Group #28502969 — subgroups

Subgroups — #28502969

Order 15104.

212 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 67 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	#21426349	C₂	1	1	yes	maximal, characteristic
2	#21426004	C₂	1	1	yes	maximal, characteristic
3	#21423467	C₂	1	2	yes	maximal
4	#21422144	C₂	1	2	yes	maximal
5	C₅₉×#1372	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₇₂	—	2	1	no	—
7	C₂×Dic₄₇₂	—	2	1	no	—
8	#16837782	C₂²	1	2	yes	—
9	#16837688	—	2	2	no	—
10	C₈×Dic₁₁₈	C₂²	1	1	yes	characteristic
11	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	1	yes	characteristic
12	Dic₅₉⋊C₁₆	—	2	1	no	—
13	#16837177	—	2	1	no	—
14	#16837147	—	2	2	no	—
15	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
16	C₅₉×#335	C₂²	1	2	yes	—
17	C₂×Dic₂₃₆	—	2	2	no	—
18	C₂×Dic₂₃₆	—	2	2	no	—
19	#14273727	—	2	2	no	—
20	C₄×Dic₁₁₈	D₄	1	1	yes	characteristic
21	Dic₄₇₂	—	4	2	no	—
22	Dic₄₇₂	—	4	2	no	—
23	C₂×C₅₉⋊C₁₆	—	2	1	no	—
24	#14273659	—	4	2	no	—
25	C₄⋊(C₅₉⋊C₈)	D₄	1	1	yes	characteristic
26	Dic₅₉⋊C₈	—	2	1	no	—
27	#14273652	—	4	2	no	—
28	C₈×Dic₅₉	—	2	1	no	—
29	C₂×C₁₆×C₅₉	D₄	1	1	yes	characteristic
30	C₂×C₁₆×C₅₉	D₄	1	1	yes	characteristic
31	C₄⋊C₈×C₅₉	—	2	2	no	—
32	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
33	C₂×Dic₁₁₈	—	4	2	no	—
34	C₂×Dic₁₁₈	D₈	1	1	yes	characteristic
35	C₂×C₈×C₅₉	—	4	2	no	—
36	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
37	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
38	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
39	C₄⋊Dic₅₉	—	4	2	no	—
40	C₄⋊Dic₅₉	—	4	2	no	—
41	C₄⋊Dic₅₉	D₈	1	1	yes	characteristic
42	Dic₅₉⋊C₄	—	2	1	no	—
43	C₄×Dic₅₉	—	2	1	no	—
44	C₂×C₅₉⋊C₈	—	2	1	no	—
45	Dic₂₃₆	—	4	2	no	—
46	Dic₂₃₆	—	4	2	no	—
47	Dic₂₃₆	—	8	2	no	—
48	C₁₆×C₅₉	—	2	1	no	—
49	C₁₆×C₅₉	SD₁₆	1	2	yes	—
50	C₅₉⋊C₁₆	—	4	1	no	—
51	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
52	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
53	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
54	C₂×Dic₅₉	—	2	1	no	—
55	C₂×Dic₅₉	—	2	1	no	—
56	C₂×Dic₅₉	—	8	2	no	—
57	Dic₁₁₈	—	2	2	no	—
58	Dic₁₁₈	—	8	2	no	—
59	Dic₁₁₈	—	8	2	no	—
60	C₈×C₅₉	—	2	1	no	—
61	C₈×C₅₉	—	8	2	no	—
62	C₈×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
63	C₈×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
64	C₅₉⋊C₈	—	4	1	no	—
65	#12443	—	59	1	no	maximal, Sylow
66	C₂²×C₅₉	#466	1	1	yes	characteristic
67	C₄×C₅₉	—	2	1	no	—
68	C₄×C₅₉	#460	1	1	yes	characteristic
69	C₄×C₅₉	#472	1	1	yes	characteristic
70	C₄×C₅₉	#510	1	2	yes	—
71	Dic₅₉	—	4	1	no	—
72	Dic₅₉	—	4	1	no	—
73	Dic₅₉	—	16	2	no	—
74	#2247	—	59	1	no	—
75	#2183	—	59	1	no	—
76	#1670	—	59	2	no	—
77	#1372	D₅₉	1	1	yes	characteristic
78	#1362	—	59	2	no	—
79	C₂×C₅₉	#1668	1	1	yes	characteristic
80	C₂×C₅₉	#2219	1	1	yes	characteristic
81	C₂×C₅₉	#2222	1	1	yes	characteristic
82	C₂×Q₃₂	—	118	1	no	—
83	C₂×Q₃₂	—	118	1	no	—
84	#494	—	59	2	no	—
85	#475	—	118	2	no	—
86	C₈×Q₈	—	59	1	no	—
87	C₄⋊C₁₆	—	59	1	no	—
88	C₄⋊C₁₆	—	118	1	no	—
89	#346	—	118	1	no	—
90	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
91	#335	C₂×D₅₉	1	2	yes	—
92	#332	—	118	2	no	—
93	C₅₉	#12443	1	1	yes	characteristic, complemented, Sylow
94	C₂×Q₁₆	—	118	2	no	—
95	C₂×Q₁₆	—	118	2	no	—
96	C₄⋊Q₈	—	118	2	no	—
97	C₄×Q₈	—	59	1	no	—
98	Q₃₂	—	236	2	no	—
99	Q₃₂	—	236	2	no	—
100	C₂×C₁₆	—	118	1	no	—
101	C₂×C₁₆	D₂₃₆	1	1	yes	characteristic
102	C₂×C₁₆	D₂₃₆	1	1	yes	characteristic
103	#106	—	236	2	no	—
104	C₄⋊C₈	—	2	2	no	—
105	C₄⋊C₈	—	59	1	no	—
106	C₄⋊C₈	—	118	1	no	—
107	#103	—	236	2	no	—
108	C₄×C₈	—	118	1	no	—
109	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
110	C₂×Q₈	—	59	1	no	—
111	C₂×Q₈	—	236	2	no	—
112	Q₁₆	—	472	2	no	—
113	Q₁₆	—	236	2	no	—
114	Q₁₆	—	236	2	no	—
115	C₂×C₈	—	4	2	no	—
116	C₂×C₈	—	118	1	no	—
117	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
118	C₂×C₈	#11745689	1	1	yes	characteristic
119	C₄⋊C₄	—	59	1	no	—
120	C₄⋊C₄	—	118	1	no	—
121	C₄⋊C₄	—	236	2	no	—
122	C₄⋊C₄	—	236	2	no	—
123	C₄²	—	118	1	no	—
124	C₄²	D₄×D₅₉	1	1	yes	characteristic
125	C₁₆	—	2	1	no	—
126	C₁₆	—	236	1	no	—
127	C₁₆	#11745660	1	2	yes	—
128	Q₈	—	118	2	no	—
129	Q₈	—	472	2	no	—
130	Q₈	—	472	2	no	—
131	C₂×C₄	—	118	1	no	—
132	C₂×C₄	—	118	1	no	—
133	C₂×C₄	—	472	2	no	—
134	C₂×C₄	#14273702	1	1	yes	characteristic
135	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
136	C₂×C₄	#14273750	1	1	yes	characteristic
137	C₈	—	2	1	no	—
138	C₈	—	8	2	no	—
139	C₈	—	236	1	no	—
140	C₈	C₂×#11745660	1	1	yes	characteristic
141	C₈	#14273755	1	1	yes	characteristic
142	C₂²	#16837639	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	944	2	no	—
145	C₄	—	236	1	no	—
146	C₄	—	236	1	no	—
147	C₄	#16837603	1	1	yes	characteristic
148	C₄	#16837673	1	1	yes	characteristic
149	C₄	#16837841	1	2	yes	—
150	C₂	#21423457	1	1	yes	characteristic
151	C₂	#21426205	1	1	yes	characteristic
152	C₂	#21426222	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21426349

C₂

yes

maximal, characteristic

#21426004

C₂

yes

maximal, characteristic

#21423467

C₂

yes

maximal

#21422144

C₂

yes

maximal

C₅₉×#1372

C₂

yes

maximal, characteristic

C₂×Dic₄₇₂

—

C₂×Dic₄₇₂

—

#16837782

C₂²

yes

—

#16837688

—

C₈×Dic₁₁₈

C₂²

yes

characteristic

C₄⋊(C₅₉⋊C₁₆)

C₂²

yes

characteristic

Dic₅₉⋊C₁₆

—

#16837177

—

#16837147

—

C₄×C₁₆×C₅₉

C₂²

yes

characteristic

C₅₉×#335

C₂²

yes

—

C₂×Dic₂₃₆

—

C₂×Dic₂₃₆

—

#14273727

—

C₄×Dic₁₁₈

D₄

yes

characteristic

Dic₄₇₂

—

Dic₄₇₂

—

C₂×C₅₉⋊C₁₆

—

#14273659

—

C₄⋊(C₅₉⋊C₈)

D₄

yes

characteristic

Dic₅₉⋊C₈

—

#14273652

—

C₈×Dic₅₉

—

C₂×C₁₆×C₅₉

D₄

yes

characteristic

C₂×C₁₆×C₅₉

D₄

yes

characteristic

C₄⋊C₈×C₅₉

—

C₄×C₈×C₅₉

C₂³

yes

characteristic

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

D₈

yes

characteristic

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

#41

yes

characteristic

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

D₈

yes

characteristic

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

Dic₂₃₆

—

Dic₂₃₆

—

Dic₂₃₆

—

C₁₆×C₅₉

—

C₁₆×C₅₉

SD₁₆

yes

—

C₅₉⋊C₁₆

—

C₂×C₄×C₅₉

#121

yes

characteristic

C₂×C₄×C₅₉

C₂×D₈

yes

characteristic

C₂×C₄×C₅₉

#135

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

C₂×SD₁₆

yes

characteristic

C₈×C₅₉

AGL(1,ℤ/8)

yes

characteristic

C₅₉⋊C₈

—

—

C₂²×C₅₉

yes

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

—

—

—

D₅₉

yes

characteristic

#1362

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×Q₃₂

—

118

—

C₂×Q₃₂

—

118

—

#494

—

#475

—

118

—

C₈×Q₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

118

—

#346

—

118

—

C₄×C₁₆

C₂×D₅₉

yes

characteristic

#335

C₂×D₅₉

yes

—

#332

—

118

—

C₅₉

#12443

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

118

—

C₂×Q₁₆

—

118

—

C₄⋊Q₈

—

118

—

C₄×Q₈

—

Q₃₂

—

236

—

Q₃₂

—

236

—

100

C₂×C₁₆

—

118

—

101

C₂×C₁₆

D₂₃₆

yes

characteristic

102

C₂×C₁₆

D₂₃₆

yes

characteristic

103

#106

—

236

—

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

C₄⋊C₈

—

118

—

107

#103

—

236

—

108

C₄×C₈

—

118

—

109

C₄×C₈

C₂²×D₅₉

yes

characteristic

110

C₂×Q₈

—

111

C₂×Q₈

—

236

—

112

Q₁₆

—

472

—

113

Q₁₆

—

236

—

114

Q₁₆

—

236

—

115

C₂×C₈

—

116

C₂×C₈

—

118

—

117

C₂×C₈

C₂×D₂₃₆

yes

characteristic

118

C₂×C₈

#11745689

yes

characteristic

119

C₄⋊C₄

—

120

C₄⋊C₄

—

118

—

121

C₄⋊C₄

—

236

—

122

C₄⋊C₄

—

236

—

123

C₄²

—

118

—

124

C₄²

D₄×D₅₉

yes

characteristic

125

C₁₆

—

126

C₁₆

—

236

—

127

C₁₆

#11745660

yes

—

128

Q₈

—

118

—

129

Q₈

—

472

—

130

Q₈

—

472

—

131

C₂×C₄

—

118

—

132

C₂×C₄

—

118

—

133

C₂×C₄

—

472

—

134

C₂×C₄

#14273702

yes

characteristic

135

C₂×C₄

D₈×D₅₉

yes

characteristic

136

C₂×C₄

#14273750

yes

characteristic

137

C₈

—

138

C₈

—

139

C₈

—

236

—

140

C₈

C₂×#11745660

yes

characteristic

141

C₈

#14273755

yes

characteristic

142

C₂²

#16837639

yes

characteristic

143

C₄

—

144

C₄

—

944

—

145

C₄

—

236

—

146

C₄

—

236

—

147

C₄

#16837603

yes

characteristic

148

C₄

#16837673

yes

characteristic

149

C₄

#16837841

yes

—

150

C₂

#21423457

yes

characteristic

151

C₂

#21426205

yes

characteristic

152

C₂

#21426222

yes

characteristic

	subgroup	G/H
center	C₂²	#16837639
commutator, cosocle	C₄×C₈×C₅₉	C₂³
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₅₉×#1372	C₂
socle, prime radical	C₂²×C₅₉	#466
hypercenter	#1372	D₅₉
hypocenter	C₅₉	#12443
LCS	G ▷ C₄×C₈×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1372

subgroup

G/H

center

C₂²

#16837639

commutator, cosocle

C₄×C₈×C₅₉

C₂³

Frattini

C₄×C₈

C₂²×D₅₉

Fitting

C₅₉×#1372

C₂

C₂²×C₅₉

D₅₉

C₅₉

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1372

N	H ≅ G/N	type
93. C₅₉	#12443	semidirect

H ≅ G/N

type

93. C₅₉

#12443

semidirect

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

subgroup	index
#21426349	2
#21426004	2
#21423467	2
#21422144	2
C₅₉×#1372	2
#12443	59

subgroup

index

C₅₉×#1372

Subgroups — #28502969

Conjugacy classes of subgroups

Complements

Maximal subgroups