Group #21434518 — subgroups

Subgroups — C₂×#16837787 #21434518

Order 7552.

592 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 165 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×#11745671	C₂	1	2	yes	maximal, complemented
2	C₂²×D₄₇₂	C₂	1	1	yes	maximal, characteristic
3	C₂×#14273716	C₂	1	2	yes	maximal, complemented
4	#16837787	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₈×Dic₅₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Q₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₂₃₆	C₂²	1	2	yes	—
8	C₂×#11745671	—	2	8	no	—
9	C₂×#11745671	C₂²	1	8	yes	complemented
10	C₂×D₄₇₂	—	2	4	no	—
11	C₂×D₄₇₂	C₂²	1	4	yes	—
12	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
13	#14273716	—	2	4	no	—
14	#14273716	C₂²	1	8	yes	complemented
15	C₂×(C₂×D₅₉)⋊C₄	—	2	4	no	—
16	C₂×C₄×Dic₅₉	C₂²	1	1	yes	characteristic
17	C₈×Dic₅₉	C₂²	1	4	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	—	2	2	no	—
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	D₄	1	2	yes	—
30	C₂×D₂₃₆	—	2	4	no	—
31	C₂×D₂₃₆	—	2	4	no	—
32	C₂×D₂₃₆	—	2	2	no	—
33	C₂×D₂₃₆	C₂³	1	4	yes	—
34	Q₁₆×C₅₉	—	2	8	no	—
35	C₂×C₈×C₅₉	D₄	1	4	yes	—
36	C₂×C₈×C₅₉	C₂³	1	2	yes	—
37	#11745671	—	2	16	no	—
38	(C₂×D₅₉)⋊C₄	—	2	16	no	—
39	C₄×Dic₅₉	D₄	1	2	yes	—
40	C₄×Dic₅₉	C₂³	1	2	yes	—
41	C₂×C₅₉⋊C₈	D₄	1	4	yes	—
42	C₂×C₅₉⋊C₈	C₂³	1	2	yes	—
43	D₄₇₂	—	2	8	no	—
44	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
45	C₂²×D₅₉	—	2	4	no	—
46	C₂²×D₅₉	—	4	4	no	—
47	C₂²×D₅₉	—	4	8	no	—
48	Q₈×C₅₉	—	2	4	no	—
49	Q₈×C₅₉	—	2	4	no	—
50	Q₈×C₅₉	—	4	4	no	—
51	C₂×C₄×C₅₉	—	2	4	no	—
52	C₂×C₄×C₅₉	—	4	4	no	—
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	2	yes	—
56	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
57	C₂×Dic₅₉	—	2	4	no	—
58	C₂×Dic₅₉	C₂×D₄	1	4	yes	—
59	D₂₃₆	—	2	4	no	—
60	D₂₃₆	—	2	4	no	—
61	D₂₃₆	—	4	4	no	—
62	C₈×C₅₉	C₂×D₄	1	4	yes	—
63	C₅₉⋊C₈	C₂×D₄	1	4	yes	—
64	C₂²×C₅₉	#127	1	2	yes	—
65	C₂²×C₅₉	#135	1	4	yes	—
66	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
67	C₂×D₅₉	—	4	4	no	—
68	C₂×D₅₉	—	4	8	no	—
69	C₂×D₅₉	—	8	8	no	—
70	C₄×C₅₉	—	4	4	no	—
71	C₄×C₅₉	#127	1	2	yes	—
72	C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
73	C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
74	Dic₅₉	—	2	4	no	—
75	C₂×#495	—	59	1	no	maximal, Sylow
76	C₂×C₅₉	#495	1	4	yes	complemented
77	C₂×C₅₉	C₂×#127	1	1	yes	characteristic
78	C₂×C₅₉	C₂×#135	1	2	yes	—
79	D₅₉	—	8	4	no	—
80	C₂²×Q₁₆	D₅₉	1	1	yes	characteristic, complemented
81	C₂²×SD₁₆	—	59	2	no	—
82	C₂²×D₈	—	59	1	no	—
83	C₂×#124	—	59	2	no	—
84	#495	—	59	8	no	—
85	C₂×C₄×C₈	—	59	1	no	—
86	C₅₉	C₂×#495	1	1	yes	characteristic, complemented, Sylow
87	C₂²×Q₈	C₂×D₅₉	1	2	yes	—
88	C₂²×D₄	—	59	2	no	—
89	C₂×Q₁₆	—	2	4	no	—
90	C₂×Q₁₆	C₂×D₅₉	1	4	yes	complemented
91	C₂×SD₁₆	—	59	8	no	—
92	C₂×SD₁₆	—	118	8	no	—
93	C₂×D₈	—	59	4	no	—
94	C₂×D₈	—	118	4	no	—
95	C₂²×C₈	—	59	1	no	—
96	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
97	#124	—	59	8	no	—
98	#124	—	118	4	no	—
99	C₂×C₂²⋊C₄	—	118	4	no	—
100	C₂×C₄²	—	59	1	no	—
101	C₄×C₈	—	59	4	no	—
102	C₂⁴	—	118	2	no	—
103	C₂×Q₈	—	2	2	no	—
104	C₂×Q₈	—	2	4	no	—
105	C₂×Q₈	—	2	4	no	—
106	C₂×Q₈	C₂²×D₅₉	1	4	yes	—
107	C₂×D₄	—	59	4	no	—
108	C₂×D₄	—	118	4	no	—
109	C₂×D₄	—	118	2	no	—
110	C₂×D₄	—	118	4	no	—
111	C₂²×C₄	—	2	2	no	—
112	C₂²×C₄	—	59	2	no	—
113	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
114	Q₁₆	—	2	8	no	—
115	SD₁₆	—	118	16	no	—
116	D₈	—	118	8	no	—
117	C₂×C₈	—	59	4	no	—
118	C₂×C₈	—	59	2	no	—
119	C₂×C₈	C₂²⋊D₅₉	1	4	yes	—
120	C₂×C₈	C₂²×D₅₉	1	2	yes	—
121	C₂²⋊C₄	—	118	16	no	—
122	C₄²	—	59	2	no	—
123	C₄²	—	59	2	no	—
124	C₂³	—	118	4	no	—
125	C₂³	—	236	4	no	—
126	C₂³	—	236	8	no	—
127	C₂³	D₄×D₅₉	1	1	yes	characteristic
128	Q₈	—	2	4	no	—
129	Q₈	—	2	4	no	—
130	Q₈	—	4	4	no	—
131	D₄	—	118	4	no	—
132	D₄	—	118	4	no	—
133	D₄	—	236	4	no	—
134	C₂×C₄	—	2	4	no	—
135	C₂×C₄	—	4	4	no	—
136	C₂×C₄	—	59	4	no	—
137	C₂×C₄	—	118	4	no	—
138	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
139	C₂×C₄	D₄×D₅₉	1	2	yes	—
140	C₂×C₄	C₂×C₂²⋊D₅₉	1	2	yes	—
141	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
142	C₈	—	59	4	no	—
143	C₈	C₂×C₂²⋊D₅₉	1	4	yes	—
144	C₂²	—	472	8	no	—
145	C₂²	—	236	8	no	—
146	C₂²	—	236	4	no	—
147	C₂²	#14273723	1	2	yes	—
148	C₂²	#14273750	1	4	yes	—
149	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
150	C₄	—	4	4	no	—
151	C₄	—	118	4	no	—
152	C₄	#14273723	1	2	yes	—
153	C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
154	C₄	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
155	C₂	—	472	4	no	—
156	C₂	#16837787	1	4	yes	direct product factor
157	C₂	C₂×#14273723	1	1	yes	characteristic
158	C₂	C₂×#14273750	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11745671

C₂

yes

maximal, complemented

C₂²×D₄₇₂

C₂

yes

maximal, characteristic

C₂×#14273716

C₂

yes

maximal, complemented

#16837787

C₂

yes

maximal, direct product factor

C₂×C₈×Dic₅₉

C₂

yes

maximal, characteristic, complemented

C₂²×Q₁₆×C₅₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₃₆

C₂²

yes

—

C₂×#11745671

—

C₂×#11745671

C₂²

yes

complemented

C₂×D₄₇₂

—

C₂×D₄₇₂

C₂²

yes

—

C₂²×C₅₉⋊C₈

C₂²

yes

characteristic

#14273716

—

#14273716

C₂²

yes

complemented

C₂×(C₂×D₅₉)⋊C₄

—

C₂×C₄×Dic₅₉

C₂²

yes

characteristic

C₈×Dic₅₉

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

—

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

D₄

yes

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

C₂³

yes

—

Q₁₆×C₅₉

—

C₂×C₈×C₅₉

D₄

yes

—

C₂×C₈×C₅₉

C₂³

yes

—

#11745671

—

(C₂×D₅₉)⋊C₄

—

C₄×Dic₅₉

D₄

yes

—

C₄×Dic₅₉

C₂³

yes

—

C₂×C₅₉⋊C₈

D₄

yes

—

C₂×C₅₉⋊C₈

C₂³

yes

—

D₄₇₂

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

C₂²×D₅₉

—

C₂²×D₅₉

—

C₂²×D₅₉

—

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

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂⁴

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

C₂×D₄

yes

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

C₈×C₅₉

C₂×D₄

yes

—

C₅₉⋊C₈

C₂×D₄

yes

—

C₂²×C₅₉

#127

yes

—

C₂²×C₅₉

#135

yes

—

C₂²×C₅₉

C₂²×D₄

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#127

yes

—

C₄×C₅₉

C₂²×D₄

yes

characteristic

C₄×C₅₉

C₂²×D₄

yes

characteristic

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

D₅₉

—

C₂²×Q₁₆

D₅₉

yes

characteristic, complemented

C₂²×SD₁₆

—

C₂²×D₈

—

C₂×#124

—

#495

—

C₂×C₄×C₈

—

C₅₉

C₂×#495

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₅₉

yes

—

C₂²×D₄

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂×D₅₉

yes

complemented

C₂×SD₁₆

—

C₂×SD₁₆

—

118

—

C₂×D₈

—

C₂×D₈

—

118

—

C₂²×C₈

—

C₂²×C₈

C₂×D₅₉

yes

characteristic

#124

—

#124

—

118

—

C₂×C₂²⋊C₄

—

118

—

100

C₂×C₄²

—

101

C₄×C₈

—

102

C₂⁴

—

118

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

C₂²×D₅₉

yes

—

107

C₂×D₄

—

108

C₂×D₄

—

118

—

109

C₂×D₄

—

118

—

110

C₂×D₄

—

118

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

C₂²×D₅₉

yes

characteristic

114

Q₁₆

—

115

SD₁₆

—

118

—

116

D₈

—

118

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

C₂²⋊D₅₉

yes

—

120

C₂×C₈

C₂²×D₅₉

yes

—

121

C₂²⋊C₄

—

118

—

122

C₄²

—

123

C₄²

—

124

C₂³

—

118

—

125

C₂³

—

236

—

126

C₂³

—

236

—

127

C₂³

D₄×D₅₉

yes

characteristic

128

Q₈

—

129

Q₈

—

130

Q₈

—

131

D₄

—

118

—

132

D₄

—

118

—

133

D₄

—

236

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

118

—

138

C₂×C₄

D₄×D₅₉

yes

characteristic

139

C₂×C₄

D₄×D₅₉

yes

—

140

C₂×C₄

C₂×C₂²⋊D₅₉

yes

—

141

C₂×C₄

C₂³×D₅₉

yes

characteristic

142

C₈

—

143

C₈

C₂×C₂²⋊D₅₉

yes

—

144

C₂²

—

472

—

145

C₂²

—

236

—

146

C₂²

—

236

—

147

C₂²

#14273723

yes

—

148

C₂²

#14273750

yes

—

149

C₂²

C₂×D₄×D₅₉

yes

characteristic

150

C₄

—

151

C₄

—

118

—

152

C₄

#14273723

yes

—

153

C₄

C₂×D₄×D₅₉

yes

characteristic

154

C₄

C₂²×C₂²⋊D₅₉

yes

characteristic

155

C₂

—

472

—

156

C₂

#16837787

yes

direct product factor

157

C₂

C₂×#14273723

yes

characteristic

158

C₂

C₂×#14273750

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×#495
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×#495

LCS

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
1. C₂²×#11745671	C₂	semidirect
3. C₂×#14273716	C₂	semidirect
4. #16837787	C₂	direct (N × H)
5. C₂×C₈×Dic₅₉	C₂	semidirect
6. C₂²×Q₁₆×C₅₉	C₂	semidirect
9. C₂×#11745671	C₂²	semidirect
14. #14273716	C₂²	semidirect
17. C₈×Dic₅₉	C₂²	semidirect
20. C₂×Q₁₆×C₅₉	C₂²	semidirect
76. C₂×C₅₉	#495	semidirect
80. C₂²×Q₁₆	D₅₉	semidirect
86. C₅₉	C₂×#495	semidirect
90. C₂×Q₁₆	C₂×D₅₉	semidirect
156. C₂	#16837787	direct (N × H)

H ≅ G/N

type

1. C₂²×#11745671

C₂

semidirect

3. C₂×#14273716

C₂

semidirect

4. #16837787

C₂

direct (N × H)

5. C₂×C₈×Dic₅₉

C₂

semidirect

6. C₂²×Q₁₆×C₅₉

C₂

semidirect

9. C₂×#11745671

C₂²

semidirect

14. #14273716

C₂²

semidirect

17. C₈×Dic₅₉

C₂²

semidirect

20. C₂×Q₁₆×C₅₉

C₂²

semidirect

76. C₂×C₅₉

#495

semidirect

80. C₂²×Q₁₆

D₅₉

semidirect

86. C₅₉

C₂×#495

semidirect

90. C₂×Q₁₆

C₂×D₅₉

semidirect

156. C₂

#16837787

direct (N × H)

Subgroups — C₂×#16837787 #21434518

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16837787 #21434518

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16837787 #21434518