Group #21433695 — subgroups

Subgroups — C₂×#16837657 #21433695

Order 7552.

492 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 157 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×#14273703	C₂	1	2	yes	maximal
3	#16837657	C₂	1	8	yes	maximal, direct product factor
4	C₂×#14273659	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14273651	C₂	1	2	yes	maximal, complemented
6	C₂²×Q₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₄×D₅₉	C₂²	1	1	yes	characteristic
8	C₂×#11745659	—	2	4	no	—
9	C₂×#11745659	C₂²	1	4	yes	—
10	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
11	#14273703	—	2	4	no	—
12	#14273703	C₂²	1	8	yes	—
13	C₂×C₄⋊Dic₅₉	C₂²	1	2	yes	—
14	C₂×Dic₅₉⋊C₄	—	2	2	no	—
15	C₂×(C₂×D₅₉)⋊C₄	—	2	2	no	—
16	#14273659	C₂²	1	4	yes	complemented
17	#14273651	C₂²	1	8	yes	complemented
18	C₂²×Q₈×C₅₉	C₂²	1	2	yes	—
19	C₂×Q₁₆×C₅₉	—	2	4	no	—
20	C₂×Q₁₆×C₅₉	C₂²	1	4	yes	complemented
21	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
22	C₂³×D₅₉	D₄	1	1	yes	characteristic
23	C₂×Q₈×C₅₉	—	2	2	no	—
24	C₂×Q₈×C₅₉	—	2	4	no	—
25	C₂×Q₈×C₅₉	—	2	4	no	—
26	C₂×Q₈×C₅₉	C₂³	1	4	yes	—
27	C₂²×C₄×C₅₉	—	2	2	no	—
28	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
29	C₂²×Dic₅₉	—	2	2	no	—
30	C₂²×Dic₅₉	D₄	1	1	yes	characteristic
31	C₂×C₄×D₅₉	—	2	2	no	—
32	C₂×C₄×D₅₉	—	2	2	no	—
33	C₂×C₄×D₅₉	D₄	1	2	yes	—
34	C₂×C₄×D₅₉	C₂³	1	2	yes	—
35	Q₁₆×C₅₉	—	2	8	no	—
36	C₂×C₈×C₅₉	D₄	1	4	yes	—
37	C₂×C₈×C₅₉	C₂³	1	2	yes	—
38	(C₂×D₅₉)⋊C₄	—	2	8	no	—
39	C₄⋊Dic₅₉	—	2	2	no	—
40	C₄⋊Dic₅₉	C₂³	1	4	yes	—
41	Dic₅₉⋊C₄	—	2	8	no	—
42	C₂×C₅₉⋊C₈	—	2	2	no	—
43	C₂×C₅₉⋊C₈	C₂³	1	2	yes	—
44	#11745659	—	2	8	no	—
45	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
46	C₂²×D₅₉	—	2	2	no	—
47	C₂²×D₅₉	—	2	4	no	—
48	C₂²×D₅₉	C₂×D₄	1	2	yes	—
49	Q₈×C₅₉	—	2	4	no	—
50	Q₈×C₅₉	—	2	4	no	—
51	Q₈×C₅₉	—	4	4	no	—
52	C₂×C₄×C₅₉	—	2	4	no	—
53	C₂×C₄×C₅₉	—	4	4	no	—
54	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₅₉	#41	1	2	yes	—
57	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
58	C₂×Dic₅₉	—	2	2	no	—
59	C₂×Dic₅₉	—	2	4	no	—
60	C₂×Dic₅₉	—	4	4	no	—
61	C₂×Dic₅₉	C₂×D₄	1	2	yes	—
62	C₄×D₅₉	—	2	4	no	—
63	C₄×D₅₉	—	2	2	no	—
64	C₄×D₅₉	—	2	2	no	—
65	C₈×C₅₉	C₂×D₄	1	4	yes	—
66	C₅₉⋊C₈	—	2	2	no	—
67	C₂²×C₅₉	#121	1	2	yes	—
68	C₂²×C₅₉	#137	1	4	yes	—
69	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
70	C₂×D₅₉	—	2	2	no	—
71	C₂×D₅₉	—	2	4	no	—
72	C₂×D₅₉	—	4	4	no	—
73	C₄×C₅₉	—	4	4	no	—
74	C₄×C₅₉	#121	1	2	yes	—
75	C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
76	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
77	Dic₅₉	—	2	2	no	—
78	Dic₅₉	—	4	4	no	—
79	C₂×#470	—	59	1	no	maximal, Sylow
80	C₂×C₅₉	#470	1	4	yes	complemented
81	C₂×C₅₉	C₂×#121	1	1	yes	characteristic
82	C₂×C₅₉	C₂×#137	1	2	yes	—
83	D₅₉	—	4	2	no	—
84	C₂²×Q₁₆	D₅₉	1	1	yes	characteristic, complemented
85	C₂²×M₁₆	—	59	1	no	—
86	C₂×C₂²⋊Q₈	—	59	2	no	—
87	#470	—	59	8	no	—
88	C₂×#106	—	59	1	no	—
89	C₂×#103	—	59	2	no	—
90	C₅₉	C₂×#470	1	1	yes	characteristic, complemented, Sylow
91	C₂²×Q₈	C₂×D₅₉	1	2	yes	—
92	C₂³×C₄	—	59	1	no	—
93	C₂×Q₁₆	—	2	4	no	—
94	C₂×Q₁₆	C₂×D₅₉	1	4	yes	complemented
95	C₂×M₁₆	—	59	4	no	—
96	C₂×M₁₆	—	118	4	no	—
97	C₂²×C₈	—	59	1	no	—
98	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
99	C₂²⋊Q₈	—	59	8	no	—
100	C₂²⋊Q₈	—	118	4	no	—
101	C₂×C₄⋊C₄	—	59	2	no	—
102	C₂×C₄⋊C₄	—	118	2	no	—
103	C₂×C₂²⋊C₄	—	118	2	no	—
104	#106	—	59	4	no	—
105	#103	—	59	8	no	—
106	C₂⁴	—	59	1	no	—
107	C₂×Q₈	—	2	2	no	—
108	C₂×Q₈	—	2	4	no	—
109	C₂×Q₈	—	2	4	no	—
110	C₂×Q₈	C₂²×D₅₉	1	4	yes	—
111	C₂²×C₄	—	2	2	no	—
112	C₂²×C₄	—	59	1	no	—
113	C₂²×C₄	—	59	2	no	—
114	C₂²×C₄	—	59	2	no	—
115	C₂²×C₄	—	118	2	no	—
116	C₂²×C₄	—	118	2	no	—
117	C₂²×C₄	—	118	2	no	—
118	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
119	Q₁₆	—	2	8	no	—
120	M₁₆	—	118	8	no	—
121	C₂×C₈	—	59	2	no	—
122	C₂×C₈	—	118	2	no	—
123	C₂×C₈	C₂²⋊D₅₉	1	4	yes	—
124	C₂×C₈	C₂²×D₅₉	1	2	yes	—
125	C₄⋊C₄	—	59	4	no	—
126	C₄⋊C₄	—	118	8	no	—
127	C₄⋊C₄	—	118	2	no	—
128	C₂²⋊C₄	—	118	8	no	—
129	C₂³	—	59	2	no	—
130	C₂³	—	118	2	no	—
131	C₂³	—	118	4	no	—
132	C₂³	D₄×D₅₉	1	1	yes	characteristic
133	Q₈	—	2	4	no	—
134	Q₈	—	2	4	no	—
135	Q₈	—	4	4	no	—
136	C₂×C₄	—	2	4	no	—
137	C₂×C₄	—	4	4	no	—
138	C₂×C₄	—	59	2	no	—
139	C₂×C₄	—	118	2	no	—
140	C₂×C₄	—	118	4	no	—
141	C₂×C₄	—	118	2	no	—
142	C₂×C₄	—	118	2	no	—
143	C₂×C₄	—	118	4	no	—
144	C₂×C₄	—	236	4	no	—
145	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
146	C₂×C₄	#11745687	1	2	yes	—
147	C₂×C₄	C₂×C₂²⋊D₅₉	1	2	yes	—
148	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
149	C₈	—	118	2	no	—
150	C₈	C₂×C₂²⋊D₅₉	1	4	yes	—
151	C₂²	—	118	2	no	—
152	C₂²	—	118	4	no	—
153	C₂²	—	236	4	no	—
154	C₂²	#14273698	1	2	yes	—
155	C₂²	#14273759	1	4	yes	—
156	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
157	C₄	—	4	4	no	—
158	C₄	—	118	2	no	—
159	C₄	—	236	4	no	—
160	C₄	#14273698	1	2	yes	—
161	C₄	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
162	C₄	C₂×#11745687	1	1	yes	characteristic
163	C₂	—	236	2	no	—
164	C₂	#16837657	1	4	yes	direct product factor
165	C₂	C₂×#14273698	1	1	yes	characteristic
166	C₂	C₂×#14273759	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11745659

C₂

yes

maximal, characteristic

C₂×#14273703

C₂

yes

maximal

#16837657

C₂

yes

maximal, direct product factor

C₂×#14273659

C₂

yes

maximal, characteristic, complemented

C₂×#14273651

C₂

yes

maximal, complemented

C₂²×Q₁₆×C₅₉

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₅₉

C₂²

yes

characteristic

C₂×#11745659

—

C₂×#11745659

C₂²

yes

—

C₂²×C₅₉⋊C₈

C₂²

yes

characteristic

#14273703

—

#14273703

C₂²

yes

—

C₂×C₄⋊Dic₅₉

C₂²

yes

—

C₂×Dic₅₉⋊C₄

—

C₂×(C₂×D₅₉)⋊C₄

—

#14273659

C₂²

yes

complemented

#14273651

C₂²

yes

complemented

C₂²×Q₈×C₅₉

C₂²

yes

—

C₂×Q₁₆×C₅₉

—

C₂×Q₁₆×C₅₉

C₂²

yes

complemented

C₂²×C₈×C₅₉

C₂²

yes

characteristic

C₂³×D₅₉

D₄

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

C₂³

yes

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₂²×Dic₅₉

D₄

yes

characteristic

C₂×C₄×D₅₉

—

C₂×C₄×D₅₉

—

C₂×C₄×D₅₉

D₄

yes

—

C₂×C₄×D₅₉

C₂³

yes

—

Q₁₆×C₅₉

—

C₂×C₈×C₅₉

D₄

yes

—

C₂×C₈×C₅₉

C₂³

yes

—

(C₂×D₅₉)⋊C₄

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

C₂³

yes

—

Dic₅₉⋊C₄

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₂³

yes

—

#11745659

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

C₂²×D₅₉

—

C₂²×D₅₉

—

C₂²×D₅₉

C₂×D₄

yes

—

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

#41

yes

—

C₂×C₄×C₅₉

C₂⁴

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

C₂×D₄

yes

—

C₄×D₅₉

—

C₄×D₅₉

—

C₄×D₅₉

—

C₈×C₅₉

C₂×D₄

yes

—

C₅₉⋊C₈

—

C₂²×C₅₉

#121

yes

—

C₂²×C₅₉

#137

yes

—

C₂²×C₅₉

C₂²×D₄

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#121

yes

—

C₄×C₅₉

C₂²×D₄

yes

characteristic

C₄×C₅₉

C₂×#41

yes

characteristic

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

D₅₉

—

C₂²×Q₁₆

D₅₉

yes

characteristic, complemented

C₂²×M₁₆

—

C₂×C₂²⋊Q₈

—

#470

—

C₂×#106

—

C₂×#103

—

C₅₉

C₂×#470

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₅₉

yes

—

C₂³×C₄

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂×D₅₉

yes

complemented

C₂×M₁₆

—

C₂×M₁₆

—

118

—

C₂²×C₈

—

C₂²×C₈

C₂×D₅₉

yes

characteristic

C₂²⋊Q₈

—

100

C₂²⋊Q₈

—

118

—

101

C₂×C₄⋊C₄

—

102

C₂×C₄⋊C₄

—

118

—

103

C₂×C₂²⋊C₄

—

118

—

104

#106

—

105

#103

—

106

C₂⁴

—

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂×Q₈

C₂²×D₅₉

yes

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

118

—

116

C₂²×C₄

—

118

—

117

C₂²×C₄

—

118

—

118

C₂²×C₄

C₂²×D₅₉

yes

characteristic

119

Q₁₆

—

120

M₁₆

—

118

—

121

C₂×C₈

—

122

C₂×C₈

—

118

—

123

C₂×C₈

C₂²⋊D₅₉

yes

—

124

C₂×C₈

C₂²×D₅₉

yes

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

118

—

127

C₄⋊C₄

—

118

—

128

C₂²⋊C₄

—

118

—

129

C₂³

—

130

C₂³

—

118

—

131

C₂³

—

118

—

132

C₂³

D₄×D₅₉

yes

characteristic

133

Q₈

—

134

Q₈

—

135

Q₈

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

118

—

140

C₂×C₄

—

118

—

141

C₂×C₄

—

118

—

142

C₂×C₄

—

118

—

143

C₂×C₄

—

118

—

144

C₂×C₄

—

236

—

145

C₂×C₄

D₄×D₅₉

yes

characteristic

146

C₂×C₄

#11745687

yes

—

147

C₂×C₄

C₂×C₂²⋊D₅₉

yes

—

148

C₂×C₄

C₂³×D₅₉

yes

characteristic

149

C₈

—

118

—

150

C₈

C₂×C₂²⋊D₅₉

yes

—

151

C₂²

—

118

—

152

C₂²

—

118

—

153

C₂²

—

236

—

154

C₂²

#14273698

yes

—

155

C₂²

#14273759

yes

—

156

C₂²

C₂×D₄×D₅₉

yes

characteristic

157

C₄

—

158

C₄

—

118

—

159

C₄

—

236

—

160

C₄

#14273698

yes

—

161

C₄

C₂²×C₂²⋊D₅₉

yes

characteristic

162

C₄

C₂×#11745687

yes

characteristic

163

C₂

—

236

—

164

C₂

#16837657

yes

direct product factor

165

C₂

C₂×#14273698

yes

characteristic

166

C₂

C₂×#14273759

yes

—

	subgroup	G/H
center	C₂³	D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₂²×Q₁₆×C₅₉	C₂
socle	C₂³×C₅₉	C₂×D₄
hypercenter	C₂²×Q₁₆	D₅₉
hypocenter	C₅₉	C₂×#470
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

subgroup

G/H

center

C₂³

D₄×D₅₉

commutator

C₂×C₄×C₅₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₉

Fitting

C₂²×Q₁₆×C₅₉

C₂

socle

C₂³×C₅₉

C₂×D₄

hypercenter

C₂²×Q₁₆

D₅₉

hypocenter

C₅₉

C₂×#470

LCS

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
3. #16837657	C₂	direct (N × H)
4. C₂×#14273659	C₂	semidirect
5. C₂×#14273651	C₂	semidirect
6. C₂²×Q₁₆×C₅₉	C₂	semidirect
16. #14273659	C₂²	semidirect
17. #14273651	C₂²	semidirect
20. C₂×Q₁₆×C₅₉	C₂²	semidirect
80. C₂×C₅₉	#470	semidirect
84. C₂²×Q₁₆	D₅₉	semidirect
90. C₅₉	C₂×#470	semidirect
94. C₂×Q₁₆	C₂×D₅₉	semidirect
164. C₂	#16837657	direct (N × H)

H ≅ G/N

type

3. #16837657

C₂

direct (N × H)

4. C₂×#14273659

C₂

semidirect

5. C₂×#14273651

C₂

semidirect

6. C₂²×Q₁₆×C₅₉

C₂

semidirect

16. #14273659

C₂²

semidirect

17. #14273651

C₂²

semidirect

20. C₂×Q₁₆×C₅₉

C₂²

semidirect

80. C₂×C₅₉

#470

semidirect

84. C₂²×Q₁₆

D₅₉

semidirect

90. C₅₉

C₂×#470

semidirect

94. C₂×Q₁₆

C₂×D₅₉

semidirect

164. C₂

#16837657

direct (N × H)

Subgroups — C₂×#16837657 #21433695

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16837657 #21433695

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16837657 #21433695