Group #28505052 — subgroups

Subgroups — #28505052

Order 15104.

302 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 135 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	#21425852	C₂	1	1	yes	maximal, characteristic
2	#21423782	C₂	1	1	yes	maximal, characteristic
3	#21422177	C₂	1	4	yes	maximal
4	C₅₉×#2150	C₂	1	1	yes	maximal, characteristic
5	C₂×#14273667	C₂²	1	2	yes	—
6	C₂×#14273638	C₄	1	1	yes	characteristic
7	C₂×#14273637	C₄	1	2	yes	—
8	C₂×C₈×Dic₅₉	C₂²	1	2	yes	—
9	#16837248	C₄	1	1	yes	characteristic
10	C₄⋊(C₅₉⋊C₁₆)	—	2	2	no	—
11	C₂×M₃₂×C₅₉	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
13	C₅₉×C₄⋊C₁₆	—	2	2	no	—
14	C₂²×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
15	C₂×C₄×Dic₅₉	C₂×C₄	1	2	yes	—
16	#14273667	—	2	4	no	—
17	C₂×C₅₉⋊C₁₆	—	2	2	no	—
18	#14273638	C₈	1	4	yes	—
19	#14273637	—	2	4	no	—
20	C₈×Dic₅₉	—	2	2	no	—
21	C₈×Dic₅₉	—	2	2	no	—
22	C₅₉⋊#95	—	2	2	no	—
23	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
25	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
26	M₃₂×C₅₉	—	2	4	no	—
27	C₂×C₁₆×C₅₉	—	2	2	no	—
28	C₄×C₈×C₅₉	D₄	1	2	yes	—
29	C₄×C₈×C₅₉	Q₈	1	2	yes	—
30	C₂²×C₄×C₅₉	M₁₆	1	2	yes	—
31	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
32	C₂²×Dic₅₉	—	2	2	no	—
33	C₂×C₈×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	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
40	C₄²×C₅₉	C₂×C₈	1	2	yes	—
41	C₄×Dic₅₉	—	2	2	no	—
42	C₄×Dic₅₉	M₁₆	1	4	yes	—
43	C₂×C₅₉⋊C₈	—	2	2	no	—
44	C₂×C₅₉⋊C₈	—	2	2	no	—
45	C₂×C₅₉⋊C₈	C₂×C₈	1	4	yes	—
46	C₁₆×C₅₉	—	4	2	no	—
47	C₅₉⋊C₁₆	—	4	2	no	—
48	C₂³×C₅₉	C₂×M₁₆	1	1	yes	characteristic
49	C₂×C₄×C₅₉	—	4	2	no	—
50	C₂×C₄×C₅₉	C₄⋊C₈	1	4	yes	—
51	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₅₉	#117	1	2	yes	—
53	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
54	C₂×C₄×C₅₉	C₂×M₁₆	1	2	yes	—
55	C₂×Dic₅₉	—	2	4	no	—
56	C₂×Dic₅₉	—	4	4	no	—
57	C₈×C₅₉	—	2	2	no	—
58	C₈×C₅₉	#124	1	1	yes	characteristic
59	C₈×C₅₉	#124	1	1	yes	characteristic
60	C₈×C₅₉	#125	1	2	yes	—
61	C₅₉⋊C₈	—	2	4	no	—
62	#12884	—	59	1	no	maximal, Sylow
63	C₂²×C₅₉	—	2	2	no	—
64	C₂²×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
65	C₂²×C₅₉	#432	1	2	yes	—
66	C₄×C₅₉	—	2	4	no	—
67	C₄×C₅₉	#382	1	1	yes	characteristic
68	C₄×C₅₉	#423	1	1	yes	characteristic
69	C₄×C₅₉	#431	1	2	yes	—
70	Dic₅₉	—	4	4	no	—
71	#2150	—	59	1	no	—
72	#2150	D₅₉	1	1	yes	characteristic
73	#1726	—	59	1	no	—
74	#1376	—	59	4	no	—
75	C₂×C₅₉	—	2	2	no	—
76	C₂×C₅₉	#1831	1	1	yes	characteristic
77	C₂×C₅₉	#2153	1	2	yes	—
78	C₂×M₃₂	—	59	2	no	—
79	C₂×M₃₂	C₂×D₅₉	1	2	yes	—
80	C₂×#97	—	59	1	no	—
81	C₂×#97	—	59	2	no	—
82	C₂×C₄×C₈	—	59	2	no	—
83	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
84	#376	—	59	1	no	—
85	C₄⋊C₁₆	—	2	2	no	—
86	C₄⋊C₁₆	—	118	2	no	—
87	C₅₉	#12884	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₈	—	59	2	no	—
89	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
90	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
91	C₂×C₄²	—	59	2	no	—
92	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
93	M₃₂	—	2	4	no	—
94	M₃₂	—	118	4	no	—
95	C₂×C₁₆	—	2	2	no	—
96	C₂×C₁₆	—	118	2	no	—
97	#97	—	59	4	no	—
98	#97	—	118	4	no	—
99	C₄×C₈	—	118	2	no	—
100	C₄×C₈	—	118	2	no	—
101	C₄×C₈	Dic₁₁₈	1	2	yes	—
102	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
103	#95	—	118	2	no	—
104	C₂²×C₄	—	118	2	no	—
105	C₂²×C₄	#11745659	1	2	yes	—
106	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	59	4	no	—
109	C₂×C₈	—	118	2	no	—
110	C₂×C₈	—	118	2	no	—
111	C₂×C₈	Dic₅₉⋊C₄	1	2	yes	—
112	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
113	C₂×C₈	#11745687	1	2	yes	—
114	C₂×C₈	#11745689	1	2	yes	—
115	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
116	C₄²	—	59	4	no	—
117	C₄²	—	118	2	no	—
118	C₄²	C₈×D₅₉	1	2	yes	—
119	C₄²	Dic₅₉⋊C₄	1	2	yes	—
120	C₁₆	—	4	2	no	—
121	C₁₆	—	236	2	no	—
122	C₂³	C₂×#11745659	1	1	yes	characteristic
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	118	4	no	—
125	C₂×C₄	—	236	4	no	—
126	C₂×C₄	Dic₅₉⋊C₈	1	4	yes	—
127	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
128	C₂×C₄	#14273685	1	2	yes	—
129	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
130	C₂×C₄	M₁₆×D₅₉	1	2	yes	—
131	C₈	—	2	2	no	—
132	C₈	—	118	4	no	—
133	C₈	#14273714	1	1	yes	characteristic
134	C₈	#14273716	1	1	yes	characteristic
135	C₈	#14273718	1	2	yes	—
136	C₂²	—	2	2	no	—
137	C₂²	C₂×Dic₅₉⋊C₈	1	1	yes	characteristic
138	C₂²	#16837458	1	2	yes	—
139	C₄	—	2	4	no	—
140	C₄	—	236	4	no	—
141	C₄	#16837267	1	1	yes	characteristic
142	C₄	#16837425	1	1	yes	characteristic
143	C₄	#16837454	1	2	yes	—
144	C₂	—	2	2	no	—
145	C₂	#21424212	1	1	yes	characteristic
146	C₂	#21425861	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21425852

C₂

yes

maximal, characteristic

#21423782

C₂

yes

maximal, characteristic

#21422177

C₂

yes

maximal

C₅₉×#2150

C₂

yes

maximal, characteristic

C₂×#14273667

C₂²

yes

—

C₂×#14273638

C₄

yes

characteristic

C₂×#14273637

C₄

yes

—

C₂×C₈×Dic₅₉

C₂²

yes

—

#16837248

C₄

yes

characteristic

C₄⋊(C₅₉⋊C₁₆)

—

C₂×M₃₂×C₅₉

C₂²

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₅₉×C₄⋊C₁₆

—

C₂²×C₅₉⋊C₈

C₂×C₄

yes

—

C₂×C₄×Dic₅₉

C₂×C₄

yes

—

#14273667

—

C₂×C₅₉⋊C₁₆

—

#14273638

C₈

yes

—

#14273637

—

C₈×Dic₅₉

—

C₈×Dic₅₉

—

C₅₉⋊#95

—

C₂²×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₄²×C₅₉

C₂×C₄

yes

characteristic

M₃₂×C₅₉

—

C₂×C₁₆×C₅₉

—

C₄×C₈×C₅₉

D₄

yes

—

C₄×C₈×C₅₉

Q₈

yes

—

C₂²×C₄×C₅₉

M₁₆

yes

—

C₂²×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂²×Dic₅₉

—

C₂×C₈×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

—

C₄²×C₅₉

C₄⋊C₄

yes

—

C₄²×C₅₉

C₂×C₈

yes

—

C₄×Dic₅₉

—

C₄×Dic₅₉

M₁₆

yes

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₂×C₈

yes

—

C₁₆×C₅₉

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

C₂×M₁₆

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₄⋊C₈

yes

—

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

#117

yes

—

C₂×C₄×C₅₉

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₉

C₂×M₁₆

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₈×C₅₉

—

C₈×C₅₉

yes

C₈×C₅₉

yes

C₈×C₅₉

yes

—

C₅₉⋊C₈

—

#12884

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₅₉

#432

yes

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

Dic₅₉

—

#2150

—

#2150

D₅₉

yes

characteristic

#1726

—

#1376

—

C₂×C₅₉

—

C₂×C₅₉

#1831

yes

characteristic

C₂×C₅₉

#2153

yes

—

C₂×M₃₂

—

C₂×M₃₂

C₂×D₅₉

yes

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

#376

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

118

—

C₅₉

#12884

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₄×D₅₉

yes

characteristic

C₂²×C₈

C₂²×D₅₉

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₄×D₅₉

yes

characteristic

M₃₂

—

M₃₂

—

118

—

C₂×C₁₆

—

C₂×C₁₆

—

118

—

#97

—

#97

—

118

—

C₄×C₈

—

118

—

100

C₄×C₈

—

118

—

101

C₄×C₈

Dic₁₁₈

yes

—

102

C₄×C₈

C₂²⋊D₅₉

yes

—

103

#95

—

118

—

104

C₂²×C₄

—

118

—

105

C₂²×C₄

#11745659

yes

—

106

C₂²×C₄

C₂×C₄×D₅₉

yes

characteristic

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

118

—

110

C₂×C₈

—

118

—

111

C₂×C₈

Dic₅₉⋊C₄

yes

—

112

C₂×C₈

C₂×Dic₁₁₈

yes

characteristic

113

C₂×C₈

#11745687

yes

—

114

C₂×C₈

#11745689

yes

—

115

C₂×C₈

C₂×C₂²⋊D₅₉

yes

characteristic

116

C₄²

—

117

C₄²

—

118

—

118

C₄²

C₈×D₅₉

yes

—

119

C₄²

Dic₅₉⋊C₄

yes

—

120

C₁₆

—

121

C₁₆

—

236

—

122

C₂³

C₂×#11745659

yes

characteristic

123

C₂×C₄

—

124

C₂×C₄

—

118

—

125

C₂×C₄

—

236

—

126

C₂×C₄

Dic₅₉⋊C₈

yes

—

127

C₂×C₄

C₂×Dic₅₉⋊C₄

yes

characteristic

128

C₂×C₄

#14273685

yes

—

129

C₂×C₄

C₂×C₈×D₅₉

yes

characteristic

130

C₂×C₄

M₁₆×D₅₉

yes

—

131

C₈

—

132

C₈

—

118

—

133

C₈

#14273714

yes

characteristic

134

C₈

#14273716

yes

characteristic

135

C₈

#14273718

yes

—

136

C₂²

—

137

C₂²

C₂×Dic₅₉⋊C₈

yes

characteristic

138

C₂²

#16837458

yes

—

139

C₄

—

140

C₄

—

236

—

141

C₄

#16837267

yes

characteristic

142

C₄

#16837425

yes

characteristic

143

C₄

#16837454

yes

—

144

C₂

—

145

C₂

#21424212

yes

characteristic

146

C₂

#21425861

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#2150	C₂
socle	C₂²×C₅₉	C₂×C₄⋊C₈
prime radical	C₂³×C₅₉	C₂×M₁₆
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#2150	D₅₉
hypocenter	C₅₉	#12884
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

subgroup

G/H

center

C₂×C₈

C₂×C₂²⋊D₅₉

commutator

C₂×C₄×C₅₉

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₅₉

Fitting

C₅₉×#2150

C₂

socle

C₂²×C₅₉

C₂×C₄⋊C₈

prime radical

C₂³×C₅₉

C₂×M₁₆

cosocle

C₂²×C₈×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

N	H ≅ G/N	type
87. C₅₉	#12884	semidirect

H ≅ G/N

type

87. C₅₉

#12884

semidirect

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

subgroup	index
#21425852	2
#21423782	2
#21422177	2
C₅₉×#2150	2
#12884	59

subgroup

index

C₅₉×#2150

Subgroups — #28505052

Conjugacy classes of subgroups

Complements

Maximal subgroups