Group #19134642 — subgroups

Subgroups — #19134642

Order 5504.

252 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 89 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	1	yes	maximal, characteristic
2	C₂×#13063261	C₂	1	1	yes	maximal, characteristic, complemented
3	#14582881	C₂	1	2	yes	maximal
4	C₄₃⋊#338	C₂	1	2	yes	maximal, complemented
5	C₂×#108×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10621898	C₂²	1	1	yes	characteristic
7	#13063331	—	2	2	no	—
8	#13063331	—	2	2	no	—
9	C₂×#10621877	C₂²	1	2	yes	—
10	C₂×#10621877	C₂²	1	1	yes	characteristic
11	C₂×#10621872	C₄	1	2	yes	—
12	C₂×C₈×D₄₃	C₄	1	2	yes	—
13	#13063261	—	2	2	no	—
14	#13063243	—	2	2	no	—
15	#13063241	—	2	2	no	—
16	C₂×M₁₆×C₄₃	C₂²	1	2	yes	—
17	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic
18	#108×C₄₃	—	2	2	no	—
19	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
20	C₂×C₂²⋊D₄₃	Q₈	1	2	yes	—
21	#10621898	—	2	1	no	—
22	#10621898	—	2	1	no	—
23	#10621898	—	2	1	no	—
24	#10621898	—	2	1	no	—
25	C₂×D₁₇₂	D₄	1	1	yes	characteristic
26	C₂×C₄×D₄₃	C₂×C₄	1	2	yes	—
27	C₂×Dic₈₆	D₄	1	1	yes	characteristic
28	M₁₆×C₄₃	—	2	2	no	—
29	M₁₆×C₄₃	—	2	2	no	—
30	C₂×C₈×C₄₃	—	2	2	no	—
31	C₂×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×C₄₃	D₄	1	2	yes	—
34	C₂×C₈×C₄₃	D₄	1	2	yes	—
35	#10621877	—	2	1	no	—
36	#10621877	—	2	1	no	—
37	#10621877	—	2	2	no	—
38	#10621877	—	2	2	no	—
39	C₂×C₄₃⋊C₈	—	2	2	no	—
40	C₂×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
41	#10621872	—	2	4	no	—
42	C₈×D₄₃	—	2	4	no	—
43	C₂³×C₄₃	C₂×Q₈	1	1	yes	characteristic
44	C₂²×D₄₃	C₄⋊C₄	1	2	yes	—
45	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
50	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
51	C₂²⋊D₄₃	—	2	2	no	—
52	C₂²⋊D₄₃	—	2	2	no	—
53	C₂×Dic₄₃	C₄⋊C₄	1	2	yes	—
54	D₁₇₂	—	2	1	no	—
55	D₁₇₂	—	2	1	no	—
56	C₄×D₄₃	—	2	2	no	—
57	C₄×D₄₃	—	2	2	no	—
58	Dic₈₆	—	2	1	no	—
59	Dic₈₆	—	2	1	no	—
60	C₈×C₄₃	—	4	2	no	—
61	C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
62	C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
63	C₄₃⋊C₈	—	2	2	no	—
64	C₄₃⋊C₈	—	4	2	no	—
65	C₂²×C₄₃	—	4	1	no	—
66	C₂²×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
68	C₂²×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
69	C₂×D₄₃	—	2	2	no	—
70	C₂×D₄₃	—	4	2	no	—
71	C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₄×C₄₃	#117	1	1	yes	characteristic
73	C₄×C₄₃	#121	1	1	yes	characteristic
74	C₄×C₄₃	#121	1	1	yes	characteristic
75	Dic₄₃	—	2	2	no	—
76	#1813	—	43	1	no	maximal, Sylow
77	C₂×C₄₃	—	2	1	no	—
78	C₂×C₄₃	—	2	1	no	—
79	C₂×C₄₃	—	2	1	no	—
80	C₂×C₄₃	#380	1	1	yes	characteristic
81	D₄₃	—	4	2	no	—
82	C₂×#131	—	43	1	no	—
83	C₂×#108	—	43	1	no	—
84	C₂×#108	D₄₃	1	1	yes	characteristic, complemented
85	#413	—	43	2	no	—
86	#338	—	43	2	no	—
87	C₄₃	#1813	1	1	yes	characteristic, complemented, Sylow
88	C₂×#41	—	43	1	no	—
89	#131	—	86	2	no	—
90	#131	—	86	2	no	—
91	C₂×M₁₆	—	43	1	no	—
92	C₂×M₁₆	—	43	2	no	—
93	C₂×M₁₆	—	43	2	no	—
94	C₂×M₁₆	C₂×D₄₃	1	2	yes	—
95	C₂²×C₈	—	43	2	no	—
96	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic
97	#108	—	2	2	no	—
98	#108	—	86	2	no	—
99	#101	—	86	2	no	—
100	#100	—	86	2	no	—
101	#41	—	86	1	no	—
102	#41	—	86	1	no	—
103	#41	—	86	1	no	—
104	#41	—	86	1	no	—
105	C₂×Q₈	—	43	1	no	—
106	C₂×D₄	—	43	2	no	—
107	C₂×D₄	—	43	1	no	—
108	C₂²×C₄	—	43	2	no	—
109	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
110	M₁₆	—	2	2	no	—
111	M₁₆	—	2	2	no	—
112	M₁₆	—	86	1	no	—
113	M₁₆	—	86	1	no	—
114	M₁₆	—	86	4	no	—
115	M₁₆	—	86	2	no	—
116	M₁₆	—	86	2	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	43	2	no	—
119	C₂×C₈	—	86	4	no	—
120	C₂×C₈	—	86	2	no	—
121	C₂×C₈	C₄×D₄₃	1	1	yes	characteristic
122	C₂×C₈	C₄×D₄₃	1	1	yes	characteristic
123	C₂×C₈	D₁₇₂	1	2	yes	—
124	C₂×C₈	C₂²⋊D₄₃	1	2	yes	—
125	C₂³	—	43	2	no	—
126	C₂³	Q₈×D₄₃	1	1	yes	characteristic
127	Q₈	—	86	1	no	—
128	Q₈	—	86	1	no	—
129	D₄	—	86	2	no	—
130	D₄	—	86	2	no	—
131	D₄	—	86	1	no	—
132	D₄	—	86	1	no	—
133	C₂×C₄	—	43	2	no	—
134	C₂×C₄	—	86	2	no	—
135	C₂×C₄	—	86	2	no	—
136	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
137	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
138	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
139	C₂×C₄	#10621900	1	1	yes	characteristic
140	C₂×C₄	#10621902	1	1	yes	characteristic
141	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
142	C₈	—	4	2	no	—
143	C₈	—	86	2	no	—
144	C₈	—	172	2	no	—
145	C₈	(C₂×D₄₃)⋊C₄	1	2	yes	—
146	C₈	(C₂×D₄₃)⋊C₄	1	2	yes	—
147	C₂²	—	4	1	no	—
148	C₂²	—	86	2	no	—
149	C₂²	—	172	2	no	—
150	C₂²	C₄⋊C₄×D₄₃	1	1	yes	characteristic
151	C₂²	#13063301	1	1	yes	characteristic
152	C₂²	#13063305	1	1	yes	characteristic
153	C₄	—	86	2	no	—
154	C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
155	C₄	#13063283	1	1	yes	characteristic
156	C₄	#13063296	1	1	yes	characteristic
157	C₄	#13063300	1	1	yes	characteristic
158	C₂	—	2	1	no	—
159	C₂	—	2	1	no	—
160	C₂	—	2	1	no	—
161	C₂	—	172	2	no	—
162	C₂	#14582756	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13063331

C₂

yes

maximal, characteristic

C₂×#13063261

C₂

yes

maximal, characteristic, complemented

#14582881

C₂

yes

maximal

C₄₃⋊#338

C₂

yes

maximal, complemented

C₂×#108×C₄₃

C₂

yes

maximal, characteristic, complemented

C₂×#10621898

C₂²

yes

—

—

C₂²

yes

—

C₂×#10621877

C₂²

yes

characteristic

C₂×#10621872

C₄

yes

—

C₂×C₈×D₄₃

C₄

yes

—

#13063261

—

#13063243

—

#13063241

—

C₂×M₁₆×C₄₃

C₂²

yes

—

C₂²×C₈×C₄₃

C₂²

yes

characteristic

#108×C₄₃

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂×C₂²⋊D₄₃

Q₈

yes

—

—

—

—

—

C₂×D₁₇₂

D₄

yes

characteristic

C₂×C₄×D₄₃

C₂×C₄

yes

—

C₂×Dic₈₆

D₄

yes

characteristic

M₁₆×C₄₃

—

M₁₆×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₂×C₄

yes

characteristic

C₂×C₈×C₄₃

C₂×C₄

yes

characteristic

C₂×C₈×C₄₃

D₄

yes

—

C₂×C₈×C₄₃

D₄

yes

—

—

—

—

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

C₂×C₄

yes

—

#10621872

—

C₈×D₄₃

—

C₂³×C₄₃

C₂×Q₈

yes

characteristic

C₂²×D₄₃

C₄⋊C₄

yes

—

C₂×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂²⋊D₄₃

—

C₂²⋊D₄₃

—

C₂×Dic₄₃

C₄⋊C₄

yes

—

D₁₇₂

—

D₁₇₂

—

C₄×D₄₃

—

C₄×D₄₃

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

C₂²⋊C₄

yes

—

C₈×C₄₃

C₂²⋊C₄

yes

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₄₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₄₃

C₂²⋊Q₈

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

C₂×C₂²⋊C₄

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

#1813

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

#380

yes

characteristic

D₄₃

—

C₂×#131

—

C₂×#108

—

C₂×#108

D₄₃

yes

characteristic, complemented

#413

—

#338

—

C₄₃

#1813

yes

characteristic, complemented, Sylow

C₂×#41

—

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₄₃

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₄₃

yes

—

—

—

100

#100

—

101

#41

—

102

#41

—

103

#41

—

104

#41

—

105

C₂×Q₈

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂²×C₄

—

109

C₂²×C₄

C₂²×D₄₃

yes

characteristic

110

M₁₆

—

111

M₁₆

—

112

M₁₆

—

113

M₁₆

—

114

M₁₆

—

115

M₁₆

—

116

M₁₆

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

C₄×D₄₃

yes

characteristic

122

C₂×C₈

C₄×D₄₃

yes

characteristic

123

C₂×C₈

D₁₇₂

yes

—

124

C₂×C₈

C₂²⋊D₄₃

yes

—

125

C₂³

—

126

C₂³

Q₈×D₄₃

yes

characteristic

127

Q₈

—

128

Q₈

—

129

D₄

—

130

D₄

—

131

D₄

—

132

D₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

C₂×C₄×D₄₃

yes

characteristic

137

C₂×C₄

C₂×D₁₇₂

yes

characteristic

138

C₂×C₄

D₄×D₄₃

yes

characteristic

139

C₂×C₄

#10621900

yes

characteristic

140

C₂×C₄

#10621902

yes

characteristic

141

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

142

C₈

—

143

C₈

—

144

C₈

—

172

—

145

C₈

(C₂×D₄₃)⋊C₄

yes

—

146

C₈

(C₂×D₄₃)⋊C₄

yes

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

172

—

150

C₂²

C₄⋊C₄×D₄₃

yes

characteristic

151

C₂²

#13063301

yes

characteristic

152

C₂²

#13063305

yes

characteristic

153

C₄

—

154

C₄

C₂×(C₂×D₄₃)⋊C₄

yes

characteristic

155

C₄

#13063283

yes

characteristic

156

C₄

#13063296

yes

characteristic

157

C₄

#13063300

yes

characteristic

158

C₂

—

159

C₂

—

160

C₂

—

161

C₂

—

172

—

162

C₂

#14582756

yes

characteristic

	subgroup	G/H
center	C₄	#13063283
commutator	C₂×C₄×C₄₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₃
Fitting	C₂×#108×C₄₃	C₂
socle	C₂×C₄₃	#380
hypercenter	C₂×#108	D₄₃
hypocenter	C₄₃	#1813
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₄

#13063283

commutator

C₂×C₄×C₄₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₃

Fitting

C₂×#108×C₄₃

C₂

C₂×C₄₃

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
2. C₂×#13063261	C₂	semidirect
4. C₄₃⋊#338	C₂	semidirect
5. C₂×#108×C₄₃	C₂	semidirect
84. C₂×#108	D₄₃	semidirect
87. C₄₃	#1813	semidirect

H ≅ G/N

type

2. C₂×#13063261

C₂

semidirect

4. C₄₃⋊#338

C₂

semidirect

5. C₂×#108×C₄₃

C₂

semidirect

84. C₂×#108

D₄₃

semidirect

87. C₄₃

#1813

semidirect

Subgroups — #19134642

Conjugacy classes of subgroups

Complements