Group #14572724 — subgroups

Subgroups — #14572724

Order 2432.

364 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 221 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	#11926675	C₂	1	6	yes	maximal, complemented
2	Q₈×C₁₉⋊C₈	C₂	1	2	yes	maximal, complemented
3	#11926601	C₂	1	3	yes	maximal
4	#11926597	C₂	1	3	yes	maximal
5	C₂×C₄×Q₈×C₁₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₁₉	C₄	1	1	yes	characteristic
7	C₄×Q₈×C₁₉	C₄	1	3	yes	—
8	C₄×Q₈×C₁₉	C₄	1	1	yes	characteristic
9	C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
10	C₄×Q₈×C₁₉	C₂²	1	3	yes	—
11	C₂×C₄⋊C₄×C₁₉	C₄	1	3	yes	—
12	C₂×C₄²×C₁₉	C₂²	1	3	yes	—
13	C₂²⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂²⋊(C₁₉⋊C₈)	C₂²	1	3	yes	—
15	C₄⋊(C₁₉⋊C₈)	C₂²	1	6	yes	—
16	C₄⋊(C₁₉⋊C₈)	C₂²	1	6	yes	—
17	#10597260	C₂²	1	6	yes	—
18	C₄×C₁₉⋊C₈	C₂²	1	6	yes	—
19	C₂×Q₈×C₁₉	—	2	4	no	—
20	C₂×Q₈×C₁₉	C₂×C₄	1	3	yes	—
21	C₂×Q₈×C₁₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₉	C₂×C₄	1	3	yes	—
23	C₂²×C₄×C₁₉	C₂×C₄	1	3	yes	—
24	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
25	C₄⋊C₄×C₁₉	C₂×C₄	1	6	yes	—
26	C₄⋊C₄×C₁₉	C₂×C₄	1	3	yes	—
27	C₄⋊C₄×C₁₉	C₂×C₄	1	3	yes	—
28	C₄²×C₁₉	C₂×C₄	1	3	yes	—
29	C₄²×C₁₉	C₂×C₄	1	3	yes	—
30	C₄²×C₁₉	C₂³	1	3	yes	—
31	C₄²×C₁₉	C₂³	1	3	yes	—
32	C₂×C₁₉⋊C₈	C₂³	1	2	yes	—
33	C₂×C₁₉⋊C₈	C₂³	1	6	yes	—
34	C₂³×C₁₉	C₂²×C₄	1	1	yes	characteristic
35	Q₈×C₁₉	—	2	6	no	—
36	Q₈×C₁₉	M₁₆	1	4	yes	—
37	C₂×C₄×C₁₉	—	2	6	no	—
38	C₂×C₄×C₁₉	—	2	6	no	—
39	C₂×C₄×C₁₉	—	2	2	no	—
40	C₂×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₉	C₂²×C₄	1	3	yes	—
42	C₂×C₄×C₁₉	C₂²×C₄	1	3	yes	—
43	C₂×C₄×C₁₉	C₂²×C₄	1	3	yes	—
44	C₂×C₄×C₁₉	C₂²×C₄	1	3	yes	—
45	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
46	C₁₉⋊C₈	—	2	2	no	—
47	C₁₉⋊C₈	—	2	6	no	—
48	#2984	—	19	1	no	maximal, Sylow
49	C₂²×C₁₉	—	2	1	no	—
50	C₂²×C₁₉	—	2	1	no	—
51	C₂²×C₁₉	#131	1	2	yes	—
52	C₂²×C₁₉	C₂³×C₄	1	1	yes	characteristic
53	C₄×C₁₉	—	2	1	no	—
54	C₄×C₁₉	—	2	3	no	—
55	C₄×C₁₉	—	2	3	no	—
56	C₄×C₁₉	—	2	3	no	—
57	C₄×C₁₉	C₂×M₁₆	1	6	yes	—
58	C₄×C₁₉	2₋¹⁺⁴	1	2	yes	—
59	C₂×C₄×Q₈	D₁₉	1	1	yes	characteristic
60	#446	—	19	6	no	—
61	C₈×Q₈	—	19	2	no	—
62	#432	—	19	3	no	—
63	#431	—	19	3	no	—
64	C₂×C₁₉	—	2	2	no	—
65	C₂×C₁₉	#519	1	1	yes	characteristic
66	C₂×C₁₉	C₂²×M₁₆	1	1	yes	characteristic
67	C₂×C₁₉	C₂×#131	1	1	yes	characteristic
68	C₂²×Q₈	Dic₁₉	1	1	yes	characteristic
69	C₄×Q₈	Dic₁₉	1	3	yes	—
70	C₄×Q₈	Dic₁₉	1	1	yes	characteristic
71	C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic
72	C₄×Q₈	C₂×D₁₉	1	3	yes	—
73	C₂×C₄⋊C₄	Dic₁₉	1	3	yes	—
74	C₂×C₄²	C₂×D₁₉	1	3	yes	—
75	C₄⋊C₈	—	19	6	no	—
76	C₄⋊C₈	—	19	6	no	—
77	C₂²⋊C₈	—	19	1	no	—
78	C₂²⋊C₈	—	19	3	no	—
79	#97	—	19	6	no	—
80	C₄×C₈	—	19	6	no	—
81	C₁₉	#2984	1	1	yes	characteristic, complemented, Sylow
82	C₂×Q₈	—	2	4	no	—
83	C₂×Q₈	C₂×Dic₁₉	1	3	yes	—
84	C₂×Q₈	C₂×Dic₁₉	1	1	yes	characteristic
85	C₂²×C₄	C₂×Dic₁₉	1	3	yes	—
86	C₂²×C₄	C₂×Dic₁₉	1	3	yes	—
87	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
88	C₂×C₈	—	19	2	no	—
89	C₂×C₈	—	19	6	no	—
90	C₄⋊C₄	C₂×Dic₁₉	1	6	yes	—
91	C₄⋊C₄	C₂×Dic₁₉	1	3	yes	—
92	C₄⋊C₄	C₂×Dic₁₉	1	3	yes	—
93	C₄²	C₂×Dic₁₉	1	3	yes	—
94	C₄²	C₂×Dic₁₉	1	3	yes	—
95	C₄²	C₂²×D₁₉	1	3	yes	—
96	C₄²	C₂²×D₁₉	1	3	yes	—
97	C₂³	C₂²×Dic₁₉	1	1	yes	characteristic
98	Q₈	—	2	6	no	—
99	Q₈	#64520	1	4	yes	—
100	C₂×C₄	—	2	6	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	2	6	no	—
103	C₂×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
104	C₂×C₄	C₂²×Dic₁₉	1	3	yes	—
105	C₂×C₄	C₂²×Dic₁₉	1	3	yes	—
106	C₂×C₄	C₂²×Dic₁₉	1	3	yes	—
107	C₂×C₄	C₂²×Dic₁₉	1	3	yes	—
108	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
109	C₈	—	38	2	no	—
110	C₈	—	38	6	no	—
111	C₂²	—	2	1	no	—
112	C₂²	—	2	1	no	—
113	C₂²	#10597394	1	2	yes	—
114	C₂²	C₂³×Dic₁₉	1	1	yes	characteristic
115	C₄	—	2	3	no	—
116	C₄	—	2	3	no	—
117	C₄	—	2	3	no	—
118	C₄	—	2	1	no	—
119	C₄	C₂×#64520	1	6	yes	—
120	C₄	#10597433	1	2	yes	—
121	C₂	—	2	2	no	—
122	C₂	#11927022	1	1	yes	characteristic
123	C₂	C₂²×#64520	1	1	yes	characteristic
124	C₂	C₂×#10597394	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11926675

C₂

yes

maximal, complemented

Q₈×C₁₉⋊C₈

C₂

yes

maximal, complemented

#11926601

C₂

yes

maximal

#11926597

C₂

yes

maximal

C₂×C₄×Q₈×C₁₉

C₂

yes

maximal, characteristic

C₂²×Q₈×C₁₉

C₄

yes

characteristic

C₄×Q₈×C₁₉

C₄

yes

—

C₄×Q₈×C₁₉

C₄

yes

characteristic

C₄×Q₈×C₁₉

C₂²

yes

characteristic

C₄×Q₈×C₁₉

C₂²

yes

—

C₂×C₄⋊C₄×C₁₉

C₄

yes

—

C₂×C₄²×C₁₉

C₂²

yes

—

C₂²⋊(C₁₉⋊C₈)

C₂²

yes

characteristic

C₂²⋊(C₁₉⋊C₈)

C₂²

yes

—

C₄⋊(C₁₉⋊C₈)

C₂²

yes

—

C₄⋊(C₁₉⋊C₈)

C₂²

yes

—

#10597260

C₂²

yes

—

C₄×C₁₉⋊C₈

C₂²

yes

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

C₂×C₄

yes

—

C₂×Q₈×C₁₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉

C₂×C₄

yes

—

C₂²×C₄×C₁₉

C₂×C₄

yes

—

C₂²×C₄×C₁₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₉

C₂×C₄

yes

—

C₄⋊C₄×C₁₉

C₂×C₄

yes

—

C₄⋊C₄×C₁₉

C₂×C₄

yes

—

C₄²×C₁₉

C₂×C₄

yes

—

C₄²×C₁₉

C₂×C₄

yes

—

C₄²×C₁₉

C₂³

yes

—

C₄²×C₁₉

C₂³

yes

—

C₂×C₁₉⋊C₈

C₂³

yes

—

C₂×C₁₉⋊C₈

C₂³

yes

—

C₂³×C₁₉

C₂²×C₄

yes

characteristic

Q₈×C₁₉

—

Q₈×C₁₉

M₁₆

yes

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉

C₂²×C₄

yes

—

C₂×C₄×C₁₉

C₂²×C₄

yes

—

C₂×C₄×C₁₉

C₂²×C₄

yes

—

C₂×C₄×C₁₉

C₂²×C₄

yes

—

C₂×C₄×C₁₉

C₂⁴

yes

characteristic

C₁₉⋊C₈

—

C₁₉⋊C₈

—

#2984

—

maximal, Sylow

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#131

yes

—

C₂²×C₁₉

C₂³×C₄

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

C₂×M₁₆

yes

—

C₄×C₁₉

2₋¹⁺⁴

yes

—

C₂×C₄×Q₈

D₁₉

yes

characteristic

#446

—

C₈×Q₈

—

#432

—

#431

—

C₂×C₁₉

—

C₂×C₁₉

#519

yes

characteristic

C₂×C₁₉

C₂²×M₁₆

yes

characteristic

C₂×C₁₉

C₂×#131

yes

characteristic

C₂²×Q₈

Dic₁₉

yes

characteristic

C₄×Q₈

Dic₁₉

yes

—

C₄×Q₈

Dic₁₉

yes

characteristic

C₄×Q₈

C₂×D₁₉

yes

characteristic

C₄×Q₈

C₂×D₁₉

yes

—

C₂×C₄⋊C₄

Dic₁₉

yes

—

C₂×C₄²

C₂×D₁₉

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₁₉

#2984

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

C₂×Dic₁₉

yes

—

C₂×Q₈

C₂×Dic₁₉

yes

characteristic

C₂²×C₄

C₂×Dic₁₉

yes

—

C₂²×C₄

C₂×Dic₁₉

yes

—

C₂²×C₄

C₂²×D₁₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

C₂×Dic₁₉

yes

—

C₄⋊C₄

C₂×Dic₁₉

yes

—

C₄⋊C₄

C₂×Dic₁₉

yes

—

C₄²

C₂×Dic₁₉

yes

—

C₄²

C₂×Dic₁₉

yes

—

C₄²

C₂²×D₁₉

yes

—

C₄²

C₂²×D₁₉

yes

—

C₂³

C₂²×Dic₁₉

yes

characteristic

Q₈

—

Q₈

#64520

yes

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

C₂²×Dic₁₉

yes

characteristic

104

C₂×C₄

C₂²×Dic₁₉

yes

—

105

C₂×C₄

C₂²×Dic₁₉

yes

—

106

C₂×C₄

C₂²×Dic₁₉

yes

—

107

C₂×C₄

C₂²×Dic₁₉

yes

—

108

C₂×C₄

C₂³×D₁₉

yes

characteristic

109

C₈

—

110

C₈

—

111

C₂²

—

112

C₂²

—

113

C₂²

#10597394

yes

—

114

C₂²

C₂³×Dic₁₉

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

C₂×#64520

yes

—

120

C₄

#10597433

yes

—

121

C₂

—

122

C₂

#11927022

yes

characteristic

123

C₂

C₂²×#64520

yes

characteristic

124

C₂

C₂×#10597394

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₉
commutator, socle	C₂²×C₁₉	C₂³×C₄
Fitting	C₂×C₄×Q₈×C₁₉	C₂
prime radical	C₂³×C₁₉	C₂²×C₄
cosocle	C₂×C₄×C₁₉	C₂⁴
hypercenter	C₂×C₄×Q₈	D₁₉
hypocenter	C₁₉	#2984
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₉

commutator, socle

C₂²×C₁₉

C₂³×C₄

Fitting

C₂×C₄×Q₈×C₁₉

C₂

prime radical

C₂³×C₁₉

C₂²×C₄

cosocle

C₂×C₄×C₁₉

C₂⁴

hypercenter

C₂×C₄×Q₈

D₁₉

C₁₉

G ▷ C₂²×C₁₉ ▷ C₁₉

C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
1. #11926675	C₂	semidirect
2. Q₈×C₁₉⋊C₈	C₂	semidirect
81. C₁₉	#2984	semidirect

H ≅ G/N

type

1. #11926675

C₂

semidirect

2. Q₈×C₁₉⋊C₈

C₂

semidirect

81. C₁₉

#2984

semidirect

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

subgroup	index
#11926675	2
Q₈×C₁₉⋊C₈	2
#11926601	2
#11926597	2
C₂×C₄×Q₈×C₁₉	2
#2984	19

subgroup

index

Q₈×C₁₉⋊C₈

C₂×C₄×Q₈×C₁₉

Subgroups — #14572724

Conjugacy classes of subgroups

Complements

Maximal subgroups