Group #21424135 — subgroups

Subgroups — #21424135

Order 7552.

272 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 93 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	C₂²×#11745659	C₂	1	1	yes	maximal, characteristic
2	C₂×#14273663	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837365	C₂	1	2	yes	maximal
4	#16837166	C₂	1	2	yes	maximal, complemented
5	C₂×#108×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₅₉	C₂²	1	1	yes	characteristic
7	C₂×#11745664	C₂²	1	2	yes	—
8	C₂×#11745659	—	2	2	no	—
9	C₂×#11745659	—	2	2	no	—
10	C₂×#11745659	C₄	1	4	yes	—
11	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
12	#14273663	—	2	2	no	—
13	(C₂×D₅₉)⋊C₈	—	2	4	no	—
14	C₂×M₁₆×C₅₉	C₂²	1	2	yes	—
15	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	#108×C₅₉	—	2	2	no	—
17	C₂³×D₅₉	Q₈	1	1	yes	characteristic
18	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
19	C₂²×Dic₅₉	Q₈	1	1	yes	characteristic
20	C₂×C₄×D₅₉	—	2	1	no	—
21	C₂×C₄×D₅₉	—	2	1	no	—
22	C₂×C₄×D₅₉	—	2	1	no	—
23	C₂×C₄×D₅₉	—	2	1	no	—
24	C₂×C₄×D₅₉	C₂×C₄	1	2	yes	—
25	C₂×C₄×D₅₉	D₄	1	2	yes	—
26	M₁₆×C₅₉	—	2	2	no	—
27	M₁₆×C₅₉	—	2	2	no	—
28	C₂×C₈×C₅₉	—	2	2	no	—
29	C₂×C₈×C₅₉	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₅₉	D₄	1	2	yes	—
31	C₂×C₈×C₅₉	D₄	1	2	yes	—
32	#11745664	—	2	2	no	—
33	#11745664	—	2	2	no	—
34	C₂×C₅₉⋊C₈	—	2	1	no	—
35	C₂×C₅₉⋊C₈	—	2	1	no	—
36	C₂×C₅₉⋊C₈	—	2	2	no	—
37	C₂×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
38	#11745659	—	2	8	no	—
39	C₂³×C₅₉	C₂×Q₈	1	1	yes	characteristic
40	C₂²×D₅₉	—	2	1	no	—
41	C₂²×D₅₉	—	2	1	no	—
42	C₂²×D₅₉	—	4	2	no	—
43	C₂²×D₅₉	C₄⋊C₄	1	2	yes	—
44	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
49	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
50	C₂×Dic₅₉	—	2	1	no	—
51	C₂×Dic₅₉	—	2	1	no	—
52	C₂×Dic₅₉	C₄⋊C₄	1	2	yes	—
53	C₄×D₅₉	—	2	2	no	—
54	C₄×D₅₉	—	2	2	no	—
55	C₄×D₅₉	—	2	2	no	—
56	C₄×D₅₉	—	2	2	no	—
57	C₈×C₅₉	—	4	2	no	—
58	C₈×C₅₉	C₂²⋊C₄	1	4	yes	—
59	C₅₉⋊C₈	—	2	2	no	—
60	C₅₉⋊C₈	—	4	2	no	—
61	C₂²×C₅₉	—	2	2	no	—
62	C₂²×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
64	C₂²×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
65	C₂×D₅₉	—	2	2	no	—
66	C₂×D₅₉	—	2	4	no	—
67	C₂×D₅₉	—	4	2	no	—
68	C₂×D₅₉	—	4	2	no	—
69	C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₄×C₅₉	#117	1	1	yes	characteristic
71	C₄×C₅₉	#121	1	1	yes	characteristic
72	C₄×C₅₉	#121	1	1	yes	characteristic
73	Dic₅₉	—	2	2	no	—
74	#1811	—	59	1	no	maximal, Sylow
75	C₂×C₅₉	—	2	1	no	—
76	C₂×C₅₉	—	2	1	no	—
77	C₂×C₅₉	#380	1	1	yes	characteristic
78	C₂×C₅₉	#430	1	2	yes	—
79	D₅₉	—	4	2	no	—
80	C₂²×M₁₆	—	59	1	no	—
81	C₂×#108	—	59	1	no	—
82	C₂×#108	D₅₉	1	1	yes	characteristic, complemented
83	#407	—	59	2	no	—
84	#341	—	59	2	no	—
85	C₅₉	#1811	1	1	yes	characteristic, complemented, Sylow
86	C₂³×C₄	—	59	1	no	—
87	C₂×M₁₆	—	59	4	no	—
88	C₂×M₁₆	—	59	2	no	—
89	C₂×M₁₆	—	118	2	no	—
90	C₂×M₁₆	—	118	2	no	—
91	C₂×M₁₆	C₂×D₅₉	1	2	yes	—
92	C₂²×C₈	—	59	1	no	—
93	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
94	#108	—	2	2	no	—
95	#108	—	118	2	no	—
96	C₂²⋊C₈	—	118	4	no	—
97	C₂⁴	—	59	1	no	—
98	C₂²×C₄	—	59	1	no	—
99	C₂²×C₄	—	59	2	no	—
100	C₂²×C₄	—	59	2	no	—
101	C₂²×C₄	—	118	1	no	—
102	C₂²×C₄	—	118	1	no	—
103	C₂²×C₄	—	118	1	no	—
104	C₂²×C₄	—	118	1	no	—
105	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
106	M₁₆	—	2	2	no	—
107	M₁₆	—	2	2	no	—
108	M₁₆	—	118	8	no	—
109	M₁₆	—	118	2	no	—
110	M₁₆	—	118	2	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	59	2	no	—
113	C₂×C₈	—	118	1	no	—
114	C₂×C₈	—	118	1	no	—
115	C₂×C₈	—	118	2	no	—
116	C₂×C₈	C₄×D₅₉	1	2	yes	—
117	C₂×C₈	D₂₃₆	1	2	yes	—
118	C₂×C₈	C₂²⋊D₅₉	1	2	yes	—
119	C₂³	—	59	2	no	—
120	C₂³	—	118	1	no	—
121	C₂³	—	118	1	no	—
122	C₂³	—	236	2	no	—
123	C₂³	Q₈×D₅₉	1	1	yes	characteristic
124	C₂×C₄	—	59	2	no	—
125	C₂×C₄	—	118	1	no	—
126	C₂×C₄	—	118	1	no	—
127	C₂×C₄	—	118	2	no	—
128	C₂×C₄	—	118	2	no	—
129	C₂×C₄	—	118	2	no	—
130	C₂×C₄	—	118	2	no	—
131	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
132	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
133	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
134	C₂×C₄	#11745687	1	1	yes	characteristic
135	C₂×C₄	#11745689	1	1	yes	characteristic
136	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
137	C₈	—	4	2	no	—
138	C₈	—	118	2	no	—
139	C₈	—	236	2	no	—
140	C₈	(C₂×D₅₉)⋊C₄	1	4	yes	—
141	C₂²	—	2	2	no	—
142	C₂²	—	118	4	no	—
143	C₂²	—	118	2	no	—
144	C₂²	—	236	2	no	—
145	C₂²	—	236	2	no	—
146	C₂²	C₄⋊C₄×D₅₉	1	1	yes	characteristic
147	C₂²	#14273703	1	1	yes	characteristic
148	C₂²	#14273707	1	1	yes	characteristic
149	C₄	—	118	2	no	—
150	C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
151	C₄	#14273685	1	1	yes	characteristic
152	C₄	#14273698	1	1	yes	characteristic
153	C₄	#14273702	1	1	yes	characteristic
154	C₂	—	2	1	no	—
155	C₂	—	2	1	no	—
156	C₂	—	236	2	no	—
157	C₂	#16837259	1	1	yes	characteristic
158	C₂	#16837449	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11745659

C₂

yes

maximal, characteristic

C₂×#14273663

C₂

yes

maximal, characteristic, complemented

#16837365

C₂

yes

maximal

#16837166

C₂

yes

maximal, complemented

C₂×#108×C₅₉

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₅₉

C₂²

yes

characteristic

C₂×#11745664

C₂²

yes

—

C₂×#11745659

—

C₂×#11745659

—

C₂×#11745659

C₄

yes

—

C₂²×C₅₉⋊C₈

C₂²

yes

characteristic

#14273663

—

(C₂×D₅₉)⋊C₈

—

C₂×M₁₆×C₅₉

C₂²

yes

—

C₂²×C₈×C₅₉

C₂²

yes

characteristic

#108×C₅₉

—

C₂³×D₅₉

Q₈

yes

characteristic

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

Q₈

yes

characteristic

C₂×C₄×D₅₉

—

C₂×C₄×D₅₉

—

C₂×C₄×D₅₉

—

C₂×C₄×D₅₉

—

C₂×C₄×D₅₉

C₂×C₄

yes

—

C₂×C₄×D₅₉

D₄

yes

—

M₁₆×C₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂×C₄

yes

—

C₂×C₈×C₅₉

D₄

yes

—

C₂×C₈×C₅₉

D₄

yes

—

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₂×C₄

yes

—

#11745659

—

C₂³×C₅₉

C₂×Q₈

yes

characteristic

C₂²×D₅₉

—

C₂²×D₅₉

—

C₂²×D₅₉

—

C₂²×D₅₉

C₄⋊C₄

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

C₄⋊C₄

yes

—

C₄×D₅₉

—

C₄×D₅₉

—

C₄×D₅₉

—

C₄×D₅₉

—

C₈×C₅₉

—

C₈×C₅₉

C₂²⋊C₄

yes

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₅₉

C₂²⋊Q₈

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

C₂×C₂²⋊C₄

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

#1811

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#380

yes

characteristic

C₂×C₅₉

#430

yes

—

D₅₉

—

C₂²×M₁₆

—

C₂×#108

—

C₂×#108

D₅₉

yes

characteristic, complemented

#407

—

#341

—

C₅₉

#1811

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

118

—

C₂×M₁₆

—

118

—

C₂×M₁₆

C₂×D₅₉

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₅₉

yes

characteristic

#108

—

#108

—

118

—

C₂²⋊C₈

—

118

—

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

118

—

102

C₂²×C₄

—

118

—

103

C₂²×C₄

—

118

—

104

C₂²×C₄

—

118

—

105

C₂²×C₄

C₂²×D₅₉

yes

characteristic

106

M₁₆

—

107

M₁₆

—

108

M₁₆

—

118

—

109

M₁₆

—

118

—

110

M₁₆

—

118

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

118

—

114

C₂×C₈

—

118

—

115

C₂×C₈

—

118

—

116

C₂×C₈

C₄×D₅₉

yes

—

117

C₂×C₈

D₂₃₆

yes

—

118

C₂×C₈

C₂²⋊D₅₉

yes

—

119

C₂³

—

120

C₂³

—

118

—

121

C₂³

—

118

—

122

C₂³

—

236

—

123

C₂³

Q₈×D₅₉

yes

characteristic

124

C₂×C₄

—

125

C₂×C₄

—

118

—

126

C₂×C₄

—

118

—

127

C₂×C₄

—

118

—

128

C₂×C₄

—

118

—

129

C₂×C₄

—

118

—

130

C₂×C₄

—

118

—

131

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

132

C₂×C₄

C₂×D₂₃₆

yes

characteristic

133

C₂×C₄

D₄×D₅₉

yes

characteristic

134

C₂×C₄

#11745687

yes

characteristic

135

C₂×C₄

#11745689

yes

characteristic

136

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

137

C₈

—

138

C₈

—

118

—

139

C₈

—

236

—

140

C₈

(C₂×D₅₉)⋊C₄

yes

—

141

C₂²

—

142

C₂²

—

118

—

143

C₂²

—

118

—

144

C₂²

—

236

—

145

C₂²

—

236

—

146

C₂²

C₄⋊C₄×D₅₉

yes

characteristic

147

C₂²

#14273703

yes

characteristic

148

C₂²

#14273707

yes

characteristic

149

C₄

—

118

—

150

C₄

C₂×(C₂×D₅₉)⋊C₄

yes

characteristic

151

C₄

#14273685

yes

characteristic

152

C₄

#14273698

yes

characteristic

153

C₄

#14273702

yes

characteristic

154

C₂

—

155

C₂

—

156

C₂

—

236

—

157

C₂

#16837259

yes

characteristic

158

C₂

#16837449

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₂×#108×C₅₉	C₂
socle	C₂²×C₅₉	C₂×C₄⋊C₄
hypercenter	C₂×#108	D₅₉
hypocenter	C₅₉	#1811
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂×C₄

D₄×D₅₉

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₂×#108×C₅₉

C₂

socle

C₂²×C₅₉

C₂×C₄⋊C₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
2. C₂×#14273663	C₂	semidirect
4. #16837166	C₂	semidirect
5. C₂×#108×C₅₉	C₂	semidirect
82. C₂×#108	D₅₉	semidirect
85. C₅₉	#1811	semidirect

H ≅ G/N

type

2. C₂×#14273663

C₂

semidirect

4. #16837166

C₂

semidirect

5. C₂×#108×C₅₉

C₂

semidirect

82. C₂×#108

D₅₉

semidirect

85. C₅₉

#1811

semidirect

Subgroups — #21424135

Conjugacy classes of subgroups

Complements