Group #28509221 — subgroups

Subgroups — #28509221

Order 15104.

180 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 91 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	#21425691	C₂	1	1	yes	maximal, characteristic
2	Dic₅₉⋊C₃₂	C₂	1	2	yes	maximal, complemented
3	(C₂×D₅₉)⋊C₃₂	C₂	1	2	yes	maximal
4	#21422230	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×C₃₂×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14273666	C₄	1	1	yes	characteristic
7	C₂×C₁₆×D₅₉	C₂²	1	1	yes	characteristic
8	#16837356	C₄	1	1	yes	characteristic
9	C₂×C₅₉⋊C₃₂	C₂²	1	2	yes	—
10	#16837178	C₄	1	1	yes	characteristic
11	#16837177	C₄	1	1	yes	characteristic
12	C₁₆×Dic₅₉	C₂²	1	1	yes	characteristic
13	C₂×C₃₂×C₅₉	C₂²	1	2	yes	—
14	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
15	C₂×#11745659	C₈	1	1	yes	characteristic
16	C₂×C₈×D₅₉	C₂×C₄	1	1	yes	characteristic
17	#14273684	C₈	1	1	yes	characteristic
18	#14273666	—	2	2	no	—
19	C₂×C₅₉⋊C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₂×C₅₉⋊C₁₆	C₂×C₄	1	1	yes	characteristic
21	C₁₆×D₅₉	—	2	2	no	—
22	#14273638	C₈	1	1	yes	characteristic
23	#14273637	C₈	1	1	yes	characteristic
24	C₈×Dic₅₉	C₂×C₄	1	1	yes	characteristic
25	C₅₉⋊C₃₂	—	2	2	no	—
26	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
28	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
29	C₃₂×C₅₉	—	2	2	no	—
30	C₂×C₄×D₅₉	C₂×C₈	1	1	yes	characteristic
31	C₂×C₈×C₅₉	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
33	C₄²×C₅₉	C₂×C₈	1	1	yes	characteristic
34	(C₂×D₅₉)⋊C₄	C₁₆	1	2	yes	—
35	Dic₅₉⋊C₄	C₁₆	1	2	yes	—
36	C₄×Dic₅₉	C₂×C₈	1	1	yes	characteristic
37	C₂×C₅₉⋊C₈	C₂×C₈	1	1	yes	characteristic
38	C₂×C₅₉⋊C₈	C₂×C₈	1	1	yes	characteristic
39	#11745659	—	2	2	no	—
40	C₈×D₅₉	—	2	1	no	—
41	C₈×D₅₉	—	2	1	no	—
42	C₁₆×C₅₉	M₁₆	1	2	yes	—
43	C₁₆×C₅₉	#41	1	2	yes	—
44	C₅₉⋊C₁₆	—	2	1	no	—
45	C₅₉⋊C₁₆	—	2	1	no	—
46	C₂²×D₅₉	C₂×C₁₆	1	1	yes	characteristic
47	C₂×C₄×C₅₉	C₂×C₁₆	1	2	yes	—
48	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
49	C₂×Dic₅₉	C₂×C₁₆	1	2	yes	—
50	C₂×Dic₅₉	C₂×C₁₆	1	1	yes	characteristic
51	C₄×D₅₉	—	2	1	no	—
52	C₄×D₅₉	—	2	1	no	—
53	C₈×C₅₉	M₃₂	1	2	yes	—
54	C₈×C₅₉	#117	1	1	yes	characteristic
55	C₈×C₅₉	C₂×M₁₆	1	1	yes	characteristic
56	C₅₉⋊C₈	—	2	1	no	—
57	C₅₉⋊C₈	—	2	1	no	—
58	#13641	—	59	1	no	maximal, Sylow
59	C₂²×C₅₉	C₂²×C₁₆	1	1	yes	characteristic
60	C₂×D₅₉	—	2	1	no	—
61	C₂×D₅₉	—	2	2	no	—
62	C₄×C₅₉	—	2	2	no	—
63	C₄×C₅₉	#431	1	1	yes	characteristic
64	C₄×C₅₉	C₂×M₃₂	1	1	yes	characteristic
65	Dic₅₉	—	2	1	no	—
66	Dic₅₉	—	2	2	no	—
67	#2108	—	59	1	no	—
68	C₄⋊C₃₂	—	59	2	no	—
69	C₂²⋊C₃₂	—	59	2	no	—
70	#1397	—	59	1	no	—
71	C₄×C₃₂	D₅₉	1	1	yes	characteristic, complemented
72	C₂×C₅₉	#2162	1	1	yes	characteristic
73	C₂×C₅₉	#2258	1	2	yes	—
74	D₅₉	—	4	1	no	—
75	C₂×M₃₂	—	59	1	no	—
76	C₂²×C₁₆	—	59	1	no	—
77	#405	—	59	1	no	—
78	C₂×C₃₂	—	59	2	no	—
79	C₂×C₃₂	C₂×D₅₉	1	2	yes	—
80	#346	—	59	1	no	—
81	#346	—	59	1	no	—
82	C₄×C₁₆	—	59	1	no	—
83	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
84	C₅₉	#13641	1	1	yes	characteristic, complemented, Sylow
85	C₂×M₁₆	—	59	1	no	—
86	C₂²×C₈	—	59	1	no	—
87	#117	—	59	1	no	—
88	M₃₂	—	118	2	no	—
89	C₂×C₁₆	—	59	1	no	—
90	C₂×C₁₆	—	59	1	no	—
91	C₂×C₁₆	—	118	2	no	—
92	C₂×C₁₆	C₄×D₅₉	1	1	yes	characteristic
93	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
94	#97	—	59	1	no	—
95	#97	—	59	1	no	—
96	C₄×C₈	—	59	1	no	—
97	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
98	C₃₂	—	2	2	no	—
99	C₃₂	—	118	2	no	—
100	C₂²×C₄	—	59	1	no	—
101	M₁₆	—	118	2	no	—
102	C₂×C₈	—	59	1	no	—
103	C₂×C₈	—	59	1	no	—
104	C₂×C₈	—	118	1	no	—
105	C₂×C₈	—	118	1	no	—
106	C₂×C₈	C₈×D₅₉	1	1	yes	characteristic
107	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
108	C₄⋊C₄	—	59	2	no	—
109	C₂²⋊C₄	—	59	2	no	—
110	C₄²	—	59	1	no	—
111	C₄²	C₈×D₅₉	1	1	yes	characteristic
112	C₁₆	—	118	1	no	—
113	C₁₆	—	118	1	no	—
114	C₁₆	#11745659	1	2	yes	—
115	C₁₆	#11745685	1	2	yes	—
116	C₂³	—	59	1	no	—
117	C₂×C₄	—	59	2	no	—
118	C₂×C₄	—	59	1	no	—
119	C₂×C₄	—	118	1	no	—
120	C₂×C₄	—	118	1	no	—
121	C₂×C₄	C₁₆×D₅₉	1	2	yes	—
122	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
123	C₈	—	118	1	no	—
124	C₈	—	118	1	no	—
125	C₈	#14273666	1	2	yes	—
126	C₈	#14273684	1	1	yes	characteristic
127	C₈	C₂×#11745659	1	1	yes	characteristic
128	C₂²	—	118	2	no	—
129	C₂²	—	118	1	no	—
130	C₂²	C₂×C₁₆×D₅₉	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	118	2	no	—
133	C₄	—	118	1	no	—
134	C₄	#16837453	1	1	yes	characteristic
135	C₄	C₂×#14273666	1	1	yes	characteristic
136	C₂	—	236	1	no	—
137	C₂	#21425893	1	1	yes	characteristic
138	C₂	#21426389	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21425691

C₂

yes

maximal, characteristic

Dic₅₉⋊C₃₂

C₂

yes

maximal, complemented

(C₂×D₅₉)⋊C₃₂

C₂

yes

maximal

#21422230

C₂

yes

maximal, characteristic, complemented

C₄×C₃₂×C₅₉

C₂

yes

maximal, characteristic, complemented

C₂×#14273666

C₄

yes

characteristic

C₂×C₁₆×D₅₉

C₂²

yes

characteristic

#16837356

C₄

yes

characteristic

C₂×C₅₉⋊C₃₂

C₂²

yes

—

#16837178

C₄

yes

characteristic

#16837177

C₄

yes

characteristic

C₁₆×Dic₅₉

C₂²

yes

characteristic

C₂×C₃₂×C₅₉

C₂²

yes

—

C₄×C₁₆×C₅₉

C₂²

yes

characteristic

C₂×#11745659

C₈

yes

characteristic

C₂×C₈×D₅₉

C₂×C₄

yes

characteristic

#14273684

C₈

yes

characteristic

#14273666

—

C₂×C₅₉⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₅₉⋊C₁₆

C₂×C₄

yes

characteristic

C₁₆×D₅₉

—

#14273638

C₈

yes

characteristic

#14273637

C₈

yes

characteristic

C₈×Dic₅₉

C₂×C₄

yes

characteristic

C₅₉⋊C₃₂

—

C₂×C₁₆×C₅₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₉

C₂³

yes

characteristic

C₄×C₈×C₅₉

C₂×C₄

yes

characteristic

C₃₂×C₅₉

—

C₂×C₄×D₅₉

C₂×C₈

yes

characteristic

C₂×C₈×C₅₉

C₂×C₈

yes

characteristic

C₂×C₈×C₅₉

C₂²×C₄

yes

characteristic

C₄²×C₅₉

C₂×C₈

yes

characteristic

(C₂×D₅₉)⋊C₄

C₁₆

yes

—

Dic₅₉⋊C₄

C₁₆

yes

—

C₄×Dic₅₉

C₂×C₈

yes

characteristic

C₂×C₅₉⋊C₈

C₂×C₈

yes

characteristic

C₂×C₅₉⋊C₈

C₂×C₈

yes

characteristic

#11745659

—

C₈×D₅₉

—

C₈×D₅₉

—

C₁₆×C₅₉

M₁₆

yes

—

C₁₆×C₅₉

#41

yes

—

C₅₉⋊C₁₆

—

C₅₉⋊C₁₆

—

C₂²×D₅₉

C₂×C₁₆

yes

characteristic

C₂×C₄×C₅₉

C₂×C₁₆

yes

—

C₂×C₄×C₅₉

C₂²×C₈

yes

characteristic

C₂×Dic₅₉

C₂×C₁₆

yes

—

C₂×Dic₅₉

C₂×C₁₆

yes

characteristic

C₄×D₅₉

—

C₄×D₅₉

—

C₈×C₅₉

M₃₂

yes

—

C₈×C₅₉

#117

yes

characteristic

C₈×C₅₉

C₂×M₁₆

yes

characteristic

C₅₉⋊C₈

—

C₅₉⋊C₈

—

#13641

—

maximal, Sylow

C₂²×C₅₉

C₂²×C₁₆

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#431

yes

characteristic

C₄×C₅₉

C₂×M₃₂

yes

characteristic

Dic₅₉

—

Dic₅₉

—

#2108

—

C₄⋊C₃₂

—

C₂²⋊C₃₂

—

#1397

—

C₄×C₃₂

D₅₉

yes

characteristic, complemented

C₂×C₅₉

#2162

yes

characteristic

C₂×C₅₉

#2258

yes

—

D₅₉

—

C₂×M₃₂

—

C₂²×C₁₆

—

#405

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₅₉

yes

—

#346

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₅₉

yes

characteristic

C₅₉

#13641

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

#117

—

M₃₂

—

118

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

118

—

C₂×C₁₆

C₄×D₅₉

yes

characteristic

C₂×C₁₆

C₂²×D₅₉

yes

characteristic

#97

—

#97

—

C₄×C₈

—

C₄×C₈

C₄×D₅₉

yes

characteristic

C₃₂

—

C₃₂

—

118

—

100

C₂²×C₄

—

101

M₁₆

—

118

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

118

—

105

C₂×C₈

—

118

—

106

C₂×C₈

C₈×D₅₉

yes

characteristic

107

C₂×C₈

C₂×C₄×D₅₉

yes

characteristic

108

C₄⋊C₄

—

109

C₂²⋊C₄

—

110

C₄²

—

111

C₄²

C₈×D₅₉

yes

characteristic

112

C₁₆

—

118

—

113

C₁₆

—

118

—

114

C₁₆

#11745659

yes

—

115

C₁₆

#11745685

yes

—

116

C₂³

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

118

—

120

C₂×C₄

—

118

—

121

C₂×C₄

C₁₆×D₅₉

yes

—

122

C₂×C₄

C₂×C₈×D₅₉

yes

characteristic

123

C₈

—

118

—

124

C₈

—

118

—

125

C₈

#14273666

yes

—

126

C₈

#14273684

yes

characteristic

127

C₈

C₂×#11745659

yes

characteristic

128

C₂²

—

118

—

129

C₂²

—

118

—

130

C₂²

C₂×C₁₆×D₅₉

yes

characteristic

131

C₄

—

132

C₄

—

118

—

133

C₄

—

118

—

134

C₄

#16837453

yes

characteristic

135

C₄

C₂×#14273666

yes

characteristic

136

C₂

—

236

—

137

C₂

#21425893

yes

characteristic

138

C₂

#21426389

yes

—

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₅₉
commutator, socle	C₂²×C₅₉	C₂²×C₁₆
Fitting	C₄×C₃₂×C₅₉	C₂
prime radical	C₂²×D₅₉	C₂×C₁₆
cosocle	C₂×C₁₆×C₅₉	C₂³
hypercenter	C₄×C₃₂	D₅₉
hypocenter	C₅₉	#13641
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₁₆ ◁ C₄×C₃₂

subgroup

G/H

center, Frattini

C₂×C₁₆

C₂²×D₅₉

commutator, socle

C₂²×C₅₉

C₂²×C₁₆

Fitting

C₄×C₃₂×C₅₉

C₂

prime radical

C₂²×D₅₉

C₂×C₁₆

cosocle

C₂×C₁₆×C₅₉

C₂³

hypercenter

C₄×C₃₂

D₅₉

C₅₉

G ▷ C₂²×C₅₉ ▷ C₅₉

C₁ ◁ C₂×C₁₆ ◁ C₄×C₃₂

N	H ≅ G/N	type
2. Dic₅₉⋊C₃₂	C₂	semidirect
4. #21422230	C₂	semidirect
5. C₄×C₃₂×C₅₉	C₂	semidirect
71. C₄×C₃₂	D₅₉	semidirect
84. C₅₉	#13641	semidirect

H ≅ G/N

type

2. Dic₅₉⋊C₃₂

C₂

semidirect

4. #21422230

C₂

semidirect

5. C₄×C₃₂×C₅₉

C₂

semidirect

71. C₄×C₃₂

D₅₉

semidirect

84. C₅₉

#13641

semidirect

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

subgroup	index
#21425691	2
Dic₅₉⋊C₃₂	2
(C₂×D₅₉)⋊C₃₂	2
#21422230	2
C₄×C₃₂×C₅₉	2
#13641	59

subgroup

index

#21425691

Dic₅₉⋊C₃₂

(C₂×D₅₉)⋊C₃₂

#21422230

C₄×C₃₂×C₅₉

#13641

Subgroups — #28509221

Conjugacy classes of subgroups

Complements

Maximal subgroups