Group #28505105 — subgroups

Subgroups — #28505105

Order 15104.

382 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 167 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×#16837222	C₂	1	4	yes	maximal
2	#21425852	C₂	1	1	yes	maximal, characteristic
3	C₄×C₈×Dic₅₉	C₂	1	1	yes	maximal, characteristic
4	C₅₉×#2150	C₂	1	1	yes	maximal, characteristic
5	C₂×#14273667	C₂²	1	2	yes	—
6	C₂×C₄×C₅₉⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₅₉	C₄	1	2	yes	—
8	C₂×C₈×Dic₅₉	C₂²	1	2	yes	—
9	C₄²×Dic₅₉	C₄	1	1	yes	characteristic
10	#16837222	—	2	8	no	—
11	C₄⋊(C₅₉⋊C₁₆)	—	2	2	no	—
12	C₂×M₃₂×C₅₉	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₅₉×C₄⋊C₁₆	—	2	2	no	—
15	C₂²×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
16	C₂×C₄×Dic₅₉	—	2	2	no	—
17	C₂×C₄×Dic₅₉	C₂×C₄	1	2	yes	—
18	#14273667	—	2	4	no	—
19	C₂×C₅₉⋊C₁₆	—	2	2	no	—
20	C₄×C₅₉⋊C₈	C₈	1	2	yes	—
21	C₄×C₅₉⋊C₈	C₈	1	2	yes	—
22	C₈×Dic₅₉	—	2	4	no	—
23	C₈×Dic₅₉	D₄	1	4	yes	—
24	C₈×Dic₅₉	Q₈	1	4	yes	—
25	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
27	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
28	M₃₂×C₅₉	—	2	4	no	—
29	C₂×C₁₆×C₅₉	—	2	2	no	—
30	C₄×C₈×C₅₉	D₄	1	2	yes	—
31	C₄×C₈×C₅₉	Q₈	1	2	yes	—
32	C₂²×C₄×C₅₉	M₁₆	1	2	yes	—
33	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
34	C₂²×Dic₅₉	—	2	2	no	—
35	C₂×C₈×C₅₉	—	2	2	no	—
36	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
37	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
38	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
41	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
42	C₄²×C₅₉	C₂×C₈	1	1	yes	characteristic
43	C₄²×C₅₉	C₂×C₈	1	1	yes	characteristic
44	C₄×Dic₅₉	—	2	8	no	—
45	C₄×Dic₅₉	—	2	2	no	—
46	C₄×Dic₅₉	C₄⋊C₄	1	4	yes	—
47	C₂×C₅₉⋊C₈	C₄⋊C₄	1	4	yes	—
48	C₂×C₅₉⋊C₈	C₂×C₈	1	4	yes	—
49	C₂×C₅₉⋊C₈	M₁₆	1	4	yes	—
50	C₁₆×C₅₉	—	4	2	no	—
51	C₅₉⋊C₁₆	—	4	2	no	—
52	C₂³×C₅₉	C₂×M₁₆	1	1	yes	characteristic
53	C₂×C₄×C₅₉	—	2	4	no	—
54	C₂×C₄×C₅₉	C₄⋊C₈	1	2	yes	—
55	C₂×C₄×C₅₉	C₄⋊C₈	1	2	yes	—
56	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
57	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
59	C₂×C₄×C₅₉	C₂×M₁₆	1	2	yes	—
60	C₂×Dic₅₉	—	2	8	no	—
61	C₂×Dic₅₉	—	2	4	no	—
62	C₈×C₅₉	—	2	2	no	—
63	C₈×C₅₉	#127	1	1	yes	characteristic
64	C₈×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
65	C₈×C₅₉	C₄⋊Q₈	1	2	yes	—
66	C₅₉⋊C₈	C₄⋊C₈	1	8	yes	—
67	#12900	—	59	1	no	maximal, Sylow
68	C₂²×C₅₉	—	2	2	no	—
69	C₂²×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
70	C₂²×C₅₉	#423	1	2	yes	—
71	C₄×C₅₉	—	2	2	no	—
72	C₄×C₅₉	#364	1	4	yes	complemented
73	C₄×C₅₉	#384	1	1	yes	characteristic
74	C₄×C₅₉	C₂×C₄⋊C₈	1	2	yes	—
75	C₄×C₅₉	#423	1	1	yes	characteristic
76	Dic₅₉	—	2	8	no	—
77	C₂×#364	—	59	4	no	—
78	#2150	—	59	1	no	—
79	#2150	D₅₉	1	1	yes	characteristic
80	C₄²×C₈	—	59	1	no	—
81	C₂×C₅₉	—	2	2	no	—
82	C₂×C₅₉	#1842	1	1	yes	characteristic
83	C₂×C₅₉	C₂×#364	1	2	yes	—
84	C₂×M₃₂	—	59	2	no	—
85	C₂×M₃₂	C₂×D₅₉	1	2	yes	—
86	C₂×C₄×C₈	—	59	2	no	—
87	C₂×C₄×C₈	—	59	1	no	—
88	C₂×C₄×C₈	—	59	2	no	—
89	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
90	C₄³	—	59	1	no	—
91	#364	—	118	8	no	—
92	C₄⋊C₁₆	—	2	2	no	—
93	C₄⋊C₁₆	—	118	2	no	—
94	C₅₉	#12900	1	1	yes	characteristic, complemented, Sylow
95	C₂²×C₈	—	59	2	no	—
96	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
97	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
98	C₂×C₄²	—	59	2	no	—
99	C₂×C₄²	—	118	2	no	—
100	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
101	M₃₂	—	2	4	no	—
102	M₃₂	—	118	4	no	—
103	C₂×C₁₆	—	2	2	no	—
104	C₂×C₁₆	—	118	2	no	—
105	C₄×C₈	—	59	4	no	—
106	C₄×C₈	—	59	4	no	—
107	C₄×C₈	—	59	2	no	—
108	C₄×C₈	—	59	2	no	—
109	C₄×C₈	—	118	4	no	—
110	C₄×C₈	Dic₁₁₈	1	2	yes	—
111	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
112	C₂²×C₄	—	118	2	no	—
113	C₂²×C₄	#11745659	1	2	yes	—
114	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	59	4	no	—
117	C₂×C₈	—	59	4	no	—
118	C₂×C₈	—	59	4	no	—
119	C₂×C₈	Dic₅₉⋊C₄	1	2	yes	—
120	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
121	C₂×C₈	D₄×D₅₉	1	2	yes	—
122	C₂×C₈	Q₈×D₅₉	1	2	yes	—
123	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
124	C₄²	—	59	4	no	—
125	C₄²	—	118	8	no	—
126	C₄²	—	118	2	no	—
127	C₄²	C₈×D₅₉	1	1	yes	characteristic
128	C₄²	C₈×D₅₉	1	1	yes	characteristic
129	C₄²	Dic₅₉⋊C₄	1	2	yes	—
130	C₁₆	—	4	2	no	—
131	C₁₆	—	236	2	no	—
132	C₂³	C₂×#11745659	1	1	yes	characteristic
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	118	8	no	—
135	C₂×C₄	—	118	4	no	—
136	C₂×C₄	Dic₅₉⋊C₈	1	2	yes	—
137	C₂×C₄	Dic₅₉⋊C₈	1	2	yes	—
138	C₂×C₄	C₄⋊C₄×D₅₉	1	2	yes	—
139	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
140	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
141	C₂×C₄	M₁₆×D₅₉	1	2	yes	—
142	C₈	—	2	2	no	—
143	C₈	—	59	8	no	—
144	C₈	#14273723	1	1	yes	characteristic
145	C₈	#14273725	1	1	yes	characteristic
146	C₈	#14273726	1	2	yes	—
147	C₂²	—	2	2	no	—
148	C₂²	C₂×Dic₅₉⋊C₈	1	1	yes	characteristic
149	C₂²	#16837424	1	2	yes	—
150	C₄	—	2	2	no	—
151	C₄	—	118	8	no	—
152	C₄	#16837222	1	4	yes	complemented
153	C₄	#16837271	1	1	yes	characteristic
154	C₄	C₄⋊C₈×D₅₉	1	2	yes	—
155	C₄	#16837425	1	1	yes	characteristic
156	C₂	—	2	2	no	—
157	C₂	#21424247	1	1	yes	characteristic
158	C₂	C₂×#16837222	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16837222

C₂

yes

maximal

#21425852

C₂

yes

maximal, characteristic

C₄×C₈×Dic₅₉

C₂

yes

maximal, characteristic

C₅₉×#2150

C₂

yes

maximal, characteristic

C₂×#14273667

C₂²

yes

—

C₂×C₄×C₅₉⋊C₈

C₄

yes

characteristic

C₂×C₈×Dic₅₉

C₄

yes

—

C₂×C₈×Dic₅₉

C₂²

yes

—

C₄²×Dic₅₉

C₄

yes

characteristic

#16837222

—

C₄⋊(C₅₉⋊C₁₆)

—

C₂×M₃₂×C₅₉

C₂²

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₅₉×C₄⋊C₁₆

—

C₂²×C₅₉⋊C₈

C₂×C₄

yes

—

C₂×C₄×Dic₅₉

—

C₂×C₄×Dic₅₉

C₂×C₄

yes

—

#14273667

—

C₂×C₅₉⋊C₁₆

—

C₄×C₅₉⋊C₈

C₈

yes

—

C₄×C₅₉⋊C₈

C₈

yes

—

C₈×Dic₅₉

—

C₈×Dic₅₉

D₄

yes

—

C₈×Dic₅₉

Q₈

yes

—

C₂²×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₄²×C₅₉

C₂×C₄

yes

characteristic

M₃₂×C₅₉

—

C₂×C₁₆×C₅₉

—

C₄×C₈×C₅₉

D₄

yes

—

C₄×C₈×C₅₉

Q₈

yes

—

C₂²×C₄×C₅₉

M₁₆

yes

—

C₂²×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂²×Dic₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₄⋊C₄

yes

—

C₂×C₈×C₅₉

C₂×D₄

yes

—

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₉

C₂×Q₈

yes

—

C₄²×C₅₉

C₄⋊C₄

yes

—

C₄²×C₅₉

C₂×C₈

yes

characteristic

C₄²×C₅₉

C₂×C₈

yes

characteristic

C₄×Dic₅₉

—

C₄×Dic₅₉

—

C₄×Dic₅₉

C₄⋊C₄

yes

—

C₂×C₅₉⋊C₈

C₄⋊C₄

yes

—

C₂×C₅₉⋊C₈

C₂×C₈

yes

—

C₂×C₅₉⋊C₈

M₁₆

yes

—

C₁₆×C₅₉

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

C₂×M₁₆

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₄⋊C₈

yes

—

C₂×C₄×C₅₉

C₄⋊C₈

yes

—

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₉

C₂×M₁₆

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₈×C₅₉

—

C₈×C₅₉

#127

yes

characteristic

C₈×C₅₉

C₄⋊Q₈

yes

characteristic

C₈×C₅₉

C₄⋊Q₈

yes

—

C₅₉⋊C₈

C₄⋊C₈

yes

—

#12900

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₅₉

#423

yes

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

C₂×C₄⋊C₈

yes

—

C₄×C₅₉

#423

yes

characteristic

Dic₅₉

—

C₂×#364

—

#2150

—

#2150

D₅₉

yes

characteristic

C₄²×C₈

—

C₂×C₅₉

—

C₂×C₅₉

#1842

yes

characteristic

C₂×C₅₉

C₂×#364

yes

—

C₂×M₃₂

—

C₂×M₃₂

C₂×D₅₉

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

C₄³

—

#364

—

118

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

118

—

C₅₉

#12900

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₄×D₅₉

yes

characteristic

C₂²×C₈

C₂²×D₅₉

yes

characteristic

C₂×C₄²

—

C₂×C₄²

—

118

—

100

C₂×C₄²

C₄×D₅₉

yes

characteristic

101

M₃₂

—

102

M₃₂

—

118

—

103

C₂×C₁₆

—

104

C₂×C₁₆

—

118

—

105

C₄×C₈

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

—

118

—

110

C₄×C₈

Dic₁₁₈

yes

—

111

C₄×C₈

C₂²⋊D₅₉

yes

—

112

C₂²×C₄

—

118

—

113

C₂²×C₄

#11745659

yes

—

114

C₂²×C₄

C₂×C₄×D₅₉

yes

characteristic

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

Dic₅₉⋊C₄

yes

—

120

C₂×C₈

C₂×Dic₁₁₈

yes

characteristic

121

C₂×C₈

D₄×D₅₉

yes

—

122

C₂×C₈

Q₈×D₅₉

yes

—

123

C₂×C₈

C₂×C₂²⋊D₅₉

yes

characteristic

124

C₄²

—

125

C₄²

—

118

—

126

C₄²

—

118

—

127

C₄²

C₈×D₅₉

yes

characteristic

128

C₄²

C₈×D₅₉

yes

characteristic

129

C₄²

Dic₅₉⋊C₄

yes

—

130

C₁₆

—

131

C₁₆

—

236

—

132

C₂³

C₂×#11745659

yes

characteristic

133

C₂×C₄

—

134

C₂×C₄

—

118

—

135

C₂×C₄

—

118

—

136

C₂×C₄

Dic₅₉⋊C₈

yes

—

137

C₂×C₄

Dic₅₉⋊C₈

yes

—

138

C₂×C₄

C₄⋊C₄×D₅₉

yes

—

139

C₂×C₄

C₂×Dic₅₉⋊C₄

yes

characteristic

140

C₂×C₄

C₂×C₈×D₅₉

yes

characteristic

141

C₂×C₄

M₁₆×D₅₉

yes

—

142

C₈

—

143

C₈

—

144

C₈

#14273723

yes

characteristic

145

C₈

#14273725

yes

characteristic

146

C₈

#14273726

yes

—

147

C₂²

—

148

C₂²

C₂×Dic₅₉⋊C₈

yes

characteristic

149

C₂²

#16837424

yes

—

150

C₄

—

151

C₄

—

118

—

152

C₄

#16837222

yes

complemented

153

C₄

#16837271

yes

characteristic

154

C₄

C₄⋊C₈×D₅₉

yes

—

155

C₄

#16837425

yes

characteristic

156

C₂

—

157

C₂

#21424247

yes

characteristic

158

C₂

C₂×#16837222

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#2150	C₂
socle	C₂²×C₅₉	C₂×C₄⋊C₈
prime radical	C₂³×C₅₉	C₂×M₁₆
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#2150	D₅₉
hypocenter	C₅₉	#12900
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

subgroup

G/H

center

C₂×C₈

C₂×C₂²⋊D₅₉

commutator

C₂×C₄×C₅₉

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₅₉

Fitting

C₅₉×#2150

C₂

socle

C₂²×C₅₉

C₂×C₄⋊C₈

prime radical

C₂³×C₅₉

C₂×M₁₆

cosocle

C₂²×C₈×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

N	H ≅ G/N	type
72. C₄×C₅₉	#364	semidirect
94. C₅₉	#12900	semidirect
152. C₄	#16837222	semidirect

H ≅ G/N

type

72. C₄×C₅₉

#364

semidirect

94. C₅₉

#12900

semidirect

152. C₄

#16837222

semidirect

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

subgroup	index
C₂×#16837222	2
#21425852	2
C₄×C₈×Dic₅₉	2
C₅₉×#2150	2
#12900	59

subgroup

index

C₂×#16837222

#21425852

C₄×C₈×Dic₅₉

C₅₉×#2150

#12900

Subgroups — #28505105

Conjugacy classes of subgroups

Complements

Maximal subgroups