Group #21424648 — subgroups

Subgroups — #21424648

Order 7552.

308 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 89 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	#16837973	C₂	1	1	yes	maximal, characteristic
2	#16837386	C₂	1	1	yes	maximal, characteristic
3	C₂×#14273645	C₂	1	2	yes	maximal, complemented
4	#16837174	C₂	1	2	yes	maximal, complemented
5	C₅₉×#568	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₁₈	C₂²	1	1	yes	characteristic
7	C₂×#11745664	C₂²	1	2	yes	—
8	#14273725	—	2	1	no	—
9	#14273714	—	2	1	no	—
10	#14273708	C₄	1	4	yes	—
11	#14273706	—	2	2	no	—
12	#14273686	C₂²	1	1	yes	characteristic
13	#14273646	—	2	1	no	—
14	#14273645	—	2	4	no	—
15	#14273644	—	2	1	no	—
16	C₂×#41×C₅₉	C₂²	1	1	yes	characteristic
17	#131×C₅₉	—	2	4	no	—
18	C₂×M₁₆×C₅₉	—	2	2	no	—
19	C₂×M₁₆×C₅₉	C₂²	1	2	yes	—
20	#41×C₅₉	—	2	2	no	—
21	#41×C₅₉	—	2	2	no	—
22	C₂×Q₈×C₅₉	D₄	1	1	yes	characteristic
23	C₂×D₄×C₅₉	C₂×C₄	1	2	yes	—
24	C₂×D₄×C₅₉	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₉	D₄	1	2	yes	—
26	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
27	C₂²×Dic₅₉	C₂×C₄	1	2	yes	—
28	C₂×Dic₁₁₈	—	2	2	no	—
29	C₂×Dic₁₁₈	—	2	1	no	—
30	C₂×Dic₁₁₈	—	2	1	no	—
31	C₂×Dic₁₁₈	—	2	2	no	—
32	M₁₆×C₅₉	—	2	4	no	—
33	M₁₆×C₅₉	—	2	2	no	—
34	M₁₆×C₅₉	D₄	1	4	yes	—
35	C₂×C₈×C₅₉	—	2	4	no	—
36	C₂×C₈×C₅₉	—	2	2	no	—
37	C₂²⋊Dic₅₉	—	2	4	no	—
38	C₂²⋊Dic₅₉	C₂×C₄	1	2	yes	—
39	C₄⋊Dic₅₉	—	2	1	no	—
40	Dic₅₉⋊C₄	—	2	4	no	—
41	C₄×Dic₅₉	—	2	1	no	—
42	#11745664	—	2	2	no	—
43	#11745664	—	2	2	no	—
44	C₂×C₅₉⋊C₈	—	2	2	no	—
45	C₂³×C₅₉	C₂²⋊C₄	1	2	yes	—
46	C₂³×C₅₉	C₂²×C₄	1	1	yes	characteristic
47	Q₈×C₅₉	—	2	2	no	—
48	D₄×C₅₉	—	2	4	no	—
49	D₄×C₅₉	—	2	2	no	—
50	C₂×C₄×C₅₉	—	2	2	no	—
51	C₂×C₄×C₅₉	—	2	2	no	—
52	C₂×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
53	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₅₉	#41	1	2	yes	—
57	C₂×Dic₅₉	—	2	2	no	—
58	C₂×Dic₅₉	—	2	2	no	—
59	C₂×Dic₅₉	—	2	4	no	—
60	Dic₁₁₈	—	4	1	no	—
61	Dic₁₁₈	—	4	2	no	—
62	Dic₁₁₈	—	4	1	no	—
63	C₈×C₅₉	—	2	4	no	—
64	C₅₉⋊C₈	—	4	2	no	—
65	C₂²×C₅₉	—	2	2	no	—
66	C₂²×C₅₉	—	4	1	no	—
67	C₂²×C₅₉	—	4	2	no	—
68	C₂²×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₅₉	C₄×D₄	1	2	yes	—
70	C₄×C₅₉	—	2	2	no	—
71	C₄×C₅₉	#120	1	1	yes	characteristic
72	C₄×C₅₉	#121	1	2	yes	—
73	C₄×C₅₉	#123	1	1	yes	characteristic
74	Dic₅₉	—	4	2	no	—
75	Dic₅₉	—	4	2	no	—
76	#1915	—	59	1	no	maximal, Sylow
77	C₂×C₅₉	—	2	1	no	—
78	C₂×C₅₉	—	2	2	no	—
79	C₂×C₅₉	—	4	2	no	—
80	C₂×C₅₉	#386	1	1	yes	characteristic
81	#568	D₅₉	1	1	yes	characteristic
82	#536	—	59	1	no	—
83	#413	—	59	1	no	—
84	C₂×#101	—	59	2	no	—
85	#344	—	59	2	no	—
86	C₅₉	#1915	1	1	yes	characteristic, complemented, Sylow
87	C₂×#41	C₂×D₅₉	1	1	yes	characteristic
88	C₂²×Q₈	—	59	1	no	—
89	#131	—	2	4	no	—
90	C₂×M₁₆	—	2	2	no	—
91	C₂×M₁₆	—	59	2	no	—
92	C₂×M₁₆	C₂×D₅₉	1	2	yes	—
93	C₄⋊Q₈	—	118	1	no	—
94	#124	—	118	1	no	—
95	#123	—	59	4	no	—
96	C₂²⋊Q₈	—	118	2	no	—
97	#117	—	59	1	no	—
98	#101	—	118	1	no	—
99	#101	—	118	4	no	—
100	#100	—	118	1	no	—
101	#41	—	2	2	no	—
102	#41	—	2	2	no	—
103	C₂×Q₈	—	118	1	no	—
104	C₂×Q₈	—	118	2	no	—
105	C₂×Q₈	—	118	2	no	—
106	C₂×Q₈	—	118	1	no	—
107	C₂×Q₈	D₂₃₆	1	1	yes	characteristic
108	C₂×D₄	C₄×D₅₉	1	2	yes	—
109	C₂×D₄	D₂₃₆	1	1	yes	characteristic
110	C₂²×C₄	—	59	2	no	—
111	C₂²×C₄	C₂²⋊D₅₉	1	2	yes	—
112	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
113	M₁₆	—	2	4	no	—
114	M₁₆	—	2	2	no	—
115	M₁₆	—	118	2	no	—
116	M₁₆	—	118	2	no	—
117	M₁₆	C₂²⋊D₅₉	1	4	yes	—
118	C₂×C₈	—	2	4	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	118	2	no	—
121	C₄⋊C₄	—	118	4	no	—
122	C₄⋊C₄	—	118	1	no	—
123	C₂²⋊C₄	—	59	2	no	—
124	C₂²⋊C₄	—	118	4	no	—
125	C₄²	—	118	1	no	—
126	C₂³	(C₂×D₅₉)⋊C₄	1	2	yes	—
127	C₂³	C₂×C₄×D₅₉	1	1	yes	characteristic
128	Q₈	—	2	2	no	—
129	Q₈	—	236	1	no	—
130	Q₈	—	236	2	no	—
131	Q₈	—	236	1	no	—
132	D₄	—	2	4	no	—
133	D₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	118	2	no	—
137	C₂×C₄	—	118	2	no	—
138	C₂×C₄	—	118	4	no	—
139	C₂×C₄	(C₂×D₅₉)⋊C₄	1	2	yes	—
140	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
141	C₂×C₄	#11745685	1	2	yes	—
142	C₂×C₄	C₂×C₂²⋊D₅₉	1	2	yes	—
143	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
144	C₈	—	2	4	no	—
145	C₈	—	236	2	no	—
146	C₂²	—	2	2	no	—
147	C₂²	—	4	1	no	—
148	C₂²	—	4	2	no	—
149	C₂²	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
150	C₂²	C₄×C₂²⋊D₅₉	1	2	yes	—
151	C₄	—	2	2	no	—
152	C₄	—	236	2	no	—
153	C₄	—	236	2	no	—
154	C₄	#14273696	1	1	yes	characteristic
155	C₄	#14273701	1	2	yes	—
156	C₄	#14273710	1	1	yes	characteristic
157	C₂	—	2	1	no	—
158	C₂	—	2	2	no	—
159	C₂	—	4	2	no	—
160	C₂	#16837281	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837973

C₂

yes

maximal, characteristic

#16837386

C₂

yes

maximal, characteristic

C₂×#14273645

C₂

yes

maximal, complemented

#16837174

C₂

yes

maximal, complemented

C₅₉×#568

C₂

yes

maximal, characteristic

C₂²×Dic₁₁₈

C₂²

yes

characteristic

C₂×#11745664

C₂²

yes

—

#14273725

—

#14273714

—

#14273708

C₄

yes

—

#14273706

—

#14273686

C₂²

yes

—

—

—

C₂×#41×C₅₉

C₂²

yes

characteristic

#131×C₅₉

—

C₂×M₁₆×C₅₉

—

C₂×M₁₆×C₅₉

C₂²

yes

—

#41×C₅₉

—

#41×C₅₉

—

C₂×Q₈×C₅₉

D₄

yes

characteristic

C₂×D₄×C₅₉

C₂×C₄

yes

—

C₂×D₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

C₂×C₄

yes

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

M₁₆×C₅₉

—

M₁₆×C₅₉

—

M₁₆×C₅₉

D₄

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂²⋊Dic₅₉

—

C₂²⋊Dic₅₉

C₂×C₄

yes

—

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

C₂²⋊C₄

yes

—

C₂³×C₅₉

C₂²×C₄

yes

characteristic

Q₈×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²⋊C₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₉

C₄×D₄

yes

—

C₄×C₅₉

—

C₄×C₅₉

#120

yes

characteristic

C₄×C₅₉

#121

yes

—

C₄×C₅₉

#123

yes

characteristic

Dic₅₉

—

Dic₅₉

—

#1915

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#386

yes

characteristic

#568

D₅₉

yes

—

—

—

—

C₅₉

#1915

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₅₉

yes

characteristic

C₂²×Q₈

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₉

yes

—

C₄⋊Q₈

—

118

—

#124

—

118

—

#123

—

C₂²⋊Q₈

—

118

—

#117

—

#101

—

118

—

#101

—

118

—

100

#100

—

118

—

101

#41

—

102

#41

—

103

C₂×Q₈

—

118

—

104

C₂×Q₈

—

118

—

105

C₂×Q₈

—

118

—

106

C₂×Q₈

—

118

—

107

C₂×Q₈

D₂₃₆

yes

characteristic

108

C₂×D₄

C₄×D₅₉

yes

—

109

C₂×D₄

D₂₃₆

yes

characteristic

110

C₂²×C₄

—

111

C₂²×C₄

C₂²⋊D₅₉

yes

—

112

C₂²×C₄

C₂²×D₅₉

yes

characteristic

113

M₁₆

—

114

M₁₆

—

115

M₁₆

—

118

—

116

M₁₆

—

118

—

117

M₁₆

C₂²⋊D₅₉

yes

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

118

—

121

C₄⋊C₄

—

118

—

122

C₄⋊C₄

—

118

—

123

C₂²⋊C₄

—

124

C₂²⋊C₄

—

118

—

125

C₄²

—

118

—

126

C₂³

(C₂×D₅₉)⋊C₄

yes

—

127

C₂³

C₂×C₄×D₅₉

yes

characteristic

128

Q₈

—

129

Q₈

—

236

—

130

Q₈

—

236

—

131

Q₈

—

236

—

132

D₄

—

133

D₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

118

—

137

C₂×C₄

—

118

—

138

C₂×C₄

—

118

—

139

C₂×C₄

(C₂×D₅₉)⋊C₄

yes

—

140

C₂×C₄

C₂×D₂₃₆

yes

characteristic

141

C₂×C₄

#11745685

yes

—

142

C₂×C₄

C₂×C₂²⋊D₅₉

yes

—

143

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

144

C₈

—

145

C₈

—

236

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

C₂×(C₂×D₅₉)⋊C₄

yes

characteristic

150

C₂²

C₄×C₂²⋊D₅₉

yes

—

151

C₄

—

152

C₄

—

236

—

153

C₄

—

236

—

154

C₄

#14273696

yes

characteristic

155

C₄

#14273701

yes

—

156

C₄

#14273710

yes

characteristic

157

C₂

—

158

C₂

—

159

C₂

—

160

C₂

#16837281

yes

characteristic

	subgroup	G/H
center	C₂	#16837281
commutator	C₂³×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₅₉×#568	C₂
socle	C₂×C₅₉	#386
hypercenter	#568	D₅₉
hypocenter	C₅₉	#1915
LCS	G ▷ C₂³×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #568

subgroup

G/H

center

C₂

#16837281

commutator

C₂³×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₅₉×#568

C₂

C₂×C₅₉

D₅₉

C₅₉

G ▷ C₂³×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #568

N	H ≅ G/N	type
3. C₂×#14273645	C₂	semidirect
4. #16837174	C₂	semidirect
86. C₅₉	#1915	semidirect

H ≅ G/N

type

3. C₂×#14273645

C₂

semidirect

4. #16837174

C₂

semidirect

86. C₅₉

#1915

semidirect

Subgroups — #21424648

Conjugacy classes of subgroups

Complements