Group #19143314 — subgroups

Subgroups — #19143314

Order 5504.

484 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 189 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×#13063331	C₂	1	2	yes	maximal, complemented
2	#14583449	C₂	1	1	yes	maximal, characteristic
3	#14582972	C₂	1	8	yes	maximal, complemented
4	#14582923	C₂	1	1	yes	maximal, characteristic, complemented
5	#14582865	C₂	1	2	yes	maximal, complemented
6	C₂×#97×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#10621898	C₂²	1	2	yes	—
8	#13063331	—	2	8	no	—
9	C₂×#10621877	C₂²	1	2	yes	—
10	C₂×#10621872	C₂²	1	4	yes	—
11	C₂×C₈×D₄₃	C₂²	1	4	yes	—
12	#13063325	C₄	1	1	yes	characteristic
13	#13063322	C₄	1	1	yes	characteristic
14	#13063313	C₄	1	2	yes	—
15	#13063299	C₄	1	4	yes	—
16	C₄×D₁₇₂	C₂²	1	4	yes	—
17	C₄⋊(C₄₃⋊C₈)	C₂²	1	4	yes	—
18	(C₂×D₄₃)⋊C₈	C₂²	1	8	yes	—
19	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic
20	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic
21	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
22	#97×C₄₃	C₂²	1	4	yes	—
23	C₂²×C₄×C₄₃	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
25	C₂×C₂²⋊D₄₃	C₂×C₄	1	4	yes	—
26	#10621898	—	2	4	no	—
27	#10621898	—	2	4	no	—
28	C₂×D₁₇₂	C₂×C₄	1	2	yes	—
29	C₂×D₁₇₂	C₂×C₄	1	4	yes	—
30	C₂×C₄×D₄₃	C₂³	1	4	yes	—
31	C₂×Dic₈₆	C₂×C₄	1	2	yes	—
32	C₂×C₈×C₄₃	—	2	2	no	—
33	C₂×C₈×C₄₃	D₄	1	4	yes	complemented
34	C₂×C₈×C₄₃	C₂³	1	2	yes	—
35	C₂×C₈×C₄₃	C₂³	1	2	yes	—
36	C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
37	C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
38	C₄²×C₄₃	C₂³	1	2	yes	—
39	(C₂×D₄₃)⋊C₄	C₂×C₄	1	8	yes	—
40	C₄⋊Dic₄₃	C₂×C₄	1	4	yes	—
41	#10621877	—	2	4	no	—
42	C₂×C₄₃⋊C₈	C₂³	1	4	yes	—
43	#10621872	—	2	8	no	—
44	C₈×D₄₃	—	2	8	no	—
45	C₂³×C₄₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×D₄₃	C₂²×C₄	1	4	yes	—
47	C₂×C₄×C₄₃	—	4	2	no	—
48	C₂×C₄×C₄₃	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₄₃	C₂²×C₄	1	2	yes	—
50	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₄₃	#41	1	2	yes	—
53	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
54	C₂²⋊D₄₃	—	2	8	no	—
55	C₂×Dic₄₃	C₂²×C₄	1	4	yes	—
56	D₁₇₂	—	2	4	no	—
57	D₁₇₂	—	4	4	no	—
58	C₄×D₄₃	—	2	4	no	—
59	C₄×D₄₃	—	2	4	no	—
60	Dic₈₆	—	2	4	no	—
61	C₈×C₄₃	—	2	2	no	—
62	C₈×C₄₃	C₂×D₄	1	4	yes	—
63	C₄₃⋊C₈	—	2	4	no	—
64	C₂²×C₄₃	—	2	2	no	—
65	C₂²×C₄₃	C₄×D₄	1	2	yes	—
66	C₂²×C₄₃	C₂³×C₄	1	1	yes	characteristic
67	C₂×D₄₃	—	2	4	no	—
68	C₂×D₄₃	—	2	8	no	—
69	C₄×C₄₃	—	2	2	no	—
70	C₄×C₄₃	—	2	2	no	—
71	C₄×C₄₃	C₄×D₄	1	2	yes	—
72	C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
73	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
74	Dic₄₃	—	2	4	no	—
75	#2932	—	43	1	no	maximal, Sylow
76	C₂×C₄₃	—	2	2	no	—
77	C₂×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
78	C₂×C₄₃	#568	1	2	yes	—
79	D₄₃	—	4	4	no	—
80	C₂×#131	—	43	2	no	—
81	#532	—	43	1	no	—
82	#435	—	43	8	no	—
83	#423	—	43	1	no	—
84	#408	—	43	2	no	—
85	C₂×#97	D₄₃	1	1	yes	characteristic, complemented
86	C₄₃	#2932	1	1	yes	characteristic, complemented, Sylow
87	C₂×#41	—	43	2	no	—
88	#131	—	86	8	no	—
89	C₂×M₁₆	—	43	2	no	—
90	C₂×M₁₆	—	43	4	no	—
91	C₂²×C₈	—	43	4	no	—
92	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic
93	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic
94	C₄⋊Q₈	—	43	1	no	—
95	#127	—	43	1	no	—
96	#124	—	43	2	no	—
97	#121	—	43	4	no	—
98	C₄×D₄	—	43	4	no	—
99	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
100	C₄⋊C₈	—	43	4	no	—
101	C₂²⋊C₈	—	43	8	no	—
102	#97	C₂×D₄₃	1	4	yes	—
103	#41	—	86	4	no	—
104	#41	—	86	4	no	—
105	C₂×Q₈	—	43	2	no	—
106	C₂×D₄	—	43	4	no	—
107	C₂×D₄	—	43	2	no	—
108	C₂×D₄	—	43	4	no	—
109	C₂²×C₄	—	43	4	no	—
110	C₂²×C₄	C₄×D₄₃	1	2	yes	—
111	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
112	M₁₆	—	86	4	no	—
113	M₁₆	—	86	8	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	43	4	no	—
116	C₂×C₈	—	86	8	no	—
117	C₂×C₈	D₁₇₂	1	4	yes	complemented
118	C₂×C₈	C₂²×D₄₃	1	2	yes	—
119	C₂×C₈	C₂²×D₄₃	1	2	yes	—
120	C₄⋊C₄	—	43	4	no	—
121	C₂²⋊C₄	—	43	8	no	—
122	C₄²	C₄×D₄₃	1	1	yes	characteristic
123	C₄²	C₄×D₄₃	1	1	yes	characteristic
124	C₄²	C₂²×D₄₃	1	2	yes	—
125	C₂³	—	43	4	no	—
126	C₂³	C₂×C₄×D₄₃	1	1	yes	characteristic
127	Q₈	—	86	4	no	—
128	D₄	—	86	8	no	—
129	D₄	—	86	4	no	—
130	D₄	—	172	4	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	43	4	no	—
133	C₂×C₄	—	86	4	no	—
134	C₂×C₄	—	86	4	no	—
135	C₂×C₄	C₂×C₄×D₄₃	1	2	yes	—
136	C₂×C₄	C₂×C₄×D₄₃	1	2	yes	—
137	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
138	C₂×C₄	C₂×D₁₇₂	1	2	yes	—
139	C₂×C₄	#10621898	1	2	yes	—
140	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
141	C₈	—	2	2	no	—
142	C₈	—	86	4	no	—
143	C₈	C₂×D₁₇₂	1	4	yes	—
144	C₂²	—	2	2	no	—
145	C₂²	—	86	4	no	—
146	C₂²	—	86	8	no	—
147	C₂²	C₄×D₁₇₂	1	2	yes	—
148	C₂²	C₂²×C₄×D₄₃	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	2	2	no	—
151	C₄	—	86	4	no	—
152	C₄	C₄×D₁₇₂	1	2	yes	—
153	C₄	C₂²×D₁₇₂	1	1	yes	characteristic
154	C₄	C₂×#10621898	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	—	172	4	no	—
157	C₂	C₂×C₄×D₁₇₂	1	1	yes	characteristic
158	C₂	#14583665	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13063331

C₂

yes

maximal, complemented

#14583449

C₂

yes

maximal, characteristic

#14582972

C₂

yes

maximal, complemented

#14582923

C₂

yes

maximal, characteristic, complemented

#14582865

C₂

yes

maximal, complemented

C₂×#97×C₄₃

C₂

yes

maximal, characteristic, complemented

C₂×#10621898

C₂²

yes

—

#13063331

—

C₂×#10621877

C₂²

yes

—

C₂×#10621872

C₂²

yes

—

C₂×C₈×D₄₃

C₂²

yes

—

#13063325

C₄

yes

characteristic

#13063322

C₄

yes

characteristic

#13063313

C₄

yes

—

#13063299

C₄

yes

—

C₄×D₁₇₂

C₂²

yes

—

C₄⋊(C₄₃⋊C₈)

C₂²

yes

—

(C₂×D₄₃)⋊C₈

C₂²

yes

—

C₂²×C₈×C₄₃

C₂²

yes

characteristic

C₂²×C₈×C₄₃

C₂²

yes

characteristic

C₂×C₄²×C₄₃

C₂²

yes

characteristic

#97×C₄₃

C₂²

yes

—

C₂²×C₄×C₄₃

C₂×C₄

yes

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂×C₂²⋊D₄₃

C₂×C₄

yes

—

#10621898

—

#10621898

—

C₂×D₁₇₂

C₂×C₄

yes

—

C₂×D₁₇₂

C₂×C₄

yes

—

C₂×C₄×D₄₃

C₂³

yes

—

C₂×Dic₈₆

C₂×C₄

yes

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

D₄

yes

complemented

C₂×C₈×C₄₃

C₂³

yes

—

C₂×C₈×C₄₃

C₂³

yes

—

C₄²×C₄₃

C₂×C₄

yes

characteristic

C₄²×C₄₃

C₂×C₄

yes

characteristic

C₄²×C₄₃

C₂³

yes

—

(C₂×D₄₃)⋊C₄

C₂×C₄

yes

—

C₄⋊Dic₄₃

C₂×C₄

yes

—

#10621877

—

C₂×C₄₃⋊C₈

C₂³

yes

—

#10621872

—

C₈×D₄₃

—

C₂³×C₄₃

C₂²×C₄

yes

characteristic

C₂²×D₄₃

C₂²×C₄

yes

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²×C₄

yes

—

C₂×C₄×C₄₃

C₂²×C₄

yes

—

C₂×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

#41

yes

—

C₂×C₄×C₄₃

C₂⁴

yes

characteristic

C₂²⋊D₄₃

—

C₂×Dic₄₃

C₂²×C₄

yes

—

D₁₇₂

—

D₁₇₂

—

C₄×D₄₃

—

C₄×D₄₃

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

C₂×D₄

yes

—

C₄₃⋊C₈

—

C₂²×C₄₃

—

C₂²×C₄₃

C₄×D₄

yes

—

C₂²×C₄₃

C₂³×C₄

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₄×D₄

yes

—

C₄×C₄₃

C₂²×D₄

yes

characteristic

C₄×C₄₃

C₂×#41

yes

characteristic

Dic₄₃

—

#2932

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

C₂×C₄×D₄

yes

characteristic

C₂×C₄₃

#568

yes

—

D₄₃

—

—

—

—

—

—

C₂×#97

D₄₃

yes

characteristic, complemented

C₄₃

#2932

yes

characteristic, complemented, Sylow

C₂×#41

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₄₃

yes

characteristic

C₂²×C₈

C₂×D₄₃

yes

characteristic

C₄⋊Q₈

—

#127

—

#124

—

#121

—

C₄×D₄

—

C₂×C₄²

C₂×D₄₃

yes

characteristic

100

C₄⋊C₈

—

101

C₂²⋊C₈

—

102

#97

C₂×D₄₃

yes

—

103

#41

—

104

#41

—

105

C₂×Q₈

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂²×C₄

—

110

C₂²×C₄

C₄×D₄₃

yes

—

111

C₂²×C₄

C₂²×D₄₃

yes

characteristic

112

M₁₆

—

113

M₁₆

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

D₁₇₂

yes

complemented

118

C₂×C₈

C₂²×D₄₃

yes

—

119

C₂×C₈

C₂²×D₄₃

yes

—

120

C₄⋊C₄

—

121

C₂²⋊C₄

—

122

C₄²

C₄×D₄₃

yes

characteristic

123

C₄²

C₄×D₄₃

yes

characteristic

124

C₄²

C₂²×D₄₃

yes

—

125

C₂³

—

126

C₂³

C₂×C₄×D₄₃

yes

characteristic

127

Q₈

—

128

D₄

—

129

D₄

—

130

D₄

—

172

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

C₂×C₄×D₄₃

yes

—

136

C₂×C₄

C₂×C₄×D₄₃

yes

—

137

C₂×C₄

C₂×C₄×D₄₃

yes

characteristic

138

C₂×C₄

C₂×D₁₇₂

yes

—

139

C₂×C₄

#10621898

yes

—

140

C₂×C₄

C₂³×D₄₃

yes

characteristic

141

C₈

—

142

C₈

—

143

C₈

C₂×D₁₇₂

yes

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

C₄×D₁₇₂

yes

—

148

C₂²

C₂²×C₄×D₄₃

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

C₄×D₁₇₂

yes

—

153

C₄

C₂²×D₁₇₂

yes

characteristic

154

C₄

C₂×#10621898

yes

characteristic

155

C₂

—

156

C₂

—

172

—

157

C₂

C₂×C₄×D₁₇₂

yes

characteristic

158

C₂

#14583665

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₄₃
commutator, socle	C₂²×C₄₃	C₂³×C₄
Fitting	C₂×#97×C₄₃	C₂
hypercenter	C₂×#97	D₄₃
hypocenter	C₄₃	#2932
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₄₃

commutator, socle

C₂²×C₄₃

C₂³×C₄

Fitting

C₂×#97×C₄₃

C₂

hypercenter

C₂×#97

D₄₃

C₄₃

G ▷ C₂²×C₄₃ ▷ C₄₃

C₁ ◁ C₂×C₄ ◁ C₂×#97

N	H ≅ G/N	type
1. C₂×#13063331	C₂	semidirect
3. #14582972	C₂	semidirect
4. #14582923	C₂	semidirect
5. #14582865	C₂	semidirect
6. C₂×#97×C₄₃	C₂	semidirect
33. C₂×C₈×C₄₃	D₄	semidirect
85. C₂×#97	D₄₃	semidirect
86. C₄₃	#2932	semidirect
117. C₂×C₈	D₁₇₂	semidirect

H ≅ G/N

type

1. C₂×#13063331

C₂

semidirect

3. #14582972

C₂

semidirect

4. #14582923

C₂

semidirect

5. #14582865

C₂

semidirect

6. C₂×#97×C₄₃

C₂

semidirect

33. C₂×C₈×C₄₃

D₄

semidirect

85. C₂×#97

D₄₃

semidirect

86. C₄₃

#2932

semidirect

117. C₂×C₈

D₁₇₂

semidirect

Subgroups — #19143314

Conjugacy classes of subgroups

Complements