Group #19145135 — subgroups

Subgroups — #19145135

Order 5504.

340 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 149 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	#14583527	C₂	1	2	yes	maximal
2	#14583306	C₂	1	4	yes	maximal, complemented
3	#14583303	C₂	1	2	yes	maximal, complemented
4	#14583300	C₂	1	2	yes	maximal, complemented
5	#14582942	C₂	1	2	yes	maximal
6	C₂×#13063259	C₂	1	1	yes	maximal, characteristic
7	#14582854	C₂	1	1	yes	maximal, characteristic
8	C₄₃×#427	C₂	1	1	yes	maximal, characteristic
9	C₂×#10621877	C₂²	1	1	yes	characteristic
10	#13063325	—	2	2	no	—
11	#13063324	C₂²	1	4	yes	—
12	#13063317	—	2	2	no	—
13	#13063316	C₂²	1	4	yes	—
14	#13063302	—	2	4	no	—
15	#13063284	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
17	#13063259	C₂²	1	2	yes	—
18	#13063259	C₂²	1	2	yes	—
19	#13063258	C₂²	1	4	yes	—
20	#13063256	C₂²	1	4	yes	—
21	#13063235	C₂²	1	2	yes	—
22	C₈×Dic₄₃	C₂²	1	2	yes	—
23	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic
24	#117×C₄₃	C₂²	1	2	yes	—
25	#107×C₄₃	C₂²	1	1	yes	characteristic
26	#107×C₄₃	C₂²	1	1	yes	characteristic
27	#106×C₄₃	C₂²	1	2	yes	—
28	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
29	C₂²×Dic₄₃	—	2	2	no	—
30	C₂×Dic₈₆	—	2	4	no	—
31	C₂×C₈×C₄₃	Q₈	1	4	yes	complemented
32	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
35	C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
36	C₂²⋊C₄×C₄₃	—	2	2	no	—
37	C₄²×C₄₃	—	2	2	no	—
38	C₂²⋊Dic₄₃	D₄	1	2	yes	—
39	C₄⋊Dic₄₃	—	2	2	no	—
40	C₄⋊Dic₄₃	—	2	4	no	—
41	C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
42	C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
43	C₄⋊Dic₄₃	C₂³	1	4	yes	—
44	Dic₄₃⋊C₄	—	2	4	no	—
45	Dic₄₃⋊C₄	—	2	4	no	—
46	C₄×Dic₄₃	C₂³	1	2	yes	—
47	#10621877	Q₈	1	4	yes	complemented
48	C₂×C₄₃⋊C₈	C₂³	1	2	yes	—
49	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₄₃	—	2	2	no	—
51	C₂×C₄×C₄₃	—	2	2	no	—
52	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₄₃	C₂×Q₈	1	2	yes	—
54	C₂×C₄×C₄₃	C₂×Q₈	1	2	yes	—
55	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
56	C₂×Dic₄₃	—	2	4	no	—
57	C₂×Dic₄₃	—	8	2	no	—
58	C₂×Dic₄₃	C₂×D₄	1	2	yes	—
59	C₂×Dic₄₃	C₂×D₄	1	2	yes	—
60	Dic₈₆	—	8	4	no	—
61	C₈×C₄₃	C₂×Q₈	1	2	yes	—
62	C₈×C₄₃	C₂×Q₈	1	2	yes	—
63	C₄₃⋊C₈	C₂×Q₈	1	4	yes	—
64	C₂²×C₄₃	—	2	1	no	—
65	C₂²×C₄₃	—	2	1	no	—
66	C₂²×C₄₃	C₄⋊Q₈	1	2	yes	—
67	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
68	C₄×C₄₃	—	4	2	no	—
69	C₄×C₄₃	—	4	2	no	—
70	C₄×C₄₃	C₄⋊Q₈	1	2	yes	—
71	C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
72	C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
73	Dic₄₃	—	2	2	no	—
74	Dic₄₃	—	2	2	no	—
75	Dic₄₃	—	4	4	no	—
76	#3164	—	43	1	no	maximal, Sylow
77	C₂×C₄₃	—	2	2	no	—
78	C₂×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
79	C₂×C₄₃	#576	1	1	yes	characteristic
80	C₂×C₄₃	#578	1	1	yes	characteristic
81	#544	—	43	2	no	—
82	#501	—	43	4	no	—
83	#500	—	43	2	no	—
84	#499	—	43	2	no	—
85	#428	—	43	2	no	—
86	#427	D₄₃	1	1	yes	characteristic
87	C₂×#107	—	43	1	no	—
88	#405	—	43	1	no	—
89	C₄₃	#3164	1	1	yes	characteristic, complemented, Sylow
90	C₂×M₁₆	—	43	1	no	—
91	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic
92	C₄⋊Q₈	—	43	4	no	—
93	C₄⋊Q₈	—	86	2	no	—
94	#125	—	43	4	no	—
95	#125	—	86	2	no	—
96	C₂²⋊Q₈	—	86	4	no	—
97	#117	—	43	1	no	—
98	#117	C₂×D₄₃	1	2	yes	—
99	C₂×C₄⋊C₄	—	43	2	no	—
100	#107	—	43	2	no	—
101	#107	—	43	2	no	—
102	#107	—	43	4	no	—
103	#107	C₂×D₄₃	1	1	yes	characteristic
104	#107	C₂×D₄₃	1	1	yes	characteristic
105	#106	—	43	4	no	—
106	#106	C₂×D₄₃	1	2	yes	—
107	#97	—	43	2	no	—
108	C₄×C₈	—	43	2	no	—
109	C₂×Q₈	—	86	4	no	—
110	C₂²×C₄	—	86	2	no	—
111	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
112	M₁₆	—	43	4	no	—
113	C₂×C₈	—	43	2	no	—
114	C₂×C₈	Dic₈₆	1	4	yes	complemented
115	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
116	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
117	C₄⋊C₄	—	43	1	no	—
118	C₄⋊C₄	—	43	1	no	—
119	C₄⋊C₄	—	43	4	no	—
120	C₄⋊C₄	—	86	4	no	—
121	C₄⋊C₄	—	86	4	no	—
122	C₄⋊C₄	—	86	4	no	—
123	C₄⋊C₄	—	86	2	no	—
124	C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
125	C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
126	C₂²⋊C₄	—	2	2	no	—
127	C₂²⋊C₄	—	43	2	no	—
128	C₄²	—	2	2	no	—
129	C₄²	—	43	2	no	—
130	C₂³	D₄×D₄₃	1	1	yes	characteristic
131	Q₈	—	344	4	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	43	2	no	—
135	C₂×C₄	—	43	2	no	—
136	C₂×C₄	—	86	4	no	—
137	C₂×C₄	—	344	2	no	—
138	C₂×C₄	C₂×Dic₈₆	1	2	yes	—
139	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
140	C₂×C₄	Q₈×D₄₃	1	2	yes	—
141	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
142	C₈	—	43	4	no	—
143	C₈	C₂×Dic₈₆	1	2	yes	—
144	C₈	C₂×Dic₈₆	1	2	yes	—
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	#13063324	1	2	yes	—
148	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
149	C₄	—	4	2	no	—
150	C₄	—	4	2	no	—
151	C₄	—	86	2	no	—
152	C₄	—	86	2	no	—
153	C₄	—	172	4	no	—
154	C₄	#13063324	1	2	yes	—
155	C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
156	C₄	C₂²×Dic₈₆	1	1	yes	characteristic
157	C₂	—	2	2	no	—
158	C₂	C₂×#13063324	1	1	yes	characteristic
159	C₂	#14583701	1	1	yes	characteristic
160	C₂	#14583713	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14583527

C₂

yes

maximal

#14583306

C₂

yes

maximal, complemented

#14583303

C₂

yes

maximal, complemented

#14583300

C₂

yes

maximal, complemented

#14582942

C₂

yes

maximal

C₂×#13063259

C₂

yes

maximal, characteristic

#14582854

C₂

yes

maximal, characteristic

C₄₃×#427

C₂

yes

maximal, characteristic

C₂×#10621877

C₂²

yes

characteristic

#13063325

—

#13063324

C₂²

yes

—

#13063317

—

#13063316

C₂²

yes

—

#13063302

—

#13063284

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₃

C₂²

yes

—

#13063259

C₂²

yes

—

#13063259

C₂²

yes

—

#13063258

C₂²

yes

—

#13063256

C₂²

yes

—

#13063235

C₂²

yes

—

C₈×Dic₄₃

C₂²

yes

—

C₂²×C₈×C₄₃

C₂²

yes

characteristic

#117×C₄₃

C₂²

yes

—

#107×C₄₃

C₂²

yes

characteristic

#107×C₄₃

C₂²

yes

characteristic

#106×C₄₃

C₂²

yes

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

Q₈

yes

complemented

C₂×C₈×C₄₃

C₂³

yes

characteristic

C₂×C₈×C₄₃

C₂³

yes

characteristic

C₄⋊C₄×C₄₃

C₂³

yes

—

C₄⋊C₄×C₄₃

C₂³

yes

—

C₂²⋊C₄×C₄₃

—

C₄²×C₄₃

—

C₂²⋊Dic₄₃

D₄

yes

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

D₄

yes

characteristic

C₄⋊Dic₄₃

D₄

yes

characteristic

C₄⋊Dic₄₃

C₂³

yes

—

Dic₄₃⋊C₄

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

C₂³

yes

—

#10621877

Q₈

yes

complemented

C₂×C₄₃⋊C₈

C₂³

yes

—

C₂³×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×Q₈

yes

—

C₂×C₄×C₄₃

C₂×Q₈

yes

—

C₂×C₄×C₄₃

C₂⁴

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

C₂×D₄

yes

—

C₂×Dic₄₃

C₂×D₄

yes

—

Dic₈₆

—

C₈×C₄₃

C₂×Q₈

yes

—

C₈×C₄₃

C₂×Q₈

yes

—

C₄₃⋊C₈

C₂×Q₈

yes

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₄⋊Q₈

yes

—

C₂²×C₄₃

C₂²×D₄

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₄⋊Q₈

yes

—

C₄×C₄₃

C₂²×Q₈

yes

characteristic

C₄×C₄₃

C₂²×Q₈

yes

characteristic

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

#3164

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

C₂×C₄⋊Q₈

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

—

—

—

—

D₄₃

yes

characteristic

C₂×#107

—

#405

—

C₄₃

#3164

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

C₂×D₄₃

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

#125

—

C₂²⋊Q₈

—

#117

—

#117

C₂×D₄₃

yes

—

C₂×C₄⋊C₄

—

100

#107

—

101

#107

—

102

#107

—

103

#107

C₂×D₄₃

yes

characteristic

104

#107

C₂×D₄₃

yes

characteristic

105

#106

—

106

#106

C₂×D₄₃

yes

—

107

#97

—

108

C₄×C₈

—

109

C₂×Q₈

—

110

C₂²×C₄

—

111

C₂²×C₄

C₂²×D₄₃

yes

characteristic

112

M₁₆

—

113

C₂×C₈

—

114

C₂×C₈

Dic₈₆

yes

complemented

115

C₂×C₈

C₂²×D₄₃

yes

characteristic

116

C₂×C₈

C₂²×D₄₃

yes

characteristic

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

C₂²×D₄₃

yes

—

125

C₄⋊C₄

C₂²×D₄₃

yes

—

126

C₂²⋊C₄

—

127

C₂²⋊C₄

—

128

C₄²

—

129

C₄²

—

130

C₂³

D₄×D₄₃

yes

characteristic

131

Q₈

—

344

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

344

—

138

C₂×C₄

C₂×Dic₈₆

yes

—

139

C₂×C₄

D₄×D₄₃

yes

characteristic

140

C₂×C₄

Q₈×D₄₃

yes

—

141

C₂×C₄

C₂³×D₄₃

yes

characteristic

142

C₈

—

143

C₈

C₂×Dic₈₆

yes

—

144

C₈

C₂×Dic₈₆

yes

—

145

C₂²

—

146

C₂²

—

147

C₂²

#13063324

yes

—

148

C₂²

C₂×D₄×D₄₃

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

172

—

154

C₄

#13063324

yes

—

155

C₄

C₂×Q₈×D₄₃

yes

characteristic

156

C₄

C₂²×Dic₈₆

yes

characteristic

157

C₂

—

158

C₂

C₂×#13063324

yes

characteristic

159

C₂

#14583701

yes

characteristic

160

C₂

#14583713

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#427	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#427	D₄₃
hypocenter	C₄₃	#3164
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂²

C₂×D₄×D₄₃

commutator

C₂×C₄×C₄₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₃

Fitting

C₄₃×#427

C₂

socle

C₂²×C₄₃

C₂²×D₄

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
2. #14583306	C₂	semidirect
3. #14583303	C₂	semidirect
4. #14583300	C₂	semidirect
31. C₂×C₈×C₄₃	Q₈	semidirect
47. #10621877	Q₈	semidirect
89. C₄₃	#3164	semidirect
114. C₂×C₈	Dic₈₆	semidirect

H ≅ G/N

type

2. #14583306

C₂

semidirect

3. #14583303

C₂

semidirect

4. #14583300

C₂

semidirect

31. C₂×C₈×C₄₃

Q₈

semidirect

47. #10621877

Q₈

semidirect

89. C₄₃

#3164

semidirect

114. C₂×C₈

Dic₈₆

semidirect

Subgroups — #19145135

Conjugacy classes of subgroups

Complements