Group #28509601 — subgroups

Subgroups — #28509601

Order 15104.

158 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 55 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	#21426373	C₂	1	1	yes	maximal, characteristic
2	#21425954	C₂	1	1	yes	maximal, characteristic
3	#21422295	C₂	1	1	yes	maximal, characteristic
4	Dic₅₉⋊C₃₂	C₂	1	1	yes	maximal, characteristic
5	#21422278	C₂	1	1	yes	maximal, characteristic
6	#21422276	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#1424	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₄₇₂	C₂²	1	1	yes	characteristic
9	#16837803	—	2	1	no	—
10	#16837496	—	2	1	no	—
11	C₂×C₅₉⋊C₃₂	C₂²	1	1	yes	characteristic
12	#16837229	—	2	1	no	—
13	#16837228	C₂²	1	1	yes	characteristic
14	#16837205	—	2	1	no	—
15	C₁₆×Dic₅₉	C₂²	1	1	yes	characteristic
16	C₂×C₃₂×C₅₉	C₂²	1	1	yes	characteristic
17	C₅₉×#366	C₂²	1	1	yes	characteristic
18	C₅₉×#366	C₂²	1	1	yes	characteristic
19	C₂×Dic₂₃₆	—	2	1	no	—
20	#14273718	—	4	1	no	—
21	#14273692	—	4	1	no	—
22	Dic₄₇₂	—	2	1	no	—
23	Dic₄₇₂	Q₈	1	2	yes	—
24	C₂×C₅₉⋊C₁₆	D₄	1	1	yes	characteristic
25	#14273661	—	2	1	no	—
26	#14273658	—	4	1	no	—
27	#14273650	—	4	1	no	—
28	C₈×Dic₅₉	D₄	1	1	yes	characteristic
29	C₅₉⋊C₃₂	—	2	1	no	—
30	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
31	#107×C₅₉	—	2	1	no	—
32	#107×C₅₉	—	2	1	no	—
33	C₃₂×C₅₉	—	2	1	no	—
34	C₂×Dic₁₁₈	—	4	1	no	—
35	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
36	C₄⋊C₄×C₅₉	—	4	1	no	—
37	C₄⋊C₄×C₅₉	—	4	1	no	—
38	C₄⋊Dic₅₉	—	4	1	no	—
39	C₄⋊Dic₅₉	—	8	1	no	—
40	Dic₅₉⋊C₄	—	8	1	no	—
41	Dic₅₉⋊C₄	—	8	1	no	—
42	C₄×Dic₅₉	—	8	1	no	—
43	C₄×Dic₅₉	D₈	1	1	yes	characteristic
44	C₂×C₅₉⋊C₈	D₈	1	1	yes	characteristic
45	Dic₂₃₆	—	2	2	no	—
46	Dic₂₃₆	—	4	1	no	—
47	C₁₆×C₅₉	C₂×Q₈	1	1	yes	characteristic
48	C₁₆×C₅₉	#41	1	1	yes	characteristic
49	C₅₉⋊C₁₆	—	2	1	no	—
50	C₂×C₄×C₅₉	—	8	1	no	—
51	C₂×C₄×C₅₉	—	8	1	no	—
52	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
53	C₂×Dic₅₉	—	8	1	no	—
54	C₂×Dic₅₉	—	8	1	no	—
55	C₂×Dic₅₉	D₁₆	1	1	yes	characteristic
56	C₂×Dic₅₉	D₁₆	1	1	yes	characteristic
57	Dic₁₁₈	—	4	2	no	—
58	Dic₁₁₈	—	8	1	no	—
59	C₈×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
60	C₈×C₅₉	#137	1	1	yes	characteristic
61	C₅₉⋊C₈	—	2	1	no	—
62	#13708	—	59	1	no	maximal, Sylow
63	C₂²×C₅₉	C₂×D₁₆	1	1	yes	characteristic
64	C₄×C₅₉	—	16	1	no	—
65	C₄×C₅₉	—	16	1	no	—
66	C₄×C₅₉	#474	1	1	yes	characteristic
67	C₄×C₅₉	#510	1	1	yes	characteristic
68	Dic₅₉	—	2	1	no	—
69	Dic₅₉	—	2	1	no	—
70	Dic₅₉	—	8	2	no	—
71	Dic₅₉	—	16	1	no	—
72	#2253	—	59	1	no	—
73	C₄×Q₃₂	—	59	1	no	—
74	#1424	D₅₉	1	1	yes	characteristic
75	#1423	—	59	1	no	—
76	C₄⋊C₃₂	—	59	1	no	—
77	#1416	—	59	1	no	—
78	#1416	—	59	1	no	—
79	C₂×C₅₉	#2224	1	1	yes	characteristic
80	C₂×C₅₉	#2262	1	1	yes	characteristic
81	C₂×C₅₉	#2264	1	1	yes	characteristic
82	C₂×Q₃₂	—	59	1	no	—
83	#499	—	118	1	no	—
84	C₄×Q₁₆	—	118	1	no	—
85	C₂×C₃₂	—	59	1	no	—
86	C₂×C₃₂	C₂×D₅₉	1	1	yes	characteristic
87	#367	—	118	1	no	—
88	#366	—	59	1	no	—
89	#366	C₂×D₅₉	1	1	yes	characteristic
90	#366	C₂×D₅₉	1	1	yes	characteristic
91	#358	—	118	1	no	—
92	C₄×C₁₆	—	59	1	no	—
93	C₅₉	#13708	1	1	yes	characteristic, complemented, Sylow
94	C₂×Q₁₆	—	118	1	no	—
95	#125	—	236	1	no	—
96	C₄×Q₈	—	236	1	no	—
97	Q₃₂	—	59	2	no	—
98	Q₃₂	—	118	1	no	—
99	C₂×C₁₆	—	59	1	no	—
100	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
101	#107	—	2	1	no	—
102	#107	—	2	1	no	—
103	#107	—	118	1	no	—
104	#106	—	236	1	no	—
105	#103	—	236	1	no	—
106	C₄×C₈	—	59	1	no	—
107	C₃₂	—	2	1	no	—
108	C₃₂	—	118	1	no	—
109	C₂×Q₈	—	236	1	no	—
110	Q₁₆	—	118	2	no	—
111	Q₁₆	—	236	1	no	—
112	C₂×C₈	—	59	1	no	—
113	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
114	C₄⋊C₄	—	4	1	no	—
115	C₄⋊C₄	—	4	1	no	—
116	C₄⋊C₄	—	472	1	no	—
117	C₄⋊C₄	—	472	1	no	—
118	C₄⋊C₄	—	472	1	no	—
119	C₄⋊C₄	—	236	1	no	—
120	C₄²	—	59	1	no	—
121	C₄²	—	472	1	no	—
122	C₁₆	—	118	1	no	—
123	C₁₆	#11745685	1	1	yes	characteristic
124	C₁₆	Q₈×D₅₉	1	1	yes	characteristic
125	Q₈	—	472	1	no	—
126	Q₈	—	236	2	no	—
127	C₂×C₄	—	8	1	no	—
128	C₂×C₄	—	8	1	no	—
129	C₂×C₄	—	59	1	no	—
130	C₂×C₄	—	59	1	no	—
131	C₂×C₄	—	472	1	no	—
132	C₂×C₄	—	472	1	no	—
133	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
134	C₈	—	118	1	no	—
135	C₈	#14273705	1	1	yes	characteristic
136	C₈	#14273759	1	1	yes	characteristic
137	C₂²	D₁₆×D₅₉	1	1	yes	characteristic
138	C₄	—	16	1	no	—
139	C₄	—	16	1	no	—
140	C₄	—	118	1	no	—
141	C₄	—	118	1	no	—
142	C₄	—	944	1	no	—
143	C₄	—	472	2	no	—
144	C₄	#16837681	1	1	yes	characteristic
145	C₄	#16837844	1	1	yes	characteristic
146	C₂	#21426232	1	1	yes	characteristic
147	C₂	#21426404	1	1	yes	characteristic
148	C₂	#21426412	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21426373

C₂

yes

maximal, characteristic

#21425954

C₂

yes

maximal, characteristic

#21422295

C₂

yes

maximal, characteristic

Dic₅₉⋊C₃₂

C₂

yes

maximal, characteristic

#21422278

C₂

yes

maximal, characteristic

#21422276

C₂

yes

maximal, characteristic

C₅₉×#1424

C₂

yes

maximal, characteristic

C₂×Dic₄₇₂

C₂²

yes

characteristic

#16837803

—

#16837496

—

C₂×C₅₉⋊C₃₂

C₂²

yes

characteristic

#16837229

—

#16837228

C₂²

yes

characteristic

#16837205

—

C₁₆×Dic₅₉

C₂²

yes

characteristic

C₂×C₃₂×C₅₉

C₂²

yes

characteristic

C₅₉×#366

C₂²

yes

characteristic

C₅₉×#366

C₂²

yes

characteristic

C₂×Dic₂₃₆

—

#14273718

—

#14273692

—

Dic₄₇₂

—

Dic₄₇₂

Q₈

yes

—

C₂×C₅₉⋊C₁₆

D₄

yes

—

—

—

C₈×Dic₅₉

D₄

yes

characteristic

C₅₉⋊C₃₂

—

C₂×C₁₆×C₅₉

C₂³

yes

characteristic

#107×C₅₉

—

#107×C₅₉

—

C₃₂×C₅₉

—

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₄×Dic₅₉

D₈

yes

characteristic

C₂×C₅₉⋊C₈

D₈

yes

characteristic

Dic₂₃₆

—

Dic₂₃₆

—

C₁₆×C₅₉

C₂×Q₈

yes

characteristic

C₁₆×C₅₉

#41

yes

characteristic

C₅₉⋊C₁₆

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₈

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

D₁₆

yes

characteristic

C₂×Dic₅₉

D₁₆

yes

characteristic

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

C₂²⋊Q₈

yes

characteristic

C₈×C₅₉

#137

yes

characteristic

C₅₉⋊C₈

—

#13708

—

maximal, Sylow

C₂²×C₅₉

C₂×D₁₆

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

#2253

—

C₄×Q₃₂

—

#1424

D₅₉

yes

characteristic

#1423

—

C₄⋊C₃₂

—

#1416

—

#1416

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×Q₃₂

—

#499

—

118

—

C₄×Q₁₆

—

118

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₅₉

yes

characteristic

#367

—

118

—

#366

—

#366

C₂×D₅₉

yes

characteristic

#366

C₂×D₅₉

yes

characteristic

#358

—

118

—

C₄×C₁₆

—

C₅₉

#13708

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

118

—

#125

—

236

—

C₄×Q₈

—

236

—

Q₃₂

—

Q₃₂

—

118

—

C₂×C₁₆

—

100

C₂×C₁₆

C₂²×D₅₉

yes

characteristic

101

#107

—

102

#107

—

103

#107

—

118

—

104

#106

—

236

—

105

#103

—

236

—

106

C₄×C₈

—

107

C₃₂

—

108

C₃₂

—

118

—

109

C₂×Q₈

—

236

—

110

Q₁₆

—

118

—

111

Q₁₆

—

236

—

112

C₂×C₈

—

113

C₂×C₈

D₄×D₅₉

yes

characteristic

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

472

—

117

C₄⋊C₄

—

472

—

118

C₄⋊C₄

—

472

—

119

C₄⋊C₄

—

236

—

120

C₄²

—

121

C₄²

—

472

—

122

C₁₆

—

118

—

123

C₁₆

#11745685

yes

characteristic

124

C₁₆

Q₈×D₅₉

yes

characteristic

125

Q₈

—

472

—

126

Q₈

—

236

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

472

—

132

C₂×C₄

—

472

—

133

C₂×C₄

D₈×D₅₉

yes

characteristic

134

C₈

—

118

—

135

C₈

#14273705

yes

characteristic

136

C₈

#14273759

yes

characteristic

137

C₂²

D₁₆×D₅₉

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

118

—

141

C₄

—

118

—

142

C₄

—

944

—

143

C₄

—

472

—

144

C₄

#16837681

yes

characteristic

145

C₄

#16837844

yes

characteristic

146

C₂

#21426232

yes

characteristic

147

C₂

#21426404

yes

characteristic

148

C₂

#21426412

yes

characteristic

	subgroup	G/H
center	C₂²	D₁₆×D₅₉
commutator, cosocle	C₂×C₁₆×C₅₉	C₂³
Frattini	C₂×C₁₆	C₂²×D₅₉
Fitting	C₅₉×#1424	C₂
socle, prime radical	C₂²×C₅₉	C₂×D₁₆
hypercenter	#1424	D₅₉
hypocenter	C₅₉	#13708
LCS	G ▷ C₂×C₁₆×C₅₉ ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1424

subgroup

G/H

center

C₂²

D₁₆×D₅₉

commutator, cosocle

C₂×C₁₆×C₅₉

C₂³

Frattini

C₂×C₁₆

C₂²×D₅₉

Fitting

C₅₉×#1424

C₂

socle, prime radical

C₂²×C₅₉

C₂×D₁₆

D₅₉

C₅₉

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

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1424

N	H ≅ G/N	type
93. C₅₉	#13708	semidirect

H ≅ G/N

type

93. C₅₉

#13708

semidirect

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

subgroup	index
#21426373	2
#21425954	2
#21422295	2
Dic₅₉⋊C₃₂	2
#21422278	2
#21422276	2
C₅₉×#1424	2
#13708	59

subgroup

index

Dic₅₉⋊C₃₂

C₅₉×#1424

Subgroups — #28509601

Conjugacy classes of subgroups

Complements

Maximal subgroups