Group #21432662 — subgroups

Subgroups — #21432662

Order 7552.

484 conjugacy classes of subgroups; only the 255 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 255 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×#14273734	C₂	1	1	yes	maximal, characteristic
2	#41×Dic₅₉	C₂	1	1	yes	maximal, characteristic
3	#16837430	C₂	1	3	yes	maximal, complemented
4	#16837428	C₂	1	3	yes	maximal, complemented
5	#16837425	C₂	1	3	yes	maximal, complemented
6	C₂×Dic₅₉⋊C₈	C₂	1	3	yes	maximal, complemented
7	C₂×#131×C₅₉	C₂	1	1	yes	maximal, characteristic
8	#14273734	—	2	4	no	—
9	C₂×#11745664	C₂²	1	3	yes	—
10	C₂²×C₅₉⋊C₈	C₂²	1	3	yes	—
11	Q₈×Dic₅₉	C₄	1	1	yes	characteristic
12	Q₈×Dic₅₉	C₄	1	1	yes	characteristic
13	D₄×Dic₅₉	C₄	1	3	yes	—
14	D₄×Dic₅₉	C₄	1	3	yes	—
15	#14273686	C₂²	1	3	yes	—
16	C₂×C₄×Dic₅₉	C₂²	1	3	yes	—
17	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
18	Dic₅₉⋊C₈	C₂²	1	3	yes	—
19	Dic₅₉⋊C₈	C₂²	1	3	yes	—
20	Dic₅₉⋊C₈	C₂²	1	3	yes	—
21	Dic₅₉⋊C₈	C₂²	1	6	yes	—
22	C₂×#41×C₅₉	C₂²	1	1	yes	characteristic
23	#131×C₅₉	—	2	4	no	—
24	C₂×M₁₆×C₅₉	C₂²	1	3	yes	—
25	C₂²×C₈×C₅₉	C₂²	1	3	yes	—
26	#41×C₅₉	D₄	1	4	yes	—
27	#41×C₅₉	Q₈	1	4	yes	—
28	C₂×Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
29	C₂×D₄×C₅₉	C₂×C₄	1	3	yes	—
30	C₂²×C₄×C₅₉	C₂³	1	3	yes	—
31	C₂²×Dic₅₉	C₂×C₄	1	3	yes	—
32	C₂²×Dic₅₉	C₂×C₄	1	3	yes	—
33	M₁₆×C₅₉	—	2	6	no	—
34	C₂×C₈×C₅₉	—	2	6	no	—
35	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
36	C₂×C₈×C₅₉	C₂³	1	3	yes	—
37	C₂²⋊Dic₅₉	C₂×C₄	1	6	yes	—
38	C₄⋊Dic₅₉	C₂×C₄	1	3	yes	—
39	C₄⋊Dic₅₉	C₂×C₄	1	3	yes	—
40	C₄×Dic₅₉	C₂×C₄	1	3	yes	—
41	C₄×Dic₅₉	C₂×C₄	1	3	yes	—
42	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
43	C₄×Dic₅₉	C₂³	1	3	yes	—
44	#11745664	—	2	6	no	—
45	C₂×C₅₉⋊C₈	—	2	6	no	—
46	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
47	C₂×C₅₉⋊C₈	C₂³	1	3	yes	—
48	C₂³×C₅₉	C₂²×C₄	1	3	yes	—
49	Q₈×C₅₉	C₄⋊C₄	1	4	yes	—
50	D₄×C₅₉	C₄⋊C₄	1	12	yes	—
51	C₂×C₄×C₅₉	C₂²×C₄	1	3	yes	—
52	C₂×C₄×C₅₉	C₂×D₄	1	6	yes	—
53	C₂×C₄×C₅₉	C₂×Q₈	1	6	yes	—
54	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
55	C₂×Dic₅₉	—	2	6	no	—
56	C₂×Dic₅₉	—	4	3	no	—
57	C₂×Dic₅₉	C₂²×C₄	1	1	yes	characteristic
58	C₂×Dic₅₉	C₂²×C₄	1	3	yes	—
59	C₂×Dic₅₉	C₂²×C₄	1	3	yes	—
60	C₂×Dic₅₉	C₂²×C₄	1	1	yes	characteristic
61	C₈×C₅₉	—	2	1	no	—
62	C₈×C₅₉	—	2	3	no	—
63	C₅₉⋊C₈	—	2	1	no	—
64	C₅₉⋊C₈	—	2	3	no	—
65	C₂²×C₅₉	—	2	3	no	—
66	C₂²×C₅₉	—	2	3	no	—
67	C₂²×C₅₉	C₂×C₄⋊C₄	1	6	yes	—
68	C₂²×C₅₉	C₂³×C₄	1	1	yes	characteristic
69	C₄×C₅₉	C₂×C₄⋊C₄	1	6	yes	—
70	C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
71	C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
72	Dic₅₉	—	2	3	no	—
73	Dic₅₉	—	2	1	no	—
74	Dic₅₉	—	2	3	no	—
75	Dic₅₉	#131	1	2	yes	complemented
76	#2906	—	59	1	no	maximal, Sylow
77	C₂×C₅₉	—	2	6	no	—
78	C₂×C₅₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₅₉	C₂×#131	1	1	yes	characteristic
80	C₂×C₅₉	#568	1	1	yes	characteristic
81	C₂×#131	—	59	1	no	—
82	C₂×#131	D₅₉	1	1	yes	characteristic
83	C₄×#41	—	59	1	no	—
84	#424	—	59	3	no	—
85	#424	—	59	3	no	—
86	#423	—	59	3	no	—
87	C₂×C₄⋊C₈	—	59	3	no	—
88	C₅₉	#2906	1	1	yes	characteristic, complemented, Sylow
89	C₂×#41	C₂×D₅₉	1	1	yes	characteristic
90	#131	—	2	4	no	—
91	#131	—	118	4	no	—
92	C₂×M₁₆	—	59	3	no	—
93	C₂×M₁₆	C₂×D₅₉	1	3	yes	—
94	C₂²×C₈	—	59	3	no	—
95	C₂²×C₈	C₂×D₅₉	1	3	yes	—
96	C₄×Q₈	—	59	1	no	—
97	C₄×Q₈	—	59	1	no	—
98	C₄×D₄	—	59	3	no	—
99	C₄×D₄	—	59	3	no	—
100	#117	—	59	3	no	—
101	C₂×C₄²	—	59	3	no	—
102	C₄⋊C₈	—	59	1	no	—
103	C₄⋊C₈	—	59	3	no	—
104	C₄⋊C₈	—	59	3	no	—
105	C₄⋊C₈	—	59	3	no	—
106	C₄⋊C₈	—	59	6	no	—
107	#41	Dic₁₁₈	1	4	yes	—
108	#41	C₂²⋊D₅₉	1	4	yes	—
109	C₂×Q₈	C₄×D₅₉	1	1	yes	characteristic
110	C₂×D₄	C₄×D₅₉	1	3	yes	—
111	C₂²×C₄	—	59	3	no	—
112	C₂²×C₄	—	59	3	no	—
113	C₂²×C₄	C₂²×D₅₉	1	3	yes	—
114	M₁₆	—	2	6	no	—
115	M₁₆	—	118	6	no	—
116	C₂×C₈	—	2	6	no	—
117	C₂×C₈	—	59	1	no	—
118	C₂×C₈	—	59	3	no	—
119	C₂×C₈	—	118	6	no	—
120	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
121	C₂×C₈	C₂²×D₅₉	1	3	yes	—
122	C₄⋊C₄	—	59	3	no	—
123	C₄⋊C₄	—	59	3	no	—
124	C₂²⋊C₄	—	59	6	no	—
125	C₄²	—	59	3	no	—
126	C₄²	—	59	3	no	—
127	C₄²	—	59	1	no	—
128	C₄²	—	59	3	no	—
129	C₂³	C₂×C₄×D₅₉	1	3	yes	—
130	Q₈	Dic₅₉⋊C₄	1	4	yes	—
131	D₄	Dic₅₉⋊C₄	1	12	yes	—
132	C₂×C₄	—	59	1	no	—
133	C₂×C₄	—	59	1	no	—
134	C₂×C₄	—	59	3	no	—
135	C₂×C₄	—	59	3	no	—
136	C₂×C₄	—	118	6	no	—
137	C₂×C₄	—	236	3	no	—
138	C₂×C₄	C₂×Dic₁₁₈	1	6	yes	—
139	C₂×C₄	C₂×C₄×D₅₉	1	3	yes	—
140	C₂×C₄	C₂×C₂²⋊D₅₉	1	6	yes	—
141	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
142	C₈	—	2	1	no	—
143	C₈	—	2	3	no	—
144	C₈	—	118	1	no	—
145	C₈	—	118	3	no	—
146	C₂²	—	2	3	no	—
147	C₂²	—	2	3	no	—
148	C₂²	C₂×Dic₅₉⋊C₄	1	6	yes	—
149	C₂²	C₂²×C₄×D₅₉	1	1	yes	characteristic
150	C₄	—	59	2	no	—
151	C₄	—	118	3	no	—
152	C₄	—	118	1	no	—
153	C₄	—	118	3	no	—
154	C₄	C₂×Dic₅₉⋊C₄	1	6	yes	—
155	C₄	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
156	C₄	C₂²×Dic₁₁₈	1	1	yes	characteristic
157	C₂	—	2	6	no	—
158	C₂	C₂²×Dic₅₉⋊C₄	1	1	yes	characteristic
159	C₂	#131×D₅₉	1	1	yes	characteristic
160	C₂	#16838167	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273734

C₂

yes

maximal, characteristic

#41×Dic₅₉

C₂

yes

maximal, characteristic

#16837430

C₂

yes

maximal, complemented

#16837428

C₂

yes

maximal, complemented

#16837425

C₂

yes

maximal, complemented

C₂×Dic₅₉⋊C₈

C₂

yes

maximal, complemented

C₂×#131×C₅₉

C₂

yes

maximal, characteristic

#14273734

—

C₂×#11745664

C₂²

yes

—

C₂²×C₅₉⋊C₈

C₂²

yes

—

Q₈×Dic₅₉

C₄

yes

characteristic

Q₈×Dic₅₉

C₄

yes

characteristic

D₄×Dic₅₉

C₄

yes

—

D₄×Dic₅₉

C₄

yes

—

#14273686

C₂²

yes

—

C₂×C₄×Dic₅₉

C₂²

yes

—

Dic₅₉⋊C₈

C₂²

yes

characteristic

Dic₅₉⋊C₈

C₂²

yes

—

Dic₅₉⋊C₈

C₂²

yes

—

Dic₅₉⋊C₈

C₂²

yes

—

Dic₅₉⋊C₈

C₂²

yes

—

C₂×#41×C₅₉

C₂²

yes

characteristic

#131×C₅₉

—

C₂×M₁₆×C₅₉

C₂²

yes

—

C₂²×C₈×C₅₉

C₂²

yes

—

#41×C₅₉

D₄

yes

—

#41×C₅₉

Q₈

yes

—

C₂×Q₈×C₅₉

C₂×C₄

yes

characteristic

C₂×D₄×C₅₉

C₂×C₄

yes

—

C₂²×C₄×C₅₉

C₂³

yes

—

C₂²×Dic₅₉

C₂×C₄

yes

—

C₂²×Dic₅₉

C₂×C₄

yes

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₈×C₅₉

C₂³

yes

—

C₂²⋊Dic₅₉

C₂×C₄

yes

—

C₄⋊Dic₅₉

C₂×C₄

yes

—

C₄⋊Dic₅₉

C₂×C₄

yes

—

C₄×Dic₅₉

C₂×C₄

yes

—

C₄×Dic₅₉

C₂×C₄

yes

—

C₄×Dic₅₉

C₂³

yes

characteristic

C₄×Dic₅₉

C₂³

yes

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₂³

yes

characteristic

C₂×C₅₉⋊C₈

C₂³

yes

—

C₂³×C₅₉

C₂²×C₄

yes

—

Q₈×C₅₉

C₄⋊C₄

yes

—

D₄×C₅₉

C₄⋊C₄

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×Q₈

yes

—

C₂×C₄×C₅₉

C₂⁴

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

C₂²×C₄

yes

characteristic

C₂×Dic₅₉

C₂²×C₄

yes

—

C₂×Dic₅₉

C₂²×C₄

yes

—

C₂×Dic₅₉

C₂²×C₄

yes

characteristic

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×C₄⋊C₄

yes

—

C₂²×C₅₉

C₂³×C₄

yes

characteristic

C₄×C₅₉

C₂×C₄⋊C₄

yes

—

C₄×C₅₉

C₂²×D₄

yes

characteristic

C₄×C₅₉

C₂²×Q₈

yes

characteristic

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

yes

—

C₂×C₅₉

—

C₂×C₅₉

C₂²×C₄⋊C₄

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

D₅₉

yes

characteristic

C₄×#41

—

#424

—

#424

—

#423

—

C₂×C₄⋊C₈

—

C₅₉

#2906

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₅₉

yes

characteristic

#131

—

#131

—

118

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₉

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₅₉

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

—

100

#117

—

101

C₂×C₄²

—

102

C₄⋊C₈

—

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

C₄⋊C₈

—

107

#41

Dic₁₁₈

yes

—

108

#41

C₂²⋊D₅₉

yes

—

109

C₂×Q₈

C₄×D₅₉

yes

characteristic

110

C₂×D₄

C₄×D₅₉

yes

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

C₂²×D₅₉

yes

—

114

M₁₆

—

115

M₁₆

—

118

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

118

—

120

C₂×C₈

C₂²×D₅₉

yes

characteristic

121

C₂×C₈

C₂²×D₅₉

yes

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₂²⋊C₄

—

125

C₄²

—

126

C₄²

—

127

C₄²

—

128

C₄²

—

129

C₂³

C₂×C₄×D₅₉

yes

—

130

Q₈

Dic₅₉⋊C₄

yes

—

131

D₄

Dic₅₉⋊C₄

yes

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

118

—

137

C₂×C₄

—

236

—

138

C₂×C₄

C₂×Dic₁₁₈

yes

—

139

C₂×C₄

C₂×C₄×D₅₉

yes

—

140

C₂×C₄

C₂×C₂²⋊D₅₉

yes

—

141

C₂×C₄

C₂³×D₅₉

yes

characteristic

142

C₈

—

143

C₈

—

144

C₈

—

118

—

145

C₈

—

118

—

146

C₂²

—

147

C₂²

—

148

C₂²

C₂×Dic₅₉⋊C₄

yes

—

149

C₂²

C₂²×C₄×D₅₉

yes

characteristic

150

C₄

—

151

C₄

—

118

—

152

C₄

—

118

—

153

C₄

—

118

—

154

C₄

C₂×Dic₅₉⋊C₄

yes

—

155

C₄

C₂²×C₂²⋊D₅₉

yes

characteristic

156

C₄

C₂²×Dic₁₁₈

yes

characteristic

157

C₂

—

158

C₂

C₂²×Dic₅₉⋊C₄

yes

characteristic

159

C₂

#131×D₅₉

yes

characteristic

160

C₂

#16838167

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₉
commutator, socle	C₂²×C₅₉	C₂³×C₄
Fitting	C₂×#131×C₅₉	C₂
hypercenter	C₂×#131	D₅₉
hypocenter	C₅₉	#2906
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#131

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₅₉

commutator, socle

C₂²×C₅₉

C₂³×C₄

Fitting

C₂×#131×C₅₉

C₂

D₅₉

C₅₉

G ▷ C₂²×C₅₉ ▷ C₅₉

C₁ ◁ C₂×C₄ ◁ C₂×#131

N	H ≅ G/N	type
3. #16837430	C₂	semidirect
4. #16837428	C₂	semidirect
5. #16837425	C₂	semidirect
6. C₂×Dic₅₉⋊C₈	C₂	semidirect
75. Dic₅₉	#131	semidirect
88. C₅₉	#2906	semidirect

H ≅ G/N

type

3. #16837430

C₂

semidirect

4. #16837428

C₂

semidirect

5. #16837425

C₂

semidirect

6. C₂×Dic₅₉⋊C₈

C₂

semidirect

75. Dic₅₉

#131

semidirect

88. C₅₉

#2906

semidirect

Subgroups — #21432662

Conjugacy classes of subgroups

Complements