Group #21425012 — subgroups

Subgroups — #21425012

Order 7552.

276 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 97 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×#14273663	C₂	1	1	yes	maximal, characteristic
2	#16837411	C₂	1	2	yes	maximal
3	C₂×#14273646	C₂	1	2	yes	maximal
4	#16837357	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#11745664	C₂²	1	2	yes	—
7	C₂×#11745664	C₂²	1	1	yes	characteristic
8	#14273686	C₂²	1	1	yes	characteristic
9	#14273663	—	2	2	no	—
10	#14273651	—	2	4	no	—
11	#14273646	—	2	4	no	—
12	#14273637	—	2	1	no	—
13	C₈×Dic₅₉	—	2	1	no	—
14	C₂²×Q₈×C₅₉	C₂²	1	2	yes	—
15	C₂×Q₁₆×C₅₉	—	2	1	no	—
16	C₂×Q₁₆×C₅₉	—	2	1	no	—
17	C₂×Q₁₆×C₅₉	—	2	2	no	—
18	C₂×Q₁₆×C₅₉	C₄	1	4	yes	complemented
19	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
20	C₂×Q₈×C₅₉	—	2	2	no	—
21	C₂×Q₈×C₅₉	—	2	2	no	—
22	C₂×Q₈×C₅₉	—	2	2	no	—
23	C₂×Q₈×C₅₉	—	2	2	no	—
24	C₂×Q₈×C₅₉	—	2	2	no	—
25	C₂×Q₈×C₅₉	C₂×C₄	1	4	yes	—
26	C₂²×C₄×C₅₉	—	2	2	no	—
27	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
28	Q₁₆×C₅₉	—	2	4	no	—
29	Q₁₆×C₅₉	—	2	4	no	—
30	C₂×C₈×C₅₉	C₂×C₄	1	2	yes	—
31	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
32	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
33	C₂×C₈×C₅₉	D₄	1	2	yes	—
34	C₂²⋊Dic₅₉	—	2	1	no	—
35	C₄⋊Dic₅₉	D₄	1	2	yes	—
36	C₄×Dic₅₉	—	2	1	no	—
37	#11745664	—	2	1	no	—
38	#11745664	—	2	2	no	—
39	#11745664	—	2	2	no	—
40	#11745664	D₄	1	2	yes	—
41	C₂×C₅₉⋊C₈	—	2	1	no	—
42	C₂×C₅₉⋊C₈	—	2	2	no	—
43	C₂³×C₅₉	#41	1	1	yes	characteristic
44	Q₈×C₅₉	—	2	4	no	—
45	Q₈×C₅₉	—	2	4	no	—
46	Q₈×C₅₉	—	4	2	no	—
47	Q₈×C₅₉	—	4	2	no	—
48	C₂×C₄×C₅₉	—	2	4	no	—
49	C₂×C₄×C₅₉	—	4	2	no	—
50	C₂×C₄×C₅₉	—	4	2	no	—
51	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
57	C₂×Dic₅₉	—	2	2	no	—
58	C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
59	C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
60	C₅₉⋊C₈	—	2	2	no	—
61	C₅₉⋊C₈	—	4	2	no	—
62	C₂²×C₅₉	—	2	2	no	—
63	C₂²×C₅₉	C₄×D₄	1	1	yes	characteristic
64	C₂²×C₅₉	#121	1	1	yes	characteristic
65	C₂²×C₅₉	#124	1	1	yes	characteristic
66	C₄×C₅₉	—	4	4	no	—
67	C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
69	C₄×C₅₉	#121	1	1	yes	characteristic
70	C₄×C₅₉	#127	1	1	yes	characteristic
71	Dic₅₉	—	4	2	no	—
72	#1979	—	59	1	no	maximal, Sylow
73	C₂×C₅₉	—	2	1	no	—
74	C₂×C₅₉	—	2	1	no	—
75	C₂×C₅₉	#390	1	1	yes	characteristic
76	C₂×C₅₉	#473	1	2	yes	—
77	C₂²×Q₁₆	D₅₉	1	1	yes	characteristic
78	C₂×#108	—	59	1	no	—
79	#419	—	59	2	no	—
80	C₂×#101	—	59	2	no	—
81	#405	—	59	1	no	—
82	C₅₉	#1979	1	1	yes	characteristic, complemented, Sylow
83	C₂²×Q₈	C₂×D₅₉	1	2	yes	—
84	C₂×Q₁₆	—	2	1	no	—
85	C₂×Q₁₆	—	2	1	no	—
86	C₂×Q₁₆	—	2	2	no	—
87	C₂×Q₁₆	Dic₅₉	1	4	yes	complemented
88	C₂×M₁₆	—	59	1	no	—
89	C₂×M₁₆	—	59	2	no	—
90	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
91	#117	—	59	1	no	—
92	#108	—	118	2	no	—
93	#103	—	118	4	no	—
94	#101	—	118	4	no	—
95	#97	—	118	1	no	—
96	C₄×C₈	—	118	1	no	—
97	C₂×Q₈	—	2	2	no	—
98	C₂×Q₈	—	2	2	no	—
99	C₂×Q₈	—	2	2	no	—
100	C₂×Q₈	—	2	2	no	—
101	C₂×Q₈	—	2	2	no	—
102	C₂×Q₈	C₂×Dic₅₉	1	4	yes	—
103	C₂²×C₄	—	2	2	no	—
104	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
105	Q₁₆	—	2	4	no	—
106	Q₁₆	—	2	4	no	—
107	M₁₆	—	59	2	no	—
108	M₁₆	—	118	1	no	—
109	M₁₆	—	118	2	no	—
110	M₁₆	—	118	2	no	—
111	C₂×C₈	—	118	1	no	—
112	C₂×C₈	—	118	2	no	—
113	C₂×C₈	C₂×Dic₅₉	1	2	yes	—
114	C₂×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
115	C₂×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
116	C₂×C₈	C₂²⋊D₅₉	1	2	yes	—
117	C₄⋊C₄	—	59	2	no	—
118	C₂²⋊C₄	—	118	1	no	—
119	C₄²	—	118	1	no	—
120	C₂³	#11745687	1	1	yes	characteristic
121	Q₈	—	2	4	no	—
122	Q₈	—	2	4	no	—
123	Q₈	—	4	2	no	—
124	Q₈	—	4	2	no	—
125	C₂×C₄	—	2	4	no	—
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	—	4	2	no	—
128	C₂×C₄	—	118	2	no	—
129	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
130	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
131	C₂×C₄	#11745687	1	1	yes	characteristic
132	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
133	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
134	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
135	C₈	—	118	2	no	—
136	C₈	—	236	2	no	—
137	C₈	C₂²⋊Dic₅₉	1	2	yes	—
138	C₈	C₂²⋊Dic₅₉	1	2	yes	—
139	C₂²	—	2	2	no	—
140	C₂²	D₄×Dic₅₉	1	1	yes	characteristic
141	C₂²	#14273698	1	1	yes	characteristic
142	C₂²	#14273714	1	1	yes	characteristic
143	C₄	—	4	4	no	—
144	C₄	—	236	2	no	—
145	C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
146	C₄	D₄×Dic₅₉	1	1	yes	characteristic
147	C₄	#14273698	1	1	yes	characteristic
148	C₄	#14273723	1	1	yes	characteristic
149	C₂	—	2	1	no	—
150	C₂	—	2	1	no	—
151	C₂	#16837305	1	1	yes	characteristic
152	C₂	#16837674	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273663

C₂

yes

maximal, characteristic

#16837411

C₂

yes

maximal

C₂×#14273646

C₂

yes

maximal

#16837357

C₂

yes

maximal, characteristic

C₂²×Q₁₆×C₅₉

C₂

yes

maximal, characteristic

C₂×#11745664

C₂²

yes

—

C₂×#11745664

C₂²

yes

characteristic

#14273686

C₂²

yes

—

—

—

—

C₈×Dic₅₉

—

C₂²×Q₈×C₅₉

C₂²

yes

—

C₂×Q₁₆×C₅₉

—

C₂×Q₁₆×C₅₉

—

C₂×Q₁₆×C₅₉

—

C₂×Q₁₆×C₅₉

C₄

yes

complemented

C₂²×C₈×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

C₂×C₄

yes

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

Q₁₆×C₅₉

—

Q₁₆×C₅₉

—

C₂×C₈×C₅₉

C₂×C₄

yes

—

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

D₄

yes

—

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

D₄

yes

—

C₄×Dic₅₉

—

—

—

—

D₄

yes

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

#41

yes

characteristic

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

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

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

characteristic

C₂×Dic₅₉

—

C₈×C₅₉

C₂²⋊C₄

yes

—

C₈×C₅₉

C₂²⋊C₄

yes

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

C₄×D₄

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₄×C₅₉

—

C₄×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₉

C₄×D₄

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

#1979

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#390

yes

characteristic

C₂×C₅₉

#473

yes

—

C₂²×Q₁₆

D₅₉

yes

—

—

—

—

C₅₉

#1979

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₅₉

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

Dic₅₉

yes

complemented

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₅₉

yes

characteristic

#117

—

#108

—

118

—

#103

—

118

—

#101

—

118

—

#97

—

118

—

C₄×C₈

—

118

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

C₂×Dic₅₉

yes

—

103

C₂²×C₄

—

104

C₂²×C₄

C₂²×D₅₉

yes

characteristic

105

Q₁₆

—

106

Q₁₆

—

107

M₁₆

—

108

M₁₆

—

118

—

109

M₁₆

—

118

—

110

M₁₆

—

118

—

111

C₂×C₈

—

118

—

112

C₂×C₈

—

118

—

113

C₂×C₈

C₂×Dic₅₉

yes

—

114

C₂×C₈

C₂²⋊D₅₉

yes

characteristic

115

C₂×C₈

C₂²⋊D₅₉

yes

characteristic

116

C₂×C₈

C₂²⋊D₅₉

yes

—

117

C₄⋊C₄

—

118

C₂²⋊C₄

—

118

—

119

C₄²

—

118

—

120

C₂³

#11745687

yes

characteristic

121

Q₈

—

122

Q₈

—

123

Q₈

—

124

Q₈

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

118

—

129

C₂×C₄

D₄×D₅₉

yes

characteristic

130

C₂×C₄

D₄×D₅₉

yes

characteristic

131

C₂×C₄

#11745687

yes

characteristic

132

C₂×C₄

C₂²×Dic₅₉

yes

characteristic

133

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

134

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

135

C₈

—

118

—

136

C₈

—

236

—

137

C₈

C₂²⋊Dic₅₉

yes

—

138

C₈

C₂²⋊Dic₅₉

yes

—

139

C₂²

—

140

C₂²

D₄×Dic₅₉

yes

characteristic

141

C₂²

#14273698

yes

characteristic

142

C₂²

#14273714

yes

characteristic

143

C₄

—

144

C₄

—

236

—

145

C₄

C₂×C₂²⋊Dic₅₉

yes

characteristic

146

C₄

D₄×Dic₅₉

yes

characteristic

147

C₄

#14273698

yes

characteristic

148

C₄

#14273723

yes

characteristic

149

C₂

—

150

C₂

—

151

C₂

#16837305

yes

characteristic

152

C₂

#16837674

yes

—

	subgroup	G/H
center	C₂²	#14273698
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₂²×Q₁₆×C₅₉	C₂
socle	C₂²×C₅₉	#121
hypercenter	C₂²×Q₁₆	D₅₉
hypocenter	C₅₉	#1979
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

subgroup

G/H

center

C₂²

#14273698

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₂²×Q₁₆×C₅₉

C₂

socle

C₂²×C₅₉

#121

hypercenter

C₂²×Q₁₆

D₅₉

hypocenter

C₅₉

#1979

LCS

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
18. C₂×Q₁₆×C₅₉	C₄	semidirect
82. C₅₉	#1979	semidirect
87. C₂×Q₁₆	Dic₅₉	semidirect

H ≅ G/N

type

18. C₂×Q₁₆×C₅₉

C₄

semidirect

82. C₅₉

#1979

semidirect

87. C₂×Q₁₆

Dic₅₉

semidirect

Subgroups — #21425012

Conjugacy classes of subgroups

Complements