Group #21424619 — subgroups

Subgroups — #21424619

Order 7552.

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

normal subgroups only

This group has 312 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	#16838169	C₂	1	1	yes	maximal, characteristic
2	#16837411	C₂	1	2	yes	maximal, complemented
3	#16837376	C₂	1	1	yes	maximal, characteristic
4	#16837158	C₂	1	2	yes	maximal, complemented
5	C₅₉×#536	C₂	1	1	yes	maximal, characteristic
6	#14273734	—	2	4	no	—
7	C₂×#11745664	—	2	2	no	—
8	C₂×#11745664	C₂²	1	2	yes	—
9	#14273686	C₂²	1	2	yes	—
10	#14273651	—	2	4	no	—
11	#14273642	—	2	2	no	—
12	C₂×#41×C₅₉	C₂²	1	1	yes	characteristic
13	C₂²×Q₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₄⋊Q₈×C₅₉	C₄	1	2	yes	complemented
15	#124×C₅₉	C₄	1	2	yes	complemented
16	#123×C₅₉	—	2	2	no	—
17	C₂²⋊Q₈×C₅₉	—	2	2	no	—
18	#117×C₅₉	C₂²	1	1	yes	characteristic
19	#41×C₅₉	—	2	2	no	—
20	#41×C₅₉	—	2	2	no	—
21	C₂×Q₈×C₅₉	—	2	1	no	—
22	C₂×Q₈×C₅₉	—	2	2	no	—
23	C₂×Q₈×C₅₉	—	2	2	no	—
24	C₂×Q₈×C₅₉	C₂×C₄	1	2	yes	—
25	C₂×Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
26	C₂×D₄×C₅₉	C₂×C₄	1	1	yes	characteristic
27	C₂×D₄×C₅₉	D₄	1	2	yes	—
28	C₂²×C₄×C₅₉	—	2	1	no	—
29	C₂²×C₄×C₅₉	D₄	1	2	yes	—
30	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
31	C₄⋊C₄×C₅₉	—	2	4	no	—
32	C₄⋊C₄×C₅₉	—	2	1	no	—
33	C₂²⋊C₄×C₅₉	—	2	4	no	—
34	C₂²⋊C₄×C₅₉	—	2	1	no	—
35	C₄²×C₅₉	C₂×C₄	1	2	yes	—
36	C₂²⋊Dic₅₉	—	2	2	no	—
37	C₄⋊Dic₅₉	—	2	2	no	—
38	C₄×Dic₅₉	—	2	2	no	—
39	#11745664	—	2	4	no	—
40	#11745664	—	2	2	no	—
41	#11745664	—	2	2	no	—
42	C₂×C₅₉⋊C₈	—	2	4	no	—
43	C₂×C₅₉⋊C₈	D₄	1	4	yes	—
44	C₂³×C₅₉	C₂²⋊C₄	1	2	yes	—
45	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
46	Q₈×C₅₉	—	2	2	no	—
47	Q₈×C₅₉	—	2	4	no	—
48	Q₈×C₅₉	—	4	1	no	—
49	Q₈×C₅₉	—	4	1	no	—
50	D₄×C₅₉	—	2	4	no	—
51	D₄×C₅₉	—	2	2	no	—
52	C₂×C₄×C₅₉	—	2	2	no	—
53	C₂×C₄×C₅₉	—	2	2	no	—
54	C₂×C₄×C₅₉	—	2	2	no	—
55	C₂×C₄×C₅₉	—	2	2	no	—
56	C₂×C₄×C₅₉	—	4	2	no	—
57	C₂×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
58	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₅₉	#41	1	2	yes	—
62	C₂×Dic₅₉	—	4	2	no	—
63	C₅₉⋊C₈	—	2	4	no	—
64	C₂²×C₅₉	—	2	2	no	—
65	C₂²×C₅₉	—	4	1	no	—
66	C₂²×C₅₉	—	4	2	no	—
67	C₂²×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₅₉	#121	1	2	yes	—
69	C₄×C₅₉	—	2	2	no	—
70	C₄×C₅₉	—	4	2	no	—
71	C₄×C₅₉	—	4	2	no	—
72	C₄×C₅₉	C₄×D₄	1	2	yes	—
73	C₄×C₅₉	#120	1	1	yes	characteristic
74	C₄×C₅₉	#123	1	1	yes	characteristic
75	Dic₅₉	—	8	2	no	—
76	#1909	—	59	1	no	maximal, Sylow
77	C₂×C₅₉	—	2	2	no	—
78	C₂×C₅₉	—	2	1	no	—
79	C₂×C₅₉	—	4	2	no	—
80	C₂×C₅₉	#386	1	1	yes	characteristic
81	#568	—	59	1	no	—
82	#536	D₅₉	1	1	yes	characteristic
83	#419	—	59	2	no	—
84	#410	—	59	1	no	—
85	H(3,ℤ/4)	—	59	2	no	—
86	C₅₉	#1909	1	1	yes	characteristic, complemented, Sylow
87	C₂×#41	C₂×D₅₉	1	1	yes	characteristic
88	C₂²×Q₈	C₂×D₅₉	1	1	yes	characteristic
89	#131	—	118	4	no	—
90	C₂×M₁₆	—	59	2	no	—
91	C₂×M₁₆	—	118	2	no	—
92	C₄⋊Q₈	Dic₅₉	1	2	yes	complemented
93	#124	Dic₅₉	1	2	yes	complemented
94	#123	—	2	2	no	—
95	C₂²⋊Q₈	—	2	2	no	—
96	#117	—	59	2	no	—
97	#117	C₂×D₅₉	1	1	yes	characteristic
98	#103	—	118	4	no	—
99	#99	—	118	2	no	—
100	#41	—	2	2	no	—
101	#41	—	2	2	no	—
102	C₂×Q₈	—	2	1	no	—
103	C₂×Q₈	—	2	2	no	—
104	C₂×Q₈	—	2	2	no	—
105	C₂×Q₈	C₂×Dic₅₉	1	2	yes	—
106	C₂×Q₈	C₂×Dic₅₉	1	1	yes	characteristic
107	C₂×D₄	C₂×Dic₅₉	1	1	yes	characteristic
108	C₂×D₄	C₂²⋊D₅₉	1	2	yes	—
109	C₂²×C₄	—	2	1	no	—
110	C₂²×C₄	C₂²⋊D₅₉	1	2	yes	—
111	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
112	M₁₆	—	118	4	no	—
113	M₁₆	—	118	2	no	—
114	M₁₆	—	118	2	no	—
115	C₂×C₈	—	59	4	no	—
116	C₂×C₈	—	118	4	no	—
117	C₄⋊C₄	—	2	4	no	—
118	C₄⋊C₄	—	2	1	no	—
119	C₄⋊C₄	—	118	2	no	—
120	C₂²⋊C₄	—	2	4	no	—
121	C₂²⋊C₄	—	2	1	no	—
122	C₂²⋊C₄	—	118	2	no	—
123	C₄²	—	118	2	no	—
124	C₄²	C₂×Dic₅₉	1	2	yes	—
125	C₂³	C₂²⋊Dic₅₉	1	2	yes	—
126	C₂³	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
127	Q₈	—	2	2	no	—
128	Q₈	—	2	4	no	—
129	Q₈	—	4	1	no	—
130	Q₈	—	4	1	no	—
131	D₄	—	2	4	no	—
132	D₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	4	2	no	—
138	C₂×C₄	—	236	2	no	—
139	C₂×C₄	C₂²⋊Dic₅₉	1	2	yes	—
140	C₂×C₄	D₄×D₅₉	1	2	yes	—
141	C₂×C₄	#11745687	1	2	yes	—
142	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
143	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
144	C₈	—	118	4	no	—
145	C₂²	—	2	2	no	—
146	C₂²	—	4	1	no	—
147	C₂²	—	4	2	no	—
148	C₂²	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
149	C₂²	#14273699	1	2	yes	—
150	C₄	—	2	2	no	—
151	C₄	—	4	2	no	—
152	C₄	—	4	2	no	—
153	C₄	—	472	2	no	—
154	C₄	D₄×Dic₅₉	1	2	yes	—
155	C₄	#14273695	1	1	yes	characteristic
156	C₄	#14273708	1	1	yes	characteristic
157	C₂	—	2	1	no	—
158	C₂	—	2	2	no	—
159	C₂	—	4	2	no	—
160	C₂	#16837282	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16838169

C₂

yes

maximal, characteristic

#16837411

C₂

yes

maximal, complemented

#16837376

C₂

yes

maximal, characteristic

#16837158

C₂

yes

maximal, complemented

C₅₉×#536

C₂

yes

maximal, characteristic

#14273734

—

C₂×#11745664

—

C₂×#11745664

C₂²

yes

—

#14273686

C₂²

yes

—

#14273651

—

#14273642

—

C₂×#41×C₅₉

C₂²

yes

characteristic

C₂²×Q₈×C₅₉

C₂²

yes

characteristic

C₄⋊Q₈×C₅₉

C₄

yes

complemented

#124×C₅₉

C₄

yes

complemented

#123×C₅₉

—

C₂²⋊Q₈×C₅₉

—

#117×C₅₉

C₂²

yes

characteristic

#41×C₅₉

—

#41×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

C₂×C₄

yes

—

C₂×Q₈×C₅₉

C₂×C₄

yes

characteristic

C₂×D₄×C₅₉

C₂×C₄

yes

characteristic

C₂×D₄×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₂²⋊C₄×C₅₉

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

C₂×C₄

yes

—

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄×Dic₅₉

—

#11745664

—

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

D₄

yes

—

C₂³×C₅₉

C₂²⋊C₄

yes

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²⋊C₄

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

—

C₂×Dic₅₉

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₉

#121

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₄×D₄

yes

—

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

#1909

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

yes

—

D₅₉

yes

—

—

—

C₅₉

#1909

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₅₉

yes

characteristic

C₂²×Q₈

C₂×D₅₉

yes

characteristic

#131

—

118

—

C₂×M₁₆

—

C₂×M₁₆

—

118

—

C₄⋊Q₈

Dic₅₉

yes

complemented

#124

Dic₅₉

yes

complemented

#123

—

C₂²⋊Q₈

—

#117

—

#117

C₂×D₅₉

yes

characteristic

#103

—

118

—

#99

—

118

—

100

#41

—

101

#41

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

C₂×Dic₅₉

yes

—

106

C₂×Q₈

C₂×Dic₅₉

yes

characteristic

107

C₂×D₄

C₂×Dic₅₉

yes

characteristic

108

C₂×D₄

C₂²⋊D₅₉

yes

—

109

C₂²×C₄

—

110

C₂²×C₄

C₂²⋊D₅₉

yes

—

111

C₂²×C₄

C₂²×D₅₉

yes

characteristic

112

M₁₆

—

118

—

113

M₁₆

—

118

—

114

M₁₆

—

118

—

115

C₂×C₈

—

116

C₂×C₈

—

118

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

118

—

120

C₂²⋊C₄

—

121

C₂²⋊C₄

—

122

C₂²⋊C₄

—

118

—

123

C₄²

—

118

—

124

C₄²

C₂×Dic₅₉

yes

—

125

C₂³

C₂²⋊Dic₅₉

yes

—

126

C₂³

C₂×C₂²⋊D₅₉

yes

characteristic

127

Q₈

—

128

Q₈

—

129

Q₈

—

130

Q₈

—

131

D₄

—

132

D₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

236

—

139

C₂×C₄

C₂²⋊Dic₅₉

yes

—

140

C₂×C₄

D₄×D₅₉

yes

—

141

C₂×C₄

#11745687

yes

—

142

C₂×C₄

C₂²×Dic₅₉

yes

characteristic

143

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

144

C₈

—

118

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

C₂×C₂²⋊Dic₅₉

yes

characteristic

149

C₂²

#14273699

yes

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

472

—

154

C₄

D₄×Dic₅₉

yes

—

155

C₄

#14273695

yes

characteristic

156

C₄

#14273708

yes

characteristic

157

C₂

—

158

C₂

—

159

C₂

—

160

C₂

#16837282

yes

characteristic

	subgroup	G/H
center	C₂	#16837282
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₅₉×#536	C₂
socle	C₂×C₅₉	#386
hypercenter	#536	D₅₉
hypocenter	C₅₉	#1909
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #536

subgroup

G/H

center

C₂

#16837282

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₅₉×#536

C₂

C₂×C₅₉

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #536

N	H ≅ G/N	type
2. #16837411	C₂	semidirect
4. #16837158	C₂	semidirect
14. C₄⋊Q₈×C₅₉	C₄	semidirect
15. #124×C₅₉	C₄	semidirect
86. C₅₉	#1909	semidirect
92. C₄⋊Q₈	Dic₅₉	semidirect
93. #124	Dic₅₉	semidirect

H ≅ G/N

type

2. #16837411

C₂

semidirect

4. #16837158

C₂

semidirect

14. C₄⋊Q₈×C₅₉

C₄

semidirect

15. #124×C₅₉

C₄

semidirect

86. C₅₉

#1909

semidirect

92. C₄⋊Q₈

Dic₅₉

semidirect

93. #124

Dic₅₉

semidirect

Subgroups — #21424619

Conjugacy classes of subgroups

Complements