Group #21424217 — subgroups

Subgroups — #21424217

Order 7552.

172 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 73 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×#14273662	C₂	1	1	yes	maximal, characteristic
2	#16837438	C₂	1	1	yes	maximal, characteristic
3	#16837360	C₂	1	1	yes	maximal, characteristic
4	#16837173	C₂	1	1	yes	maximal, characteristic
5	#16837159	C₂	1	1	yes	maximal, characteristic
6	#16837158	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#344	C₂	1	1	yes	maximal, characteristic
8	C₂×#11745664	C₂²	1	1	yes	characteristic
9	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
10	#14273686	C₂²	1	1	yes	characteristic
11	#14273686	C₂²	1	1	yes	characteristic
12	#14273662	—	2	1	no	—
13	#14273662	—	2	1	no	—
14	#14273660	—	2	1	no	—
15	#14273658	—	2	1	no	—
16	#14273637	C₄	1	2	yes	complemented
17	C₈×Dic₅₉	C₄	1	2	yes	complemented
18	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
19	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
20	#117×C₅₉	C₂²	1	1	yes	characteristic
21	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
22	M₁₆×C₅₉	—	2	1	no	—
23	M₁₆×C₅₉	—	2	1	no	—
24	M₁₆×C₅₉	—	2	1	no	—
25	M₁₆×C₅₉	—	2	1	no	—
26	C₂×C₈×C₅₉	—	2	1	no	—
27	C₂×C₈×C₅₉	C₂×C₄	1	2	yes	—
28	C₄⋊C₄×C₅₉	—	2	1	no	—
29	C₂²⋊C₄×C₅₉	—	2	1	no	—
30	C₄²×C₅₉	—	2	1	no	—
31	C₂²⋊Dic₅₉	—	2	1	no	—
32	C₂²⋊Dic₅₉	—	2	1	no	—
33	C₄⋊Dic₅₉	—	2	1	no	—
34	C₄⋊Dic₅₉	—	2	1	no	—
35	C₄×Dic₅₉	—	2	1	no	—
36	C₄×Dic₅₉	C₂×C₄	1	2	yes	—
37	#11745664	—	2	1	no	—
38	#11745664	—	2	1	no	—
39	C₂×C₅₉⋊C₈	—	2	1	no	—
40	C₂×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
41	C₂×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
42	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
43	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
44	C₂×C₅₉⋊C₈	Q₈	1	1	yes	characteristic
45	C₂×C₅₉⋊C₈	Q₈	1	1	yes	characteristic
46	C₂³×C₅₉	#41	1	1	yes	characteristic
47	C₂×C₄×C₅₉	—	4	1	no	—
48	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
52	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
53	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
54	C₂×Dic₅₉	—	2	2	no	—
55	C₂×Dic₅₉	—	4	1	no	—
56	C₈×C₅₉	—	2	2	no	—
57	C₈×C₅₉	—	4	1	no	—
58	C₅₉⋊C₈	—	4	1	no	—
59	C₅₉⋊C₈	C₄⋊C₄	1	1	yes	characteristic
60	C₅₉⋊C₈	C₄⋊C₄	1	1	yes	characteristic
61	C₅₉⋊C₈	C₄⋊C₄	1	1	yes	characteristic
62	C₅₉⋊C₈	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₅₉	—	4	1	no	—
64	C₂²×C₅₉	#117	1	1	yes	characteristic
65	C₂²×C₅₉	#124	1	1	yes	characteristic
66	C₂²×C₅₉	#125	1	1	yes	characteristic
67	C₄×C₅₉	—	8	1	no	—
68	C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₄×C₅₉	#117	1	1	yes	characteristic
70	C₄×C₅₉	#124	1	1	yes	characteristic
71	C₄×C₅₉	#125	1	1	yes	characteristic
72	Dic₅₉	—	4	2	no	—
73	Dic₅₉	—	8	1	no	—
74	#1833	—	59	1	no	maximal, Sylow
75	C₂×C₅₉	—	2	1	no	—
76	C₂×C₅₉	—	2	1	no	—
77	C₂×C₅₉	—	2	1	no	—
78	C₂×C₅₉	#382	1	1	yes	characteristic
79	C₂×#108	—	59	1	no	—
80	#427	—	59	1	no	—
81	#405	—	59	1	no	—
82	#344	—	59	1	no	—
83	#344	D₅₉	1	1	yes	characteristic
84	H(3,ℤ/4)	—	59	1	no	—
85	H(3,ℤ/4)	—	59	1	no	—
86	C₅₉	#1833	1	1	yes	characteristic, complemented, Sylow
87	C₂×M₁₆	—	59	1	no	—
88	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
89	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
90	C₂²×C₈	—	59	1	no	—
91	#117	—	59	1	no	—
92	#117	—	59	1	no	—
93	#117	C₂×D₅₉	1	1	yes	characteristic
94	#108	—	118	1	no	—
95	#108	—	118	1	no	—
96	#107	—	118	1	no	—
97	#106	—	118	1	no	—
98	#97	—	59	2	no	—
99	C₄×C₈	—	59	2	no	—
100	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
101	M₁₆	—	2	1	no	—
102	M₁₆	—	2	1	no	—
103	M₁₆	—	2	1	no	—
104	M₁₆	—	2	1	no	—
105	M₁₆	—	118	1	no	—
106	M₁₆	—	118	1	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	59	1	no	—
109	C₂×C₈	—	59	1	no	—
110	C₂×C₈	—	59	1	no	—
111	C₂×C₈	—	59	1	no	—
112	C₂×C₈	—	59	1	no	—
113	C₂×C₈	—	59	1	no	—
114	C₂×C₈	—	118	1	no	—
115	C₂×C₈	C₄×D₅₉	1	2	yes	—
116	C₄⋊C₄	—	2	1	no	—
117	C₄⋊C₄	—	118	1	no	—
118	C₄⋊C₄	—	118	1	no	—
119	C₂²⋊C₄	—	2	1	no	—
120	C₂²⋊C₄	—	118	1	no	—
121	C₂²⋊C₄	—	118	1	no	—
122	C₄²	—	2	1	no	—
123	C₄²	—	59	2	no	—
124	C₄²	—	118	1	no	—
125	C₂³	#11745685	1	1	yes	characteristic
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	118	2	no	—
128	C₂×C₄	—	236	1	no	—
129	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
130	C₂×C₄	#11745685	1	1	yes	characteristic
131	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
132	C₂×C₄	#11745687	1	1	yes	characteristic
133	C₂×C₄	#11745687	1	1	yes	characteristic
134	C₂×C₄	Q₈×D₅₉	1	1	yes	characteristic
135	C₈	—	2	2	no	—
136	C₈	—	4	1	no	—
137	C₈	—	59	1	no	—
138	C₈	—	59	1	no	—
139	C₈	—	59	1	no	—
140	C₈	—	59	1	no	—
141	C₈	—	236	1	no	—
142	C₂²	—	4	1	no	—
143	C₂²	#14273684	1	1	yes	characteristic
144	C₂²	#14273713	1	1	yes	characteristic
145	C₂²	#14273717	1	1	yes	characteristic
146	C₄	—	8	1	no	—
147	C₄	—	472	1	no	—
148	C₄	—	236	2	no	—
149	C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
150	C₄	#14273683	1	1	yes	characteristic
151	C₄	#14273713	1	1	yes	characteristic
152	C₄	#14273717	1	1	yes	characteristic
153	C₂	—	2	1	no	—
154	C₂	—	2	1	no	—
155	C₂	—	2	1	no	—
156	C₂	#16837266	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273662

C₂

yes

maximal, characteristic

#16837438

C₂

yes

maximal, characteristic

#16837360

C₂

yes

maximal, characteristic

#16837173

C₂

yes

maximal, characteristic

#16837159

C₂

yes

maximal, characteristic

#16837158

C₂

yes

maximal, characteristic

C₅₉×#344

C₂

yes

maximal, characteristic

C₂×#11745664

C₂²

yes

characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

characteristic

#14273686

C₂²

yes

characteristic

#14273686

C₂²

yes

—

—

—

—

C₄

yes

complemented

C₈×Dic₅₉

C₄

yes

complemented

C₂×M₁₆×C₅₉

C₂²

yes

characteristic

C₂×M₁₆×C₅₉

C₂²

yes

characteristic

#117×C₅₉

C₂²

yes

characteristic

C₂²×C₄×C₅₉

C₂³

yes

characteristic

M₁₆×C₅₉

—

M₁₆×C₅₉

—

M₁₆×C₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂×C₄

yes

—

C₄⋊C₄×C₅₉

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

—

C₂²⋊Dic₅₉

—

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄×Dic₅₉

—

C₄×Dic₅₉

C₂×C₄

yes

—

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₂×C₄

yes

characteristic

C₂×C₅₉⋊C₈

C₂×C₄

yes

characteristic

C₂×C₅₉⋊C₈

D₄

yes

characteristic

C₂×C₅₉⋊C₈

D₄

yes

characteristic

C₂×C₅₉⋊C₈

Q₈

yes

characteristic

C₂×C₅₉⋊C₈

Q₈

yes

characteristic

C₂³×C₅₉

#41

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

C₄⋊C₄

yes

characteristic

C₅₉⋊C₈

C₄⋊C₄

yes

characteristic

C₅₉⋊C₈

C₄⋊C₄

yes

characteristic

C₅₉⋊C₈

C₄⋊C₄

yes

characteristic

C₂²×C₅₉

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₄×C₅₉

—

C₄×C₅₉

C₂×C₄⋊C₄

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

#1833

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

yes

—

—

—

—

D₅₉

yes

characteristic

H(3,ℤ/4)

—

H(3,ℤ/4)

—

C₅₉

#1833

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₉

yes

characteristic

C₂×M₁₆

C₂×D₅₉

yes

characteristic

C₂²×C₈

—

#117

—

#117

—

#117

C₂×D₅₉

yes

characteristic

#108

—

118

—

#108

—

118

—

#107

—

118

—

#106

—

118

—

#97

—

C₄×C₈

—

100

C₂²×C₄

C₂²×D₅₉

yes

characteristic

101

M₁₆

—

102

M₁₆

—

103

M₁₆

—

104

M₁₆

—

105

M₁₆

—

118

—

106

M₁₆

—

118

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

118

—

115

C₂×C₈

C₄×D₅₉

yes

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

—

118

C₄⋊C₄

—

118

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

118

—

121

C₂²⋊C₄

—

118

—

122

C₄²

—

123

C₄²

—

124

C₄²

—

118

—

125

C₂³

#11745685

yes

characteristic

126

C₂×C₄

—

127

C₂×C₄

—

118

—

128

C₂×C₄

—

236

—

129

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

130

C₂×C₄

#11745685

yes

characteristic

131

C₂×C₄

D₄×D₅₉

yes

characteristic

132

C₂×C₄

#11745687

yes

characteristic

133

C₂×C₄

#11745687

yes

characteristic

134

C₂×C₄

Q₈×D₅₉

yes

characteristic

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

—

139

C₈

—

140

C₈

—

141

C₈

—

236

—

142

C₂²

—

143

C₂²

#14273684

yes

characteristic

144

C₂²

#14273713

yes

characteristic

145

C₂²

#14273717

yes

characteristic

146

C₄

—

147

C₄

—

472

—

148

C₄

—

236

—

149

C₄

C₄⋊C₄×D₅₉

yes

characteristic

150

C₄

#14273683

yes

characteristic

151

C₄

#14273713

yes

characteristic

152

C₄

#14273717

yes

characteristic

153

C₂

—

154

C₂

—

155

C₂

—

156

C₂

#16837266

yes

characteristic

	subgroup	G/H
center	C₄	#14273683
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₅₉×#344	C₂
socle	C₂×C₅₉	#382
hypercenter	#344	D₅₉
hypocenter	C₅₉	#1833
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #344

subgroup

G/H

center

C₄

#14273683

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₅₉×#344

C₂

C₂×C₅₉

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #344

N	H ≅ G/N	type
16. #14273637	C₄	semidirect
17. C₈×Dic₅₉	C₄	semidirect
86. C₅₉	#1833	semidirect

H ≅ G/N

type

16. #14273637

C₄

semidirect

17. C₈×Dic₅₉

C₄

semidirect

86. C₅₉

#1833

semidirect

Subgroups — #21424217

Conjugacy classes of subgroups

Complements