Group #19144901 — subgroups

Subgroups — C₂×#14583246 #19144901

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	C₂×#13063322	C₂	1	1	yes	maximal, characteristic
3	#14583246	C₂	1	8	yes	maximal, direct product factor
4	C₂×#13063247	C₂	1	4	yes	maximal, complemented
5	C₂×C₄×C₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Dic₈₆	—	2	1	no	—
7	C₂²×D₁₇₂	—	2	1	no	—
8	C₂²×D₁₇₂	—	2	1	no	—
9	C₂²×D₁₇₂	C₂²	1	2	yes	—
10	#13063325	—	2	2	no	—
11	#13063325	C₂²	1	4	yes	complemented
12	#13063322	—	2	2	no	—
13	#13063322	C₂²	1	4	yes	—
14	C₂×C₄⋊Dic₄₃	—	2	1	no	—
15	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
16	#13063247	C₂²	1	16	yes	complemented
17	C₂²×C₈×C₄₃	C₂²	1	2	yes	—
18	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
19	C₄×C₈×C₄₃	C₂²	1	4	yes	complemented
20	C₂³×D₄₃	—	2	2	no	—
21	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
24	C₂²×Dic₄₃	—	2	2	no	—
25	C₂×D₁₇₂	—	2	2	no	—
26	C₂×D₁₇₂	—	2	4	no	—
27	C₂×D₁₇₂	—	2	4	no	—
28	C₂×D₁₇₂	—	2	8	no	—
29	C₂×D₁₇₂	—	4	4	no	—
30	C₂×D₁₇₂	—	4	4	no	—
31	C₂×D₁₇₂	C₂³	1	4	yes	—
32	C₂×Dic₈₆	—	2	4	no	—
33	C₂×Dic₈₆	—	4	4	no	—
34	C₂×C₈×C₄₃	—	2	4	no	—
35	C₂×C₈×C₄₃	C₂³	1	4	yes	—
36	C₄²×C₄₃	D₄	1	2	yes	—
37	C₄²×C₄₃	C₂³	1	2	yes	—
38	C₄⋊Dic₄₃	—	2	4	no	—
39	C₄⋊Dic₄₃	—	2	2	no	—
40	C₄⋊Dic₄₃	C₂³	1	4	yes	—
41	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×D₄₃	—	2	4	no	—
43	C₂²×D₄₃	—	4	4	no	—
44	C₂²×D₄₃	—	4	4	no	—
45	C₂²×D₄₃	—	4	4	no	—
46	C₂×C₄×C₄₃	D₈	1	4	yes	—
47	C₂×C₄×C₄₃	SD₁₆	1	4	yes	—
48	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₄₃	#41	1	4	yes	—
52	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
53	C₂×Dic₄₃	—	2	4	no	—
54	C₂×Dic₄₃	—	4	4	no	—
55	D₁₇₂	—	2	8	no	—
56	D₁₇₂	—	4	4	no	—
57	D₁₇₂	—	4	8	no	—
58	D₁₇₂	—	4	8	no	—
59	Dic₈₆	—	4	8	no	—
60	C₈×C₄₃	—	2	4	no	—
61	C₂²×C₄₃	#124	1	2	yes	—
62	C₂²×C₄₃	C₂×D₈	1	2	yes	—
63	C₂²×C₄₃	C₂×SD₁₆	1	2	yes	—
64	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
65	C₂×D₄₃	—	4	4	no	—
66	C₂×D₄₃	—	4	4	no	—
67	C₂×D₄₃	—	4	4	no	—
68	C₂×D₄₃	—	8	8	no	—
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	#486	1	4	yes	complemented
76	C₂×C₄₃	C₂×#124	1	1	yes	characteristic
77	C₂×C₄₃	C₂²×D₈	1	1	yes	characteristic
78	C₂×C₄₃	C₂²×SD₁₆	1	1	yes	characteristic
79	D₄₃	—	8	4	no	—
80	C₂×C₄⋊Q₈	—	43	1	no	—
81	C₂×#127	—	43	1	no	—
82	#486	—	43	8	no	—
83	C₂×#102	—	43	4	no	—
84	C₂×C₄×C₈	D₄₃	1	1	yes	characteristic, complemented
85	C₄₃	C₂×#486	1	1	yes	characteristic, complemented, Sylow
86	C₂²×Q₈	—	86	1	no	—
87	C₂²×D₄	—	43	2	no	—
88	C₂²×D₄	—	86	1	no	—
89	C₂²×D₄	—	86	1	no	—
90	C₂²×C₈	C₂×D₄₃	1	2	yes	—
91	C₄⋊Q₈	—	43	4	no	—
92	C₄⋊Q₈	—	86	2	no	—
93	#127	—	43	4	no	—
94	#127	—	86	2	no	—
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	C₂×D₄₃	1	4	yes	complemented
100	C₂⁴	—	86	2	no	—
101	C₂×Q₈	—	86	4	no	—
102	C₂×Q₈	—	172	4	no	—
103	C₂×D₄	—	43	4	no	—
104	C₂×D₄	—	86	4	no	—
105	C₂×D₄	—	86	4	no	—
106	C₂×D₄	—	86	2	no	—
107	C₂×D₄	—	86	8	no	—
108	C₂×D₄	—	172	4	no	—
109	C₂×D₄	—	172	4	no	—
110	C₂²×C₄	—	86	2	no	—
111	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
112	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
113	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
114	C₂×C₈	—	2	4	no	—
115	C₂×C₈	C₂²×D₄₃	1	4	yes	—
116	C₄⋊C₄	—	43	4	no	—
117	C₄⋊C₄	—	86	2	no	—
118	C₄⋊C₄	—	86	4	no	—
119	C₄²	D₁₇₂	1	2	yes	—
120	C₄²	C₂²×D₄₃	1	2	yes	—
121	C₂³	—	86	4	no	—
122	C₂³	—	172	4	no	—
123	C₂³	—	172	4	no	—
124	C₂³	—	172	4	no	—
125	C₂³	C₂×D₁₇₂	1	1	yes	characteristic
126	Q₈	—	172	8	no	—
127	D₄	—	86	8	no	—
128	D₄	—	172	4	no	—
129	D₄	—	172	8	no	—
130	D₄	—	172	8	no	—
131	C₂×C₄	—	86	4	no	—
132	C₂×C₄	—	172	4	no	—
133	C₂×C₄	#10621873	1	4	yes	—
134	C₂×C₄	D₃₄₄	1	4	yes	—
135	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
136	C₂×C₄	C₂×D₁₇₂	1	2	yes	—
137	C₂×C₄	C₂×D₁₇₂	1	2	yes	—
138	C₂×C₄	#10621898	1	4	yes	—
139	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
140	C₈	—	2	4	no	—
141	C₂²	—	344	8	no	—
142	C₂²	—	172	4	no	—
143	C₂²	—	172	4	no	—
144	C₂²	—	172	4	no	—
145	C₂²	#13063313	1	2	yes	—
146	C₂²	C₂×D₃₄₄	1	2	yes	—
147	C₂²	C₂×#10621873	1	2	yes	—
148	C₂²	C₂²×D₁₇₂	1	1	yes	characteristic
149	C₄	—	172	4	no	—
150	C₄	#13063313	1	2	yes	—
151	C₄	C₂×D₃₄₄	1	4	yes	—
152	C₄	C₂×#10621873	1	4	yes	—
153	C₄	C₂×#10621898	1	2	yes	—
154	C₂	—	344	4	no	—
155	C₂	#14583246	1	4	yes	direct product factor
156	C₂	C₂×#13063313	1	1	yes	characteristic
157	C₂	C₂²×D₃₄₄	1	1	yes	characteristic
158	C₂	C₂²×#10621873	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13063325

C₂

yes

maximal, characteristic, complemented

C₂×#13063322

C₂

yes

maximal, characteristic

#14583246

C₂

yes

maximal, direct product factor

C₂×#13063247

C₂

yes

maximal, complemented

C₂×C₄×C₈×C₄₃

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₈₆

—

C₂²×D₁₇₂

—

C₂²×D₁₇₂

—

C₂²×D₁₇₂

C₂²

yes

—

#13063325

—

#13063325

C₂²

yes

complemented

#13063322

—

#13063322

C₂²

yes

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

C₂²

yes

—

#13063247

C₂²

yes

complemented

C₂²×C₈×C₄₃

C₂²

yes

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₄×C₈×C₄₃

C₂²

yes

complemented

C₂³×D₄₃

—

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

C₂³

yes

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₂³

yes

—

C₄²×C₄₃

D₄

yes

—

C₄²×C₄₃

C₂³

yes

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

C₂³

yes

—

C₂³×C₄₃

C₂×D₄

yes

characteristic

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

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

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

Dic₈₆

—

C₈×C₄₃

—

C₂²×C₄₃

#124

yes

—

C₂²×C₄₃

C₂×D₈

yes

—

C₂²×C₄₃

C₂×SD₁₆

yes

—

C₂²×C₄₃

C₂²×D₄

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

#124

yes

—

C₄×C₄₃

C₂×D₈

yes

—

C₄×C₄₃

C₂×SD₁₆

yes

—

C₄×C₄₃

C₂×#41

yes

—

Dic₄₃

—

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

C₂²×D₈

yes

characteristic

C₂×C₄₃

C₂²×SD₁₆

yes

characteristic

D₄₃

—

C₂×C₄⋊Q₈

—

C₂×#127

—

#486

—

C₂×#102

—

C₂×C₄×C₈

D₄₃

yes

characteristic, complemented

C₄₃

C₂×#486

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×D₄

—

C₂²×D₄

—

C₂²×D₄

—

C₂²×C₈

C₂×D₄₃

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#127

—

#127

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄₃

yes

characteristic

#102

—

C₄×C₈

C₂×D₄₃

yes

complemented

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

—

172

—

109

C₂×D₄

—

172

—

110

C₂²×C₄

—

111

C₂²×C₄

D₁₇₂

yes

characteristic

112

C₂²×C₄

D₁₇₂

yes

characteristic

113

C₂²×C₄

C₂²×D₄₃

yes

characteristic

114

C₂×C₈

—

115

C₂×C₈

C₂²×D₄₃

yes

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄²

D₁₇₂

yes

—

120

C₄²

C₂²×D₄₃

yes

—

121

C₂³

—

122

C₂³

—

172

—

123

C₂³

—

172

—

124

C₂³

—

172

—

125

C₂³

C₂×D₁₇₂

yes

characteristic

126

Q₈

—

172

—

127

D₄

—

128

D₄

—

172

—

129

D₄

—

172

—

130

D₄

—

172

—

131

C₂×C₄

—

132

C₂×C₄

—

172

—

133

C₂×C₄

#10621873

yes

—

134

C₂×C₄

D₃₄₄

yes

—

135

C₂×C₄

C₂×D₁₇₂

yes

characteristic

136

C₂×C₄

C₂×D₁₇₂

yes

—

137

C₂×C₄

C₂×D₁₇₂

yes

—

138

C₂×C₄

#10621898

yes

—

139

C₂×C₄

C₂³×D₄₃

yes

characteristic

140

C₈

—

141

C₂²

—

344

—

142

C₂²

—

172

—

143

C₂²

—

172

—

144

C₂²

—

172

—

145

C₂²

#13063313

yes

—

146

C₂²

C₂×D₃₄₄

yes

—

147

C₂²

C₂×#10621873

yes

—

148

C₂²

C₂²×D₁₇₂

yes

characteristic

149

C₄

—

172

—

150

C₄

#13063313

yes

—

151

C₄

C₂×D₃₄₄

yes

—

152

C₄

C₂×#10621873

yes

—

153

C₄

C₂×#10621898

yes

—

154

C₂

—

344

—

155

C₂

#14583246

yes

direct product factor

156

C₂

C₂×#13063313

yes

characteristic

157

C₂

C₂²×D₃₄₄

yes

characteristic

158

C₂

C₂²×#10621873

yes

characteristic

	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×#486
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×#486

LCS

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
1. C₂×#13063325	C₂	semidirect
3. #14583246	C₂	direct (N × H)
4. C₂×#13063247	C₂	semidirect
5. C₂×C₄×C₈×C₄₃	C₂	semidirect
11. #13063325	C₂²	semidirect
16. #13063247	C₂²	semidirect
19. C₄×C₈×C₄₃	C₂²	semidirect
75. C₂×C₄₃	#486	semidirect
84. C₂×C₄×C₈	D₄₃	semidirect
85. C₄₃	C₂×#486	semidirect
99. C₄×C₈	C₂×D₄₃	semidirect
155. C₂	#14583246	direct (N × H)

H ≅ G/N

type

1. C₂×#13063325

C₂

semidirect

3. #14583246

C₂

direct (N × H)

4. C₂×#13063247

C₂

semidirect

5. C₂×C₄×C₈×C₄₃

C₂

semidirect

11. #13063325

C₂²

semidirect

16. #13063247

C₂²

semidirect

19. C₄×C₈×C₄₃

C₂²

semidirect

75. C₂×C₄₃

#486

semidirect

84. C₂×C₄×C₈

D₄₃

semidirect

85. C₄₃

C₂×#486

semidirect

99. C₄×C₈

C₂×D₄₃

semidirect

155. C₂

#14583246

direct (N × H)

Subgroups — C₂×#14583246 #19144901

Conjugacy classes of subgroups

Complements

Subgroups — C2×#14583246 #19144901

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#14583246 #19144901