Subgroups — #18070211

Order 4864.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₁₉⋊C₃₂)	C₂	1	1	yes	maximal, characteristic
2	C₁₆×C₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₁₉×C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₉⋊C₃₂	—	2	1	no	—
5	C₄×C₁₉⋊C₁₆	C₄	1	1	yes	characteristic
6	C₈×C₁₉⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₁₉×C₃₂	—	2	1	no	—
8	C₄×C₁₆×C₁₉	C₂²	1	1	yes	characteristic
9	C₂×C₁₆×C₁₉	D₄	1	1	yes	characteristic
10	C₂×C₁₆×C₁₉	Q₈	1	1	yes	characteristic
11	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
12	C₂×C₁₉⋊C₁₆	—	2	1	no	—
13	C₄×C₁₉⋊C₈	C₈	1	1	yes	characteristic
14	C₄×C₁₉⋊C₈	C₈	1	1	yes	characteristic
15	C₁₉×C₃₂	—	4	1	no	—
16	C₁₉⋊C₃₂	—	4	1	no	—
17	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
18	C₂×C₈×C₁₉	M₁₆	1	1	yes	characteristic
19	C₄²×C₁₉	C₂×C₈	1	1	yes	characteristic
20	C₂×C₁₉⋊C₈	—	2	1	no	—
21	C₂×C₁₉⋊C₈	C₁₆	1	1	yes	characteristic
22	C₂×C₁₉⋊C₈	C₁₆	1	1	yes	characteristic
23	C₁₆×C₁₉	—	2	1	no	—
24	C₁₆×C₁₉	D₈	1	1	yes	characteristic
25	C₁₆×C₁₉	Q₁₆	1	1	yes	characteristic
26	C₁₉⋊C₁₆	—	2	2	no	—
27	#7458	—	19	1	no	maximal, Sylow
28	C₂×C₄×C₁₉	C₄⋊C₈	1	1	yes	characteristic
29	C₂×C₄×C₁₉	C₂×C₁₆	1	1	yes	characteristic
30	C₂×C₄×C₁₉	M₃₂	1	1	yes	characteristic
31	C₈×C₁₉	—	2	1	no	—
32	C₈×C₁₉	#107	1	1	yes	characteristic
33	C₈×C₁₉	#108	1	1	yes	characteristic
34	C₁₉⋊C₈	—	2	2	no	—
35	C₁₉⋊C₈	—	2	1	no	—
36	C₁₉⋊C₈	C₃₂	1	2	yes	complemented
37	C₄⋊C₃₂	—	19	1	no	—
38	C₄⋊C₃₂	D₁₉	1	1	yes	characteristic
39	C₈×C₁₆	—	19	1	no	—
40	C₂²×C₁₉	C₄⋊C₁₆	1	1	yes	characteristic
41	C₄×C₁₉	—	2	1	no	—
42	C₄×C₁₉	#335	1	1	yes	characteristic
43	C₄×C₁₉	#364	1	1	yes	characteristic
44	C₄×C₁₉	C₂×C₃₂	1	1	yes	characteristic
45	C₄×C₁₉	M₆₄	1	1	yes	characteristic
46	C₂×C₃₂	—	2	1	no	—
47	C₂×C₃₂	—	38	1	no	—
48	C₄×C₁₆	—	19	1	no	—
49	C₄×C₁₆	C₂×D₁₉	1	1	yes	characteristic
50	C₈²	—	19	1	no	—
51	C₂×C₁₉	#1370	1	1	yes	characteristic
52	C₂×C₁₉	C₄⋊C₃₂	1	1	yes	characteristic
53	C₂×C₁₉	#1422	1	1	yes	characteristic
54	C₂×C₁₆	—	38	1	no	—
55	C₂×C₁₆	Dic₃₈	1	1	yes	characteristic
56	C₂×C₁₆	C₂²⋊D₁₉	1	1	yes	characteristic
57	C₄×C₈	—	19	1	no	—
58	C₄×C₈	—	19	1	no	—
59	C₄×C₈	C₄×D₁₉	1	1	yes	characteristic
60	C₃₂	—	4	1	no	—
61	C₃₂	—	76	1	no	—
62	C₁₉	#7458	1	1	yes	characteristic, complemented, Sylow
63	C₂×C₈	—	19	1	no	—
64	C₂×C₈	—	19	1	no	—
65	C₂×C₈	—	38	1	no	—
66	C₂×C₈	#64515	1	1	yes	characteristic
67	C₂×C₈	Dic₁₉⋊C₄	1	1	yes	characteristic
68	C₄²	C₈×D₁₉	1	1	yes	characteristic
69	C₁₆	—	2	1	no	—
70	C₁₆	—	38	2	no	—
71	C₁₆	#64525	1	1	yes	characteristic
72	C₁₆	#64528	1	1	yes	characteristic
73	C₂×C₄	C₁₆×D₁₉	1	1	yes	characteristic
74	C₂×C₄	#10597255	1	1	yes	characteristic
75	C₂×C₄	Dic₁₉⋊C₈	1	1	yes	characteristic
76	C₈	—	2	1	no	—
77	C₈	—	19	2	no	—
78	C₈	—	38	2	no	—
79	C₈	—	38	1	no	—
80	C₈	#10597264	1	1	yes	characteristic
81	C₈	#10597279	1	1	yes	characteristic
82	C₂²	Dic₁₉⋊C₁₆	1	1	yes	characteristic
83	C₄	—	2	1	no	—
84	C₄	#11926296	1	1	yes	characteristic
85	C₄	#11926364	1	1	yes	characteristic
86	C₄	C₃₂×D₁₉	1	1	yes	characteristic
87	C₄	#11926377	1	1	yes	characteristic
88	C₂	#14561704	1	1	yes	characteristic
89	C₂	Dic₁₉⋊C₃₂	1	1	yes	characteristic
90	C₂	#14561832	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂²⋊D₁₉
commutator	C₄×C₁₉	C₂×C₃₂
Frattini	C₄×C₁₆	C₂×D₁₉
Fitting	C₁₉×C₄⋊C₃₂	C₂
socle, prime radical	C₂²×C₁₉	C₄⋊C₁₆
cosocle	C₄×C₁₆×C₁₉	C₂²
hypercenter	C₄⋊C₃₂	D₁₉
hypocenter	C₁₉	#7458
LCS	G ▷ C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₁₆ ◁ C₄×C₁₆ ◁ C₄⋊C₃₂

Complements

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

N	H ≅ G/N	type
36. C₁₉⋊C₈	C₃₂	semidirect
62. C₁₉	#7458	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊(C₁₉⋊C₃₂)	2
C₁₆×C₁₉⋊C₈	2
C₁₉×C₄⋊C₃₂	2
#7458	19