Group #28509136 — subgroups

Subgroups — #28509136

Order 15104.

290 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 115 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	C₂×#16837236	C₂	1	1	yes	maximal, characteristic
2	#21425880	C₂	1	1	yes	maximal, characteristic
3	#21422277	C₂	1	4	yes	maximal, complemented
4	C₂²×Q₃₂×C₅₉	C₂	1	1	yes	maximal, characteristic
5	#16837440	—	2	1	no	—
6	#16837236	—	2	2	no	—
7	C₂×C₅₉⋊C₃₂	C₂²	1	2	yes	—
8	#16837230	—	2	1	no	—
9	#16837228	C₂²	1	2	yes	—
10	C₂²×Q₁₆×C₅₉	—	2	1	no	—
11	C₂×Q₃₂×C₅₉	—	2	1	no	—
12	C₂×Q₃₂×C₅₉	—	2	2	no	—
13	C₂×Q₃₂×C₅₉	C₄	1	4	yes	complemented
14	C₂×Q₃₂×C₅₉	C₂²	1	2	yes	—
15	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
16	#14273686	—	4	1	no	—
17	#14273661	—	2	2	no	—
18	#14273659	—	4	1	no	—
19	C₅₉⋊C₃₂	—	2	2	no	—
20	C₂²×Q₈×C₅₉	—	4	1	no	—
21	C₂×Q₁₆×C₅₉	—	2	4	no	—
22	C₂×Q₁₆×C₅₉	—	2	2	no	—
23	C₂×Q₁₆×C₅₉	—	4	1	no	—
24	C₂×Q₁₆×C₅₉	—	4	2	no	—
25	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
26	Q₃₂×C₅₉	—	2	4	no	—
27	Q₃₂×C₅₉	—	2	2	no	—
28	Q₃₂×C₅₉	C₂×C₄	1	4	yes	—
29	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
30	C₂×C₁₆×C₅₉	D₄	1	1	yes	characteristic
31	C₂×C₁₆×C₅₉	D₄	1	2	yes	—
32	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
33	C₂×Q₈×C₅₉	—	4	4	no	—
34	C₂×Q₈×C₅₉	—	4	2	no	—
35	C₂×Q₈×C₅₉	—	8	1	no	—
36	C₂×Q₈×C₅₉	—	8	2	no	—
37	C₂²×C₄×C₅₉	—	8	1	no	—
38	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
39	Q₁₆×C₅₉	—	2	4	no	—
40	Q₁₆×C₅₉	—	4	4	no	—
41	Q₁₆×C₅₉	—	4	2	no	—
42	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
43	C₂×C₈×C₅₉	D₈	1	1	yes	characteristic
44	C₂×C₈×C₅₉	SD₁₆	1	2	yes	—
45	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
46	C₂²⋊Dic₅₉	—	8	1	no	—
47	C₄⋊Dic₅₉	—	4	2	no	—
48	C₄×Dic₅₉	—	8	1	no	—
49	C₁₆×C₅₉	C₂²⋊C₄	1	2	yes	—
50	C₁₆×C₅₉	C₂²×C₄	1	1	yes	characteristic
51	C₁₆×C₅₉	C₂×D₄	1	1	yes	characteristic
52	C₂³×C₅₉	D₁₆	1	1	yes	characteristic
53	Q₈×C₅₉	—	4	4	no	—
54	Q₈×C₅₉	—	8	4	no	—
55	Q₈×C₅₉	—	8	2	no	—
56	C₂×C₄×C₅₉	—	8	4	no	—
57	C₂×C₄×C₅₉	—	8	2	no	—
58	C₂×C₄×C₅₉	#102	1	2	yes	—
59	C₂×C₄×C₅₉	D₁₆	1	1	yes	characteristic
60	C₂×C₄×C₅₉	SD₃₂	1	2	yes	—
61	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
62	C₂×Dic₅₉	—	8	2	no	—
63	C₈×C₅₉	#102	1	2	yes	—
64	C₈×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₈×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
66	#13618	—	59	1	no	maximal, Sylow
67	C₂²×C₅₉	—	2	2	no	—
68	C₂²×C₅₉	#357	1	2	yes	—
69	C₂²×C₅₉	C₂×D₁₆	1	1	yes	characteristic
70	C₄×C₅₉	—	8	4	no	—
71	C₄×C₅₉	#357	1	2	yes	—
72	C₄×C₅₉	C₂×#102	1	1	yes	characteristic
73	C₄×C₅₉	C₂×SD₃₂	1	1	yes	characteristic
74	Dic₅₉	—	16	2	no	—
75	C₂²×Q₃₂	D₅₉	1	1	yes	characteristic
76	C₂×M₆₄	—	59	1	no	—
77	#2158	—	59	1	no	—
78	#1416	—	59	4	no	—
79	C₂×C₅₉	—	2	2	no	—
80	C₂×C₅₉	C₂×#357	1	1	yes	characteristic
81	C₂×C₅₉	#2264	1	2	yes	—
82	C₂²×Q₁₆	—	2	1	no	—
83	C₂×Q₃₂	—	2	1	no	—
84	C₂×Q₃₂	—	2	2	no	—
85	C₂×Q₃₂	Dic₅₉	1	4	yes	complemented
86	C₂×Q₃₂	C₂×D₅₉	1	2	yes	—
87	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
88	#427	—	118	1	no	—
89	M₆₄	—	118	2	no	—
90	C₂×C₃₂	—	59	2	no	—
91	#367	—	118	1	no	—
92	#366	—	59	2	no	—
93	C₅₉	#13618	1	1	yes	characteristic, complemented, Sylow
94	C₂²×Q₈	—	4	1	no	—
95	C₂×Q₁₆	—	2	4	no	—
96	C₂×Q₁₆	—	2	2	no	—
97	C₂×Q₁₆	—	4	1	no	—
98	C₂×Q₁₆	—	4	2	no	—
99	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
100	#117	—	236	1	no	—
101	Q₃₂	—	2	4	no	—
102	Q₃₂	—	2	2	no	—
103	Q₃₂	C₂×Dic₅₉	1	4	yes	—
104	C₂×C₁₆	C₂×Dic₅₉	1	2	yes	—
105	C₂×C₁₆	C₂²⋊D₅₉	1	1	yes	characteristic
106	C₂×C₁₆	C₂²⋊D₅₉	1	2	yes	—
107	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
108	#107	—	118	2	no	—
109	#106	—	236	1	no	—
110	C₃₂	—	118	2	no	—
111	C₂×Q₈	—	4	4	no	—
112	C₂×Q₈	—	4	2	no	—
113	C₂×Q₈	—	8	1	no	—
114	C₂×Q₈	—	8	2	no	—
115	C₂²×C₄	—	8	1	no	—
116	C₂²×C₄	#11745669	1	1	yes	characteristic
117	Q₁₆	—	2	4	no	—
118	Q₁₆	—	4	4	no	—
119	Q₁₆	—	4	2	no	—
120	C₂×C₈	#11745669	1	1	yes	characteristic
121	C₂×C₈	#11745670	1	2	yes	—
122	C₂×C₈	C₂²⋊Dic₅₉	1	2	yes	—
123	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
124	C₄⋊C₄	—	236	2	no	—
125	C₂²⋊C₄	—	472	1	no	—
126	C₄²	—	472	1	no	—
127	C₁₆	C₂²⋊Dic₅₉	1	2	yes	—
128	C₁₆	C₂²×Dic₅₉	1	1	yes	characteristic
129	C₁₆	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
130	C₂³	#14273668	1	1	yes	characteristic
131	Q₈	—	4	4	no	—
132	Q₈	—	8	4	no	—
133	Q₈	—	8	2	no	—
134	C₂×C₄	—	8	4	no	—
135	C₂×C₄	—	8	2	no	—
136	C₂×C₄	—	472	2	no	—
137	C₂×C₄	#14273648	1	2	yes	—
138	C₂×C₄	#14273668	1	1	yes	characteristic
139	C₂×C₄	#14273671	1	2	yes	—
140	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
141	C₈	#14273648	1	2	yes	—
142	C₈	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
143	C₈	C₂×#11745670	1	1	yes	characteristic
144	C₂²	—	2	2	no	—
145	C₂²	#16837203	1	2	yes	—
146	C₂²	C₂×#14273668	1	1	yes	characteristic
147	C₄	—	8	4	no	—
148	C₄	—	944	2	no	—
149	C₄	#16837203	1	2	yes	—
150	C₄	C₂×#14273648	1	1	yes	characteristic
151	C₄	C₂×#14273671	1	1	yes	characteristic
152	C₂	—	2	2	no	—
153	C₂	C₂×#16837203	1	1	yes	characteristic
154	C₂	#21426414	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16837236

C₂

yes

maximal, characteristic

#21425880

C₂

yes

maximal, characteristic

#21422277

C₂

yes

maximal, complemented

C₂²×Q₃₂×C₅₉

C₂

yes

maximal, characteristic

#16837440

—

#16837236

—

C₂×C₅₉⋊C₃₂

C₂²

yes

—

#16837230

—

#16837228

C₂²

yes

—

C₂²×Q₁₆×C₅₉

—

C₂×Q₃₂×C₅₉

—

C₂×Q₃₂×C₅₉

—

C₂×Q₃₂×C₅₉

C₄

yes

complemented

C₂×Q₃₂×C₅₉

C₂²

yes

—

C₂²×C₁₆×C₅₉

C₂²

yes

—

—

—

C₅₉⋊C₃₂

—

C₂²×Q₈×C₅₉

—

C₂×Q₁₆×C₅₉

—

C₂×Q₁₆×C₅₉

—

C₂×Q₁₆×C₅₉

—

C₂×Q₁₆×C₅₉

—

C₂²×C₈×C₅₉

D₄

yes

characteristic

Q₃₂×C₅₉

—

Q₃₂×C₅₉

—

Q₃₂×C₅₉

C₂×C₄

yes

—

C₂×C₁₆×C₅₉

C₂×C₄

yes

—

C₂×C₁₆×C₅₉

D₄

yes

characteristic

C₂×C₁₆×C₅₉

D₄

yes

—

C₂×C₁₆×C₅₉

C₂³

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

D₈

yes

characteristic

Q₁₆×C₅₉

—

Q₁₆×C₅₉

—

Q₁₆×C₅₉

—

C₂×C₈×C₅₉

C₂²⋊C₄

yes

—

C₂×C₈×C₅₉

D₈

yes

characteristic

C₂×C₈×C₅₉

SD₁₆

yes

—

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄×Dic₅₉

—

C₁₆×C₅₉

C₂²⋊C₄

yes

—

C₁₆×C₅₉

C₂²×C₄

yes

characteristic

C₁₆×C₅₉

C₂×D₄

yes

characteristic

C₂³×C₅₉

D₁₆

yes

characteristic

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#102

yes

—

C₂×C₄×C₅₉

D₁₆

yes

characteristic

C₂×C₄×C₅₉

SD₃₂

yes

—

C₂×C₄×C₅₉

C₂×D₈

yes

characteristic

C₂×Dic₅₉

—

C₈×C₅₉

#102

yes

—

C₈×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₅₉

C₂×SD₁₆

yes

characteristic

#13618

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

#357

yes

—

C₂²×C₅₉

C₂×D₁₆

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

#357

yes

—

C₄×C₅₉

C₂×#102

yes

characteristic

C₄×C₅₉

C₂×SD₃₂

yes

characteristic

Dic₅₉

—

C₂²×Q₃₂

D₅₉

yes

characteristic

C₂×M₆₄

—

#2158

—

#1416

—

C₂×C₅₉

—

C₂×C₅₉

C₂×#357

yes

characteristic

C₂×C₅₉

#2264

yes

—

C₂²×Q₁₆

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

Dic₅₉

yes

complemented

C₂×Q₃₂

C₂×D₅₉

yes

—

C₂²×C₁₆

C₂×D₅₉

yes

characteristic

#427

—

118

—

M₆₄

—

118

—

C₂×C₃₂

—

#367

—

118

—

#366

—

C₅₉

#13618

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

C₂²⋊D₅₉

yes

characteristic

100

#117

—

236

—

101

Q₃₂

—

102

Q₃₂

—

103

Q₃₂

C₂×Dic₅₉

yes

—

104

C₂×C₁₆

C₂×Dic₅₉

yes

—

105

C₂×C₁₆

C₂²⋊D₅₉

yes

characteristic

106

C₂×C₁₆

C₂²⋊D₅₉

yes

—

107

C₂×C₁₆

C₂²×D₅₉

yes

characteristic

108

#107

—

118

—

109

#106

—

236

—

110

C₃₂

—

118

—

111

C₂×Q₈

—

112

C₂×Q₈

—

113

C₂×Q₈

—

114

C₂×Q₈

—

115

C₂²×C₄

—

116

C₂²×C₄

#11745669

yes

characteristic

117

Q₁₆

—

118

Q₁₆

—

119

Q₁₆

—

120

C₂×C₈

#11745669

yes

characteristic

121

C₂×C₈

#11745670

yes

—

122

C₂×C₈

C₂²⋊Dic₅₉

yes

—

123

C₂×C₈

C₂×C₂²⋊D₅₉

yes

characteristic

124

C₄⋊C₄

—

236

—

125

C₂²⋊C₄

—

472

—

126

C₄²

—

472

—

127

C₁₆

C₂²⋊Dic₅₉

yes

—

128

C₁₆

C₂²×Dic₅₉

yes

characteristic

129

C₁₆

C₂×C₂²⋊D₅₉

yes

characteristic

130

C₂³

#14273668

yes

characteristic

131

Q₈

—

132

Q₈

—

133

Q₈

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

472

—

137

C₂×C₄

#14273648

yes

—

138

C₂×C₄

#14273668

yes

characteristic

139

C₂×C₄

#14273671

yes

—

140

C₂×C₄

C₂×#11745669

yes

characteristic

141

C₈

#14273648

yes

—

142

C₈

C₂×C₂²⋊Dic₅₉

yes

characteristic

143

C₈

C₂×#11745670

yes

characteristic

144

C₂²

—

145

C₂²

#16837203

yes

—

146

C₂²

C₂×#14273668

yes

characteristic

147

C₄

—

148

C₄

—

944

—

149

C₄

#16837203

yes

—

150

C₄

C₂×#14273648

yes

characteristic

151

C₄

C₂×#14273671

yes

characteristic

152

C₂

—

153

C₂

C₂×#16837203

yes

characteristic

154

C₂

#21426414

yes

—

	subgroup	G/H
center	C₂²	C₂×#14273668
commutator	C₁₆×C₅₉	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₅₉
Fitting	C₂²×Q₃₂×C₅₉	C₂
socle	C₂²×C₅₉	C₂×D₁₆
prime radical	C₂³×C₅₉	D₁₆
cosocle	C₂×C₁₆×C₅₉	C₂³
hypercenter	C₂²×Q₃₂	D₅₉
hypocenter	C₅₉	#13618
LCS	G ▷ C₁₆×C₅₉ ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂²×Q₃₂

subgroup

G/H

center

C₂²

C₂×#14273668

commutator

C₁₆×C₅₉

C₂²×C₄

Frattini

C₂×C₁₆

C₂²×D₅₉

Fitting

C₂²×Q₃₂×C₅₉

C₂

socle

C₂²×C₅₉

C₂×D₁₆

prime radical

C₂³×C₅₉

D₁₆

cosocle

C₂×C₁₆×C₅₉

C₂³

hypercenter

C₂²×Q₃₂

D₅₉

hypocenter

C₅₉

#13618

LCS

G ▷ C₁₆×C₅₉ ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂²×Q₃₂

N	H ≅ G/N	type
3. #21422277	C₂	semidirect
13. C₂×Q₃₂×C₅₉	C₄	semidirect
85. C₂×Q₃₂	Dic₅₉	semidirect
93. C₅₉	#13618	semidirect

H ≅ G/N

type

3. #21422277

C₂

semidirect

13. C₂×Q₃₂×C₅₉

C₄

semidirect

85. C₂×Q₃₂

Dic₅₉

semidirect

93. C₅₉

#13618

semidirect

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

subgroup	index
C₂×#16837236	2
#21425880	2
#21422277	2
C₂²×Q₃₂×C₅₉	2
#13618	59

subgroup

index

C₂²×Q₃₂×C₅₉

Subgroups — #28509136

Conjugacy classes of subgroups

Complements

Maximal subgroups