—

C₂×C₄×C₈

—

#346

—

C₅₉

#12892

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

118

—

C₂²×C₈

—

C₂²×C₈

C₂²×D₅₉

yes

characteristic

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

118

—

C₂×C₁₆

C₄×D₅₉

yes

—

#108

—

#108

—

#108

—

118

—

#108

—

118

—

100

#108

—

236

—

101

C₄⋊C₈

—

236

—

102

C₄×C₈

—

103

C₄×C₈

—

104

C₂²×C₄

—

105

C₂²×C₄

D₄×D₅₉

yes

characteristic

106

M₁₆

—

107

M₁₆

—

108

M₁₆

—

236

—

109

M₁₆

—

236

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

236

—

114

C₂×C₈

C₂×C₄×D₅₉

yes

characteristic

115

C₂×C₈

#11745685

yes

—

116

C₂×C₈

#11745687

yes

—

117

C₂×C₈

Q₈×D₅₉

yes

characteristic

118

C₄²

—

119

C₄²

—

120

C₁₆

—

121

C₁₆

—

118

—

122

C₂³

Q₁₆×D₅₉

yes

characteristic

123

C₂×C₄

—

124

C₂×C₄

—

236

—

125

C₂×C₄

C₄⋊C₄×D₅₉

yes

characteristic

126

C₂×C₄

#14273713

yes

—

127

C₂×C₄

D₈×D₅₉

yes

characteristic

128

C₂×C₄

SD₁₆×D₅₉

yes

—

129

C₈

—

130

C₈

—

131

C₈

—

472

—

132

C₈

#14273683

yes

characteristic

133

C₈

#14273684

yes

characteristic

134

C₈

#14273717

yes

—

135

C₂²

—

136

C₂²

#107×D₅₉

yes

characteristic

137

C₂²

#16837751

yes

—

138

C₄

—

118

—

139

C₄

#16837266

yes

characteristic

140

C₄

#106×D₅₉

yes

characteristic

141

C₄

#16837747

yes

—

142

C₂

—

143

C₂

#21424222

yes

characteristic

144

C₂

#21425889

yes

—

	subgroup	G/H
center	C₂×C₄	D₈×D₅₉
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#1381	C₂
socle	C₂²×C₅₉	C₂×#107
prime radical	C₂³×C₅₉	C₂×Q₁₆
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#1381	D₅₉
hypocenter	C₅₉	#12892
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

subgroup

G/H

center

C₂×C₄

D₈×D₅₉

commutator

C₂×C₈×C₅₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₉

Fitting

C₅₉×#1381

C₂

socle

C₂²×C₅₉

C₂×#107

prime radical

C₂³×C₅₉

C₂×Q₁₆

cosocle

C₂²×C₈×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

N	H ≅ G/N	type
86. C₅₉	#12892	semidirect

H ≅ G/N

type

86. C₅₉

#12892

semidirect

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

subgroup	index
C₂×#16837177	2
#21424266	2
#21422190	2
#21422189	2
C₅₉×#1381	2
#12892	59

subgroup

index

C₅₉×#1381

Subgroups — #28505080

Conjugacy classes of subgroups

Complements

Maximal subgroups