Group #14577209 — subgroups

Subgroups — Q₈²×D₁₉ #14577209

Order 2432.

1112 conjugacy classes of subgroups; only the 575 normal subgroups are shown.

normal subgroups only

This group has 1112 conjugacy classes of subgroups — too many to draw. Only its 575 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×Dic₃₈	C₂	1	6	yes	maximal, complemented
2	#11927274	C₂	1	9	yes	maximal, complemented
3	C₄⋊Q₈×D₁₉	C₂	1	9	yes	maximal
4	C₄×Q₈×D₁₉	C₂	1	6	yes	maximal
5	Q₈²×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈×D₁₉	C₂²	1	6	yes	—
7	C₂×Q₈×D₁₉	C₂²	1	2	yes	—
8	C₁₉×C₄⋊Q₈	C₂²	1	9	yes	—
9	C₄×Q₈×C₁₉	C₂²	1	6	yes	—
10	Q₈×Dic₁₉	C₂²	1	6	yes	—
11	#10597390	C₂²	1	18	yes	—
12	C₄⋊C₄×D₁₉	C₂²	1	18	yes	—
13	#10597334	C₂²	1	36	yes	—
14	#10597333	C₂²	1	18	yes	—
15	C₄²×D₁₉	C₂²	1	9	yes	—
16	#10597315	C₂²	1	9	yes	—
17	C₄×Dic₃₈	C₂²	1	18	yes	—
18	C₂×Q₈×C₁₉	C₂³	1	6	yes	—
19	C₂×Q₈×C₁₉	C₂³	1	2	yes	—
20	Q₈×D₁₉	—	4	12	no	—
21	Q₈×D₁₉	Q₈	1	16	yes	direct product factor
22	C₂×C₄×D₁₉	C₂³	1	9	yes	—
23	C₂×C₄×D₁₉	C₂³	1	6	yes	—
24	C₂×Dic₃₈	C₂³	1	18	yes	—
25	C₂×Dic₃₈	C₂³	1	6	yes	—
26	C₄⋊C₄×C₁₉	C₂³	1	18	yes	—
27	C₄²×C₁₉	C₂³	1	9	yes	—
28	C₄⋊Dic₁₉	C₂³	1	18	yes	—
29	Dic₁₉⋊C₄	C₂³	1	36	yes	—
30	C₄×Dic₁₉	C₂³	1	9	yes	—
31	C₄×Dic₁₉	C₂³	1	18	yes	—
32	C₂²×D₁₉	C₂⁴	1	1	yes	characteristic
33	Q₈×C₁₉	—	4	6	no	—
34	Q₈×C₁₉	C₂×Q₈	1	8	yes	complemented
35	C₂×C₄×C₁₉	C₂⁴	1	9	yes	—
36	C₂×C₄×C₁₉	C₂⁴	1	6	yes	—
37	C₂×Dic₁₉	C₂⁴	1	9	yes	—
38	C₂×Dic₁₉	C₂⁴	1	6	yes	—
39	C₄×D₁₉	—	2	18	no	—
40	C₄×D₁₉	C₂×Q₈	1	24	yes	—
41	Dic₃₈	—	4	18	no	—
42	Dic₃₈	C₂×Q₈	1	24	yes	complemented
43	C₂×Q₈²	—	19	1	no	maximal, Sylow
44	C₂²×C₁₉	C₂⁵	1	1	yes	characteristic
45	C₂×D₁₉	C₂²×Q₈	1	4	yes	—
46	C₂×D₁₉	H(5,𝔽₂)	1	2	yes	—
47	C₄×C₁₉	—	2	9	no	—
48	C₄×C₁₉	C₂²×Q₈	1	12	yes	—
49	Dic₁₉	—	2	9	no	—
50	Dic₁₉	C₂²×Q₈	1	12	yes	—
51	Q₈²	—	19	9	no	—
52	Q₈²	—	19	6	no	—
53	Q₈²	D₁₉	1	1	yes	characteristic, direct product factor
54	C₂×C₄⋊Q₈	—	19	9	no	—
55	C₂×C₄×Q₈	—	19	6	no	—
56	C₂×C₁₉	C₂³×Q₈	1	2	yes	—
57	C₂×C₁₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
58	D₁₉	Q₈²	1	4	yes	direct product factor
59	C₂²×Q₈	—	19	2	no	—
60	C₂²×Q₈	—	19	6	no	—
61	C₄⋊Q₈	—	19	18	no	—
62	C₄⋊Q₈	—	19	36	no	—
63	C₄⋊Q₈	—	19	9	no	—
64	C₄⋊Q₈	C₂×D₁₉	1	9	yes	—
65	C₄×Q₈	—	19	6	no	—
66	C₄×Q₈	—	19	18	no	—
67	C₄×Q₈	—	19	18	no	—
68	C₄×Q₈	C₂×D₁₉	1	6	yes	—
69	C₂×C₄⋊C₄	—	19	18	no	—
70	C₂×C₄²	—	19	9	no	—
71	C₁₉	C₂×Q₈²	1	1	yes	characteristic, complemented, Sylow
72	C₂×Q₈	—	19	16	no	—
73	C₂×Q₈	—	19	6	no	—
74	C₂×Q₈	—	19	18	no	—
75	C₂×Q₈	—	76	12	no	—
76	C₂×Q₈	C₂²×D₁₉	1	6	yes	—
77	C₂×Q₈	C₂²×D₁₉	1	2	yes	—
78	C₂²×C₄	—	19	6	no	—
79	C₂²×C₄	—	19	9	no	—
80	C₄⋊C₄	—	19	36	no	—
81	C₄⋊C₄	—	19	18	no	—
82	C₄⋊C₄	C₂²×D₁₉	1	18	yes	—
83	C₄²	—	19	9	no	—
84	C₄²	—	19	18	no	—
85	C₄²	C₂²×D₁₉	1	9	yes	—
86	C₂³	—	19	1	no	—
87	Q₈	—	4	6	no	—
88	Q₈	—	19	24	no	—
89	Q₈	—	76	18	no	—
90	Q₈	Q₈×D₁₉	1	8	yes	direct product factor
91	C₂×C₄	—	19	6	no	—
92	C₂×C₄	—	19	24	no	—
93	C₂×C₄	—	19	9	no	—
94	C₂×C₄	—	38	18	no	—
95	C₂×C₄	C₂³×D₁₉	1	9	yes	—
96	C₂×C₄	C₂³×D₁₉	1	6	yes	—
97	C₂²	—	19	4	no	—
98	C₂²	—	19	2	no	—
99	C₂²	C₂⁴×D₁₉	1	1	yes	characteristic
100	C₄	—	2	9	no	—
101	C₄	—	19	12	no	—
102	C₄	—	38	9	no	—
103	C₄	C₂×Q₈×D₁₉	1	12	yes	—
104	C₂	—	19	4	no	—
105	C₂	C₂²×Q₈×D₁₉	1	2	yes	—
106	C₂	H(5,𝔽₂)×D₁₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₃₈

C₂

yes

maximal, complemented

#11927274

C₂

yes

maximal, complemented

C₄⋊Q₈×D₁₉

C₂

yes

maximal

C₄×Q₈×D₁₉

C₂

yes

maximal

Q₈²×C₁₉

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₁₉

C₂²

yes

—

C₂×Q₈×D₁₉

C₂²

yes

—

C₁₉×C₄⋊Q₈

C₂²

yes

—

C₄×Q₈×C₁₉

C₂²

yes

—

Q₈×Dic₁₉

C₂²

yes

—

#10597390

C₂²

yes

—

C₄⋊C₄×D₁₉

C₂²

yes

—

#10597334

C₂²

yes

—

#10597333

C₂²

yes

—

C₄²×D₁₉

C₂²

yes

—

#10597315

C₂²

yes

—

C₄×Dic₃₈

C₂²

yes

—

C₂×Q₈×C₁₉

C₂³

yes

—

C₂×Q₈×C₁₉

C₂³

yes

—

Q₈×D₁₉

—

Q₈×D₁₉

Q₈

yes

direct product factor

C₂×C₄×D₁₉

C₂³

yes

—

C₂×C₄×D₁₉

C₂³

yes

—

C₂×Dic₃₈

C₂³

yes

—

C₂×Dic₃₈

C₂³

yes

—

C₄⋊C₄×C₁₉

C₂³

yes

—

C₄²×C₁₉

C₂³

yes

—

C₄⋊Dic₁₉

C₂³

yes

—

Dic₁₉⋊C₄

C₂³

yes

—

C₄×Dic₁₉

C₂³

yes

—

C₄×Dic₁₉

C₂³

yes

—

C₂²×D₁₉

C₂⁴

yes

characteristic

Q₈×C₁₉

—

Q₈×C₁₉

C₂×Q₈

yes

complemented

C₂×C₄×C₁₉

C₂⁴

yes

—

C₂×C₄×C₁₉

C₂⁴

yes

—

C₂×Dic₁₉

C₂⁴

yes

—

C₂×Dic₁₉

C₂⁴

yes

—

C₄×D₁₉

—

C₄×D₁₉

C₂×Q₈

yes

—

Dic₃₈

—

Dic₃₈

C₂×Q₈

yes

complemented

C₂×Q₈²

—

maximal, Sylow

C₂²×C₁₉

C₂⁵

yes

characteristic

C₂×D₁₉

C₂²×Q₈

yes

—

C₂×D₁₉

H(5,𝔽₂)

yes

—

C₄×C₁₉

—

C₄×C₁₉

C₂²×Q₈

yes

—

Dic₁₉

—

Dic₁₉

C₂²×Q₈

yes

—

Q₈²

—

Q₈²

—

Q₈²

D₁₉

yes

characteristic, direct product factor

C₂×C₄⋊Q₈

—

C₂×C₄×Q₈

—

C₂×C₁₉

C₂³×Q₈

yes

—

C₂×C₁₉

C₂×H(5,𝔽₂)

yes

characteristic

D₁₉

Q₈²

yes

direct product factor

C₂²×Q₈

—

C₂²×Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₁₉

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂×D₁₉

yes

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₁₉

C₂×Q₈²

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²×D₁₉

yes

—

C₂×Q₈

C₂²×D₁₉

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×D₁₉

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×D₁₉

yes

—

C₂³

—

Q₈

—

Q₈

—

Q₈

—

Q₈

Q₈×D₁₉

yes

direct product factor

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂³×D₁₉

yes

—

C₂×C₄

C₂³×D₁₉

yes

—

C₂²

—

C₂²

—

C₂²

C₂⁴×D₁₉

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

C₂×Q₈×D₁₉

yes

—

104

C₂

—

105

C₂

C₂²×Q₈×D₁₉

yes

—

106

C₂

H(5,𝔽₂)×D₁₉

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₉
commutator, socle, cosocle	C₂²×C₁₉	C₂⁵
Fitting	Q₈²×C₁₉	C₂
prime radical	C₂²×D₁₉	C₂⁴
hypercenter	Q₈²	D₁₉
hypocenter	C₁₉	C₂×Q₈²
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ Q₈²

subgroup

G/H

center, Frattini

C₂²

C₂⁴×D₁₉

commutator, socle, cosocle

C₂²×C₁₉

C₂⁵

Fitting

Q₈²×C₁₉

C₂

prime radical

C₂²×D₁₉

C₂⁴

hypercenter

Q₈²

D₁₉

hypocenter

C₁₉

C₂×Q₈²

LCS

G ▷ C₂²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂² ◁ Q₈²

N	H ≅ G/N	type
1. Q₈×Dic₃₈	C₂	semidirect
2. #11927274	C₂	semidirect
5. Q₈²×C₁₉	C₂	semidirect
21. Q₈×D₁₉	Q₈	direct (N × H)
34. Q₈×C₁₉	C₂×Q₈	semidirect
42. Dic₃₈	C₂×Q₈	semidirect
53. Q₈²	D₁₉	direct (N × H)
58. D₁₉	Q₈²	direct (N × H)
71. C₁₉	C₂×Q₈²	semidirect
90. Q₈	Q₈×D₁₉	direct (N × H)

H ≅ G/N

type

1. Q₈×Dic₃₈

C₂

semidirect

2. #11927274

C₂

semidirect

5. Q₈²×C₁₉

C₂

semidirect

21. Q₈×D₁₉

Q₈

direct (N × H)

34. Q₈×C₁₉

C₂×Q₈

semidirect

42. Dic₃₈

C₂×Q₈

semidirect

53. Q₈²

D₁₉

direct (N × H)

58. D₁₉

Q₈²

direct (N × H)

71. C₁₉

C₂×Q₈²

semidirect

90. Q₈

Q₈×D₁₉

direct (N × H)

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

subgroup	index
Q₈×Dic₃₈	2
#11927274	2
C₄⋊Q₈×D₁₉	2
C₄×Q₈×D₁₉	2
Q₈²×C₁₉	2
C₂×Q₈²	19

subgroup

index

Q₈×Dic₃₈

#11927274

C₄⋊Q₈×D₁₉

C₄×Q₈×D₁₉

Q₈²×C₁₉

C₂×Q₈²

Subgroups — Q₈²×D₁₉ #14577209

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q82×D19 #14577209

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈²×D₁₉ #14577209