Group #28503387 — subgroups

Subgroups — #28503387

Order 15104.

268 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 79 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	#21426323	C₂	1	2	yes	maximal, complemented
2	#21426319	C₂	1	2	yes	maximal
3	#21423714	C₂	1	1	yes	maximal, characteristic, complemented
4	#21422013	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₉×#1709	C₂	1	1	yes	maximal, characteristic
6	#16837807	—	2	1	no	—
7	#16837807	C₂²	1	2	yes	—
8	#16837782	—	2	1	no	—
9	#16837206	—	2	4	no	—
10	#16837203	—	2	4	no	—
11	C₄×C₅₉⋊C₁₆	C₂²	1	2	yes	—
12	C₅₉×#494	C₂²	1	1	yes	characteristic
13	C₅₉×#493	C₂²	1	1	yes	characteristic
14	C₅₉×#486	—	2	1	no	—
15	C₅₉×#486	—	2	1	no	—
16	C₈²×C₅₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₃₆	—	4	2	no	—
18	#14273727	—	2	2	no	—
19	C₂×C₅₉⋊C₁₆	D₄	1	4	yes	—
20	#14273661	—	2	2	no	—
21	#14273661	—	2	2	no	—
22	#14273661	—	4	2	no	—
23	C₂×Q₁₆×C₅₉	—	2	2	no	—
24	C₂×D₈×C₅₉	—	2	2	no	—
25	C₄⋊Q₈×C₅₉	—	2	1	no	—
26	#127×C₅₉	—	2	1	no	—
27	#102×C₅₉	—	4	2	no	—
28	#102×C₅₉	—	4	2	no	—
29	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
30	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
31	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
32	C₂×Q₈×C₅₉	—	4	1	no	—
33	C₂×D₄×C₅₉	—	4	1	no	—
34	C₂×D₄×C₅₉	—	4	1	no	—
35	C₂×D₄×C₅₉	—	4	1	no	—
36	C₂×Dic₁₁₈	—	4	2	no	—
37	Q₁₆×C₅₉	—	4	4	no	—
38	D₈×C₅₉	—	4	4	no	—
39	C₂×C₈×C₅₉	D₈	1	2	yes	—
40	C₂×C₈×C₅₉	D₈	1	2	yes	—
41	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
42	C₄⋊C₄×C₅₉	—	4	1	no	—
43	C₄⋊C₄×C₅₉	—	4	1	no	—
44	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
45	C₄⋊Dic₅₉	—	4	2	no	—
46	C₄⋊Dic₅₉	—	4	2	no	—
47	Dic₂₃₆	—	8	4	no	—
48	C₅₉⋊C₁₆	—	2	4	no	—
49	C₂³×C₅₉	—	8	1	no	—
50	Q₈×C₅₉	—	8	2	no	—
51	D₄×C₅₉	—	8	2	no	—
52	D₄×C₅₉	—	8	2	no	—
53	D₄×C₅₉	—	8	2	no	—
54	C₂×C₄×C₅₉	—	8	1	no	—
55	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
56	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
57	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
58	C₂×Dic₅₉	—	8	2	no	—
59	Dic₁₁₈	—	8	4	no	—
60	C₈×C₅₉	—	2	2	no	—
61	C₈×C₅₉	—	2	2	no	—
62	C₈×C₅₉	#135	1	4	yes	—
63	#12512	—	59	1	no	maximal, Sylow
64	C₂²×C₅₉	—	16	1	no	—
65	C₂²×C₅₉	—	16	1	no	—
66	C₂²×C₅₉	—	16	1	no	—
67	C₂²×C₅₉	#493	1	1	yes	characteristic
68	C₄×C₅₉	—	16	1	no	—
69	C₄×C₅₉	#495	1	2	yes	—
70	C₄×C₅₉	#509	1	2	yes	—
71	C₄×C₅₉	#510	1	2	yes	—
72	Dic₅₉	—	16	2	no	—
73	#2241	—	59	2	no	—
74	#2240	—	59	2	no	—
75	#1713	—	59	1	no	—
76	#1709	D₅₉	1	1	yes	characteristic
77	#1311	—	59	1	no	—
78	C₂×C₅₉	—	32	1	no	—
79	C₂×C₅₉	#1709	1	1	yes	characteristic
80	C₂×C₅₉	#2250	1	1	yes	characteristic
81	C₂×C₅₉	#2251	1	1	yes	characteristic
82	#500	—	59	2	no	—
83	#500	—	118	1	no	—
84	#494	—	118	1	no	—
85	#494	C₂×D₅₉	1	1	yes	characteristic
86	#493	C₂×D₅₉	1	1	yes	characteristic
87	#486	—	2	1	no	—
88	#486	—	2	1	no	—
89	#358	—	118	4	no	—
90	#357	—	118	4	no	—
91	C₄×C₁₆	—	59	2	no	—
92	C₈²	C₂×D₅₉	1	1	yes	characteristic
93	C₅₉	#12512	1	1	yes	characteristic, complemented, Sylow
94	C₂×Q₁₆	—	2	2	no	—
95	C₂×Q₁₆	—	236	2	no	—
96	C₂×D₈	—	2	2	no	—
97	C₄⋊Q₈	—	2	1	no	—
98	C₄⋊Q₈	—	118	2	no	—
99	#127	—	2	1	no	—
100	C₂×C₁₆	—	59	4	no	—
101	#107	—	118	2	no	—
102	#107	—	118	2	no	—
103	#107	—	236	2	no	—
104	#102	—	4	2	no	—
105	#102	—	4	2	no	—
106	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
107	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
108	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
109	C₂×Q₈	—	4	1	no	—
110	C₂×Q₈	—	236	2	no	—
111	C₂×D₄	—	4	1	no	—
112	C₂×D₄	—	4	1	no	—
113	C₂×D₄	—	4	1	no	—
114	Q₁₆	—	4	4	no	—
115	Q₁₆	—	472	4	no	—
116	D₈	—	4	4	no	—
117	C₂×C₈	#11745669	1	2	yes	—
118	C₂×C₈	#11745669	1	2	yes	—
119	C₂×C₈	D₄×D₅₉	1	2	yes	—
120	C₄⋊C₄	—	4	1	no	—
121	C₄⋊C₄	—	4	1	no	—
122	C₄⋊C₄	—	236	2	no	—
123	C₄⋊C₄	—	236	2	no	—
124	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
125	C₁₆	—	118	4	no	—
126	C₂³	—	8	1	no	—
127	Q₈	—	8	2	no	—
128	Q₈	—	472	4	no	—
129	D₄	—	8	2	no	—
130	D₄	—	8	2	no	—
131	D₄	—	8	2	no	—
132	C₂×C₄	—	8	1	no	—
133	C₂×C₄	—	472	2	no	—
134	C₂×C₄	#14273723	1	1	yes	characteristic
135	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
136	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
137	C₈	—	2	2	no	—
138	C₈	—	2	2	no	—
139	C₈	#14273746	1	4	yes	—
140	C₂²	—	16	1	no	—
141	C₂²	—	16	1	no	—
142	C₂²	—	16	1	no	—
143	C₂²	#16837778	1	1	yes	characteristic
144	C₄	—	16	1	no	—
145	C₄	—	944	2	no	—
146	C₄	#16837785	1	2	yes	—
147	C₄	#16837837	1	2	yes	—
148	C₄	#16837842	1	2	yes	—
149	C₂	—	32	1	no	—
150	C₂	#21423701	1	1	yes	characteristic
151	C₂	#21426363	1	1	yes	characteristic
152	C₂	#21426368	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21426323

C₂

yes

maximal, complemented

#21426319

C₂

yes

maximal

#21423714

C₂

yes

maximal, characteristic, complemented

#21422013

C₂

yes

maximal, characteristic, complemented

C₅₉×#1709

C₂

yes

maximal, characteristic

#16837807

—

#16837807

C₂²

yes

—

#16837782

—

#16837206

—

#16837203

—

C₄×C₅₉⋊C₁₆

C₂²

yes

—

C₅₉×#494

C₂²

yes

characteristic

C₅₉×#493

C₂²

yes

characteristic

C₅₉×#486

—

C₅₉×#486

—

C₈²×C₅₉

C₂²

yes

characteristic

C₂×Dic₂₃₆

—

#14273727

—

C₂×C₅₉⋊C₁₆

D₄

yes

—

#14273661

—

#14273661

—

#14273661

—

C₂×Q₁₆×C₅₉

—

C₂×D₈×C₅₉

—

C₄⋊Q₈×C₅₉

—

#127×C₅₉

—

#102×C₅₉

—

#102×C₅₉

—

C₄×C₈×C₅₉

D₄

yes

characteristic

C₄×C₈×C₅₉

D₄

yes

characteristic

C₄×C₈×C₅₉

C₂³

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×Dic₁₁₈

—

Q₁₆×C₅₉

—

D₈×C₅₉

—

C₂×C₈×C₅₉

D₈

yes

—

C₂×C₈×C₅₉

D₈

yes

—

C₂×C₈×C₅₉

C₂×D₄

yes

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

Dic₂₃₆

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

—

Q₈×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#127

yes

characteristic

C₂×C₄×C₅₉

C₂×D₈

yes

characteristic

C₂×C₄×C₅₉

C₂×D₈

yes

characteristic

C₂×Dic₅₉

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

#135

yes

—

#12512

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#493

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

#495

yes

—

C₄×C₅₉

#509

yes

—

C₄×C₅₉

#510

yes

—

Dic₅₉

—

—

—

—

D₅₉

yes

characteristic

#1311

—

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

—

118

—

#494

—

118

—

#494

C₂×D₅₉

yes

characteristic

#493

C₂×D₅₉

yes

—

—

—

118

—

#357

—

118

—

C₄×C₁₆

—

C₈²

C₂×D₅₉

yes

characteristic

C₅₉

#12512

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

236

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

118

—

#127

—

100

C₂×C₁₆

—

101

#107

—

118

—

102

#107

—

118

—

103

#107

—

236

—

104

#102

—

105

#102

—

106

C₄×C₈

C₂²⋊D₅₉

yes

characteristic

107

C₄×C₈

C₂²⋊D₅₉

yes

characteristic

108

C₄×C₈

C₂²×D₅₉

yes

characteristic

109

C₂×Q₈

—

110

C₂×Q₈

—

236

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

114

Q₁₆

—

115

Q₁₆

—

472

—

116

D₈

—

117

C₂×C₈

#11745669

yes

—

118

C₂×C₈

#11745669

yes

—

119

C₂×C₈

D₄×D₅₉

yes

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

236

—

123

C₄⋊C₄

—

236

—

124

C₄²

C₂×C₂²⋊D₅₉

yes

characteristic

125

C₁₆

—

118

—

126

C₂³

—

127

Q₈

—

128

Q₈

—

472

—

129

D₄

—

130

D₄

—

131

D₄

—

132

C₂×C₄

—

133

C₂×C₄

—

472

—

134

C₂×C₄

#14273723

yes

characteristic

135

C₂×C₄

C₂×#11745669

yes

characteristic

136

C₂×C₄

C₂×#11745669

yes

characteristic

137

C₈

—

138

C₈

—

139

C₈

#14273746

yes

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

#16837778

yes

characteristic

144

C₄

—

145

C₄

—

944

—

146

C₄

#16837785

yes

—

147

C₄

#16837837

yes

—

148

C₄

#16837842

yes

—

149

C₂

—

150

C₂

#21423701

yes

characteristic

151

C₂

#21426363

yes

characteristic

152

C₂

#21426368

yes

characteristic

	subgroup	G/H
center	C₂²	#16837778
commutator, cosocle	C₄×C₈×C₅₉	C₂³
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₅₉×#1709	C₂
socle	C₂²×C₅₉	#493
prime radical	C₅₉×#493	C₂²
hypercenter	#1709	D₅₉
hypocenter	C₅₉	#12512
LCS	G ▷ C₄×C₈×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

subgroup

G/H

center

C₂²

#16837778

commutator, cosocle

C₄×C₈×C₅₉

C₂³

Frattini

C₄×C₈

C₂²×D₅₉

Fitting

C₅₉×#1709

C₂

socle

C₂²×C₅₉

#493

prime radical

C₅₉×#493

C₂²

D₅₉

C₅₉

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

N	H ≅ G/N	type
1. #21426323	C₂	semidirect
3. #21423714	C₂	semidirect
4. #21422013	C₂	semidirect
93. C₅₉	#12512	semidirect

H ≅ G/N

type

1. #21426323

C₂

semidirect

3. #21423714

C₂

semidirect

4. #21422013

C₂

semidirect

93. C₅₉

#12512

semidirect

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

subgroup	index
#21426323	2
#21426319	2
#21423714	2
#21422013	2
C₅₉×#1709	2
#12512	59

subgroup

index

C₅₉×#1709

Subgroups — #28503387

Conjugacy classes of subgroups

Complements

Maximal subgroups