Group #14564597 — subgroups

Subgroups — #14564597

Order 2432.

300 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 145 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×C₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#11926414	C₂	1	1	yes	maximal, characteristic
3	#11926306	C₂	1	4	yes	maximal
4	C₂×C₁₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₈×C₁₉	—	2	1	no	—
6	C₁₉×C₄⋊Q₈	—	2	2	no	—
7	C₁₉×C₄⋊Q₈	C₄	1	4	yes	complemented
8	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
9	C₂×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
10	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₁₉	—	2	1	no	—
12	C₂×C₄⋊Dic₁₉	—	2	1	no	—
13	C₂×C₄⋊Dic₁₉	C₂²	1	2	yes	—
14	C₂²×C₁₉⋊C₈	C₂²	1	2	yes	—
15	C₄×C₁₉⋊C₈	—	2	2	no	—
16	C₂×Q₈×C₁₉	—	2	4	no	—
17	C₂×Q₈×C₁₉	—	4	2	no	—
18	C₂×Q₈×C₁₉	—	4	2	no	—
19	C₂²×C₄×C₁₉	—	2	2	no	—
20	C₂²×C₄×C₁₉	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₉	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
23	C₂²×Dic₁₉	—	2	2	no	—
24	C₄⋊C₄×C₁₉	—	2	4	no	—
25	C₄⋊C₄×C₁₉	—	2	2	no	—
26	C₄⋊C₄×C₁₉	C₂×C₄	1	4	yes	—
27	C₄²×C₁₉	C₂×C₄	1	2	yes	—
28	C₄²×C₁₉	D₄	1	2	yes	—
29	C₄⋊Dic₁₉	—	2	4	no	—
30	C₄⋊Dic₁₉	—	4	2	no	—
31	C₄⋊Dic₁₉	—	4	2	no	—
32	C₂×C₁₉⋊C₈	—	2	2	no	—
33	C₂×C₁₉⋊C₈	—	2	2	no	—
34	C₂×C₁₉⋊C₈	Q₈	1	4	yes	complemented
35	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
36	Q₈×C₁₉	—	4	8	no	—
37	C₂×C₄×C₁₉	—	2	4	no	—
38	C₂×C₄×C₁₉	—	4	2	no	—
39	C₂×C₄×C₁₉	—	4	2	no	—
40	C₂×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₁₉	D₈	1	2	yes	—
43	C₂×C₄×C₁₉	SD₁₆	1	2	yes	—
44	C₂×C₄×C₁₉	SD₁₆	1	2	yes	—
45	C₂×C₄×C₁₉	Q₁₆	1	2	yes	—
46	C₂×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₉	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₁₉	#41	1	2	yes	—
50	C₂×Dic₁₉	—	4	2	no	—
51	C₂×Dic₁₉	—	4	2	no	—
52	C₂×Dic₁₉	—	4	2	no	—
53	C₁₉⋊C₈	—	2	4	no	—
54	#1990	—	19	1	no	maximal, Sylow
55	C₂²×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₁₉	#124	1	1	yes	characteristic
57	C₂²×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
58	C₂²×C₁₉	C₂×D₈	1	1	yes	characteristic
59	C₂²×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
60	C₂²×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
61	C₂²×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
62	C₄×C₁₉	—	4	4	no	—
63	C₄×C₁₉	#102	1	4	yes	—
64	C₄×C₁₉	#103	1	4	yes	—
65	C₄×C₁₉	C₄×Q₈	1	2	yes	—
66	C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
67	Dic₁₉	—	8	2	no	—
68	C₂×C₄⋊Q₈	D₁₉	1	1	yes	characteristic
69	C₂×C₄×C₈	—	19	1	no	—
70	#384	—	19	1	no	—
71	#340	—	19	4	no	—
72	C₂×C₁₉	#391	1	1	yes	characteristic
73	C₂×C₁₉	C₂×#102	1	1	yes	characteristic
74	C₂×C₁₉	C₂×#103	1	1	yes	characteristic
75	C₂×C₁₉	#486	1	1	yes	characteristic
76	C₂×C₁₉	#487	1	1	yes	characteristic
77	C₂×C₁₉	#498	1	1	yes	characteristic
78	C₂×C₁₉	#500	1	1	yes	characteristic
79	C₂²×Q₈	—	2	1	no	—
80	C₂²×C₈	—	19	2	no	—
81	C₄⋊Q₈	—	2	2	no	—
82	C₄⋊Q₈	Dic₁₉	1	4	yes	complemented
83	C₂×C₄⋊C₄	—	2	1	no	—
84	C₂×C₄⋊C₄	—	19	2	no	—
85	C₂×C₄⋊C₄	—	38	1	no	—
86	C₂×C₄⋊C₄	—	38	1	no	—
87	C₂×C₄⋊C₄	C₂×D₁₉	1	2	yes	—
88	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
89	C₄×C₈	—	38	2	no	—
90	C₁₉	#1990	1	1	yes	characteristic, complemented, Sylow
91	C₂×Q₈	—	2	4	no	—
92	C₂×Q₈	—	4	2	no	—
93	C₂×Q₈	—	4	2	no	—
94	C₂²×C₄	—	2	2	no	—
95	C₂²×C₄	—	38	2	no	—
96	C₂²×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
97	C₂²×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
99	C₂×C₈	—	19	4	no	—
100	C₂×C₈	—	38	2	no	—
101	C₂×C₈	—	38	2	no	—
102	C₄⋊C₄	—	2	4	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	38	4	no	—
105	C₄⋊C₄	—	76	2	no	—
106	C₄⋊C₄	—	76	2	no	—
107	C₄⋊C₄	C₂×Dic₁₉	1	4	yes	—
108	C₄²	C₂×Dic₁₉	1	2	yes	—
109	C₄²	C₂²⋊D₁₉	1	2	yes	—
110	C₂³	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
111	Q₈	—	4	8	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	76	2	no	—
116	C₂×C₄	—	76	2	no	—
117	C₂×C₄	—	76	2	no	—
118	C₂×C₄	#64525	1	2	yes	—
119	C₂×C₄	#64526	1	2	yes	—
120	C₂×C₄	#64527	1	2	yes	—
121	C₂×C₄	#64528	1	2	yes	—
122	C₂×C₄	C₂²⋊Dic₁₉	1	2	yes	—
123	C₂×C₄	C₂²⋊Dic₁₉	1	2	yes	—
124	C₂×C₄	Q₈×D₁₉	1	2	yes	—
125	C₂×C₄	#64545	1	2	yes	—
126	C₂×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
127	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
128	C₈	—	38	4	no	—
129	C₂²	C₂×#64525	1	1	yes	characteristic
130	C₂²	C₂×#64526	1	1	yes	characteristic
131	C₂²	C₂×#64527	1	1	yes	characteristic
132	C₂²	C₂×#64528	1	1	yes	characteristic
133	C₂²	#10597390	1	1	yes	characteristic
134	C₂²	#10597393	1	1	yes	characteristic
135	C₂²	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
136	C₄	—	4	4	no	—
137	C₄	—	152	2	no	—
138	C₄	#10597289	1	4	yes	—
139	C₄	#10597292	1	4	yes	—
140	C₄	Q₈×Dic₁₉	1	2	yes	—
141	C₄	#10597392	1	2	yes	—
142	C₂	#11926451	1	1	yes	characteristic
143	C₂	C₂×#10597289	1	1	yes	characteristic
144	C₂	C₂×#10597292	1	1	yes	characteristic
145	C₂	#11926892	1	1	yes	characteristic
146	C₂	#11926896	1	1	yes	characteristic
147	C₂	#11926940	1	1	yes	characteristic
148	C₂	#11926947	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×C₁₉⋊C₈

C₂

yes

maximal, characteristic

#11926414

C₂

yes

maximal, characteristic

#11926306

C₂

yes

maximal

C₂×C₁₉×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂²×Q₈×C₁₉

—

C₁₉×C₄⋊Q₈

—

C₁₉×C₄⋊Q₈

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₉

—

C₂×C₄⋊C₄×C₁₉

C₂²

yes

—

C₂×C₄²×C₁₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₉

—

C₂×C₄⋊Dic₁₉

—

C₂×C₄⋊Dic₁₉

C₂²

yes

—

C₂²×C₁₉⋊C₈

C₂²

yes

—

C₄×C₁₉⋊C₈

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

D₄

yes

characteristic

C₂²×C₄×C₁₉

D₄

yes

characteristic

C₂²×C₄×C₁₉

C₂³

yes

characteristic

C₂²×Dic₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

C₂×C₄

yes

—

C₄²×C₁₉

C₂×C₄

yes

—

C₄²×C₁₉

D₄

yes

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

Q₈

yes

complemented

C₂³×C₁₉

C₂×D₄

yes

characteristic

Q₈×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₉

D₈

yes

—

C₂×C₄×C₁₉

SD₁₆

yes

—

C₂×C₄×C₁₉

SD₁₆

yes

—

C₂×C₄×C₁₉

Q₁₆

yes

—

C₂×C₄×C₁₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉

C₂×Q₈

yes

—

C₂×C₄×C₁₉

#41

yes

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₁₉⋊C₈

—

#1990

—

maximal, Sylow

C₂²×C₁₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₉

#124

yes

characteristic

C₂²×C₁₉

C₄⋊Q₈

yes

characteristic

C₂²×C₁₉

C₂×D₈

yes

characteristic

C₂²×C₁₉

C₂×SD₁₆

yes

characteristic

C₂²×C₁₉

C₂×SD₁₆

yes

characteristic

C₂²×C₁₉

C₂×Q₁₆

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

#102

yes

—

C₄×C₁₉

#103

yes

—

C₄×C₁₉

C₄×Q₈

yes

—

C₄×C₁₉

C₂²⋊Q₈

yes

—

Dic₁₉

—

C₂×C₄⋊Q₈

D₁₉

yes

characteristic

C₂×C₄×C₈

—

#384

—

#340

—

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₁₉

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₉

yes

—

C₂×C₄²

C₂×D₁₉

yes

characteristic

C₄×C₈

—

C₁₉

#1990

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₉

yes

characteristic

C₂²×C₄

C₂²⋊D₁₉

yes

characteristic

C₂²×C₄

C₂²×D₁₉

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

C₂×Dic₁₉

yes

—

108

C₄²

C₂×Dic₁₉

yes

—

109

C₄²

C₂²⋊D₁₉

yes

—

110

C₂³

C₂×C₂²⋊D₁₉

yes

characteristic

111

Q₈

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

#64525

yes

—

119

C₂×C₄

#64526

yes

—

120

C₂×C₄

#64527

yes

—

121

C₂×C₄

#64528

yes

—

122

C₂×C₄

C₂²⋊Dic₁₉

yes

—

123

C₂×C₄

C₂²⋊Dic₁₉

yes

—

124

C₂×C₄

Q₈×D₁₉

yes

—

125

C₂×C₄

#64545

yes

—

126

C₂×C₄

C₂²×Dic₁₉

yes

characteristic

127

C₂×C₄

C₂×C₂²⋊D₁₉

yes

characteristic

128

C₈

—

129

C₂²

C₂×#64525

yes

characteristic

130

C₂²

C₂×#64526

yes

characteristic

131

C₂²

C₂×#64527

yes

characteristic

132

C₂²

C₂×#64528

yes

characteristic

133

C₂²

#10597390

yes

characteristic

134

C₂²

#10597393

yes

characteristic

135

C₂²

C₂×C₂²⋊Dic₁₉

yes

characteristic

136

C₄

—

137

C₄

—

152

—

138

C₄

#10597289

yes

—

139

C₄

#10597292

yes

—

140

C₄

Q₈×Dic₁₉

yes

—

141

C₄

#10597392

yes

—

142

C₂

#11926451

yes

characteristic

143

C₂

C₂×#10597289

yes

characteristic

144

C₂

C₂×#10597292

yes

characteristic

145

C₂

#11926892

yes

characteristic

146

C₂

#11926896

yes

characteristic

147

C₂

#11926940

yes

characteristic

148

C₂

#11926947

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₉
commutator	C₂×C₄×C₁₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉
Fitting	C₂×C₁₉×C₄⋊Q₈	C₂
socle, prime radical	C₂³×C₁₉	C₂×D₄
cosocle	C₂²×C₄×C₁₉	C₂³
hypercenter	C₂×C₄⋊Q₈	D₁₉
hypocenter	C₁₉	#1990
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₉

commutator

C₂×C₄×C₁₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉

Fitting

C₂×C₁₉×C₄⋊Q₈

C₂

socle, prime radical

C₂³×C₁₉

C₂×D₄

cosocle

C₂²×C₄×C₁₉

C₂³

hypercenter

C₂×C₄⋊Q₈

D₁₉

hypocenter

C₁₉

#1990

LCS

G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
7. C₁₉×C₄⋊Q₈	C₄	semidirect
34. C₂×C₁₉⋊C₈	Q₈	semidirect
82. C₄⋊Q₈	Dic₁₉	semidirect
90. C₁₉	#1990	semidirect

H ≅ G/N

type

7. C₁₉×C₄⋊Q₈

C₄

semidirect

34. C₂×C₁₉⋊C₈

Q₈

semidirect

82. C₄⋊Q₈

Dic₁₉

semidirect

90. C₁₉

#1990

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄×C₁₉⋊C₈	2
#11926414	2
#11926306	2
C₂×C₁₉×C₄⋊Q₈	2
#1990	19

subgroup

index

C₂×C₄×C₁₉⋊C₈

#11926414

#11926306

C₂×C₁₉×C₄⋊Q₈

#1990

Subgroups — #14564597

Conjugacy classes of subgroups

Complements

Maximal subgroups