Group #19145016 — subgroups

Subgroups — C₂×#14583283 #19145016

Order 5504.

592 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 181 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×Dic₁₇₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#10621873	C₂	1	2	yes	maximal, complemented
3	C₂²×D₃₄₄	C₂	1	1	yes	maximal, characteristic
4	C₂×#13063313	C₂	1	2	yes	maximal, complemented
5	#14583283	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₄×C₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×Dic₈₆	C₂²	1	2	yes	—
8	C₂²×D₁₇₂	C₂²	1	2	yes	—
9	C₂×Dic₁₇₂	—	2	4	no	—
10	C₂×Dic₁₇₂	C₂²	1	4	yes	complemented
11	C₂×#10621873	—	2	8	no	—
12	C₂×#10621873	C₂²	1	8	yes	complemented
13	C₂×D₃₄₄	—	2	4	no	—
14	C₂×D₃₄₄	C₂²	1	4	yes	—
15	#13063313	—	2	4	no	—
16	#13063313	C₂²	1	8	yes	complemented
17	C₂×(C₂×D₄₃)⋊C₄	—	2	4	no	—
18	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
21	C₄×C₈×C₄₃	C₂²	1	4	yes	complemented
22	C₂³×D₄₃	—	2	2	no	—
23	C₂²×C₄×C₄₃	D₄	1	2	yes	—
24	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
25	C₂²×Dic₄₃	—	2	2	no	—
26	C₂×D₁₇₂	—	2	2	no	—
27	C₂×D₁₇₂	—	2	4	no	—
28	C₂×D₁₇₂	—	2	4	no	—
29	C₂×D₁₇₂	C₂³	1	4	yes	—
30	C₂×Dic₈₆	—	2	4	no	—
31	C₂×Dic₈₆	—	2	4	no	—
32	C₂×Dic₈₆	—	2	2	no	—
33	C₂×Dic₈₆	C₂³	1	4	yes	—
34	C₂×C₈×C₄₃	D₄	1	4	yes	—
35	C₂×C₈×C₄₃	D₄	1	4	yes	—
36	C₂×C₈×C₄₃	C₂³	1	2	yes	—
37	C₂×C₈×C₄₃	C₂³	1	2	yes	—
38	C₄²×C₄₃	D₄	1	2	yes	—
39	C₄²×C₄₃	C₂³	1	2	yes	—
40	(C₂×D₄₃)⋊C₄	—	2	16	no	—
41	Dic₁₇₂	—	2	8	no	—
42	D₃₄₄	—	2	8	no	—
43	#10621873	—	2	16	no	—
44	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×D₄₃	—	2	4	no	—
46	C₂²×D₄₃	—	4	4	no	—
47	C₂²×D₄₃	—	4	8	no	—
48	C₂×C₄×C₄₃	—	2	4	no	—
49	C₂×C₄×C₄₃	C₂×D₄	1	4	yes	—
50	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
54	C₂×Dic₄₃	—	2	4	no	—
55	C₂×Dic₄₃	—	4	4	no	—
56	D₁₇₂	—	2	4	no	—
57	D₁₇₂	—	2	4	no	—
58	D₁₇₂	—	4	4	no	—
59	Dic₈₆	—	2	4	no	—
60	Dic₈₆	—	2	4	no	—
61	Dic₈₆	—	4	4	no	—
62	C₈×C₄₃	C₂×D₄	1	4	yes	—
63	C₈×C₄₃	C₂×D₄	1	4	yes	—
64	C₂²×C₄₃	#127	1	2	yes	—
65	C₂²×C₄₃	#135	1	4	yes	—
66	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
67	C₂×D₄₃	—	4	4	no	—
68	C₂×D₄₃	—	4	8	no	—
69	C₂×D₄₃	—	8	8	no	—
70	C₄×C₄₃	—	2	4	no	—
71	C₄×C₄₃	#127	1	2	yes	—
72	C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
73	C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
74	Dic₄₃	—	4	4	no	—
75	C₂×#495	—	43	1	no	maximal, Sylow
76	C₂×C₄₃	#495	1	4	yes	complemented
77	C₂×C₄₃	C₂×#127	1	1	yes	characteristic
78	C₂×C₄₃	C₂×#135	1	2	yes	—
79	D₄₃	—	8	4	no	—
80	C₂²×Q₁₆	—	43	1	no	—
81	C₂²×SD₁₆	—	43	2	no	—
82	C₂²×D₈	—	43	1	no	—
83	C₂×#124	—	43	2	no	—
84	#495	—	43	8	no	—
85	C₂×C₄×C₈	D₄₃	1	1	yes	characteristic, complemented
86	C₄₃	C₂×#495	1	1	yes	characteristic, complemented, Sylow
87	C₂²×Q₈	—	43	2	no	—
88	C₂²×D₄	—	43	2	no	—
89	C₂×Q₁₆	—	43	4	no	—
90	C₂×Q₁₆	—	86	4	no	—
91	C₂×SD₁₆	—	43	8	no	—
92	C₂×SD₁₆	—	86	8	no	—
93	C₂×D₈	—	43	4	no	—
94	C₂×D₈	—	86	4	no	—
95	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic
96	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic
97	#124	—	43	8	no	—
98	#124	—	86	4	no	—
99	C₂×C₂²⋊C₄	—	86	4	no	—
100	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
101	C₄×C₈	C₂×D₄₃	1	4	yes	complemented
102	C₂⁴	—	86	2	no	—
103	C₂×Q₈	—	43	4	no	—
104	C₂×Q₈	—	86	2	no	—
105	C₂×Q₈	—	86	4	no	—
106	C₂×Q₈	—	86	4	no	—
107	C₂×D₄	—	43	4	no	—
108	C₂×D₄	—	86	4	no	—
109	C₂×D₄	—	86	4	no	—
110	C₂×D₄	—	86	2	no	—
111	C₂²×C₄	—	86	2	no	—
112	C₂²×C₄	D₁₇₂	1	2	yes	—
113	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
114	Q₁₆	—	86	8	no	—
115	SD₁₆	—	86	16	no	—
116	D₈	—	86	8	no	—
117	C₂×C₈	D₁₇₂	1	4	yes	—
118	C₂×C₈	D₁₇₂	1	4	yes	—
119	C₂×C₈	C₂²×D₄₃	1	2	yes	—
120	C₂×C₈	C₂²×D₄₃	1	2	yes	—
121	C₂²⋊C₄	—	86	16	no	—
122	C₄²	D₁₇₂	1	2	yes	—
123	C₄²	C₂²×D₄₃	1	2	yes	—
124	C₂³	—	86	4	no	—
125	C₂³	—	172	4	no	—
126	C₂³	—	172	8	no	—
127	C₂³	C₂×D₁₇₂	1	1	yes	characteristic
128	Q₈	—	86	4	no	—
129	Q₈	—	86	4	no	—
130	Q₈	—	172	4	no	—
131	D₄	—	86	4	no	—
132	D₄	—	86	4	no	—
133	D₄	—	172	4	no	—
134	C₂×C₄	—	2	4	no	—
135	C₂×C₄	—	86	4	no	—
136	C₂×C₄	—	172	4	no	—
137	C₂×C₄	C₂×D₁₇₂	1	4	yes	—
138	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
139	C₂×C₄	C₂×D₁₇₂	1	2	yes	—
140	C₂×C₄	C₂×D₁₇₂	1	2	yes	—
141	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
142	C₈	C₂×D₁₇₂	1	4	yes	—
143	C₈	C₂×D₁₇₂	1	4	yes	—
144	C₂²	—	344	8	no	—
145	C₂²	—	172	8	no	—
146	C₂²	—	172	4	no	—
147	C₂²	#13063322	1	2	yes	—
148	C₂²	#13063347	1	4	yes	—
149	C₂²	C₂²×D₁₇₂	1	1	yes	characteristic
150	C₄	—	2	4	no	—
151	C₄	—	172	4	no	—
152	C₄	#13063322	1	2	yes	—
153	C₄	C₂²×D₁₇₂	1	1	yes	characteristic
154	C₄	C₂²×D₁₇₂	1	1	yes	characteristic
155	C₂	—	344	4	no	—
156	C₂	#14583283	1	4	yes	direct product factor
157	C₂	C₂×#13063322	1	1	yes	characteristic
158	C₂	C₂×#13063347	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₁₇₂

C₂

yes

maximal, characteristic, complemented

C₂²×#10621873

C₂

yes

maximal, complemented

C₂²×D₃₄₄

C₂

yes

maximal, characteristic

C₂×#13063313

C₂

yes

maximal, complemented

#14583283

C₂

yes

maximal, direct product factor

C₂×C₄×C₈×C₄₃

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₈₆

C₂²

yes

—

C₂²×D₁₇₂

C₂²

yes

—

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

C₂²

yes

complemented

C₂×#10621873

—

C₂×#10621873

C₂²

yes

complemented

C₂×D₃₄₄

—

C₂×D₃₄₄

C₂²

yes

—

#13063313

—

#13063313

C₂²

yes

complemented

C₂×(C₂×D₄₃)⋊C₄

—

C₂²×C₈×C₄₃

C₂²

yes

characteristic

C₂²×C₈×C₄₃

C₂²

yes

characteristic

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₄×C₈×C₄₃

C₂²

yes

complemented

C₂³×D₄₃

—

C₂²×C₄×C₄₃

D₄

yes

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

C₂³

yes

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

C₂³

yes

—

C₂×C₈×C₄₃

D₄

yes

—

C₂×C₈×C₄₃

D₄

yes

—

C₂×C₈×C₄₃

C₂³

yes

—

C₂×C₈×C₄₃

C₂³

yes

—

C₄²×C₄₃

D₄

yes

—

C₄²×C₄₃

C₂³

yes

—

(C₂×D₄₃)⋊C₄

—

Dic₁₇₂

—

D₃₄₄

—

#10621873

—

C₂³×C₄₃

C₂×D₄

yes

characteristic

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂⁴

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

C₂×D₄

yes

—

C₈×C₄₃

C₂×D₄

yes

—

C₂²×C₄₃

#127

yes

—

C₂²×C₄₃

#135

yes

—

C₂²×C₄₃

C₂²×D₄

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#127

yes

—

C₄×C₄₃

C₂²×D₄

yes

characteristic

C₄×C₄₃

C₂²×D₄

yes

characteristic

Dic₄₃

—

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

D₄₃

—

C₂²×Q₁₆

—

C₂²×SD₁₆

—

C₂²×D₈

—

C₂×#124

—

#495

—

C₂×C₄×C₈

D₄₃

yes

characteristic, complemented

C₄₃

C₂×#495

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×D₄

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₂×D₈

—

C₂²×C₈

C₂×D₄₃

yes

characteristic

C₂²×C₈

C₂×D₄₃

yes

characteristic

#124

—

#124

—

C₂×C₂²⋊C₄

—

100

C₂×C₄²

C₂×D₄₃

yes

characteristic

101

C₄×C₈

C₂×D₄₃

yes

complemented

102

C₂⁴

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂²×C₄

—

112

C₂²×C₄

D₁₇₂

yes

—

113

C₂²×C₄

C₂²×D₄₃

yes

characteristic

114

Q₁₆

—

115

SD₁₆

—

116

D₈

—

117

C₂×C₈

D₁₇₂

yes

—

118

C₂×C₈

D₁₇₂

yes

—

119

C₂×C₈

C₂²×D₄₃

yes

—

120

C₂×C₈

C₂²×D₄₃

yes

—

121

C₂²⋊C₄

—

122

C₄²

D₁₇₂

yes

—

123

C₄²

C₂²×D₄₃

yes

—

124

C₂³

—

125

C₂³

—

172

—

126

C₂³

—

172

—

127

C₂³

C₂×D₁₇₂

yes

characteristic

128

Q₈

—

129

Q₈

—

130

Q₈

—

172

—

131

D₄

—

132

D₄

—

133

D₄

—

172

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

172

—

137

C₂×C₄

C₂×D₁₇₂

yes

—

138

C₂×C₄

C₂×D₁₇₂

yes

characteristic

139

C₂×C₄

C₂×D₁₇₂

yes

—

140

C₂×C₄

C₂×D₁₇₂

yes

—

141

C₂×C₄

C₂³×D₄₃

yes

characteristic

142

C₈

C₂×D₁₇₂

yes

—

143

C₈

C₂×D₁₇₂

yes

—

144

C₂²

—

344

—

145

C₂²

—

172

—

146

C₂²

—

172

—

147

C₂²

#13063322

yes

—

148

C₂²

#13063347

yes

—

149

C₂²

C₂²×D₁₇₂

yes

characteristic

150

C₄

—

151

C₄

—

172

—

152

C₄

#13063322

yes

—

153

C₄

C₂²×D₁₇₂

yes

characteristic

154

C₄

C₂²×D₁₇₂

yes

characteristic

155

C₂

—

344

—

156

C₂

#14583283

yes

direct product factor

157

C₂

C₂×#13063322

yes

characteristic

158

C₂

C₂×#13063347

yes

—

	subgroup	G/H
center	C₂³	C₂×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₂×C₄×C₈×C₄₃	C₂
socle	C₂³×C₄₃	C₂×D₄
hypercenter	C₂×C₄×C₈	D₄₃
hypocenter	C₄₃	C₂×#495
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₁₇₂

commutator

C₂×C₄×C₄₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₃

Fitting

C₂×C₄×C₈×C₄₃

C₂

socle

C₂³×C₄₃

C₂×D₄

hypercenter

C₂×C₄×C₈

D₄₃

hypocenter

C₄₃

C₂×#495

LCS

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
1. C₂²×Dic₁₇₂	C₂	semidirect
2. C₂²×#10621873	C₂	semidirect
4. C₂×#13063313	C₂	semidirect
5. #14583283	C₂	direct (N × H)
6. C₂×C₄×C₈×C₄₃	C₂	semidirect
10. C₂×Dic₁₇₂	C₂²	semidirect
12. C₂×#10621873	C₂²	semidirect
16. #13063313	C₂²	semidirect
21. C₄×C₈×C₄₃	C₂²	semidirect
76. C₂×C₄₃	#495	semidirect
85. C₂×C₄×C₈	D₄₃	semidirect
86. C₄₃	C₂×#495	semidirect
101. C₄×C₈	C₂×D₄₃	semidirect
156. C₂	#14583283	direct (N × H)

H ≅ G/N

type

1. C₂²×Dic₁₇₂

C₂

semidirect

2. C₂²×#10621873

C₂

semidirect

4. C₂×#13063313

C₂

semidirect

5. #14583283

C₂

direct (N × H)

6. C₂×C₄×C₈×C₄₃

C₂

semidirect

10. C₂×Dic₁₇₂

C₂²

semidirect

12. C₂×#10621873

C₂²

semidirect

16. #13063313

C₂²

semidirect

21. C₄×C₈×C₄₃

C₂²

semidirect

76. C₂×C₄₃

#495

semidirect

85. C₂×C₄×C₈

D₄₃

semidirect

86. C₄₃

C₂×#495

semidirect

101. C₄×C₈

C₂×D₄₃

semidirect

156. C₂

#14583283

direct (N × H)

Subgroups — C₂×#14583283 #19145016

Conjugacy classes of subgroups

Complements

Subgroups — C2×#14583283 #19145016

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#14583283 #19145016