Group #28500679 — subgroups

Subgroups — #28500679

Order 15104.

210 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 77 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	C₄×Dic₄₇₂	C₂	1	1	yes	maximal, characteristic
2	#21422992	C₂	1	2	yes	maximal
3	#21422059	C₂	1	2	yes	maximal
4	C₁₆×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₅₉×#1371	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₇₂	C₄	1	1	yes	characteristic
7	#16837529	—	2	2	no	—
8	C₄×Dic₂₃₆	C₂²	1	2	yes	—
9	#16837228	C₄	1	1	yes	characteristic
10	#16837207	C₄	1	2	yes	—
11	C₄×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
12	#16837132	—	2	2	no	—
13	C₈×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
15	C₅₉×#335	C₂²	1	2	yes	—
16	C₂×Dic₂₃₆	C₂×C₄	1	2	yes	—
17	C₄×Dic₁₁₈	—	2	2	no	—
18	Dic₄₇₂	—	2	2	no	—
19	C₂×C₅₉⋊C₁₆	—	2	1	no	—
20	#14273661	C₂×C₄	1	2	yes	—
21	Dic₅₉⋊C₈	—	4	2	no	—
22	#14273652	—	2	2	no	—
23	#14273637	—	4	2	no	—
24	C₄×C₅₉⋊C₈	D₄	1	1	yes	characteristic
25	C₄×C₅₉⋊C₈	D₄	1	1	yes	characteristic
26	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
28	C₄⋊C₈×C₅₉	—	2	2	no	—
29	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
30	C₂×Dic₁₁₈	—	2	2	no	—
31	C₂×C₈×C₅₉	—	4	2	no	—
32	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
34	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
35	C₄⋊Dic₅₉	—	2	2	no	—
36	Dic₅₉⋊C₄	—	4	2	no	—
37	C₄×Dic₅₉	—	4	2	no	—
38	C₂×C₅₉⋊C₈	—	2	1	no	—
39	C₂×C₅₉⋊C₈	D₈	1	1	yes	characteristic
40	C₂×C₅₉⋊C₈	D₈	1	1	yes	characteristic
41	Dic₂₃₆	—	2	2	no	—
42	Dic₂₃₆	—	2	2	no	—
43	C₁₆×C₅₉	—	2	1	no	—
44	C₁₆×C₅₉	M₁₆	1	2	yes	—
45	C₅₉⋊C₁₆	—	2	2	no	—
46	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
47	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
48	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
49	C₂×Dic₅₉	—	4	2	no	—
50	C₂×Dic₅₉	—	4	2	no	—
51	Dic₁₁₈	—	4	2	no	—
52	Dic₁₁₈	—	4	2	no	—
53	C₈×C₅₉	—	2	1	no	—
54	C₈×C₅₉	—	8	2	no	—
55	C₈×C₅₉	C₂×M₁₆	1	1	yes	characteristic
56	C₈×C₅₉	#131	1	1	yes	characteristic
57	C₅₉⋊C₈	—	2	2	no	—
58	C₅₉⋊C₈	—	2	1	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	#436	1	1	yes	characteristic
64	C₄×C₅₉	#443	1	1	yes	characteristic
65	C₄×C₅₉	C₂×Q₃₂	1	1	yes	characteristic
66	C₄×C₅₉	#508	1	1	yes	characteristic
67	Dic₅₉	—	8	2	no	—
68	Dic₅₉	—	8	2	no	—
69	C₄×Q₃₂	—	59	1	no	—
70	#1591	—	59	2	no	—
71	#1371	D₅₉	1	1	yes	characteristic
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	—	59	1	no	—
78	#446	—	118	2	no	—
79	C₄×Q₁₆	—	59	2	no	—
80	#366	—	59	1	no	—
81	#358	—	59	2	no	—
82	C₄×C₁₆	—	59	1	no	—
83	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
84	#335	C₂×D₅₉	1	2	yes	—
85	Q₈⋊C₈	—	118	2	no	—
86	C₈²	—	59	1	no	—
87	C₅₉	#12083	1	1	yes	characteristic, complemented, Sylow
88	C₂×Q₁₆	—	59	2	no	—
89	C₄×Q₈	—	118	2	no	—
90	Q₃₂	—	118	2	no	—
91	C₂×C₁₆	—	118	1	no	—
92	C₂×C₁₆	C₄×D₅₉	1	1	yes	characteristic
93	C₂×C₁₆	C₄×D₅₉	1	1	yes	characteristic
94	#107	—	59	2	no	—
95	C₄⋊C₈	—	2	2	no	—
96	C₄⋊C₈	—	236	2	no	—
97	#103	—	118	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	—	118	2	no	—
103	Q₁₆	—	118	2	no	—
104	Q₁₆	—	118	2	no	—
105	C₂×C₈	—	4	2	no	—
106	C₂×C₈	—	59	1	no	—
107	C₂×C₈	—	59	1	no	—
108	C₂×C₈	—	118	1	no	—
109	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
110	C₂×C₈	#11745689	1	1	yes	characteristic
111	C₄⋊C₄	—	118	2	no	—
112	C₄⋊C₄	—	236	2	no	—
113	C₄²	—	236	2	no	—
114	C₄²	D₄×D₅₉	1	1	yes	characteristic
115	C₁₆	—	2	1	no	—
116	C₁₆	—	118	2	no	—
117	C₁₆	#11745659	1	2	yes	—
118	Q₈	—	236	2	no	—
119	Q₈	—	236	2	no	—
120	C₂×C₄	—	236	2	no	—
121	C₂×C₄	—	236	2	no	—
122	C₂×C₄	#14273689	1	1	yes	characteristic
123	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
124	C₂×C₄	#14273750	1	1	yes	characteristic
125	C₈	—	2	1	no	—
126	C₈	—	8	2	no	—
127	C₈	—	59	2	no	—
128	C₈	—	118	2	no	—
129	C₈	—	118	1	no	—
130	C₈	C₂×#11745659	1	1	yes	characteristic
131	C₈	#14273735	1	1	yes	characteristic
132	C₂²	#16837484	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	472	2	no	—
135	C₄	—	472	2	no	—
136	C₄	#16837479	1	1	yes	characteristic
137	C₄	#16837518	1	1	yes	characteristic
138	C₄	Q₃₂×D₅₉	1	1	yes	characteristic
139	C₄	#16837831	1	1	yes	characteristic
140	C₂	#21422980	1	1	yes	characteristic
141	C₂	#21425954	1	1	yes	characteristic
142	C₂	#21425976	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×Dic₄₇₂

C₂

yes

maximal, characteristic

#21422992

C₂

yes

maximal

#21422059

C₂

yes

maximal

C₁₆×C₅₉⋊C₈

C₂

yes

maximal, characteristic

C₅₉×#1371

C₂

yes

maximal, characteristic

C₂×Dic₄₇₂

C₄

yes

characteristic

#16837529

—

C₄×Dic₂₃₆

C₂²

yes

—

#16837228

C₄

yes

characteristic

#16837207

C₄

yes

—

C₄×C₅₉⋊C₁₆

C₂²

yes

characteristic

#16837132

—

C₈×C₅₉⋊C₈

C₂²

yes

characteristic

C₄×C₁₆×C₅₉

C₂²

yes

characteristic

C₅₉×#335

C₂²

yes

—

C₂×Dic₂₃₆

C₂×C₄

yes

—

C₄×Dic₁₁₈

—

Dic₄₇₂

—

C₂×C₅₉⋊C₁₆

—

#14273661

C₂×C₄

yes

—

Dic₅₉⋊C₈

—

#14273652

—

#14273637

—

C₄×C₅₉⋊C₈

D₄

yes

characteristic

C₄×C₅₉⋊C₈

D₄

yes

characteristic

C₂×C₁₆×C₅₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₉

C₂×C₄

yes

characteristic

C₄⋊C₈×C₅₉

—

C₄×C₈×C₅₉

C₂³

yes

characteristic

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₉

#41

yes

characteristic

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

D₈

yes

characteristic

C₂×C₅₉⋊C₈

D₈

yes

characteristic

Dic₂₃₆

—

Dic₂₃₆

—

C₁₆×C₅₉

—

C₁₆×C₅₉

M₁₆

yes

—

C₅₉⋊C₁₆

—

C₂×C₄×C₅₉

C₄×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₈

yes

characteristic

C₂×C₄×C₅₉

#135

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

C₂×M₁₆

yes

characteristic

C₈×C₅₉

#131

yes

characteristic

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

Q₃₂

yes

complemented

#12083

—

maximal, Sylow

C₂²×C₅₉

C₄×D₈

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

C₂×Q₃₂

yes

characteristic

C₄×C₅₉

#508

yes

characteristic

Dic₅₉

—

Dic₅₉

—

C₄×Q₃₂

—

#1591

—

#1371

D₅₉

yes

characteristic

#1332

—

C₈×C₁₆

—

C₂×C₅₉

#1589

yes

characteristic

C₂×C₅₉

C₄×Q₃₂

yes

characteristic

C₂×C₅₉

#2178

yes

characteristic

C₂×Q₃₂

—

#446

—

118

—

C₄×Q₁₆

—

#366

—

#358

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₅₉

yes

characteristic

#335

C₂×D₅₉

yes

—

Q₈⋊C₈

—

118

—

C₈²

—

C₅₉

#12083

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄×Q₈

—

118

—

Q₃₂

—

118

—

C₂×C₁₆

—

118

—

C₂×C₁₆

C₄×D₅₉

yes

characteristic

C₂×C₁₆

C₄×D₅₉

yes

characteristic

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

236

—

#103

—

118

—

#97

—

236

—

C₄×C₈

—

100

C₄×C₈

—

101

C₄×C₈

C₂²×D₅₉

yes

characteristic

102

C₂×Q₈

—

118

—

103

Q₁₆

—

118

—

104

Q₁₆

—

118

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

118

—

109

C₂×C₈

C₂×C₄×D₅₉

yes

characteristic

110

C₂×C₈

#11745689

yes

characteristic

111

C₄⋊C₄

—

118

—

112

C₄⋊C₄

—

236

—

113

C₄²

—

236

—

114

C₄²

D₄×D₅₉

yes

characteristic

115

C₁₆

—

116

C₁₆

—

118

—

117

C₁₆

#11745659

yes

—

118

Q₈

—

236

—

119

Q₈

—

236

—

120

C₂×C₄

—

236

—

121

C₂×C₄

—

236

—

122

C₂×C₄

#14273689

yes

characteristic

123

C₂×C₄

D₈×D₅₉

yes

characteristic

124

C₂×C₄

#14273750

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

118

—

129

C₈

—

118

—

130

C₈

C₂×#11745659

yes

characteristic

131

C₈

#14273735

yes

characteristic

132

C₂²

#16837484

yes

characteristic

133

C₄

—

134

C₄

—

472

—

135

C₄

—

472

—

136

C₄

#16837479

yes

characteristic

137

C₄

#16837518

yes

characteristic

138

C₄

Q₃₂×D₅₉

yes

characteristic

139

C₄

#16837831

yes

characteristic

140

C₂

#21422980

yes

characteristic

141

C₂

#21425954

yes

characteristic

142

C₂

#21425976

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₅₉
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₅₉×#1371	C₂
socle, prime radical	C₂²×C₅₉	C₄×D₈
cosocle	C₄×C₈×C₅₉	C₂³
hypercenter	#1371	D₅₉
hypocenter	C₅₉	#12083
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #1371

subgroup

G/H

center

C₂×C₄

D₈×D₅₉

commutator

C₂×C₈×C₅₉

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₅₉

Fitting

C₅₉×#1371

C₂

socle, prime radical

C₂²×C₅₉

C₄×D₈

cosocle

C₄×C₈×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #1371

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
C₄×Dic₄₇₂	2
#21422992	2
#21422059	2
C₁₆×C₅₉⋊C₈	2
C₅₉×#1371	2
#12083	59

subgroup

index

C₄×Dic₄₇₂

#21422992

#21422059

C₁₆×C₅₉⋊C₈

C₅₉×#1371

#12083

Subgroups — #28500679

Conjugacy classes of subgroups

Complements

Maximal subgroups