Group #28505112 — subgroups

Subgroups — #28505112

Order 15104.

278 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 119 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	#21425861	C₂	1	2	yes	maximal
2	#21425856	C₂	1	2	yes	maximal
3	#21425855	C₂	1	1	yes	maximal, characteristic
4	#21423851	C₂	1	1	yes	maximal, characteristic
5	C₅₉×#2153	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273667	C₂²	1	1	yes	characteristic
7	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
8	#16837357	C₂²	1	2	yes	—
9	#16837222	—	2	4	no	—
10	C₄⋊(C₅₉⋊C₁₆)	—	2	2	no	—
11	Dic₅₉⋊C₁₆	—	2	4	no	—
12	C₅₉⋊#338	C₄	1	1	yes	characteristic
13	#16837159	C₄	1	1	yes	characteristic
14	#16837158	C₄	1	2	yes	—
15	C₂×M₃₂×C₅₉	C₂²	1	2	yes	—
16	C₅₉×#405	C₂²	1	1	yes	characteristic
17	C₅₉×#364	—	2	2	no	—
18	C₂×#11745664	C₂×C₄	1	2	yes	—
19	#14273686	C₂×C₄	1	2	yes	—
20	#14273667	—	2	2	no	—
21	C₂×C₅₉⋊C₁₆	—	2	2	no	—
22	C₂×C₅₉⋊C₁₆	—	2	1	no	—
23	C₂×C₅₉⋊C₁₆	—	2	1	no	—
24	#14273637	Q₈	1	4	yes	—
25	C₈×Dic₅₉	D₄	1	4	yes	—
26	C₂×M₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
28	#117×C₅₉	C₂×C₄	1	1	yes	characteristic
29	M₃₂×C₅₉	—	2	2	no	—
30	M₃₂×C₅₉	—	2	2	no	—
31	C₂×C₁₆×C₅₉	—	2	2	no	—
32	#97×C₅₉	Q₈	1	2	yes	—
33	C₄×C₈×C₅₉	D₄	1	2	yes	—
34	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
35	M₁₆×C₅₉	—	2	2	no	—
36	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
37	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
38	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
41	C₄⋊C₄×C₅₉	—	2	1	no	—
42	C₂²⋊C₄×C₅₉	—	2	1	no	—
43	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
44	C₂²⋊Dic₅₉	—	2	2	no	—
45	C₄⋊Dic₅₉	—	2	2	no	—
46	C₄×Dic₅₉	C₄⋊C₄	1	4	yes	—
47	#11745664	—	2	2	no	—
48	#11745664	—	2	2	no	—
49	C₂×C₅₉⋊C₈	C₄⋊C₄	1	4	yes	—
50	C₁₆×C₅₉	—	4	2	no	—
51	C₅₉⋊C₁₆	—	4	1	no	—
52	C₅₉⋊C₁₆	—	4	1	no	—
53	C₂³×C₅₉	#131	1	1	yes	characteristic
54	C₂×C₄×C₅₉	—	2	2	no	—
55	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
56	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₅₉	#131	1	2	yes	—
58	C₂×C₄×C₅₉	#131	1	1	yes	characteristic
59	C₂×Dic₅₉	—	2	4	no	—
60	C₈×C₅₉	—	2	2	no	—
61	C₈×C₅₉	#127	1	1	yes	characteristic
62	C₈×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
63	C₈×C₅₉	C₄⋊Q₈	1	2	yes	—
64	C₅₉⋊C₈	—	2	4	no	—
65	#12902	—	59	1	no	maximal, Sylow
66	C₂²×C₅₉	—	4	1	no	—
67	C₂²×C₅₉	#424	1	2	yes	—
68	C₂²×C₅₉	#424	1	1	yes	characteristic
69	C₄×C₅₉	—	4	2	no	—
70	C₄×C₅₉	#384	1	1	yes	characteristic
71	C₄×C₅₉	#424	1	2	yes	—
72	C₄×C₅₉	#424	1	1	yes	characteristic
73	Dic₅₉	—	4	4	no	—
74	#2153	—	59	2	no	—
75	#2153	D₅₉	1	1	yes	characteristic
76	#2151	—	59	2	no	—
77	#2151	—	59	1	no	—
78	#1747	—	59	1	no	—
79	C₂×C₅₉	—	2	2	no	—
80	C₂×C₅₉	—	2	1	no	—
81	C₂×C₅₉	#1843	1	1	yes	characteristic
82	C₂×M₃₂	—	59	1	no	—
83	C₂×M₃₂	C₂×D₅₉	1	2	yes	—
84	C₂²×C₁₆	—	59	1	no	—
85	#405	—	59	2	no	—
86	#405	C₂×D₅₉	1	1	yes	characteristic
87	#364	—	2	2	no	—
88	#364	—	118	4	no	—
89	C₄⋊C₁₆	—	118	4	no	—
90	C₄⋊C₁₆	—	118	2	no	—
91	#338	—	59	1	no	—
92	H(3,ℤ/4)	—	59	1	no	—
93	H(3,ℤ/4)	—	59	2	no	—
94	C₅₉	#12902	1	1	yes	characteristic, complemented, Sylow
95	C₂×M₁₆	—	59	2	no	—
96	C₂×M₁₆	C₄×D₅₉	1	1	yes	characteristic
97	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
98	#117	—	59	2	no	—
99	#117	C₄×D₅₉	1	1	yes	characteristic
100	M₃₂	—	2	2	no	—
101	M₃₂	—	2	2	no	—
102	M₃₂	—	118	2	no	—
103	C₂×C₁₆	—	2	2	no	—
104	C₂×C₁₆	—	118	2	no	—
105	C₂×C₁₆	—	118	1	no	—
106	C₂×C₁₆	—	118	1	no	—
107	#97	—	59	4	no	—
108	#97	Dic₁₁₈	1	2	yes	—
109	C₄×C₈	—	59	4	no	—
110	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
111	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
112	M₁₆	—	2	2	no	—
113	M₁₆	—	118	2	no	—
114	M₁₆	—	118	2	no	—
115	C₂×C₈	—	59	4	no	—
116	C₂×C₈	Dic₅₉⋊C₄	1	2	yes	—
117	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
118	C₂×C₈	D₄×D₅₉	1	2	yes	—
119	C₂×C₈	Q₈×D₅₉	1	2	yes	—
120	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
121	C₄⋊C₄	—	2	1	no	—
122	C₄⋊C₄	—	118	2	no	—
123	C₂²⋊C₄	—	2	1	no	—
124	C₂²⋊C₄	—	118	2	no	—
125	C₄²	—	59	4	no	—
126	C₄²	Dic₅₉⋊C₄	1	2	yes	—
127	C₁₆	—	4	2	no	—
128	C₁₆	—	236	1	no	—
129	C₁₆	—	236	1	no	—
130	C₂³	#14273735	1	1	yes	characteristic
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	118	4	no	—
133	C₂×C₄	C₄⋊C₄×D₅₉	1	2	yes	—
134	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
135	C₂×C₄	#14273733	1	2	yes	—
136	C₂×C₄	#14273735	1	1	yes	characteristic
137	C₈	—	2	2	no	—
138	C₈	—	118	4	no	—
139	C₈	#14273723	1	1	yes	characteristic
140	C₈	#14273725	1	1	yes	characteristic
141	C₈	#14273726	1	2	yes	—
142	C₂²	—	4	1	no	—
143	C₂²	#16837427	1	2	yes	—
144	C₂²	#16837430	1	1	yes	characteristic
145	C₄	—	4	2	no	—
146	C₄	—	236	4	no	—
147	C₄	#16837271	1	1	yes	characteristic
148	C₄	#16837427	1	2	yes	—
149	C₄	#16837428	1	1	yes	characteristic
150	C₂	—	2	1	no	—
151	C₂	—	2	2	no	—
152	C₂	#21424249	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21425861

C₂

yes

maximal

#21425856

C₂

yes

maximal

#21425855

C₂

yes

maximal, characteristic

#21423851

C₂

yes

maximal, characteristic

C₅₉×#2153

C₂

yes

maximal, characteristic

C₂×#14273667

C₂²

yes

characteristic

C₂²×C₅₉⋊C₁₆

C₂²

yes

characteristic

#16837357

C₂²

yes

—

#16837222

—

C₄⋊(C₅₉⋊C₁₆)

—

Dic₅₉⋊C₁₆

—

C₅₉⋊#338

C₄

yes

characteristic

#16837159

C₄

yes

characteristic

#16837158

C₄

yes

—

C₂×M₃₂×C₅₉

C₂²

yes

—

C₅₉×#405

C₂²

yes

characteristic

C₅₉×#364

—

C₂×#11745664

C₂×C₄

yes

—

#14273686

C₂×C₄

yes

—

#14273667

—

C₂×C₅₉⋊C₁₆

—

C₂×C₅₉⋊C₁₆

—

C₂×C₅₉⋊C₁₆

—

#14273637

Q₈

yes

—

C₈×Dic₅₉

D₄

yes

—

C₂×M₁₆×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₉

C₂³

yes

characteristic

#117×C₅₉

C₂×C₄

yes

characteristic

M₃₂×C₅₉

—

M₃₂×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

—

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₉

C₂×Q₈

yes

—

C₄⋊C₄×C₅₉

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

C₄⋊C₄

yes

—

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄×Dic₅₉

C₄⋊C₄

yes

—

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

C₄⋊C₄

yes

—

C₁₆×C₅₉

—

C₅₉⋊C₁₆

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

#131

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

#131

yes

—

C₂×C₄×C₅₉

#131

yes

characteristic

C₂×Dic₅₉

—

C₈×C₅₉

—

C₈×C₅₉

#127

yes

characteristic

C₈×C₅₉

C₄⋊Q₈

yes

characteristic

C₈×C₅₉

C₄⋊Q₈

yes

—

C₅₉⋊C₈

—

#12902

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

#424

yes

—

C₂²×C₅₉

#424

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

#384

yes

characteristic

C₄×C₅₉

#424

yes

—

C₄×C₅₉

#424

yes

characteristic

Dic₅₉

—

#2153

—

#2153

D₅₉

yes

—

—

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#1843

yes

characteristic

C₂×M₃₂

—

C₂×M₃₂

C₂×D₅₉

yes

—

C₂²×C₁₆

—

#405

—

#405

C₂×D₅₉

yes

characteristic

#364

—

#364

—

118

—

C₄⋊C₁₆

—

118

—

C₄⋊C₁₆

—

118

—

#338

—

H(3,ℤ/4)

—

H(3,ℤ/4)

—

C₅₉

#12902

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

C₄×D₅₉

yes

characteristic

C₂²×C₈

C₂²×D₅₉

yes

characteristic

#117

—

#117

C₄×D₅₉

yes

characteristic

100

M₃₂

—

101

M₃₂

—

102

M₃₂

—

118

—

103

C₂×C₁₆

—

104

C₂×C₁₆

—

118

—

105

C₂×C₁₆

—

118

—

106

C₂×C₁₆

—

118

—

107

#97

—

108

#97

Dic₁₁₈

yes

—

109

C₄×C₈

—

110

C₄×C₈

C₂²⋊D₅₉

yes

—

111

C₂²×C₄

C₂×C₄×D₅₉

yes

characteristic

112

M₁₆

—

113

M₁₆

—

118

—

114

M₁₆

—

118

—

115

C₂×C₈

—

116

C₂×C₈

Dic₅₉⋊C₄

yes

—

117

C₂×C₈

C₂×Dic₁₁₈

yes

characteristic

118

C₂×C₈

D₄×D₅₉

yes

—

119

C₂×C₈

Q₈×D₅₉

yes

—

120

C₂×C₈

C₂×C₂²⋊D₅₉

yes

characteristic

121

C₄⋊C₄

—

122

C₄⋊C₄

—

118

—

123

C₂²⋊C₄

—

124

C₂²⋊C₄

—

118

—

125

C₄²

—

126

C₄²

Dic₅₉⋊C₄

yes

—

127

C₁₆

—

128

C₁₆

—

236

—

129

C₁₆

—

236

—

130

C₂³

#14273735

yes

characteristic

131

C₂×C₄

—

132

C₂×C₄

—

118

—

133

C₂×C₄

C₄⋊C₄×D₅₉

yes

—

134

C₂×C₄

C₂×Dic₅₉⋊C₄

yes

characteristic

135

C₂×C₄

#14273733

yes

—

136

C₂×C₄

#14273735

yes

characteristic

137

C₈

—

138

C₈

—

118

—

139

C₈

#14273723

yes

characteristic

140

C₈

#14273725

yes

characteristic

141

C₈

#14273726

yes

—

142

C₂²

—

143

C₂²

#16837427

yes

—

144

C₂²

#16837430

yes

characteristic

145

C₄

—

146

C₄

—

236

—

147

C₄

#16837271

yes

characteristic

148

C₄

#16837427

yes

—

149

C₄

#16837428

yes

characteristic

150

C₂

—

151

C₂

—

152

C₂

#21424249

yes

characteristic

	subgroup	G/H
center	C₈	#14273723
commutator	C₂²×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#2153	C₂
socle	C₂×C₅₉	#1843
prime radical	C₂³×C₅₉	#131
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#2153	D₅₉
hypocenter	C₅₉	#12902
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2153

subgroup

G/H

center

C₈

#14273723

commutator

C₂²×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₉

Fitting

C₅₉×#2153

C₂

C₂×C₅₉

C₂³×C₅₉

C₂²×C₈×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2153

N	H ≅ G/N	type
94. C₅₉	#12902	semidirect

H ≅ G/N

type

94. C₅₉

#12902

semidirect

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

subgroup	index
#21425861	2
#21425856	2
#21425855	2
#21423851	2
C₅₉×#2153	2
#12902	59

subgroup

index

C₅₉×#2153

Subgroups — #28505112

Conjugacy classes of subgroups

Complements

Maximal subgroups