Group #19134722 — subgroups

Subgroups — #19134722

Order 5504.

248 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 133 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×C₄×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#14582769	C₂	1	1	yes	maximal, characteristic
3	#14582661	C₂	1	4	yes	maximal
4	C₄₃×#384	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄₃⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₄₃	—	2	1	no	—
7	C₂×C₄⋊Dic₄₃	—	2	1	no	—
8	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
9	C₄×C₄₃⋊C₈	C₄	1	4	yes	complemented
10	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
11	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
12	C₂×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
13	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
14	C₂²×C₄×C₄₃	—	2	2	no	—
15	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
18	C₂²×Dic₄₃	—	2	2	no	—
19	C₄⋊C₄×C₄₃	—	2	4	no	—
20	C₄⋊C₄×C₄₃	—	4	2	no	—
21	C₄⋊C₄×C₄₃	—	4	2	no	—
22	C₄²×C₄₃	C₂×C₄	1	2	yes	—
23	C₄²×C₄₃	Q₈	1	2	yes	—
24	C₄⋊Dic₄₃	—	2	4	no	—
25	C₄⋊Dic₄₃	—	4	2	no	—
26	C₄⋊Dic₄₃	—	4	2	no	—
27	C₂×C₄₃⋊C₈	—	2	2	no	—
28	C₂×C₄₃⋊C₈	—	2	2	no	—
29	C₂×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
30	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₄₃	—	4	2	no	—
32	C₂×C₄×C₄₃	—	4	2	no	—
33	C₂×C₄×C₄₃	—	4	2	no	—
34	C₂×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
35	C₂×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
36	C₂×C₄×C₄₃	D₈	1	2	yes	—
37	C₂×C₄×C₄₃	SD₁₆	1	2	yes	—
38	C₂×C₄×C₄₃	SD₁₆	1	2	yes	—
39	C₂×C₄×C₄₃	Q₁₆	1	2	yes	—
40	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₄₃	#41	1	2	yes	—
43	C₂×C₄×C₄₃	#41	1	2	yes	—
44	C₂×Dic₄₃	—	4	2	no	—
45	C₂×Dic₄₃	—	4	2	no	—
46	C₂×Dic₄₃	—	4	2	no	—
47	C₄₃⋊C₈	—	2	4	no	—
48	C₂²×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄₃	#124	1	1	yes	characteristic
50	C₂²×C₄₃	#124	1	1	yes	characteristic
51	C₂²×C₄₃	C₂×D₈	1	1	yes	characteristic
52	C₂²×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
53	C₂²×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
54	C₂²×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
55	C₄×C₄₃	—	8	2	no	—
56	C₄×C₄₃	#106	1	4	yes	—
57	C₄×C₄₃	#107	1	4	yes	—
58	C₄×C₄₃	#117	1	2	yes	—
59	C₄×C₄₃	#125	1	2	yes	—
60	Dic₄₃	—	8	2	no	—
61	#1834	—	43	1	no	maximal, Sylow
62	C₂×C₄₃	#382	1	1	yes	characteristic
63	C₂×C₄₃	C₂×#106	1	1	yes	characteristic
64	C₂×C₄₃	C₂×#107	1	1	yes	characteristic
65	C₂×C₄₃	#486	1	1	yes	characteristic
66	C₂×C₄₃	#486	1	1	yes	characteristic
67	C₂×C₄₃	#487	1	1	yes	characteristic
68	C₂×C₄₃	#487	1	1	yes	characteristic
69	C₂×C₄×C₈	—	43	1	no	—
70	#384	—	43	1	no	—
71	#384	D₄₃	1	1	yes	characteristic
72	#340	—	43	4	no	—
73	C₄₃	#1834	1	1	yes	characteristic, complemented, Sylow
74	C₂²×C₈	—	43	2	no	—
75	C₂×C₄⋊C₄	—	2	1	no	—
76	C₂×C₄⋊C₄	—	2	1	no	—
77	C₂×C₄⋊C₄	—	43	2	no	—
78	C₂×C₄⋊C₄	—	86	1	no	—
79	C₂×C₄⋊C₄	—	86	1	no	—
80	C₂×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
81	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
82	C₄×C₈	—	43	4	no	—
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	—	86	2	no	—
85	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
86	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
87	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
88	C₂×C₈	—	43	4	no	—
89	C₂×C₈	—	86	2	no	—
90	C₂×C₈	—	86	2	no	—
91	C₄⋊C₄	—	2	4	no	—
92	C₄⋊C₄	—	4	2	no	—
93	C₄⋊C₄	—	4	2	no	—
94	C₄⋊C₄	—	86	4	no	—
95	C₄⋊C₄	—	172	2	no	—
96	C₄⋊C₄	—	172	2	no	—
97	C₄²	Dic₈₆	1	2	yes	—
98	C₄²	C₄×D₄₃	1	2	yes	—
99	C₂³	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	172	2	no	—
104	C₂×C₄	—	172	2	no	—
105	C₂×C₄	—	172	2	no	—
106	C₂×C₄	Dic₄₃⋊C₄	1	2	yes	—
107	C₂×C₄	Dic₄₃⋊C₄	1	2	yes	—
108	C₂×C₄	#10621882	1	2	yes	—
109	C₂×C₄	#10621883	1	2	yes	—
110	C₂×C₄	#10621884	1	2	yes	—
111	C₂×C₄	#10621885	1	2	yes	—
112	C₂×C₄	C₂×Dic₈₆	1	1	yes	characteristic
113	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
114	C₂×C₄	#10621900	1	2	yes	—
115	C₂×C₄	#10621902	1	2	yes	—
116	C₈	—	86	4	no	—
117	C₂²	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
118	C₂²	#13063312	1	1	yes	characteristic
119	C₂²	#13063314	1	1	yes	characteristic
120	C₂²	C₂×#10621882	1	1	yes	characteristic
121	C₂²	C₂×#10621884	1	1	yes	characteristic
122	C₂²	C₂×#10621883	1	1	yes	characteristic
123	C₂²	C₂×#10621885	1	1	yes	characteristic
124	C₄	—	8	2	no	—
125	C₄	—	344	2	no	—
126	C₄	#13063256	1	4	yes	—
127	C₄	#13063258	1	4	yes	—
128	C₄	#13063283	1	2	yes	—
129	C₄	#13063316	1	2	yes	—
130	C₂	#14582764	1	1	yes	characteristic
131	C₂	C₂×#13063256	1	1	yes	characteristic
132	C₂	C₂×#13063258	1	1	yes	characteristic
133	C₂	#14583245	1	1	yes	characteristic
134	C₂	#14583247	1	1	yes	characteristic
135	C₂	#14583249	1	1	yes	characteristic
136	C₂	#14583251	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×C₄₃⋊C₈

C₂

yes

maximal, characteristic

#14582769

C₂

yes

maximal, characteristic

#14582661

C₂

yes

maximal

C₄₃×#384

C₂

yes

maximal, characteristic

C₂²×C₄₃⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

C₂²

yes

—

C₄×C₄₃⋊C₈

C₄

yes

complemented

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄⋊C₄×C₄₃

C₂²

yes

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

C₂×C₄

yes

—

C₄²×C₄₃

Q₈

yes

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

C₂×C₄

yes

—

C₂³×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₄⋊C₄

yes

—

C₂×C₄×C₄₃

C₄⋊C₄

yes

—

C₂×C₄×C₄₃

D₈

yes

—

C₂×C₄×C₄₃

SD₁₆

yes

—

C₂×C₄×C₄₃

SD₁₆

yes

—

C₂×C₄×C₄₃

Q₁₆

yes

—

C₂×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₄₃

#41

yes

—

C₂×C₄×C₄₃

#41

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₄₃⋊C₈

—

C₂²×C₄₃

C₂×C₄⋊C₄

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

C₂×D₈

yes

characteristic

C₂²×C₄₃

C₂×SD₁₆

yes

characteristic

C₂²×C₄₃

C₂×SD₁₆

yes

characteristic

C₂²×C₄₃

C₂×Q₁₆

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

#106

yes

—

C₄×C₄₃

#107

yes

—

C₄×C₄₃

#117

yes

—

C₄×C₄₃

#125

yes

—

Dic₄₃

—

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄×C₈

—

#384

—

#384

D₄₃

yes

characteristic

#340

—

C₄₃

#1834

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₄₃

yes

—

C₂×C₄²

C₂×D₄₃

yes

characteristic

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

C₂²×C₄

C₂²×D₄₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

172

—

C₄⋊C₄

—

172

—

C₄²

Dic₈₆

yes

—

C₄²

C₄×D₄₃

yes

—

C₂³

C₂×C₂²⋊D₄₃

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

172

—

104

C₂×C₄

—

172

—

105

C₂×C₄

—

172

—

106

C₂×C₄

Dic₄₃⋊C₄

yes

—

107

C₂×C₄

Dic₄₃⋊C₄

yes

—

108

C₂×C₄

#10621882

yes

—

109

C₂×C₄

#10621883

yes

—

110

C₂×C₄

#10621884

yes

—

111

C₂×C₄

#10621885

yes

—

112

C₂×C₄

C₂×Dic₈₆

yes

characteristic

113

C₂×C₄

C₂×C₄×D₄₃

yes

characteristic

114

C₂×C₄

#10621900

yes

—

115

C₂×C₄

#10621902

yes

—

116

C₈

—

117

C₂²

C₂×Dic₄₃⋊C₄

yes

characteristic

118

C₂²

#13063312

yes

characteristic

119

C₂²

#13063314

yes

characteristic

120

C₂²

C₂×#10621882

yes

characteristic

121

C₂²

C₂×#10621884

yes

characteristic

122

C₂²

C₂×#10621883

yes

characteristic

123

C₂²

C₂×#10621885

yes

characteristic

124

C₄

—

125

C₄

—

344

—

126

C₄

#13063256

yes

—

127

C₄

#13063258

yes

—

128

C₄

#13063283

yes

—

129

C₄

#13063316

yes

—

130

C₂

#14582764

yes

characteristic

131

C₂

C₂×#13063256

yes

characteristic

132

C₂

C₂×#13063258

yes

characteristic

133

C₂

#14583245

yes

characteristic

134

C₂

#14583247

yes

characteristic

135

C₂

#14583249

yes

characteristic

136

C₂

#14583251

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₄₃
commutator	C₂×C₄×C₄₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₃
Fitting	C₄₃×#384	C₂
socle	C₂³×C₄₃	C₂×D₄
hypercenter	#384	D₄₃
hypocenter	C₄₃	#1834
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₄₃

commutator

C₂×C₄×C₄₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₃

Fitting

C₄₃×#384

C₂

socle

C₂³×C₄₃

C₂×D₄

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
9. C₄×C₄₃⋊C₈	C₄	semidirect
73. C₄₃	#1834	semidirect

H ≅ G/N

type

9. C₄×C₄₃⋊C₈

C₄

semidirect

73. C₄₃

#1834

semidirect

Subgroups — #19134722

Conjugacy classes of subgroups

Complements