Group #28500283 — subgroups

Subgroups — #28500283

Order 15104.

220 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 99 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	#364×D₅₉	C₂	1	2	yes	maximal
2	C₄×#14273666	C₂	1	1	yes	maximal, characteristic
3	#21422172	C₂	1	1	yes	maximal, characteristic, complemented
4	#21422171	C₂	1	2	yes	maximal, complemented
5	C₅₉×#1373	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14273666	C₄	1	1	yes	characteristic
7	C₂×#14273666	C₄	1	1	yes	characteristic
8	M₃₂×D₅₉	—	2	2	no	—
9	C₄×C₈×D₅₉	C₂²	1	1	yes	characteristic
10	#16837223	C₂²	1	2	yes	—
11	#16837222	—	2	2	no	—
12	#16837177	C₄	1	1	yes	characteristic
13	#16837177	C₄	1	1	yes	characteristic
14	C₄×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₅₉×#364	C₂²	1	2	yes	—
16	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₅₉	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×D₅₉	D₄	1	1	yes	characteristic
19	C₄²×D₅₉	Q₈	1	1	yes	characteristic
20	#14273667	—	2	2	no	—
21	#14273666	—	2	2	no	—
22	#14273666	—	4	2	no	—
23	#14273666	C₈	1	4	yes	—
24	C₂×C₅₉⋊C₁₆	C₂×C₄	1	1	yes	characteristic
25	C₂×C₅₉⋊C₁₆	C₂×C₄	1	1	yes	characteristic
26	C₁₆×D₅₉	—	4	2	no	—
27	C₄×C₅₉⋊C₈	Q₈	1	1	yes	characteristic
28	C₈×Dic₅₉	C₂×C₄	1	1	yes	characteristic
29	C₈×Dic₅₉	D₄	1	1	yes	characteristic
30	M₃₂×C₅₉	—	2	2	no	—
31	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
34	C₂×C₄×D₅₉	—	2	1	no	—
35	C₂×C₄×D₅₉	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
38	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
39	C₄×Dic₅₉	—	2	1	no	—
40	C₄×Dic₅₉	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₅₉⋊C₈	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₅₉⋊C₈	C₄⋊C₄	1	1	yes	characteristic
43	C₈×D₅₉	C₂×C₈	1	2	yes	—
44	C₈×D₅₉	M₁₆	1	2	yes	—
45	C₈×D₅₉	D₈	1	2	yes	—
46	C₈×D₅₉	Q₁₆	1	2	yes	—
47	C₁₆×C₅₉	—	2	1	no	—
48	C₁₆×C₅₉	—	4	2	no	—
49	C₁₆×C₅₉	C₂×C₈	1	2	yes	—
50	C₅₉⋊C₁₆	—	2	1	no	—
51	C₅₉⋊C₁₆	—	4	2	no	—
52	C₅₉⋊C₁₆	C₂×C₈	1	2	yes	—
53	C₂²×D₅₉	#108	1	1	yes	characteristic
54	C₂×C₄×C₅₉	—	2	1	no	—
55	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×Dic₅₉	—	2	1	no	—
57	C₂×Dic₅₉	#108	1	1	yes	characteristic
58	C₄×D₅₉	—	4	2	no	—
59	C₄×D₅₉	C₄⋊C₈	1	2	yes	—
60	C₄×D₅₉	#107	1	2	yes	—
61	C₈×C₅₉	C₂²×C₈	1	1	yes	characteristic
62	C₈×C₅₉	C₂×M₁₆	1	1	yes	characteristic
63	C₈×C₅₉	C₂×D₈	1	1	yes	characteristic
64	C₈×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
65	C₅₉⋊C₈	C₄⋊C₈	1	2	yes	—
66	C₅₉⋊C₈	#107	1	2	yes	—
67	#12011	—	59	1	no	maximal, Sylow
68	C₂²×C₅₉	C₂×#108	1	1	yes	characteristic
69	C₂×D₅₉	—	4	1	no	—
70	C₂×D₅₉	#335	1	2	yes	—
71	C₄×C₅₉	—	2	2	no	—
72	C₄×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
73	C₄×C₅₉	C₂×#107	1	1	yes	characteristic
74	Dic₅₉	—	4	1	no	—
75	Dic₅₉	#335	1	2	yes	—
76	C₂×#364	—	59	2	no	—
77	C₄×M₃₂	—	59	1	no	—
78	#1373	—	59	2	no	—
79	#1373	—	59	1	no	—
80	#1373	D₅₉	1	1	yes	characteristic, complemented
81	C₂×C₅₉	—	2	1	no	—
82	C₂×C₅₉	C₂×#335	1	1	yes	characteristic
83	D₅₉	—	2	2	no	—
84	C₂×M₃₂	—	59	1	no	—
85	C₂×M₃₂	—	59	1	no	—
86	C₂×M₃₂	—	118	2	no	—
87	C₂×C₄×C₈	—	59	1	no	—
88	#364	—	59	2	no	—
89	#364	—	118	2	no	—
90	#364	C₂×D₅₉	1	2	yes	—
91	#346	—	59	1	no	—
92	#346	—	59	1	no	—
93	C₄×C₁₆	—	59	1	no	—
94	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
95	C₅₉	#12011	1	1	yes	characteristic, complemented, Sylow
96	C₂²×C₈	—	59	1	no	—
97	C₂²×C₈	—	59	1	no	—
98	C₂×C₄²	—	59	1	no	—
99	M₃₂	—	2	2	no	—
100	M₃₂	—	59	4	no	—
101	M₃₂	—	118	2	no	—
102	M₃₂	—	118	2	no	—
103	M₃₂	—	236	2	no	—
104	C₂×C₁₆	—	59	1	no	—
105	C₂×C₁₆	—	59	1	no	—
106	C₂×C₁₆	—	236	2	no	—
107	C₂×C₁₆	C₄×D₅₉	1	1	yes	characteristic
108	C₂×C₁₆	C₄×D₅₉	1	1	yes	characteristic
109	C₄×C₈	—	59	1	no	—
110	C₄×C₈	—	59	1	no	—
111	C₄×C₈	—	59	1	no	—
112	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
113	C₂²×C₄	—	59	1	no	—
114	C₂²×C₄	—	118	1	no	—
115	C₂×C₈	—	59	1	no	—
116	C₂×C₈	—	59	1	no	—
117	C₂×C₈	—	59	2	no	—
118	C₂×C₈	—	59	2	no	—
119	C₂×C₈	—	59	2	no	—
120	C₂×C₈	—	59	2	no	—
121	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
122	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
123	C₄²	—	59	1	no	—
124	C₄²	—	118	1	no	—
125	C₄²	Q₈×D₅₉	1	1	yes	characteristic
126	C₁₆	—	2	1	no	—
127	C₁₆	—	4	2	no	—
128	C₁₆	—	59	2	no	—
129	C₁₆	—	118	1	no	—
130	C₁₆	—	236	2	no	—
131	C₁₆	C₈×D₅₉	1	2	yes	—
132	C₂³	—	59	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	59	1	no	—
135	C₂×C₄	—	59	2	no	—
136	C₂×C₄	—	59	2	no	—
137	C₂×C₄	—	118	1	no	—
138	C₂×C₄	—	236	2	no	—
139	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
140	C₈	—	59	2	no	—
141	C₈	—	59	2	no	—
142	C₈	C₂×C₈×D₅₉	1	1	yes	characteristic
143	C₈	M₁₆×D₅₉	1	1	yes	characteristic
144	C₈	D₈×D₅₉	1	1	yes	characteristic
145	C₈	Q₁₆×D₅₉	1	1	yes	characteristic
146	C₂²	—	59	2	no	—
147	C₂²	—	236	1	no	—
148	C₂²	#108×D₅₉	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	59	2	no	—
151	C₄	—	236	1	no	—
152	C₄	C₄⋊C₈×D₅₉	1	1	yes	characteristic
153	C₄	#107×D₅₉	1	1	yes	characteristic
154	C₂	—	2	1	no	—
155	C₂	—	118	2	no	—
156	C₂	#335×D₅₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#364×D₅₉

C₂

yes

maximal

C₄×#14273666

C₂

yes

maximal, characteristic

#21422172

C₂

yes

maximal, characteristic, complemented

#21422171

C₂

yes

maximal, complemented

C₅₉×#1373

C₂

yes

maximal, characteristic, complemented

C₂×#14273666

C₄

yes

characteristic

C₂×#14273666

C₄

yes

characteristic

M₃₂×D₅₉

—

C₄×C₈×D₅₉

C₂²

yes

characteristic

#16837223

C₂²

yes

—

#16837222

—

#16837177

C₄

yes

characteristic

#16837177

C₄

yes

characteristic

C₄×C₅₉⋊C₁₆

C₂²

yes

characteristic

C₅₉×#364

C₂²

yes

—

C₄×C₁₆×C₅₉

C₂²

yes

characteristic

C₂×C₈×D₅₉

C₂×C₄

yes

characteristic

C₂×C₈×D₅₉

D₄

yes

characteristic

C₄²×D₅₉

Q₈

yes

—

—

—

C₈

yes

—

C₂×C₅₉⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₅₉⋊C₁₆

C₂×C₄

yes

characteristic

C₁₆×D₅₉

—

C₄×C₅₉⋊C₈

Q₈

yes

characteristic

C₈×Dic₅₉

C₂×C₄

yes

characteristic

C₈×Dic₅₉

D₄

yes

characteristic

M₃₂×C₅₉

—

C₂×C₁₆×C₅₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₉

C₂×C₄

yes

characteristic

C₄×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₄×D₅₉

—

C₂×C₄×D₅₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₄²×C₅₉

C₂×Q₈

yes

characteristic

C₄×Dic₅₉

—

C₄×Dic₅₉

C₄⋊C₄

yes

characteristic

C₂×C₅₉⋊C₈

C₄⋊C₄

yes

characteristic

C₂×C₅₉⋊C₈

C₄⋊C₄

yes

characteristic

C₈×D₅₉

C₂×C₈

yes

—

C₈×D₅₉

M₁₆

yes

—

C₈×D₅₉

D₈

yes

—

C₈×D₅₉

Q₁₆

yes

—

C₁₆×C₅₉

—

C₁₆×C₅₉

—

C₁₆×C₅₉

C₂×C₈

yes

—

C₅₉⋊C₁₆

—

C₅₉⋊C₁₆

—

C₅₉⋊C₁₆

C₂×C₈

yes

—

C₂²×D₅₉

#108

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

#108

yes

characteristic

C₄×D₅₉

—

C₄×D₅₉

C₄⋊C₈

yes

—

C₄×D₅₉

#107

yes

—

C₈×C₅₉

C₂²×C₈

yes

characteristic

C₈×C₅₉

C₂×M₁₆

yes

characteristic

C₈×C₅₉

C₂×D₈

yes

characteristic

C₈×C₅₉

C₂×Q₁₆

yes

characteristic

C₅₉⋊C₈

C₄⋊C₈

yes

—

C₅₉⋊C₈

#107

yes

—

—

C₂²×C₅₉

yes

C₂×D₅₉

—

C₂×D₅₉

#335

yes

—

C₄×C₅₉

—

C₄×C₅₉

C₂×C₄⋊C₈

yes

characteristic

C₄×C₅₉

C₂×#107

yes

characteristic

Dic₅₉

—

Dic₅₉

#335

yes

—

C₂×#364

—

C₄×M₃₂

—

#1373

—

#1373

—

#1373

D₅₉

yes

characteristic, complemented

C₂×C₅₉

—

C₂×C₅₉

C₂×#335

yes

characteristic

D₅₉

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂×M₃₂

—

118

—

C₂×C₄×C₈

—

#364

—

#364

—

118

—

#364

C₂×D₅₉

yes

—

#346

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₅₉

yes

characteristic

C₅₉

#12011

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

M₃₂

—

100

M₃₂

—

101

M₃₂

—

118

—

102

M₃₂

—

118

—

103

M₃₂

—

236

—

104

C₂×C₁₆

—

105

C₂×C₁₆

—

106

C₂×C₁₆

—

236

—

107

C₂×C₁₆

C₄×D₅₉

yes

characteristic

108

C₂×C₁₆

C₄×D₅₉

yes

characteristic

109

C₄×C₈

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₄×C₈

C₂²×D₅₉

yes

characteristic

113

C₂²×C₄

—

114

C₂²×C₄

—

118

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

C₂×C₄×D₅₉

yes

characteristic

122

C₂×C₈

D₄×D₅₉

yes

characteristic

123

C₄²

—

124

C₄²

—

118

—

125

C₄²

Q₈×D₅₉

yes

characteristic

126

C₁₆

—

127

C₁₆

—

128

C₁₆

—

129

C₁₆

—

118

—

130

C₁₆

—

236

—

131

C₁₆

C₈×D₅₉

yes

—

132

C₂³

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

118

—

138

C₂×C₄

—

236

—

139

C₂×C₄

C₄⋊C₄×D₅₉

yes

characteristic

140

C₈

—

141

C₈

—

142

C₈

C₂×C₈×D₅₉

yes

characteristic

143

C₈

M₁₆×D₅₉

yes

characteristic

144

C₈

D₈×D₅₉

yes

characteristic

145

C₈

Q₁₆×D₅₉

yes

characteristic

146

C₂²

—

147

C₂²

—

236

—

148

C₂²

#108×D₅₉

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

236

—

152

C₄

C₄⋊C₈×D₅₉

yes

characteristic

153

C₄

#107×D₅₉

yes

characteristic

154

C₂

—

155

C₂

—

118

—

156

C₂

#335×D₅₉

yes

characteristic

	subgroup	G/H
center	C₈	D₈×D₅₉
commutator	C₈×C₅₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₅₉×#1373	C₂
socle	C₂×C₅₉	C₂×#335
prime radical	C₂²×D₅₉	#108
cosocle	C₄×C₈×C₅₉	C₂³
hypercenter	#1373	D₅₉
hypocenter	C₅₉	#12011
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1373

subgroup

G/H

center

C₈

D₈×D₅₉

commutator

C₈×C₅₉

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₅₉

Fitting

C₅₉×#1373

C₂

C₂×C₅₉

C₂²×D₅₉

C₄×C₈×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1373

N	H ≅ G/N	type
3. #21422172	C₂	semidirect
4. #21422171	C₂	semidirect
5. C₅₉×#1373	C₂	semidirect
80. #1373	D₅₉	semidirect
95. C₅₉	#12011	semidirect

H ≅ G/N

type

3. #21422172

C₂

semidirect

4. #21422171

C₂

semidirect

5. C₅₉×#1373

C₂

semidirect

80. #1373

D₅₉

semidirect

95. C₅₉

#12011

semidirect

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

subgroup	index
#364×D₅₉	2
C₄×#14273666	2
#21422172	2
#21422171	2
C₅₉×#1373	2
#12011	59

subgroup

index

#364×D₅₉

C₅₉×#1373

Subgroups — #28500283

Conjugacy classes of subgroups

Complements

Maximal subgroups