Group #28502948 — subgroups

Subgroups — #28502948

Order 15104.

216 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 69 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	C₁₆×Dic₁₁₈	C₂	1	1	yes	maximal, characteristic
2	#21423463	C₂	1	2	yes	maximal
3	#21422168	C₂	1	1	yes	maximal, characteristic
4	#21422145	C₂	1	2	yes	maximal
5	C₅₉×#2247	C₂	1	1	yes	maximal, characteristic
6	#16837684	—	2	2	no	—
7	C₈×Dic₁₁₈	C₂²	1	1	yes	characteristic
8	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	1	yes	characteristic
9	Dic₅₉⋊C₁₆	—	2	1	no	—
10	C₁₆×Dic₅₉	—	2	1	no	—
11	#16837155	C₂²	1	2	yes	—
12	#16837148	—	2	2	no	—
13	C₂×Q₃₂×C₅₉	—	2	1	no	—
14	C₂×Q₃₂×C₅₉	—	2	1	no	—
15	C₅₉×#494	C₂²	1	2	yes	—
16	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
17	C₄×Dic₁₁₈	D₄	1	1	yes	characteristic
18	C₂×C₅₉⋊C₁₆	—	2	1	no	—
19	#14273658	—	4	2	no	—
20	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
21	C₄⋊(C₅₉⋊C₈)	D₄	1	1	yes	characteristic
22	Dic₅₉⋊C₈	—	2	1	no	—
23	#14273650	—	4	2	no	—
24	C₈×Dic₅₉	—	2	1	no	—
25	C₂×Q₁₆×C₅₉	—	2	2	no	—
26	C₂×Q₁₆×C₅₉	—	2	2	no	—
27	C₄⋊Q₈×C₅₉	—	2	2	no	—
28	Q₃₂×C₅₉	—	4	2	no	—
29	Q₃₂×C₅₉	—	4	2	no	—
30	C₂×C₁₆×C₅₉	D₄	1	1	yes	characteristic
31	C₂×C₁₆×C₅₉	D₄	1	1	yes	characteristic
32	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
33	C₂×Q₈×C₅₉	—	4	2	no	—
34	C₂×Dic₁₁₈	D₈	1	1	yes	characteristic
35	Q₁₆×C₅₉	—	4	2	no	—
36	Q₁₆×C₅₉	—	4	2	no	—
37	Q₁₆×C₅₉	—	8	2	no	—
38	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
40	C₄⋊C₄×C₅₉	—	4	2	no	—
41	C₄⋊C₄×C₅₉	—	4	2	no	—
42	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
43	C₄⋊Dic₅₉	D₈	1	1	yes	characteristic
44	Dic₅₉⋊C₄	—	2	1	no	—
45	C₄×Dic₅₉	—	2	1	no	—
46	C₂×C₅₉⋊C₈	—	2	1	no	—
47	C₂×C₅₉⋊C₈	—	4	2	no	—
48	C₁₆×C₅₉	—	2	1	no	—
49	C₁₆×C₅₉	SD₁₆	1	2	yes	—
50	C₅₉⋊C₁₆	—	4	1	no	—
51	Q₈×C₅₉	—	8	2	no	—
52	Q₈×C₅₉	—	8	2	no	—
53	C₂×C₄×C₅₉	—	8	2	no	—
54	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
55	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
56	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
57	C₂×Dic₅₉	—	2	1	no	—
58	C₂×Dic₅₉	—	2	1	no	—
59	Dic₁₁₈	—	2	1	no	—
60	Dic₁₁₈	Q₃₂	1	2	yes	complemented
61	C₈×C₅₉	—	2	1	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	C₅₉⋊C₈	—	8	2	no	—
66	#12440	—	59	1	no	maximal, Sylow
67	C₂²×C₅₉	#466	1	1	yes	characteristic
68	C₄×C₅₉	—	2	1	no	—
69	C₄×C₅₉	—	16	2	no	—
70	C₄×C₅₉	#460	1	1	yes	characteristic
71	C₄×C₅₉	#472	1	1	yes	characteristic
72	C₄×C₅₉	C₂×Q₃₂	1	1	yes	characteristic
73	C₄×C₅₉	#508	1	1	yes	characteristic
74	Dic₅₉	—	2	2	no	—
75	Dic₅₉	—	4	1	no	—
76	#2247	D₅₉	1	1	yes	characteristic
77	Q₈×C₁₆	—	59	1	no	—
78	#1670	—	59	2	no	—
79	#1371	—	59	1	no	—
80	#1362	—	59	2	no	—
81	C₂×C₅₉	#1668	1	1	yes	characteristic
82	C₂×C₅₉	#2216	1	1	yes	characteristic
83	C₂×C₅₉	#2222	1	1	yes	characteristic
84	C₂×Q₃₂	—	2	1	no	—
85	C₂×Q₃₂	—	2	1	no	—
86	#494	C₂×D₅₉	1	2	yes	—
87	#475	—	118	2	no	—
88	C₈×Q₈	—	59	1	no	—
89	C₄⋊C₁₆	—	59	1	no	—
90	C₄⋊C₁₆	—	118	1	no	—
91	C₄×C₁₆	—	118	1	no	—
92	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
93	#335	—	59	2	no	—
94	#332	—	118	2	no	—
95	C₅₉	#12440	1	1	yes	characteristic, complemented, Sylow
96	C₂×Q₁₆	—	2	2	no	—
97	C₂×Q₁₆	—	2	2	no	—
98	C₄⋊Q₈	—	2	2	no	—
99	C₄×Q₈	—	59	1	no	—
100	Q₃₂	—	4	2	no	—
101	Q₃₂	—	4	2	no	—
102	C₂×C₁₆	—	118	1	no	—
103	C₂×C₁₆	C₂²⋊D₅₉	1	1	yes	characteristic
104	C₂×C₁₆	C₂²⋊D₅₉	1	1	yes	characteristic
105	#106	—	236	2	no	—
106	C₄⋊C₈	—	59	1	no	—
107	C₄⋊C₈	—	118	1	no	—
108	C₄⋊C₈	—	118	2	no	—
109	#103	—	236	2	no	—
110	C₄×C₈	—	118	1	no	—
111	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
112	C₂×Q₈	—	4	2	no	—
113	C₂×Q₈	—	59	1	no	—
114	Q₁₆	—	4	2	no	—
115	Q₁₆	—	4	2	no	—
116	Q₁₆	—	8	2	no	—
117	C₂×C₈	—	118	1	no	—
118	C₂×C₈	—	236	2	no	—
119	C₂×C₈	#11745687	1	1	yes	characteristic
120	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
121	C₄⋊C₄	—	4	2	no	—
122	C₄⋊C₄	—	4	2	no	—
123	C₄⋊C₄	—	59	1	no	—
124	C₄⋊C₄	—	118	1	no	—
125	C₄²	—	118	1	no	—
126	C₄²	D₄×D₅₉	1	1	yes	characteristic
127	C₁₆	—	2	1	no	—
128	C₁₆	—	236	1	no	—
129	C₁₆	#11745670	1	2	yes	—
130	Q₈	—	8	2	no	—
131	Q₈	—	8	2	no	—
132	Q₈	—	59	2	no	—
133	Q₈	—	118	1	no	—
134	C₂×C₄	—	8	2	no	—
135	C₂×C₄	—	118	1	no	—
136	C₂×C₄	—	118	1	no	—
137	C₂×C₄	#14273698	1	1	yes	characteristic
138	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
139	C₂×C₄	#14273746	1	1	yes	characteristic
140	C₈	—	2	1	no	—
141	C₈	—	472	2	no	—
142	C₈	—	236	1	no	—
143	C₈	C₂×#11745670	1	1	yes	characteristic
144	C₈	#14273753	1	1	yes	characteristic
145	C₂²	#16837635	1	1	yes	characteristic
146	C₄	—	2	1	no	—
147	C₄	—	16	2	no	—
148	C₄	—	118	2	no	—
149	C₄	—	236	1	no	—
150	C₄	#16837599	1	1	yes	characteristic
151	C₄	#16837669	1	1	yes	characteristic
152	C₄	Q₃₂×D₅₉	1	1	yes	characteristic
153	C₄	#16837835	1	1	yes	characteristic
154	C₂	#21423453	1	1	yes	characteristic
155	C₂	#21426184	1	1	yes	characteristic
156	C₂	#21426218	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₆×Dic₁₁₈

C₂

yes

maximal, characteristic

#21423463

C₂

yes

maximal

#21422168

C₂

yes

maximal, characteristic

#21422145

C₂

yes

maximal

C₅₉×#2247

C₂

yes

maximal, characteristic

#16837684

—

C₈×Dic₁₁₈

C₂²

yes

characteristic

C₄⋊(C₅₉⋊C₁₆)

C₂²

yes

characteristic

Dic₅₉⋊C₁₆

—

C₁₆×Dic₅₉

—

#16837155

C₂²

yes

—

#16837148

—

C₂×Q₃₂×C₅₉

—

C₂×Q₃₂×C₅₉

—

C₅₉×#494

C₂²

yes

—

C₄×C₁₆×C₅₉

C₂²

yes

characteristic

C₄×Dic₁₁₈

D₄

yes

characteristic

C₂×C₅₉⋊C₁₆

—

#14273658

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

D₄

yes

characteristic

Dic₅₉⋊C₈

—

#14273650

—

C₈×Dic₅₉

—

C₂×Q₁₆×C₅₉

—

C₂×Q₁₆×C₅₉

—

C₄⋊Q₈×C₅₉

—

Q₃₂×C₅₉

—

Q₃₂×C₅₉

—

C₂×C₁₆×C₅₉

D₄

yes

characteristic

C₂×C₁₆×C₅₉

D₄

yes

characteristic

C₄×C₈×C₅₉

C₂³

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×Dic₁₁₈

D₈

yes

characteristic

Q₁₆×C₅₉

—

Q₁₆×C₅₉

—

Q₁₆×C₅₉

—

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

#41

yes

characteristic

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄⋊Dic₅₉

D₈

yes

characteristic

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₁₆×C₅₉

—

C₁₆×C₅₉

SD₁₆

yes

—

C₅₉⋊C₁₆

—

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#121

yes

characteristic

C₂×C₄×C₅₉

C₂×D₈

yes

characteristic

C₂×C₄×C₅₉

#135

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

Q₃₂

yes

complemented

C₈×C₅₉

—

C₈×C₅₉

C₂×SD₁₆

yes

characteristic

C₈×C₅₉

AGL(1,ℤ/8)

yes

characteristic

C₅₉⋊C₈

—

C₅₉⋊C₈

—

—

C₂²×C₅₉

yes

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

C₂×Q₃₂

yes

characteristic

C₄×C₅₉

#508

yes

characteristic

Dic₅₉

—

Dic₅₉

—

#2247

D₅₉

yes

characteristic

Q₈×C₁₆

—

#1670

—

#1371

—

#1362

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×Q₃₂

—

C₂×Q₃₂

—

#494

C₂×D₅₉

yes

—

#475

—

118

—

C₈×Q₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

118

—

C₄×C₁₆

—

118

—

C₄×C₁₆

C₂×D₅₉

yes

characteristic

#335

—

#332

—

118

—

C₅₉

#12440

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄×Q₈

—

100

Q₃₂

—

101

Q₃₂

—

102

C₂×C₁₆

—

118

—

103

C₂×C₁₆

C₂²⋊D₅₉

yes

characteristic

104

C₂×C₁₆

C₂²⋊D₅₉

yes

characteristic

105

#106

—

236

—

106

C₄⋊C₈

—

107

C₄⋊C₈

—

118

—

108

C₄⋊C₈

—

118

—

109

#103

—

236

—

110

C₄×C₈

—

118

—

111

C₄×C₈

C₂²×D₅₉

yes

characteristic

112

C₂×Q₈

—

113

C₂×Q₈

—

114

Q₁₆

—

115

Q₁₆

—

116

Q₁₆

—

117

C₂×C₈

—

118

—

118

C₂×C₈

—

236

—

119

C₂×C₈

#11745687

yes

characteristic

120

C₂×C₈

C₂×C₂²⋊D₅₉

yes

characteristic

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

118

—

125

C₄²

—

118

—

126

C₄²

D₄×D₅₉

yes

characteristic

127

C₁₆

—

128

C₁₆

—

236

—

129

C₁₆

#11745670

yes

—

130

Q₈

—

131

Q₈

—

132

Q₈

—

133

Q₈

—

118

—

134

C₂×C₄

—

135

C₂×C₄

—

118

—

136

C₂×C₄

—

118

—

137

C₂×C₄

#14273698

yes

characteristic

138

C₂×C₄

D₈×D₅₉

yes

characteristic

139

C₂×C₄

#14273746

yes

characteristic

140

C₈

—

141

C₈

—

472

—

142

C₈

—

236

—

143

C₈

C₂×#11745670

yes

characteristic

144

C₈

#14273753

yes

characteristic

145

C₂²

#16837635

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

—

118

—

149

C₄

—

236

—

150

C₄

#16837599

yes

characteristic

151

C₄

#16837669

yes

characteristic

152

C₄

Q₃₂×D₅₉

yes

characteristic

153

C₄

#16837835

yes

characteristic

154

C₂

#21423453

yes

characteristic

155

C₂

#21426184

yes

characteristic

156

C₂

#21426218

yes

characteristic

	subgroup	G/H
center	C₂²	#16837635
commutator, cosocle	C₄×C₈×C₅₉	C₂³
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₅₉×#2247	C₂
socle, prime radical	C₂²×C₅₉	#466
hypercenter	#2247	D₅₉
hypocenter	C₅₉	#12440
LCS	G ▷ C₄×C₈×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

subgroup

G/H

center

C₂²

#16837635

commutator, cosocle

C₄×C₈×C₅₉

C₂³

Frattini

C₄×C₈

C₂²×D₅₉

Fitting

C₅₉×#2247

C₂

C₂²×C₅₉

D₅₉

C₅₉

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

N	H ≅ G/N	type
60. Dic₁₁₈	Q₃₂	semidirect
95. C₅₉	#12440	semidirect

H ≅ G/N

type

60. Dic₁₁₈

Q₃₂

semidirect

95. C₅₉

#12440

semidirect

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

subgroup	index
C₁₆×Dic₁₁₈	2
#21423463	2
#21422168	2
#21422145	2
C₅₉×#2247	2
#12440	59

subgroup

index

C₁₆×Dic₁₁₈

C₅₉×#2247

Subgroups — #28502948

Conjugacy classes of subgroups

Complements

Maximal subgroups