Group #14572767 — subgroups

Subgroups — #14572767

Order 2432.

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

normal subgroups only

This group has 394 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	Q₈×C₁₉⋊C₈	C₂	1	2	yes	maximal, complemented
2	#11926622	C₂	1	6	yes	maximal, complemented
3	#11926597	C₂	1	3	yes	maximal, complemented
4	#11926568	C₂	1	3	yes	maximal, complemented
5	C₄×#41×C₁₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#41×C₁₉	C₄	1	1	yes	characteristic
7	C₄×Q₈×C₁₉	C₄	1	1	yes	characteristic
8	C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
9	C₄×D₄×C₁₉	C₄	1	3	yes	—
10	C₄×D₄×C₁₉	C₂²	1	3	yes	—
11	C₁₉×#117	C₄	1	3	yes	—
12	C₂×C₄²×C₁₉	C₂²	1	3	yes	—
13	C₂×#64520	C₂²	1	6	yes	—
14	C₂²⋊(C₁₉⋊C₈)	C₂²	1	6	yes	—
15	C₄⋊(C₁₉⋊C₈)	C₂²	1	3	yes	—
16	C₄⋊(C₁₉⋊C₈)	C₂²	1	6	yes	—
17	C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
18	C₄×C₁₉⋊C₈	C₂²	1	6	yes	—
19	#41×C₁₉	—	2	4	no	—
20	C₂×Q₈×C₁₉	C₂×C₄	1	1	yes	characteristic
21	C₂×D₄×C₁₉	C₂×C₄	1	3	yes	—
22	C₂²×C₄×C₁₉	C₂×C₄	1	3	yes	—
23	C₂²×C₄×C₁₉	C₂×C₄	1	3	yes	—
24	C₂²×C₄×C₁₉	C₂³	1	3	yes	—
25	C₄⋊C₄×C₁₉	C₂×C₄	1	3	yes	—
26	C₄⋊C₄×C₁₉	C₂×C₄	1	3	yes	—
27	C₂²⋊C₄×C₁₉	C₂×C₄	1	6	yes	—
28	C₄²×C₁₉	C₂×C₄	1	3	yes	—
29	C₄²×C₁₉	C₂×C₄	1	3	yes	—
30	C₄²×C₁₉	C₂³	1	1	yes	characteristic
31	C₄²×C₁₉	C₂³	1	3	yes	—
32	#64520	—	4	6	no	—
33	C₂×C₁₉⋊C₈	C₂³	1	2	yes	—
34	C₂×C₁₉⋊C₈	C₂³	1	6	yes	—
35	C₂³×C₁₉	C₂²×C₄	1	3	yes	—
36	Q₈×C₁₉	M₁₆	1	4	yes	complemented
37	D₄×C₁₉	—	2	6	no	—
38	C₂×C₄×C₁₉	—	2	6	no	—
39	C₂×C₄×C₁₉	—	2	6	no	—
40	C₂×C₄×C₁₉	—	2	3	no	—
41	C₂×C₄×C₁₉	—	2	3	no	—
42	C₂×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₁₉	C₂²×C₄	1	3	yes	—
44	C₂×C₄×C₁₉	C₂²×C₄	1	3	yes	—
45	C₂×C₄×C₁₉	C₂²×C₄	1	3	yes	—
46	C₂×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
48	C₁₉⋊C₈	—	2	6	no	—
49	C₁₉⋊C₈	—	2	2	no	—
50	#2991	—	19	1	no	maximal, Sylow
51	C₂²×C₁₉	—	2	3	no	—
52	C₂²×C₁₉	—	2	3	no	—
53	C₂²×C₁₉	—	2	3	no	—
54	C₂²×C₁₉	C₂³×C₄	1	1	yes	characteristic
55	C₄×C₁₉	—	2	3	no	—
56	C₄×C₁₉	—	2	1	no	—
57	C₄×C₁₉	—	2	3	no	—
58	C₄×C₁₉	C₂×M₁₆	1	6	yes	—
59	C₄×C₁₉	#131	1	2	yes	—
60	C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
61	C₄×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
62	C₄×#41	D₁₉	1	1	yes	characteristic
63	C₈×Q₈	—	19	2	no	—
64	#436	—	19	6	no	—
65	#431	—	19	3	no	—
66	#423	—	19	3	no	—
67	C₂×C₁₉	—	4	3	no	—
68	C₂×C₁₉	#520	1	1	yes	characteristic
69	C₂×C₁₉	C₂²×M₁₆	1	1	yes	characteristic
70	C₂×C₁₉	C₂×#131	1	1	yes	characteristic
71	C₂×#41	Dic₁₉	1	1	yes	characteristic
72	C₂×M₁₆	—	19	6	no	—
73	C₄×Q₈	Dic₁₉	1	1	yes	characteristic
74	C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic
75	C₄×D₄	Dic₁₉	1	3	yes	—
76	C₄×D₄	C₂×D₁₉	1	3	yes	—
77	#117	Dic₁₉	1	3	yes	—
78	C₂×C₄²	C₂×D₁₉	1	3	yes	—
79	C₄⋊C₈	—	19	3	no	—
80	C₄⋊C₈	—	19	6	no	—
81	C₄⋊C₈	—	19	1	no	—
82	C₂²⋊C₈	—	19	6	no	—
83	C₄×C₈	—	19	6	no	—
84	C₁₉	#2991	1	1	yes	characteristic, complemented, Sylow
85	#41	—	2	4	no	—
86	C₂×Q₈	C₂×Dic₁₉	1	1	yes	characteristic
87	C₂×D₄	C₂×Dic₁₉	1	3	yes	—
88	C₂²×C₄	C₂×Dic₁₉	1	3	yes	—
89	C₂²×C₄	C₂×Dic₁₉	1	3	yes	—
90	C₂²×C₄	C₂²×D₁₉	1	3	yes	—
91	M₁₆	—	76	6	no	—
92	C₂×C₈	—	19	2	no	—
93	C₂×C₈	—	19	6	no	—
94	C₄⋊C₄	C₂×Dic₁₉	1	3	yes	—
95	C₄⋊C₄	C₂×Dic₁₉	1	3	yes	—
96	C₂²⋊C₄	C₂×Dic₁₉	1	6	yes	—
97	C₄²	C₂×Dic₁₉	1	3	yes	—
98	C₄²	C₂×Dic₁₉	1	3	yes	—
99	C₄²	C₂²×D₁₉	1	3	yes	—
100	C₄²	C₂²×D₁₉	1	1	yes	characteristic
101	C₂³	C₂²×Dic₁₉	1	3	yes	—
102	Q₈	#64520	1	4	yes	complemented
103	D₄	—	2	6	no	—
104	C₂×C₄	—	2	6	no	—
105	C₂×C₄	—	2	3	no	—
106	C₂×C₄	—	2	3	no	—
107	C₂×C₄	—	2	6	no	—
108	C₂×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
109	C₂×C₄	C₂²×Dic₁₉	1	3	yes	—
110	C₂×C₄	C₂²×Dic₁₉	1	3	yes	—
111	C₂×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
112	C₂×C₄	C₂²×Dic₁₉	1	3	yes	—
113	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
114	C₈	—	38	6	no	—
115	C₈	—	38	2	no	—
116	C₂²	—	2	3	no	—
117	C₂²	—	2	3	no	—
118	C₂²	—	2	3	no	—
119	C₂²	C₂³×Dic₁₉	1	1	yes	characteristic
120	C₄	—	2	1	no	—
121	C₄	—	2	3	no	—
122	C₄	—	2	3	no	—
123	C₄	C₂×#64520	1	6	yes	—
124	C₄	#10597394	1	2	yes	—
125	C₄	#10597435	1	1	yes	characteristic
126	C₄	#10597436	1	1	yes	characteristic
127	C₂	—	4	3	no	—
128	C₂	#11927027	1	1	yes	characteristic
129	C₂	C₂²×#64520	1	1	yes	characteristic
130	C₂	C₂×#10597394	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×C₁₉⋊C₈

C₂

yes

maximal, complemented

#11926622

C₂

yes

maximal, complemented

#11926597

C₂

yes

maximal, complemented

#11926568

C₂

yes

maximal, complemented

C₄×#41×C₁₉

C₂

yes

maximal, characteristic

C₂×#41×C₁₉

C₄

yes

characteristic

C₄×Q₈×C₁₉

C₄

yes

characteristic

C₄×Q₈×C₁₉

C₂²

yes

characteristic

C₄×D₄×C₁₉

C₄

yes

—

C₄×D₄×C₁₉

C₂²

yes

—

C₁₉×#117

C₄

yes

—

C₂×C₄²×C₁₉

C₂²

yes

—

C₂×#64520

C₂²

yes

—

C₂²⋊(C₁₉⋊C₈)

C₂²

yes

—

C₄⋊(C₁₉⋊C₈)

C₂²

yes

—

C₄⋊(C₁₉⋊C₈)

C₂²

yes

—

C₄⋊(C₁₉⋊C₈)

C₂²

yes

characteristic

C₄×C₁₉⋊C₈

C₂²

yes

—

#41×C₁₉

—

C₂×Q₈×C₁₉

C₂×C₄

yes

characteristic

C₂×D₄×C₁₉

C₂×C₄

yes

—

C₂²×C₄×C₁₉

C₂×C₄

yes

—

C₂²×C₄×C₁₉

C₂×C₄

yes

—

C₂²×C₄×C₁₉

C₂³

yes

—

C₄⋊C₄×C₁₉

C₂×C₄

yes

—

C₄⋊C₄×C₁₉

C₂×C₄

yes

—

C₂²⋊C₄×C₁₉

C₂×C₄

yes

—

C₄²×C₁₉

C₂×C₄

yes

—

C₄²×C₁₉

C₂×C₄

yes

—

C₄²×C₁₉

C₂³

yes

characteristic

C₄²×C₁₉

C₂³

yes

—

#64520

—

C₂×C₁₉⋊C₈

C₂³

yes

—

C₂×C₁₉⋊C₈

C₂³

yes

—

C₂³×C₁₉

C₂²×C₄

yes

—

Q₈×C₁₉

M₁₆

yes

complemented

D₄×C₁₉

—

C₂×C₄×C₁₉

—

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

characteristic

C₂×C₄×C₁₉

C₂⁴

yes

characteristic

C₁₉⋊C₈

—

C₁₉⋊C₈

—

#2991

—

maximal, Sylow

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

C₂³×C₄

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

C₂×M₁₆

yes

—

C₄×C₁₉

#131

yes

—

C₄×C₁₉

H(5,𝔽₂)

yes

characteristic

C₄×C₁₉

2₋¹⁺⁴

yes

characteristic

C₄×#41

D₁₉

yes

characteristic

C₈×Q₈

—

#436

—

#431

—

#423

—

C₂×C₁₉

—

C₂×C₁₉

#520

yes

characteristic

C₂×C₁₉

C₂²×M₁₆

yes

characteristic

C₂×C₁₉

C₂×#131

yes

characteristic

C₂×#41

Dic₁₉

yes

characteristic

C₂×M₁₆

—

C₄×Q₈

Dic₁₉

yes

characteristic

C₄×Q₈

C₂×D₁₉

yes

characteristic

C₄×D₄

Dic₁₉

yes

—

C₄×D₄

C₂×D₁₉

yes

—

#117

Dic₁₉

yes

—

C₂×C₄²

C₂×D₁₉

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₁₉

#2991

yes

characteristic, complemented, Sylow

#41

—

C₂×Q₈

C₂×Dic₁₉

yes

characteristic

C₂×D₄

C₂×Dic₁₉

yes

—

C₂²×C₄

C₂×Dic₁₉

yes

—

C₂²×C₄

C₂×Dic₁₉

yes

—

C₂²×C₄

C₂²×D₁₉

yes

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

C₂×Dic₁₉

yes

—

C₄⋊C₄

C₂×Dic₁₉

yes

—

C₂²⋊C₄

C₂×Dic₁₉

yes

—

C₄²

C₂×Dic₁₉

yes

—

C₄²

C₂×Dic₁₉

yes

—

C₄²

C₂²×D₁₉

yes

—

100

C₄²

C₂²×D₁₉

yes

characteristic

101

C₂³

C₂²×Dic₁₉

yes

—

102

Q₈

#64520

yes

complemented

103

D₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

C₂²×Dic₁₉

yes

characteristic

109

C₂×C₄

C₂²×Dic₁₉

yes

—

110

C₂×C₄

C₂²×Dic₁₉

yes

—

111

C₂×C₄

C₂²×Dic₁₉

yes

characteristic

112

C₂×C₄

C₂²×Dic₁₉

yes

—

113

C₂×C₄

C₂³×D₁₉

yes

characteristic

114

C₈

—

115

C₈

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

C₂³×Dic₁₉

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

C₂×#64520

yes

—

124

C₄

#10597394

yes

—

125

C₄

#10597435

yes

characteristic

126

C₄

#10597436

yes

characteristic

127

C₂

—

128

C₂

#11927027

yes

characteristic

129

C₂

C₂²×#64520

yes

characteristic

130

C₂

C₂×#10597394

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₉
commutator, socle	C₂²×C₁₉	C₂³×C₄
Fitting	C₄×#41×C₁₉	C₂
prime radical	C₂×#41×C₁₉	C₄
cosocle	C₂×C₄×C₁₉	C₂⁴
hypercenter	C₄×#41	D₁₉
hypocenter	C₁₉	#2991
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₉

commutator, socle

C₂²×C₁₉

C₂³×C₄

Fitting

C₄×#41×C₁₉

C₂

prime radical

C₂×#41×C₁₉

C₄

cosocle

C₂×C₄×C₁₉

C₂⁴

hypercenter

C₄×#41

D₁₉

C₁₉

G ▷ C₂²×C₁₉ ▷ C₁₉

C₁ ◁ C₂×C₄ ◁ C₄×#41

N	H ≅ G/N	type
1. Q₈×C₁₉⋊C₈	C₂	semidirect
2. #11926622	C₂	semidirect
3. #11926597	C₂	semidirect
4. #11926568	C₂	semidirect
36. Q₈×C₁₉	M₁₆	semidirect
84. C₁₉	#2991	semidirect
102. Q₈	#64520	semidirect

H ≅ G/N

type

1. Q₈×C₁₉⋊C₈

C₂

semidirect

2. #11926622

C₂

semidirect

3. #11926597

C₂

semidirect

4. #11926568

C₂

semidirect

36. Q₈×C₁₉

M₁₆

semidirect

84. C₁₉

#2991

semidirect

102. Q₈

#64520

semidirect

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

subgroup	index
Q₈×C₁₉⋊C₈	2
#11926622	2
#11926597	2
#11926568	2
C₄×#41×C₁₉	2
#2991	19

subgroup

index

Q₈×C₁₉⋊C₈

C₄×#41×C₁₉

Subgroups — #14572767

Conjugacy classes of subgroups

Complements

Maximal subgroups