Group #19144900 — subgroups

Subgroups — C₂×#14583245 #19144900

Order 5504.

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

normal subgroups only

This group has 528 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	C₂×#13063325	C₂	1	1	yes	maximal, characteristic, complemented
2	#14583245	C₂	1	8	yes	maximal, direct product factor
3	C₂×#13063246	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#127×C₄₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₈₆	—	2	1	no	—
7	C₂²×C₄₃⋊C₈	C₂²	1	2	yes	—
8	#13063325	—	2	2	no	—
9	#13063325	C₂²	1	4	yes	complemented
10	C₂×C₄⋊Dic₄₃	—	2	1	no	—
11	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
12	#13063246	C₂²	1	16	yes	complemented
13	C₄×C₄₃⋊C₈	C₂²	1	4	yes	complemented
14	C₂²×D₄×C₄₃	—	2	1	no	—
15	C₂²×D₄×C₄₃	—	2	1	no	—
16	C₂²×D₄×C₄₃	C₂²	1	2	yes	—
17	#127×C₄₃	—	2	2	no	—
18	#127×C₄₃	C₂²	1	4	yes	—
19	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
20	C₂⁴×C₄₃	—	2	2	no	—
21	C₂×D₄×C₄₃	—	2	2	no	—
22	C₂×D₄×C₄₃	—	2	4	no	—
23	C₂×D₄×C₄₃	—	2	4	no	—
24	C₂×D₄×C₄₃	—	2	8	no	—
25	C₂×D₄×C₄₃	—	4	4	no	—
26	C₂×D₄×C₄₃	—	4	4	no	—
27	C₂×D₄×C₄₃	C₂³	1	4	yes	—
28	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
31	C₂²×Dic₄₃	—	2	2	no	—
32	C₂×Dic₈₆	—	2	4	no	—
33	C₂×Dic₈₆	—	4	4	no	—
34	C₄²×C₄₃	D₄	1	2	yes	—
35	C₄²×C₄₃	C₂³	1	2	yes	—
36	C₄⋊Dic₄₃	—	2	4	no	—
37	C₄⋊Dic₄₃	—	2	2	no	—
38	C₄⋊Dic₄₃	C₂³	1	4	yes	—
39	C₂×C₄₃⋊C₈	—	2	4	no	—
40	C₂×C₄₃⋊C₈	C₂³	1	4	yes	—
41	C₂³×C₄₃	—	2	4	no	—
42	C₂³×C₄₃	—	4	4	no	—
43	C₂³×C₄₃	—	4	4	no	—
44	C₂³×C₄₃	—	4	4	no	—
45	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
46	D₄×C₄₃	—	2	8	no	—
47	D₄×C₄₃	—	4	4	no	—
48	D₄×C₄₃	—	4	8	no	—
49	D₄×C₄₃	—	4	8	no	—
50	C₂×C₄×C₄₃	D₈	1	4	yes	—
51	C₂×C₄×C₄₃	SD₁₆	1	4	yes	—
52	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₄₃	#41	1	4	yes	—
56	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
57	C₂×Dic₄₃	—	2	4	no	—
58	C₂×Dic₄₃	—	4	4	no	—
59	Dic₈₆	—	4	8	no	—
60	C₄₃⋊C₈	—	2	4	no	—
61	C₂²×C₄₃	—	4	4	no	—
62	C₂²×C₄₃	—	4	4	no	—
63	C₂²×C₄₃	—	4	4	no	—
64	C₂²×C₄₃	—	8	8	no	—
65	C₂²×C₄₃	#124	1	2	yes	—
66	C₂²×C₄₃	C₂×D₈	1	2	yes	—
67	C₂²×C₄₃	C₂×SD₁₆	1	2	yes	—
68	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
69	C₄×C₄₃	#124	1	2	yes	—
70	C₄×C₄₃	C₂×D₈	1	4	yes	—
71	C₄×C₄₃	C₂×SD₁₆	1	4	yes	—
72	C₄×C₄₃	C₂×#41	1	2	yes	—
73	Dic₄₃	—	4	4	no	—
74	C₂×#486	—	43	1	no	maximal, Sylow
75	C₂×C₄₃	—	8	4	no	—
76	C₂×C₄₃	#486	1	4	yes	complemented
77	C₂×C₄₃	C₂×#124	1	1	yes	characteristic
78	C₂×C₄₃	C₂²×D₈	1	1	yes	characteristic
79	C₂×C₄₃	C₂²×SD₁₆	1	1	yes	characteristic
80	C₂×C₄⋊Q₈	—	43	1	no	—
81	C₂×#127	D₄₃	1	1	yes	characteristic
82	#486	—	43	8	no	—
83	C₂×#102	—	43	4	no	—
84	C₂×C₄×C₈	—	43	1	no	—
85	C₄₃	C₂×#486	1	1	yes	characteristic, complemented, Sylow
86	C₂²×Q₈	—	86	1	no	—
87	C₂²×D₄	—	2	1	no	—
88	C₂²×D₄	—	2	1	no	—
89	C₂²×D₄	C₂×D₄₃	1	2	yes	—
90	C₂²×C₈	—	43	2	no	—
91	C₄⋊Q₈	—	43	4	no	—
92	C₄⋊Q₈	—	86	2	no	—
93	#127	—	2	2	no	—
94	#127	C₂×D₄₃	1	4	yes	—
95	C₂×C₄⋊C₄	—	43	2	no	—
96	C₂×C₄⋊C₄	—	86	1	no	—
97	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
98	#102	—	43	16	no	—
99	C₄×C₈	—	43	4	no	—
100	C₂⁴	—	2	2	no	—
101	C₂×Q₈	—	86	4	no	—
102	C₂×Q₈	—	172	4	no	—
103	C₂×D₄	—	2	8	no	—
104	C₂×D₄	—	2	2	no	—
105	C₂×D₄	—	2	4	no	—
106	C₂×D₄	—	2	4	no	—
107	C₂×D₄	—	4	4	no	—
108	C₂×D₄	—	4	4	no	—
109	C₂×D₄	C₂²×D₄₃	1	4	yes	—
110	C₂²×C₄	—	86	2	no	—
111	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
112	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
113	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
114	C₂×C₈	—	43	4	no	—
115	C₂×C₈	—	86	4	no	—
116	C₄⋊C₄	—	43	4	no	—
117	C₄⋊C₄	—	86	2	no	—
118	C₄⋊C₄	—	86	4	no	—
119	C₄²	C₂²⋊D₄₃	1	2	yes	—
120	C₄²	C₂²×D₄₃	1	2	yes	—
121	C₂³	—	2	4	no	—
122	C₂³	—	4	4	no	—
123	C₂³	—	4	4	no	—
124	C₂³	—	4	4	no	—
125	C₂³	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
126	Q₈	—	172	8	no	—
127	D₄	—	2	8	no	—
128	D₄	—	4	4	no	—
129	D₄	—	4	8	no	—
130	D₄	—	4	8	no	—
131	C₂×C₄	—	86	4	no	—
132	C₂×C₄	—	172	4	no	—
133	C₂×C₄	#10621882	1	4	yes	—
134	C₂×C₄	#10621883	1	4	yes	—
135	C₂×C₄	#10621900	1	4	yes	—
136	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
137	C₂×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
138	C₂×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
139	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
140	C₈	—	86	4	no	—
141	C₂²	—	4	4	no	—
142	C₂²	—	4	4	no	—
143	C₂²	—	4	4	no	—
144	C₂²	—	8	8	no	—
145	C₂²	#13063312	1	2	yes	—
146	C₂²	C₂×#10621882	1	2	yes	—
147	C₂²	C₂×#10621883	1	2	yes	—
148	C₂²	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
149	C₄	—	172	4	no	—
150	C₄	#13063312	1	2	yes	—
151	C₄	C₂×#10621882	1	4	yes	—
152	C₄	C₂×#10621883	1	4	yes	—
153	C₄	C₂×#10621900	1	2	yes	—
154	C₂	—	8	4	no	—
155	C₂	#14583245	1	4	yes	direct product factor
156	C₂	C₂×#13063312	1	1	yes	characteristic
157	C₂	C₂²×#10621882	1	1	yes	characteristic
158	C₂	C₂²×#10621883	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13063325

C₂

yes

maximal, characteristic, complemented

#14583245

C₂

yes

maximal, direct product factor

C₂×#13063246

C₂

yes

maximal, complemented

C₂×C₄×C₄₃⋊C₈

C₂

yes

maximal, characteristic, complemented

C₂×#127×C₄₃

C₂

yes

maximal, characteristic

C₂²×Dic₈₆

—

C₂²×C₄₃⋊C₈

C₂²

yes

—

#13063325

—

#13063325

C₂²

yes

complemented

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

C₂²

yes

—

#13063246

C₂²

yes

complemented

C₄×C₄₃⋊C₈

C₂²

yes

complemented

C₂²×D₄×C₄₃

—

C₂²×D₄×C₄₃

—

C₂²×D₄×C₄₃

C₂²

yes

—

#127×C₄₃

—

#127×C₄₃

C₂²

yes

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₂⁴×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

C₂³

yes

—

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₄²×C₄₃

D₄

yes

—

C₄²×C₄₃

C₂³

yes

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

C₂³

yes

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

C₂³

yes

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

C₂×D₄

yes

characteristic

D₄×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

C₂×C₄×C₄₃

D₈

yes

—

C₂×C₄×C₄₃

SD₁₆

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

#41

yes

—

C₂×C₄×C₄₃

C₂⁴

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

C₄₃⋊C₈

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#124

yes

—

C₂²×C₄₃

C₂×D₈

yes

—

C₂²×C₄₃

C₂×SD₁₆

yes

—

C₂²×C₄₃

C₂²×D₄

yes

characteristic

C₄×C₄₃

#124

yes

—

C₄×C₄₃

C₂×D₈

yes

—

C₄×C₄₃

C₂×SD₁₆

yes

—

C₄×C₄₃

C₂×#41

yes

—

Dic₄₃

—

C₂×#486

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

C₂²×D₈

yes

characteristic

C₂×C₄₃

C₂²×SD₁₆

yes

characteristic

C₂×C₄⋊Q₈

—

C₂×#127

D₄₃

yes

characteristic

#486

—

C₂×#102

—

C₂×C₄×C₈

—

C₄₃

C₂×#486

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×D₄

—

C₂²×D₄

—

C₂²×D₄

C₂×D₄₃

yes

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#127

—

#127

C₂×D₄₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄₃

yes

characteristic

#102

—

C₄×C₈

—

100

C₂⁴

—

101

C₂×Q₈

—

102

C₂×Q₈

—

172

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

C₂²×D₄₃

yes

—

110

C₂²×C₄

—

111

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

112

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

113

C₂²×C₄

C₂²×D₄₃

yes

characteristic

114

C₂×C₈

—

115

C₂×C₈

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄²

C₂²⋊D₄₃

yes

—

120

C₄²

C₂²×D₄₃

yes

—

121

C₂³

—

122

C₂³

—

123

C₂³

—

124

C₂³

—

125

C₂³

C₂×C₂²⋊D₄₃

yes

characteristic

126

Q₈

—

172

—

127

D₄

—

128

D₄

—

129

D₄

—

130

D₄

—

131

C₂×C₄

—

132

C₂×C₄

—

172

—

133

C₂×C₄

#10621882

yes

—

134

C₂×C₄

#10621883

yes

—

135

C₂×C₄

#10621900

yes

—

136

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

137

C₂×C₄

C₂×C₂²⋊D₄₃

yes

—

138

C₂×C₄

C₂×C₂²⋊D₄₃

yes

—

139

C₂×C₄

C₂³×D₄₃

yes

characteristic

140

C₈

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

#13063312

yes

—

146

C₂²

C₂×#10621882

yes

—

147

C₂²

C₂×#10621883

yes

—

148

C₂²

C₂²×C₂²⋊D₄₃

yes

characteristic

149

C₄

—

172

—

150

C₄

#13063312

yes

—

151

C₄

C₂×#10621882

yes

—

152

C₄

C₂×#10621883

yes

—

153

C₄

C₂×#10621900

yes

—

154

C₂

—

155

C₂

#14583245

yes

direct product factor

156

C₂

C₂×#13063312

yes

characteristic

157

C₂

C₂²×#10621882

yes

characteristic

158

C₂

C₂²×#10621883

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₂×#127×C₄₃	C₂
socle	C₂³×C₄₃	C₂×D₄
hypercenter	C₂×#127	D₄₃
hypocenter	C₄₃	C₂×#486
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#127

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₄₃

commutator

C₂×C₄×C₄₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₃

Fitting

C₂×#127×C₄₃

C₂

socle

C₂³×C₄₃

C₂×D₄

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#127

N	H ≅ G/N	type
1. C₂×#13063325	C₂	semidirect
2. #14583245	C₂	direct (N × H)
3. C₂×#13063246	C₂	semidirect
4. C₂×C₄×C₄₃⋊C₈	C₂	semidirect
9. #13063325	C₂²	semidirect
12. #13063246	C₂²	semidirect
13. C₄×C₄₃⋊C₈	C₂²	semidirect
76. C₂×C₄₃	#486	semidirect
85. C₄₃	C₂×#486	semidirect
155. C₂	#14583245	direct (N × H)

H ≅ G/N

type

1. C₂×#13063325

C₂

semidirect

2. #14583245

C₂

direct (N × H)

3. C₂×#13063246

C₂

semidirect

4. C₂×C₄×C₄₃⋊C₈

C₂

semidirect

9. #13063325

C₂²

semidirect

12. #13063246

C₂²

semidirect

13. C₄×C₄₃⋊C₈

C₂²

semidirect

76. C₂×C₄₃

#486

semidirect

85. C₄₃

C₂×#486

semidirect

155. C₂

#14583245

direct (N × H)

Subgroups — C₂×#14583245 #19144900

Conjugacy classes of subgroups

Complements

Subgroups — C2×#14583245 #19144900

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#14583245 #19144900