Group #19135347 — subgroups

Subgroups — #19135347

Order 5504.

200 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 83 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×#13063260	C₂	1	1	yes	maximal, characteristic
2	#14582935	C₂	1	1	yes	maximal, characteristic
3	#14582927	C₂	1	1	yes	maximal, characteristic
4	#14582926	C₂	1	1	yes	maximal, characteristic
5	C₂×#13063236	C₂	1	1	yes	maximal, characteristic
6	#14582659	C₂	1	1	yes	maximal, characteristic
7	C₄₃×#339	C₂	1	1	yes	maximal, characteristic
8	C₂×#10621877	C₂²	1	1	yes	characteristic
9	C₂²×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
11	#13063284	C₂²	1	1	yes	characteristic
12	#13063260	—	2	2	no	—
13	#13063258	C₄	1	2	yes	complemented
14	#13063256	C₄	1	2	yes	complemented
15	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
16	Dic₄₃⋊C₈	—	2	2	no	—
17	#13063236	—	2	2	no	—
18	C₂×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
19	#117×C₄₃	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
21	C₂²×C₄×C₄₃	—	2	1	no	—
22	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
23	M₁₆×C₄₃	—	2	1	no	—
24	M₁₆×C₄₃	—	2	1	no	—
25	C₂×C₈×C₄₃	—	2	1	no	—
26	C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
27	C₂²⋊C₄×C₄₃	—	2	1	no	—
28	C₄²×C₄₃	—	2	1	no	—
29	C₄²×C₄₃	—	2	1	no	—
30	C₄²×C₄₃	Q₈	1	2	yes	—
31	C₂²⋊Dic₄₃	—	2	1	no	—
32	C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
33	C₄×Dic₄₃	—	2	1	no	—
34	#10621877	—	2	1	no	—
35	#10621877	—	2	1	no	—
36	C₂×C₄₃⋊C₈	—	2	1	no	—
37	C₂×C₄₃⋊C₈	—	2	1	no	—
38	C₂×C₄₃⋊C₈	—	2	1	no	—
39	C₂×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄₃⋊C₈	D₄	1	2	yes	—
42	C₂×C₄₃⋊C₈	D₄	1	2	yes	—
43	C₂×C₄₃⋊C₈	Q₈	1	2	yes	—
44	C₂³×C₄₃	#41	1	1	yes	characteristic
45	C₂×C₄×C₄₃	—	2	2	no	—
46	C₂×C₄×C₄₃	—	2	2	no	—
47	C₂×C₄×C₄₃	—	4	2	no	—
48	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
54	C₂×Dic₄₃	—	2	2	no	—
55	C₈×C₄₃	—	4	1	no	—
56	C₄₃⋊C₈	—	2	2	no	—
57	C₄₃⋊C₈	—	4	1	no	—
58	C₄₃⋊C₈	C₄⋊C₄	1	2	yes	—
59	C₄₃⋊C₈	C₄⋊C₄	1	2	yes	—
60	C₂²×C₄₃	—	2	2	no	—
61	C₂²×C₄₃	C₄×D₄	1	1	yes	characteristic
62	C₂²×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
63	C₂²×C₄₃	#125	1	1	yes	characteristic
64	C₄×C₄₃	—	4	2	no	—
65	C₄×C₄₃	—	4	2	no	—
66	C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₄×C₄₃	C₄×Q₈	1	1	yes	characteristic
68	C₄×C₄₃	#121	1	1	yes	characteristic
69	C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
70	Dic₄₃	—	4	2	no	—
71	#1946	—	43	1	no	maximal, Sylow
72	C₂×C₄₃	—	2	1	no	—
73	C₂×C₄₃	—	2	1	no	—
74	C₂×C₄₃	#389	1	1	yes	characteristic
75	C₂×C₄₃	#444	1	2	yes	—
76	C₂×#108	—	43	1	no	—
77	#427	—	43	1	no	—
78	#424	—	43	1	no	—
79	#424	—	43	1	no	—
80	C₂×#97	—	43	1	no	—
81	#339	—	43	1	no	—
82	#339	D₄₃	1	1	yes	characteristic
83	C₄₃	#1946	1	1	yes	characteristic, complemented, Sylow
84	C₂×M₁₆	—	43	1	no	—
85	C₂×M₁₆	C₂×D₄₃	1	1	yes	characteristic
86	C₂²×C₈	—	43	1	no	—
87	C₂²×C₈	—	43	1	no	—
88	#117	—	43	1	no	—
89	#117	C₂×D₄₃	1	1	yes	characteristic
90	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
91	#108	—	86	2	no	—
92	#107	—	43	2	no	—
93	#106	—	43	2	no	—
94	C₄⋊C₈	—	86	2	no	—
95	C₄⋊C₈	—	86	2	no	—
96	#97	—	86	2	no	—
97	C₂²×C₄	—	2	1	no	—
98	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
99	M₁₆	—	2	1	no	—
100	M₁₆	—	2	1	no	—
101	M₁₆	—	86	1	no	—
102	M₁₆	—	86	1	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	43	1	no	—
105	C₂×C₈	—	43	1	no	—
106	C₂×C₈	—	43	2	no	—
107	C₂×C₈	—	43	2	no	—
108	C₂×C₈	—	43	2	no	—
109	C₂×C₈	—	86	1	no	—
110	C₂×C₈	—	86	1	no	—
111	C₂×C₈	—	86	1	no	—
112	C₄⋊C₄	—	43	2	no	—
113	C₄⋊C₄	C₄×D₄₃	1	2	yes	—
114	C₂²⋊C₄	—	2	1	no	—
115	C₂²⋊C₄	—	86	1	no	—
116	C₄²	—	2	1	no	—
117	C₄²	—	2	1	no	—
118	C₄²	—	86	1	no	—
119	C₄²	Dic₈₆	1	2	yes	—
120	C₂³	#10621900	1	1	yes	characteristic
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	86	2	no	—
125	C₂×C₄	C₂×Dic₈₆	1	1	yes	characteristic
126	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
127	C₂×C₄	#10621898	1	1	yes	characteristic
128	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
129	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
130	C₂×C₄	Q₈×D₄₃	1	1	yes	characteristic
131	C₈	—	4	1	no	—
132	C₈	—	43	2	no	—
133	C₈	—	43	2	no	—
134	C₈	—	86	2	no	—
135	C₈	—	172	1	no	—
136	C₂²	—	2	2	no	—
137	C₂²	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
138	C₂²	#13063302	1	1	yes	characteristic
139	C₂²	#13063315	1	1	yes	characteristic
140	C₄	—	4	2	no	—
141	C₄	—	4	2	no	—
142	C₄	—	172	2	no	—
143	C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
144	C₄	C₄×Dic₈₆	1	1	yes	characteristic
145	C₄	#13063298	1	1	yes	characteristic
146	C₄	#13063324	1	1	yes	characteristic
147	C₂	—	2	1	no	—
148	C₂	—	2	1	no	—
149	C₂	#14582796	1	1	yes	characteristic
150	C₂	#14583018	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13063260

C₂

yes

maximal, characteristic

#14582935

C₂

yes

maximal, characteristic

#14582927

C₂

yes

maximal, characteristic

#14582926

C₂

yes

maximal, characteristic

C₂×#13063236

C₂

yes

maximal, characteristic

#14582659

C₂

yes

maximal, characteristic

C₄₃×#339

C₂

yes

maximal, characteristic

C₂×#10621877

C₂²

yes

characteristic

C₂²×C₄₃⋊C₈

C₂²

yes

characteristic

C₂²×C₄₃⋊C₈

C₂²

yes

characteristic

#13063284

C₂²

yes

characteristic

#13063260

—

#13063258

C₄

yes

complemented

#13063256

C₄

yes

complemented

C₄⋊(C₄₃⋊C₈)

—

Dic₄₃⋊C₈

—

#13063236

—

C₂×M₁₆×C₄₃

C₂²

yes

characteristic

#117×C₄₃

C₂²

yes

characteristic

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

M₁₆×C₄₃

—

M₁₆×C₄₃

—

C₂×C₈×C₄₃

—

C₄⋊C₄×C₄₃

C₂×C₄

yes

—

C₂²⋊C₄×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

Q₈

yes

—

C₂²⋊Dic₄₃

—

C₄⋊Dic₄₃

C₂×C₄

yes

—

C₄×Dic₄₃

—

#10621877

—

#10621877

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

C₂×C₄

yes

characteristic

C₂×C₄₃⋊C₈

C₂×C₄

yes

characteristic

C₂×C₄₃⋊C₈

D₄

yes

—

C₂×C₄₃⋊C₈

D₄

yes

—

C₂×C₄₃⋊C₈

Q₈

yes

—

C₂³×C₄₃

#41

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

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×Q₈

yes

characteristic

C₂×C₄×C₄₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₄₃

#41

yes

characteristic

C₂×Dic₄₃

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₄₃⋊C₈

C₄⋊C₄

yes

—

C₄₃⋊C₈

C₄⋊C₄

yes

—

C₂²×C₄₃

—

C₂²×C₄₃

C₄×D₄

yes

characteristic

C₂²×C₄₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₄₃

#125

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₄₃

C₄×Q₈

yes

characteristic

C₄×C₄₃

#121

yes

characteristic

C₄×C₄₃

C₄⋊Q₈

yes

characteristic

Dic₄₃

—

#1946

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

#389

yes

characteristic

C₂×C₄₃

#444

yes

—

—

—

—

—

C₂×#97

—

#339

—

#339

D₄₃

yes

characteristic

C₄₃

#1946

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

C₂×D₄₃

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

#117

—

#117

C₂×D₄₃

yes

characteristic

C₂×C₄²

C₂×D₄₃

yes

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₄₃

yes

characteristic

M₁₆

—

100

M₁₆

—

101

M₁₆

—

102

M₁₆

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₄⋊C₄

—

113

C₄⋊C₄

C₄×D₄₃

yes

—

114

C₂²⋊C₄

—

115

C₂²⋊C₄

—

116

C₄²

—

117

C₄²

—

118

C₄²

—

119

C₄²

Dic₈₆

yes

—

120

C₂³

#10621900

yes

characteristic

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

C₂×Dic₈₆

yes

characteristic

126

C₂×C₄

C₂×C₄×D₄₃

yes

characteristic

127

C₂×C₄

#10621898

yes

characteristic

128

C₂×C₄

D₄×D₄₃

yes

characteristic

129

C₂×C₄

D₄×D₄₃

yes

characteristic

130

C₂×C₄

Q₈×D₄₃

yes

characteristic

131

C₈

—

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

—

172

—

136

C₂²

—

137

C₂²

(C₂²⋊D₄₃)⋊C₄

yes

characteristic

138

C₂²

#13063302

yes

characteristic

139

C₂²

#13063315

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

172

—

143

C₄

C₄⋊C₄×D₄₃

yes

characteristic

144

C₄

C₄×Dic₈₆

yes

characteristic

145

C₄

#13063298

yes

characteristic

146

C₄

#13063324

yes

characteristic

147

C₂

—

148

C₂

—

149

C₂

#14582796

yes

characteristic

150

C₂

#14583018

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₃
Fitting	C₄₃×#339	C₂
socle	C₂²×C₄₃	C₄×D₄
hypercenter	#339	D₄₃
hypocenter	C₄₃	#1946
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #339

subgroup

G/H

center

C₂×C₄

D₄×D₄₃

commutator

C₂×C₄×C₄₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₃

Fitting

C₄₃×#339

C₂

socle

C₂²×C₄₃

C₄×D₄

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #339

N	H ≅ G/N	type
13. #13063258	C₄	semidirect
14. #13063256	C₄	semidirect
83. C₄₃	#1946	semidirect

H ≅ G/N

type

13. #13063258

C₄

semidirect

14. #13063256

C₄

semidirect

83. C₄₃

#1946

semidirect

Subgroups — #19135347

Conjugacy classes of subgroups

Complements