Group #28505055 — subgroups

Subgroups — #28505055

Order 15104.

234 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 103 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	#21425862	C₂	1	1	yes	maximal, characteristic
2	#21423851	C₂	1	1	yes	maximal, characteristic
3	#21422184	C₂	1	2	yes	maximal
4	#21422183	C₂	1	2	yes	maximal
5	C₅₉×#2151	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273667	C₂²	1	2	yes	—
7	#16837357	C₂²	1	2	yes	—
8	#16837223	—	2	2	no	—
9	C₅₉⋊#338	C₄	1	1	yes	characteristic
10	#16837159	C₄	1	1	yes	characteristic
11	#16837158	C₄	1	2	yes	—
12	C₂×M₃₂×C₅₉	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
14	C₅₉×#405	C₂²	1	1	yes	characteristic
15	C₅₉×C₄⋊C₁₆	—	2	2	no	—
16	C₂×#11745664	C₂×C₄	1	2	yes	—
17	#14273686	C₂×C₄	1	2	yes	—
18	#14273667	—	2	2	no	—
19	#14273667	—	2	2	no	—
20	C₂×C₅₉⋊C₁₆	—	2	2	no	—
21	#14273637	—	2	2	no	—
22	C₈×Dic₅₉	—	2	2	no	—
23	C₂×M₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
25	#117×C₅₉	C₂×C₄	1	1	yes	characteristic
26	M₃₂×C₅₉	—	2	2	no	—
27	C₂×C₁₆×C₅₉	—	2	1	no	—
28	C₂×C₁₆×C₅₉	—	2	1	no	—
29	C₂×C₁₆×C₅₉	—	2	2	no	—
30	#97×C₅₉	Q₈	1	2	yes	—
31	C₄×C₈×C₅₉	D₄	1	2	yes	—
32	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
33	M₁₆×C₅₉	—	2	2	no	—
34	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
35	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
36	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
37	C₂×C₈×C₅₉	#41	1	2	yes	—
38	C₂×C₈×C₅₉	#41	1	2	yes	—
39	C₄⋊C₄×C₅₉	—	2	1	no	—
40	C₂²⋊C₄×C₅₉	—	2	1	no	—
41	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
42	C₂²⋊Dic₅₉	—	2	2	no	—
43	C₄⋊Dic₅₉	—	2	2	no	—
44	C₄×Dic₅₉	—	2	2	no	—
45	#11745664	—	2	2	no	—
46	#11745664	—	2	2	no	—
47	C₂×C₅₉⋊C₈	—	2	2	no	—
48	C₁₆×C₅₉	—	4	1	no	—
49	C₁₆×C₅₉	—	4	1	no	—
50	C₅₉⋊C₁₆	—	4	2	no	—
51	C₂³×C₅₉	#131	1	1	yes	characteristic
52	C₂×C₄×C₅₉	—	2	2	no	—
53	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₅₉	#117	1	2	yes	—
55	C₂×C₄×C₅₉	#131	1	1	yes	characteristic
56	C₂×C₄×C₅₉	#131	1	2	yes	—
57	C₂×Dic₅₉	—	4	2	no	—
58	C₈×C₅₉	—	2	2	no	—
59	C₈×C₅₉	#124	1	1	yes	characteristic
60	C₈×C₅₉	#124	1	1	yes	characteristic
61	C₈×C₅₉	#125	1	2	yes	—
62	C₅₉⋊C₈	—	4	2	no	—
63	#12885	—	59	1	no	maximal, Sylow
64	C₂²×C₅₉	—	4	1	no	—
65	C₂²×C₅₉	#424	1	1	yes	characteristic
66	C₂²×C₅₉	#433	1	2	yes	—
67	C₄×C₅₉	—	4	2	no	—
68	C₄×C₅₉	#382	1	1	yes	characteristic
69	C₄×C₅₉	#424	1	1	yes	characteristic
70	C₄×C₅₉	#433	1	2	yes	—
71	Dic₅₉	—	8	2	no	—
72	#2153	—	59	1	no	—
73	#2151	D₅₉	1	1	yes	characteristic
74	#1747	—	59	1	no	—
75	#1379	—	59	2	no	—
76	#1378	—	59	2	no	—
77	C₂×C₅₉	—	2	2	no	—
78	C₂×C₅₉	—	2	1	no	—
79	C₂×C₅₉	#1832	1	1	yes	characteristic
80	C₂×M₃₂	—	59	2	no	—
81	C₂×M₃₂	C₂×D₅₉	1	1	yes	characteristic
82	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
83	#405	—	59	2	no	—
84	#405	C₂×D₅₉	1	1	yes	characteristic
85	#364	—	118	2	no	—
86	C₄⋊C₁₆	—	2	2	no	—
87	#338	—	59	1	no	—
88	H(3,ℤ/4)	—	59	1	no	—
89	H(3,ℤ/4)	—	59	2	no	—
90	C₅₉	#12885	1	1	yes	characteristic, complemented, Sylow
91	C₂×M₁₆	—	59	2	no	—
92	C₂×M₁₆	C₄×D₅₉	1	1	yes	characteristic
93	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
94	#117	—	59	2	no	—
95	#117	C₄×D₅₉	1	1	yes	characteristic
96	M₃₂	—	2	2	no	—
97	M₃₂	—	118	2	no	—
98	M₃₂	—	118	2	no	—
99	C₂×C₁₆	—	2	2	no	—
100	C₂×C₁₆	—	2	1	no	—
101	C₂×C₁₆	—	2	1	no	—
102	C₂×C₁₆	—	118	2	no	—
103	#97	—	118	2	no	—
104	#97	Dic₁₁₈	1	2	yes	—
105	C₄×C₈	—	118	2	no	—
106	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
107	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
108	M₁₆	—	2	2	no	—
109	M₁₆	—	118	2	no	—
110	M₁₆	—	118	2	no	—
111	C₂×C₈	—	118	2	no	—
112	C₂×C₈	Dic₅₉⋊C₄	1	2	yes	—
113	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
114	C₂×C₈	#11745687	1	2	yes	—
115	C₂×C₈	#11745689	1	2	yes	—
116	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	118	2	no	—
119	C₂²⋊C₄	—	2	1	no	—
120	C₂²⋊C₄	—	118	2	no	—
121	C₄²	—	118	2	no	—
122	C₄²	Dic₅₉⋊C₄	1	2	yes	—
123	C₁₆	—	4	1	no	—
124	C₁₆	—	4	1	no	—
125	C₁₆	—	236	2	no	—
126	C₂³	#14273733	1	1	yes	characteristic
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	236	2	no	—
129	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
130	C₂×C₄	#14273685	1	2	yes	—
131	C₂×C₄	#14273733	1	1	yes	characteristic
132	C₂×C₄	#14273735	1	2	yes	—
133	C₈	—	2	2	no	—
134	C₈	—	236	2	no	—
135	C₈	#14273714	1	1	yes	characteristic
136	C₈	#14273716	1	1	yes	characteristic
137	C₈	#14273718	1	2	yes	—
138	C₂²	—	4	1	no	—
139	C₂²	#16837428	1	1	yes	characteristic
140	C₂²	#16837462	1	2	yes	—
141	C₄	—	4	2	no	—
142	C₄	—	472	2	no	—
143	C₄	#16837267	1	1	yes	characteristic
144	C₄	#16837430	1	1	yes	characteristic
145	C₄	#16837462	1	2	yes	—
146	C₂	—	2	1	no	—
147	C₂	—	2	2	no	—
148	C₂	#21424215	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21425862

C₂

yes

maximal, characteristic

#21423851

C₂

yes

maximal, characteristic

#21422184

C₂

yes

maximal

#21422183

C₂

yes

maximal

C₅₉×#2151

C₂

yes

maximal, characteristic

C₂×#14273667

C₂²

yes

—

#16837357

C₂²

yes

—

#16837223

—

C₅₉⋊#338

C₄

yes

characteristic

#16837159

C₄

yes

characteristic

#16837158

C₄

yes

—

C₂×M₃₂×C₅₉

C₂²

yes

characteristic

C₂²×C₁₆×C₅₉

C₂²

yes

characteristic

C₅₉×#405

C₂²

yes

characteristic

C₅₉×C₄⋊C₁₆

—

C₂×#11745664

C₂×C₄

yes

—

#14273686

C₂×C₄

yes

—

#14273667

—

#14273667

—

C₂×C₅₉⋊C₁₆

—

#14273637

—

C₈×Dic₅₉

—

C₂×M₁₆×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₉

C₂³

yes

characteristic

#117×C₅₉

C₂×C₄

yes

characteristic

M₃₂×C₅₉

—

C₂×C₁₆×C₅₉

—

C₂×C₁₆×C₅₉

—

C₂×C₁₆×C₅₉

—

#97×C₅₉

Q₈

yes

—

C₄×C₈×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

C₂²×C₄

yes

characteristic

M₁₆×C₅₉

—

C₂×C₈×C₅₉

C₄⋊C₄

yes

—

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₉

#41

yes

—

C₂×C₈×C₅₉

#41

yes

—

C₄⋊C₄×C₅₉

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

C₄⋊C₄

yes

—

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄×Dic₅₉

—

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₁₆×C₅₉

—

C₁₆×C₅₉

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

#131

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

#117

yes

—

C₂×C₄×C₅₉

#131

yes

characteristic

C₂×C₄×C₅₉

#131

yes

—

C₂×Dic₅₉

—

C₈×C₅₉

—

C₈×C₅₉

yes

C₈×C₅₉

yes

C₈×C₅₉

yes

—

C₅₉⋊C₈

—

#12885

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

#424

yes

characteristic

C₂²×C₅₉

#433

yes

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

Dic₅₉

—

#2153

—

#2151

D₅₉

yes

—

—

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#1832

yes

characteristic

C₂×M₃₂

—

C₂×M₃₂

C₂×D₅₉

yes

characteristic

C₂²×C₁₆

C₂×D₅₉

yes

characteristic

#405

—

#405

C₂×D₅₉

yes

characteristic

#364

—

118

—

C₄⋊C₁₆

—

#338

—

H(3,ℤ/4)

—

H(3,ℤ/4)

—

C₅₉

#12885

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

C₄×D₅₉

yes

characteristic

C₂²×C₈

C₂²×D₅₉

yes

characteristic

#117

—

#117

C₄×D₅₉

yes

characteristic

M₃₂

—

M₃₂

—

118

—

M₃₂

—

118

—

C₂×C₁₆

—

100

C₂×C₁₆

—

101

C₂×C₁₆

—

102

C₂×C₁₆

—

118

—

103

#97

—

118

—

104

#97

Dic₁₁₈

yes

—

105

C₄×C₈

—

118

—

106

C₄×C₈

C₂²⋊D₅₉

yes

—

107

C₂²×C₄

C₂×C₄×D₅₉

yes

characteristic

108

M₁₆

—

109

M₁₆

—

118

—

110

M₁₆

—

118

—

111

C₂×C₈

—

118

—

112

C₂×C₈

Dic₅₉⋊C₄

yes

—

113

C₂×C₈

C₂×Dic₁₁₈

yes

characteristic

114

C₂×C₈

#11745687

yes

—

115

C₂×C₈

#11745689

yes

—

116

C₂×C₈

C₂×C₂²⋊D₅₉

yes

characteristic

117

C₄⋊C₄

—

118

C₄⋊C₄

—

118

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

118

—

121

C₄²

—

118

—

122

C₄²

Dic₅₉⋊C₄

yes

—

123

C₁₆

—

124

C₁₆

—

125

C₁₆

—

236

—

126

C₂³

#14273733

yes

characteristic

127

C₂×C₄

—

128

C₂×C₄

—

236

—

129

C₂×C₄

C₂×Dic₅₉⋊C₄

yes

characteristic

130

C₂×C₄

#14273685

yes

—

131

C₂×C₄

#14273733

yes

characteristic

132

C₂×C₄

#14273735

yes

—

133

C₈

—

134

C₈

—

236

—

135

C₈

#14273714

yes

characteristic

136

C₈

#14273716

yes

characteristic

137

C₈

#14273718

yes

—

138

C₂²

—

139

C₂²

#16837428

yes

characteristic

140

C₂²

#16837462

yes

—

141

C₄

—

142

C₄

—

472

—

143

C₄

#16837267

yes

characteristic

144

C₄

#16837430

yes

characteristic

145

C₄

#16837462

yes

—

146

C₂

—

147

C₂

—

148

C₂

#21424215

yes

characteristic

	subgroup	G/H
center	C₈	#14273716
commutator	C₂²×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#2151	C₂
socle	C₂×C₅₉	#1832
prime radical	C₂³×C₅₉	#131
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#2151	D₅₉
hypocenter	C₅₉	#12885
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2151

subgroup

G/H

center

C₈

#14273716

commutator

C₂²×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₉

Fitting

C₅₉×#2151

C₂

C₂×C₅₉

C₂³×C₅₉

C₂²×C₈×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2151

N	H ≅ G/N	type
90. C₅₉	#12885	semidirect

H ≅ G/N

type

90. C₅₉

#12885

semidirect

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

subgroup	index
#21425862	2
#21423851	2
#21422184	2
#21422183	2
C₅₉×#2151	2
#12885	59

subgroup

index

C₅₉×#2151

Subgroups — #28505055

Conjugacy classes of subgroups

Complements

Maximal subgroups