Group #21424090 — subgroups

Subgroups — #21424090

Order 7552.

272 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 95 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×Q₈×Dic₅₉	C₂	1	1	yes	maximal, characteristic
2	#16837425	C₂	1	2	yes	maximal
3	C₂×#14273646	C₂	1	1	yes	maximal, characteristic
4	#16837170	C₂	1	2	yes	maximal
5	C₂×#101×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#11745664	C₂²	1	2	yes	—
7	Q₈×Dic₅₉	—	2	2	no	—
8	Q₈×Dic₅₉	—	2	2	no	—
9	C₂×C₄⋊Dic₅₉	C₄	1	2	yes	—
10	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₅₉	C₄	1	2	yes	—
12	C₂×C₄×Dic₅₉	C₂²	1	1	yes	characteristic
13	Dic₅₉⋊C₈	—	2	2	no	—
14	Dic₅₉⋊C₈	—	2	2	no	—
15	#14273646	—	2	2	no	—
16	C₂²×Q₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₂×M₁₆×C₅₉	C₂²	1	2	yes	—
18	#101×C₅₉	—	2	2	no	—
19	C₂×Q₈×C₅₉	—	2	2	no	—
20	C₂×Q₈×C₅₉	—	2	2	no	—
21	C₂×Q₈×C₅₉	D₄	1	2	yes	—
22	C₂×Q₈×C₅₉	Q₈	1	2	yes	—
23	C₂²×C₄×C₅₉	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
25	C₂²×Dic₅₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×Dic₅₉	C₂×C₄	1	2	yes	—
27	C₂²×Dic₅₉	C₂×C₄	1	1	yes	characteristic
28	M₁₆×C₅₉	—	2	2	no	—
29	M₁₆×C₅₉	—	2	2	no	—
30	C₂×C₈×C₅₉	—	2	2	no	—
31	C₄⋊Dic₅₉	—	2	2	no	—
32	C₄⋊Dic₅₉	—	2	4	no	—
33	C₄×Dic₅₉	—	2	2	no	—
34	C₄×Dic₅₉	—	2	2	no	—
35	C₄×Dic₅₉	D₄	1	2	yes	—
36	C₄×Dic₅₉	D₄	1	2	yes	—
37	#11745664	—	2	2	no	—
38	#11745664	—	2	2	no	—
39	C₂×C₅₉⋊C₈	—	2	2	no	—
40	C₂³×C₅₉	C₂²×C₄	1	1	yes	characteristic
41	Q₈×C₅₉	—	2	4	no	—
42	Q₈×C₅₉	—	2	4	no	—
43	C₂×C₄×C₅₉	—	2	2	no	—
44	C₂×C₄×C₅₉	—	2	2	no	—
45	C₂×C₄×C₅₉	C₄⋊C₄	1	4	yes	—
46	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₄×C₅₉	#41	1	2	yes	—
50	C₂×Dic₅₉	—	2	2	no	—
51	C₂×Dic₅₉	—	2	2	no	—
52	C₂×Dic₅₉	—	4	1	no	—
53	C₂×Dic₅₉	—	4	2	no	—
54	C₂×Dic₅₉	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×Dic₅₉	C₂²⋊C₄	1	2	yes	—
56	C₂×Dic₅₉	C₂²⋊C₄	1	1	yes	characteristic
57	C₈×C₅₉	—	4	2	no	—
58	C₅₉⋊C₈	—	4	2	no	—
59	C₂²×C₅₉	—	2	1	no	—
60	C₂²×C₅₉	—	2	1	no	—
61	C₂²×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₅₉	#117	1	1	yes	characteristic
64	C₄×C₅₉	—	2	4	no	—
65	C₄×C₅₉	#121	1	2	yes	—
66	C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
67	Dic₅₉	—	2	1	no	—
68	Dic₅₉	—	2	2	no	—
69	Dic₅₉	—	4	2	no	—
70	Dic₅₉	#101	1	2	yes	complemented
71	#1801	—	59	1	no	maximal, Sylow
72	C₂×C₅₉	—	2	1	no	—
73	C₂×C₅₉	—	2	1	no	—
74	C₂×C₅₉	#380	1	1	yes	characteristic
75	C₂×C₅₉	C₂×#101	1	1	yes	characteristic
76	C₂×C₅₉	#413	1	1	yes	characteristic
77	C₂×C₄×Q₈	—	59	1	no	—
78	#423	—	59	2	no	—
79	C₂×#101	—	59	1	no	—
80	C₂×#101	D₅₉	1	1	yes	characteristic
81	#343	—	59	2	no	—
82	C₅₉	#1801	1	1	yes	characteristic, complemented, Sylow
83	C₂²×Q₈	C₂×D₅₉	1	1	yes	characteristic
84	C₂×M₁₆	—	59	2	no	—
85	C₂×M₁₆	C₂×D₅₉	1	2	yes	—
86	C₄×Q₈	—	118	2	no	—
87	C₄×Q₈	—	118	2	no	—
88	C₂×C₄⋊C₄	—	59	2	no	—
89	C₂×C₄⋊C₄	—	59	1	no	—
90	C₂×C₄²	—	59	1	no	—
91	C₂×C₄²	—	59	2	no	—
92	C₄⋊C₈	—	118	2	no	—
93	C₄⋊C₈	—	118	2	no	—
94	#101	—	2	2	no	—
95	#101	—	118	2	no	—
96	C₂×Q₈	—	2	2	no	—
97	C₂×Q₈	—	2	2	no	—
98	C₂×Q₈	Dic₁₁₈	1	2	yes	—
99	C₂×Q₈	C₂²⋊D₅₉	1	2	yes	—
100	C₂²×C₄	—	59	1	no	—
101	C₂²×C₄	—	59	1	no	—
102	C₂²×C₄	—	59	2	no	—
103	C₂²×C₄	C₄×D₅₉	1	2	yes	—
104	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
105	M₁₆	—	2	2	no	—
106	M₁₆	—	2	2	no	—
107	M₁₆	—	118	2	no	—
108	M₁₆	—	118	2	no	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	118	2	no	—
111	C₄⋊C₄	—	118	4	no	—
112	C₄⋊C₄	—	118	2	no	—
113	C₄²	—	59	2	no	—
114	C₄²	—	59	2	no	—
115	C₄²	—	118	2	no	—
116	C₄²	—	118	2	no	—
117	C₂³	C₂×C₄×D₅₉	1	1	yes	characteristic
118	Q₈	—	2	4	no	—
119	Q₈	—	2	4	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	59	1	no	—
123	C₂×C₄	—	59	1	no	—
124	C₂×C₄	—	59	2	no	—
125	C₂×C₄	—	118	2	no	—
126	C₂×C₄	—	118	2	no	—
127	C₂×C₄	—	236	1	no	—
128	C₂×C₄	—	236	2	no	—
129	C₂×C₄	Dic₅₉⋊C₄	1	4	yes	—
130	C₂×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
131	C₂×C₄	D₄×D₅₉	1	2	yes	—
132	C₂×C₄	#11745687	1	2	yes	—
133	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
134	C₈	—	4	2	no	—
135	C₈	—	236	2	no	—
136	C₂²	—	2	1	no	—
137	C₂²	—	2	1	no	—
138	C₂²	C₂²⋊C₄×D₅₉	1	1	yes	characteristic
139	C₂²	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
140	C₂²	#14273683	1	1	yes	characteristic
141	C₄	—	2	4	no	—
142	C₄	—	59	2	no	—
143	C₄	—	118	1	no	—
144	C₄	—	118	2	no	—
145	C₄	—	236	2	no	—
146	C₄	#14273699	1	2	yes	—
147	C₄	#14273704	1	2	yes	—
148	C₂	—	2	1	no	—
149	C₂	—	2	1	no	—
150	C₂	#16837260	1	1	yes	characteristic
151	C₂	#101×D₅₉	1	1	yes	characteristic
152	C₂	#16837385	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×Dic₅₉

C₂

yes

maximal, characteristic

#16837425

C₂

yes

maximal

C₂×#14273646

C₂

yes

maximal, characteristic

#16837170

C₂

yes

maximal

C₂×#101×C₅₉

C₂

yes

maximal, characteristic

C₂×#11745664

C₂²

yes

—

Q₈×Dic₅₉

—

Q₈×Dic₅₉

—

C₂×C₄⋊Dic₅₉

C₄

yes

—

C₂×C₄⋊Dic₅₉

C₂²

yes

characteristic

C₂×C₄×Dic₅₉

C₄

yes

—

C₂×C₄×Dic₅₉

C₂²

yes

characteristic

Dic₅₉⋊C₈

—

Dic₅₉⋊C₈

—

#14273646

—

C₂²×Q₈×C₅₉

C₂²

yes

characteristic

C₂×M₁₆×C₅₉

C₂²

yes

—

#101×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

D₄

yes

—

C₂×Q₈×C₅₉

Q₈

yes

—

C₂²×C₄×C₅₉

C₂×C₄

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

C₂×C₄

yes

characteristic

C₂²×Dic₅₉

C₂×C₄

yes

—

C₂²×Dic₅₉

C₂×C₄

yes

characteristic

M₁₆×C₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄×Dic₅₉

—

C₄×Dic₅₉

—

C₄×Dic₅₉

D₄

yes

—

C₄×Dic₅₉

D₄

yes

—

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

C₂²×C₄

yes

characteristic

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₄⋊C₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₉

#41

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₅₉

C₂²⋊C₄

yes

—

C₂×Dic₅₉

C₂²⋊C₄

yes

characteristic

C₈×C₅₉

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₉

#117

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

#121

yes

—

C₄×C₅₉

C₂²⋊Q₈

yes

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

yes

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₄×Q₈

—

#423

—

C₂×#101

—

C₂×#101

D₅₉

yes

characteristic

#343

—

C₅₉

#1801

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₅₉

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₉

yes

—

C₄×Q₈

—

118

—

C₄×Q₈

—

118

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₄⋊C₈

—

118

—

C₄⋊C₈

—

118

—

#101

—

#101

—

118

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Dic₁₁₈

yes

—

C₂×Q₈

C₂²⋊D₅₉

yes

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

C₄×D₅₉

yes

—

104

C₂²×C₄

C₂²×D₅₉

yes

characteristic

105

M₁₆

—

106

M₁₆

—

107

M₁₆

—

118

—

108

M₁₆

—

118

—

109

C₂×C₈

—

110

C₂×C₈

—

118

—

111

C₄⋊C₄

—

118

—

112

C₄⋊C₄

—

118

—

113

C₄²

—

114

C₄²

—

115

C₄²

—

118

—

116

C₄²

—

118

—

117

C₂³

C₂×C₄×D₅₉

yes

characteristic

118

Q₈

—

119

Q₈

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

118

—

126

C₂×C₄

—

118

—

127

C₂×C₄

—

236

—

128

C₂×C₄

—

236

—

129

C₂×C₄

Dic₅₉⋊C₄

yes

—

130

C₂×C₄

C₂×Dic₁₁₈

yes

characteristic

131

C₂×C₄

D₄×D₅₉

yes

—

132

C₂×C₄

#11745687

yes

—

133

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

134

C₈

—

135

C₈

—

236

—

136

C₂²

—

137

C₂²

—

138

C₂²

C₂²⋊C₄×D₅₉

yes

characteristic

139

C₂²

C₂×Dic₅₉⋊C₄

yes

characteristic

140

C₂²

#14273683

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

118

—

144

C₄

—

118

—

145

C₄

—

236

—

146

C₄

#14273699

yes

—

147

C₄

#14273704

yes

—

148

C₂

—

149

C₂

—

150

C₂

#16837260

yes

characteristic

151

C₂

#101×D₅₉

yes

characteristic

152

C₂

#16837385

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₅₉
commutator	C₂³×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₂×#101×C₅₉	C₂
socle	C₂²×C₅₉	C₂×C₂²⋊C₄
hypercenter	C₂×#101	D₅₉
hypocenter	C₅₉	#1801
LCS	G ▷ C₂³×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#101

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₅₉

commutator

C₂³×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₂×#101×C₅₉

C₂

socle

C₂²×C₅₉

C₂×C₂²⋊C₄

D₅₉

C₅₉

G ▷ C₂³×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#101

N	H ≅ G/N	type
70. Dic₅₉	#101	semidirect
82. C₅₉	#1801	semidirect

H ≅ G/N

type

70. Dic₅₉

#101

semidirect

82. C₅₉

#1801

semidirect

Subgroups — #21424090

Conjugacy classes of subgroups

Complements