Subgroups — #11926839

Order 1216.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Q₈×C₁₉	C₂	1	1	yes	maximal, characteristic
2	#10597315	C₂	1	1	yes	maximal, characteristic
3	#10597292	C₂	1	2	yes	maximal
4	#10597264	C₂	1	2	yes	maximal
5	C₄⋊(C₁₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₈×C₁₉	C₂²	1	1	yes	characteristic
7	C₂×Dic₃₈	—	2	1	no	—
8	C₄⋊C₄×C₁₉	—	2	1	no	—
9	C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
10	C₄²×C₁₉	—	2	1	no	—
11	C₄²×C₁₉	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₁₉	—	2	1	no	—
13	C₄⋊Dic₁₉	C₂²	1	2	yes	—
14	C₂×C₁₉⋊C₈	C₂²	1	2	yes	—
15	Q₈×C₁₉	—	2	1	no	—
16	Q₈×C₁₉	Q₈	1	2	yes	complemented
17	C₂×C₄×C₁₉	—	2	1	no	—
18	C₂×C₄×C₁₉	—	2	1	no	—
19	C₂×C₄×C₁₉	D₄	1	1	yes	characteristic
20	C₂×C₄×C₁₉	D₄	1	1	yes	characteristic
21	C₂×C₄×C₁₉	C₂³	1	1	yes	characteristic
22	C₂×Dic₁₉	—	2	2	no	—
23	Dic₃₈	—	4	2	no	—
24	C₁₉⋊C₈	—	2	2	no	—
25	C₂²×C₁₉	C₂×D₄	1	1	yes	characteristic
26	C₄×C₁₉	—	2	2	no	—
27	C₄×C₁₉	—	2	1	no	—
28	C₄×C₁₉	—	4	1	no	—
29	C₄×C₁₉	Q₁₆	1	2	yes	—
30	C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
31	C₄×C₁₉	#41	1	1	yes	characteristic
32	Dic₁₉	—	4	2	no	—
33	#477	—	19	1	no	maximal, Sylow
34	C₂×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
35	C₂×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
36	C₂×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
37	C₄⋊Q₈	—	19	1	no	—
38	C₄×Q₈	D₁₉	1	1	yes	characteristic
39	#107	—	19	2	no	—
40	C₄⋊C₈	—	19	1	no	—
41	#103	—	19	2	no	—
42	C₁₉	#477	1	1	yes	characteristic, complemented, Sylow
43	C₂×Q₈	—	38	1	no	—
44	C₂×Q₈	C₂×D₁₉	1	1	yes	characteristic
45	C₂×C₈	—	19	2	no	—
46	C₄⋊C₄	—	2	1	no	—
47	C₄⋊C₄	—	19	2	no	—
48	C₄⋊C₄	—	38	1	no	—
49	C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
50	C₄²	—	2	1	no	—
51	C₄²	C₂×D₁₉	1	1	yes	characteristic
52	Q₈	—	2	1	no	—
53	Q₈	—	76	2	no	—
54	Q₈	Dic₃₈	1	2	yes	complemented
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	38	2	no	—
58	C₂×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
59	C₂×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
60	C₂×C₄	C₂²×D₁₉	1	1	yes	characteristic
61	C₈	—	38	2	no	—
62	C₂²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
63	C₄	—	2	1	no	—
64	C₄	—	2	2	no	—
65	C₄	—	4	1	no	—
66	C₄	—	76	2	no	—
67	C₄	#64528	1	2	yes	—
68	C₄	C₂×Dic₃₈	1	1	yes	characteristic
69	C₄	#64541	1	1	yes	characteristic
70	C₂	#10597370	1	1	yes	characteristic
71	C₂	C₂×#64528	1	1	yes	characteristic
72	C₂	#10597395	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₉
commutator, cosocle	C₂×C₄×C₁₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₉
Fitting	C₄×Q₈×C₁₉	C₂
socle, prime radical	C₂²×C₁₉	C₂×D₄
hypercenter	C₄×Q₈	D₁₉
hypocenter	C₁₉	#477
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
16. Q₈×C₁₉	Q₈	semidirect
42. C₁₉	#477	semidirect
54. Q₈	Dic₃₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×Q₈×C₁₉	2
#10597315	2
#10597292	2
#10597264	2
C₄⋊(C₁₉⋊C₈)	2
#477	19