Group #28504106 — subgroups

Subgroups — #28504106

Order 15104.

306 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 151 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	C₂×#16837231	C₂	1	2	yes	maximal
2	C₂×#16837178	C₂	1	1	yes	maximal, characteristic
3	C₄×#14273663	C₂	1	1	yes	maximal, characteristic
4	#21422189	C₂	1	2	yes	maximal
5	C₄×#108×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
8	C₂×#14273663	C₂²	1	2	yes	—
9	C₄×#11745664	—	2	1	no	—
10	#16837231	C₄	1	8	yes	complemented
11	#16837178	C₄	1	4	yes	—
12	#16837167	—	2	1	no	—
13	C₂×#108×C₅₉	C₂²	1	2	yes	—
14	C₄×M₁₆×C₅₉	—	2	1	no	—
15	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	C₅₉×#341	—	2	1	no	—
17	C₂×#11745664	—	2	2	no	—
18	C₂×C₅₉⋊C₁₆	—	2	2	no	—
19	C₂×C₅₉⋊C₁₆	C₂×C₄	1	2	yes	—
20	C₂×C₅₉⋊C₁₆	C₂×C₄	1	2	yes	—
21	C₂×C₅₉⋊C₁₆	C₂×C₄	1	4	yes	—
22	#14273663	—	2	2	no	—
23	#14273663	C₂×C₄	1	4	yes	—
24	#14273638	—	4	1	no	—
25	C₄×C₅₉⋊C₈	—	4	1	no	—
26	C₂×M₁₆×C₅₉	—	2	2	no	—
27	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
29	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
30	#108×C₅₉	—	2	2	no	—
31	#108×C₅₉	C₂×C₄	1	4	yes	—
32	#97×C₅₉	—	4	1	no	—
33	C₄×C₈×C₅₉	—	4	1	no	—
34	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₅₉	Q₈	1	2	yes	—
36	C₂²×C₄×C₅₉	Q₁₆	1	2	yes	—
37	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
38	M₁₆×C₅₉	—	2	4	no	—
39	M₁₆×C₅₉	—	4	2	no	—
40	C₂×C₈×C₅₉	—	2	2	no	—
41	C₂×C₈×C₅₉	—	4	2	no	—
42	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
43	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
45	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₈×C₅₉	#41	1	2	yes	—
47	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
48	C₄²×C₅₉	D₈	1	1	yes	characteristic
49	C₄²×C₅₉	D₈	1	1	yes	characteristic
50	#11745664	—	2	4	no	—
51	#11745664	—	4	2	no	—
52	C₂×C₅₉⋊C₈	—	4	2	no	—
53	C₅₉⋊C₁₆	—	2	2	no	—
54	C₅₉⋊C₁₆	C₄²	1	4	yes	—
55	C₂³×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
56	C₂×C₄×C₅₉	—	4	2	no	—
57	C₂×C₄×C₅₉	#107	1	2	yes	—
58	C₂×C₄×C₅₉	#107	1	2	yes	—
59	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
61	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
62	C₂×C₄×C₅₉	#135	1	2	yes	—
63	C₈×C₅₉	—	2	2	no	—
64	C₈×C₅₉	—	4	4	no	—
65	C₈×C₅₉	C₂×C₄²	1	1	yes	characteristic
66	C₈×C₅₉	#117	1	1	yes	characteristic
67	C₈×C₅₉	C₄×Q₈	1	2	yes	—
68	C₅₉⋊C₈	—	4	4	no	—
69	#12676	—	59	1	no	maximal, Sylow
70	C₂²×C₅₉	—	2	2	no	—
71	C₂²×C₅₉	C₂×#107	1	1	yes	characteristic
72	C₂²×C₅₉	C₄×Q₁₆	1	2	yes	—
73	C₄×C₅₉	—	2	2	no	—
74	C₄×C₅₉	—	2	2	no	—
75	C₄×C₅₉	C₄×C₄⋊C₄	1	1	yes	characteristic
76	C₄×C₅₉	#427	1	1	yes	characteristic
77	C₄×C₅₉	C₄×D₈	1	2	yes	—
78	C₂×#368	—	59	2	no	—
79	C₂×#346	—	59	1	no	—
80	C₄×#108	—	59	1	no	—
81	C₄×#108	D₅₉	1	1	yes	characteristic
82	#1381	—	59	2	no	—
83	C₂×C₅₉	—	2	2	no	—
84	C₂×C₅₉	C₄×#107	1	1	yes	characteristic
85	C₂×C₅₉	#2161	1	2	yes	—
86	C₂²×C₁₆	—	59	1	no	—
87	C₂²×C₁₆	—	59	1	no	—
88	C₂×#108	—	59	2	no	—
89	C₂×#108	C₂×D₅₉	1	2	yes	—
90	C₄×M₁₆	—	2	1	no	—
91	C₄×M₁₆	—	118	1	no	—
92	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
93	#368	—	59	8	no	—
94	#346	—	59	4	no	—
95	#341	—	2	1	no	—
96	#341	—	118	1	no	—
97	C₅₉	#12676	1	1	yes	characteristic, complemented, Sylow
98	C₂×M₁₆	—	2	2	no	—
99	C₂×M₁₆	—	118	2	no	—
100	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
101	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
102	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
103	C₂×C₁₆	—	59	2	no	—
104	C₂×C₁₆	—	59	4	no	—
105	C₂×C₁₆	—	59	2	no	—
106	C₂×C₁₆	—	118	2	no	—
107	#108	—	2	2	no	—
108	#108	—	59	4	no	—
109	#108	—	118	2	no	—
110	#108	C₄×D₅₉	1	4	yes	—
111	#97	—	4	1	no	—
112	#97	—	236	1	no	—
113	C₄×C₈	—	4	1	no	—
114	C₄×C₈	—	236	1	no	—
115	C₄×C₈	Dic₁₁₈	1	2	yes	—
116	C₄×C₈	C₄×D₅₉	1	2	yes	—
117	C₂²×C₄	#11745672	1	2	yes	—
118	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
119	M₁₆	—	2	4	no	—
120	M₁₆	—	4	2	no	—
121	M₁₆	—	118	4	no	—
122	M₁₆	—	236	2	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	4	2	no	—
125	C₂×C₈	—	236	2	no	—
126	C₂×C₈	Dic₅₉⋊C₄	1	2	yes	—
127	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
128	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
129	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
130	C₂×C₈	#11745685	1	2	yes	—
131	C₄²	Dic₅₉⋊C₄	1	2	yes	—
132	C₄²	#11745669	1	1	yes	characteristic
133	C₄²	#11745669	1	1	yes	characteristic
134	C₁₆	—	59	4	no	—
135	C₁₆	—	118	2	no	—
136	C₂³	C₂×#11745672	1	1	yes	characteristic
137	C₂×C₄	—	4	2	no	—
138	C₂×C₄	#14273660	1	2	yes	—
139	C₂×C₄	#14273660	1	2	yes	—
140	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
141	C₂×C₄	C₄×C₂²⋊D₅₉	1	2	yes	—
142	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
143	C₂×C₄	#14273747	1	2	yes	—
144	C₈	—	2	2	no	—
145	C₈	—	4	4	no	—
146	C₈	—	236	4	no	—
147	C₈	C₄²×D₅₉	1	1	yes	characteristic
148	C₈	#14273684	1	1	yes	characteristic
149	C₈	C₄×Dic₁₁₈	1	2	yes	—
150	C₂²	—	2	2	no	—
151	C₂²	C₂×#14273660	1	1	yes	characteristic
152	C₂²	C₄×#11745672	1	2	yes	—
153	C₄	—	2	2	no	—
154	C₄	—	2	2	no	—
155	C₄	C₄×Dic₅₉⋊C₄	1	1	yes	characteristic
156	C₄	#16837438	1	1	yes	characteristic
157	C₄	C₄×#11745669	1	2	yes	—
158	C₂	—	2	2	no	—
159	C₂	C₄×#14273660	1	1	yes	characteristic
160	C₂	#21425890	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16837231

C₂

yes

maximal

C₂×#16837178

C₂

yes

maximal, characteristic

C₄×#14273663

C₂

yes

maximal, characteristic

#21422189

C₂

yes

maximal

C₄×#108×C₅₉

C₂

yes

maximal, characteristic

C₂²×C₅₉⋊C₁₆

C₂²

yes

characteristic

C₂²×C₅₉⋊C₁₆

C₂²

yes

characteristic

C₂×#14273663

C₂²

yes

—

C₄×#11745664

—

#16837231

C₄

yes

complemented

#16837178

C₄

yes

—

#16837167

—

C₂×#108×C₅₉

C₂²

yes

—

C₄×M₁₆×C₅₉

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₅₉×#341

—

C₂×#11745664

—

C₂×C₅₉⋊C₁₆

—

C₂×C₅₉⋊C₁₆

C₂×C₄

yes

—

C₂×C₅₉⋊C₁₆

C₂×C₄

yes

—

C₂×C₅₉⋊C₁₆

C₂×C₄

yes

—

#14273663

—

#14273663

C₂×C₄

yes

—

#14273638

—

C₄×C₅₉⋊C₈

—

C₂×M₁₆×C₅₉

—

C₂²×C₈×C₅₉

D₄

yes

characteristic

C₂²×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₄²×C₅₉

D₄

yes

characteristic

#108×C₅₉

—

#108×C₅₉

C₂×C₄

yes

—

#97×C₅₉

—

C₄×C₈×C₅₉

—

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₄×C₈×C₅₉

Q₈

yes

—

C₂²×C₄×C₅₉

Q₁₆

yes

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

M₁₆×C₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₄⋊C₄

yes

—

C₂×C₈×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂²×C₄

yes

—

C₂×C₈×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₉

#41

yes

—

C₄²×C₅₉

C₄⋊C₄

yes

—

C₄²×C₅₉

D₈

yes

characteristic

C₄²×C₅₉

D₈

yes

characteristic

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₅₉⋊C₁₆

—

C₅₉⋊C₁₆

C₄²

yes

—

C₂³×C₅₉

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#107

yes

—

C₂×C₄×C₅₉

#107

yes

—

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₄×D₄

yes

—

C₂×C₄×C₅₉

C₂×D₈

yes

characteristic

C₂×C₄×C₅₉

#135

yes

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

C₂×C₄²

yes

characteristic

C₈×C₅₉

#117

yes

characteristic

C₈×C₅₉

C₄×Q₈

yes

—

C₅₉⋊C₈

—

#12676

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

C₂×#107

yes

characteristic

C₂²×C₅₉

C₄×Q₁₆

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₄×C₄⋊C₄

yes

characteristic

C₄×C₅₉

#427

yes

characteristic

C₄×C₅₉

C₄×D₈

yes

—

—

—

—

D₅₉

yes

characteristic

#1381

—

C₂×C₅₉

—

C₂×C₅₉

C₄×#107

yes

characteristic

C₂×C₅₉

#2161

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂×#108

—

C₂×#108

C₂×D₅₉

yes

—

C₄×M₁₆

—

C₄×M₁₆

—

118

—

C₂×C₄×C₈

C₂×D₅₉

yes

—

—

—

—

118

—

C₅₉

#12676

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

118

—

100

C₂²×C₈

C₂²⋊D₅₉

yes

characteristic

101

C₂²×C₈

C₂²×D₅₉

yes

characteristic

102

C₂×C₄²

C₂²⋊D₅₉

yes

characteristic

103

C₂×C₁₆

—

104

C₂×C₁₆

—

105

C₂×C₁₆

—

106

C₂×C₁₆

—

118

—

107

#108

—

108

#108

—

109

#108

—

118

—

110

#108

C₄×D₅₉

yes

—

111

#97

—

112

#97

—

236

—

113

C₄×C₈

—

114

C₄×C₈

—

236

—

115

C₄×C₈

Dic₁₁₈

yes

—

116

C₄×C₈

C₄×D₅₉

yes

—

117

C₂²×C₄

#11745672

yes

—

118

C₂²×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

119

M₁₆

—

120

M₁₆

—

121

M₁₆

—

118

—

122

M₁₆

—

236

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

236

—

126

C₂×C₈

Dic₅₉⋊C₄

yes

—

127

C₂×C₈

C₂×Dic₁₁₈

yes

characteristic

128

C₂×C₈

C₂×C₄×D₅₉

yes

characteristic

129

C₂×C₈

C₂×C₄×D₅₉

yes

—

130

C₂×C₈

#11745685

yes

—

131

C₄²

Dic₅₉⋊C₄

yes

—

132

C₄²

#11745669

yes

characteristic

133

C₄²

#11745669

yes

characteristic

134

C₁₆

—

135

C₁₆

—

118

—

136

C₂³

C₂×#11745672

yes

characteristic

137

C₂×C₄

—

138

C₂×C₄

#14273660

yes

—

139

C₂×C₄

#14273660

yes

—

140

C₂×C₄

C₂×Dic₅₉⋊C₄

yes

characteristic

141

C₂×C₄

C₄×C₂²⋊D₅₉

yes

—

142

C₂×C₄

C₂×#11745669

yes

characteristic

143

C₂×C₄

#14273747

yes

—

144

C₈

—

145

C₈

—

146

C₈

—

236

—

147

C₈

C₄²×D₅₉

yes

characteristic

148

C₈

#14273684

yes

characteristic

149

C₈

C₄×Dic₁₁₈

yes

—

150

C₂²

—

151

C₂²

C₂×#14273660

yes

characteristic

152

C₂²

C₄×#11745672

yes

—

153

C₄

—

154

C₄

—

155

C₄

C₄×Dic₅₉⋊C₄

yes

characteristic

156

C₄

#16837438

yes

characteristic

157

C₄

C₄×#11745669

yes

—

158

C₂

—

159

C₂

C₄×#14273660

yes

characteristic

160

C₂

#21425890

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×#11745669
commutator	C₈×C₅₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₄×#108×C₅₉	C₂
socle	C₂²×C₅₉	C₂×#107
prime radical	C₂³×C₅₉	C₂×Q₁₆
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₄×#108	D₅₉
hypocenter	C₅₉	#12676
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#108

subgroup

G/H

center

C₂×C₄

C₂×#11745669

commutator

C₈×C₅₉

C₂×C₄²

Frattini

C₂²×C₈

C₂²×D₅₉

Fitting

C₄×#108×C₅₉

C₂

socle

C₂²×C₅₉

C₂×#107

prime radical

C₂³×C₅₉

C₂×Q₁₆

cosocle

C₂²×C₈×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#108

N	H ≅ G/N	type
10. #16837231	C₄	semidirect
97. C₅₉	#12676	semidirect

H ≅ G/N

type

10. #16837231

C₄

semidirect

97. C₅₉

#12676

semidirect

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

subgroup	index
C₂×#16837231	2
C₂×#16837178	2
C₄×#14273663	2
#21422189	2
C₄×#108×C₅₉	2
#12676	59

subgroup

index

C₄×#108×C₅₉

Subgroups — #28504106

Conjugacy classes of subgroups

Complements

Maximal subgroups