Subgroups — #11926571

Order 1216.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#117	C₂	1	1	yes	maximal, characteristic
2	C₂×#64520	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₄⋊(C₁₉⋊C₈)	C₂	1	4	yes	maximal, complemented
5	C₂²×C₄×C₁₉	C₂²	1	1	yes	characteristic
6	C₄⋊C₄×C₁₉	C₄	1	2	yes	—
7	C₂²⋊C₄×C₁₉	C₄	1	2	yes	—
8	C₄²×C₁₉	C₂²	1	2	yes	—
9	#64520	—	2	2	no	—
10	C₂×C₁₉⋊C₈	—	2	2	no	—
11	C₂×C₁₉⋊C₈	C₂²	1	2	yes	—
12	C₂×C₁₉⋊C₈	C₂²	1	2	yes	—
13	C₂³×C₁₉	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₁₉	C₂×C₄	1	4	yes	—
15	C₂×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₁₉	D₄	1	2	yes	—
17	C₂×C₄×C₁₉	Q₈	1	2	yes	—
18	C₂×C₄×C₁₉	C₂³	1	1	yes	characteristic
19	C₁₉⋊C₈	—	2	2	no	—
20	C₁₉⋊C₈	—	2	2	no	—
21	C₂²×C₁₉	—	2	2	no	—
22	C₂²×C₁₉	C₄⋊C₄	1	2	yes	—
23	C₂²×C₁₉	C₂²×C₄	1	1	yes	characteristic
24	C₄×C₁₉	—	2	4	no	—
25	C₄×C₁₉	C₄⋊C₄	1	2	yes	—
26	C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
27	C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
28	#424	—	19	1	no	maximal, Sylow
29	C₂×C₁₉	—	2	2	no	—
30	C₂×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₁₉	#131	1	2	yes	—
32	C₂×M₁₆	—	19	1	no	—
33	C₂²×C₈	—	19	1	no	—
34	#117	D₁₉	1	1	yes	characteristic
35	C₄⋊C₈	—	19	4	no	—
36	C₁₉	#424	1	1	yes	characteristic, complemented, Sylow
37	C₂²×C₄	C₂×D₁₉	1	1	yes	characteristic
38	M₁₆	—	38	2	no	—
39	C₂×C₈	—	19	2	no	—
40	C₂×C₈	—	19	2	no	—
41	C₂×C₈	—	38	2	no	—
42	C₄⋊C₄	Dic₁₉	1	2	yes	—
43	C₂²⋊C₄	Dic₁₉	1	2	yes	—
44	C₄²	C₂×D₁₉	1	2	yes	—
45	C₂³	C₂×Dic₁₉	1	1	yes	characteristic
46	C₂×C₄	Dic₃₈	1	2	yes	—
47	C₂×C₄	D₇₆	1	2	yes	—
48	C₂×C₄	C₂×Dic₁₉	1	4	yes	—
49	C₂×C₄	C₂×Dic₁₉	1	1	yes	characteristic
50	C₂×C₄	C₂²×D₁₉	1	1	yes	characteristic
51	C₈	—	38	2	no	—
52	C₈	—	38	2	no	—
53	C₂²	—	2	2	no	—
54	C₂²	C₄⋊Dic₁₉	1	2	yes	—
55	C₂²	C₂²×Dic₁₉	1	1	yes	characteristic
56	C₄	—	2	4	no	—
57	C₄	C₄⋊Dic₁₉	1	2	yes	—
58	C₄	C₂×Dic₃₈	1	1	yes	characteristic
59	C₄	C₂×D₇₆	1	1	yes	characteristic
60	C₂	—	2	2	no	—
61	C₂	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
62	C₂	#10597394	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₁₉
commutator, socle	C₂²×C₁₉	C₂²×C₄
Fitting	C₁₉×#117	C₂
prime radical	C₂³×C₁₉	C₂×C₄
cosocle	C₂×C₄×C₁₉	C₂³
hypercenter	#117	D₁₉
hypocenter	C₁₉	#424
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ #117

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
4. C₄⋊(C₁₉⋊C₈)	C₂	semidirect
36. C₁₉	#424	semidirect

Maximal subgroups

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

subgroup	index
C₁₉×#117	2
C₂×#64520	2
C₂²×C₁₉⋊C₈	2
C₄⋊(C₁₉⋊C₈)	2
#424	19