Subgroups — #14269258

Order 1824.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11744985	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#208	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#11744817	C₂	1	2	yes	maximal, complemented
5	D₅₇⋊C₈	C₂	1	2	yes	maximal, complemented
6	#10597353	—	3	1	no	maximal, Hall
7	C₂×C₃×C₄×C₁₉	C₂²	1	1	yes	characteristic
8	#90718	C₄	1	2	yes	—
9	D₂₂₈	C₄	1	1	yes	characteristic
10	C₄×D₅₇	C₂²	1	2	yes	—
11	Dic₁₁₄	C₄	1	1	yes	characteristic
12	C₃×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
13	C₃×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
14	C₁₉×C₃⋊C₈	C₂²	1	2	yes	—
15	#64541	—	3	1	no	—
16	M₁₆×C₁₉	—	3	1	no	—
17	C₂×C₁₉⋊C₈	D₃	1	1	yes	characteristic, complemented
18	#64515	—	3	2	no	—
19	C₈×D₁₉	—	3	2	no	—
20	C₂²×C₃×C₁₉	C₂×C₄	1	1	yes	characteristic
21	C₂×D₅₇	C₂×C₄	1	2	yes	—
22	C₃×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
23	C₃×C₄×C₁₉	C₂³	1	1	yes	characteristic
24	Dic₅₇	C₂×C₄	1	2	yes	—
25	C₂×C₄×C₁₉	C₂×D₃	1	1	yes	characteristic
26	C₂²⋊D₁₉	—	3	2	no	—
27	D₇₆	—	3	1	no	—
28	C₄×D₁₉	—	3	2	no	—
29	Dic₃₈	—	3	1	no	—
30	C₈×C₁₉	—	3	2	no	—
31	C₁₉⋊C₈	C₂×D₃	1	1	yes	characteristic
32	C₁₉⋊C₈	C₂×D₃	1	1	yes	characteristic
33	C₂×C₃×C₁₉	—	2	1	no	—
34	C₂×C₃×C₁₉	C₂²×C₄	1	1	yes	characteristic
35	D₅₇	—	2	2	no	—
36	#901	—	19	1	no	maximal, Hall
37	C₂²×C₁₉	C₄×D₃	1	1	yes	characteristic
38	C₂×D₁₉	—	3	2	no	—
39	C₄×C₁₉	C₄×D₃	1	1	yes	characteristic
40	C₄×C₁₉	C₂²×D₃	1	1	yes	characteristic
41	Dic₁₉	—	3	2	no	—
42	C₃×C₁₉	#131	1	1	yes	characteristic, complemented, Hall
43	#235	—	19	1	no	—
44	C₂×C₃×C₈	—	19	1	no	—
45	#208	D₁₉	1	1	yes	characteristic, complemented
46	ΓL(1,𝔽₂₅)	—	19	2	no	—
47	D₃×C₈	—	19	2	no	—
48	C₂×C₁₉	—	2	1	no	—
49	C₂×C₁₉	C₂×C₄×D₃	1	1	yes	characteristic
50	D₁₉	—	6	2	no	—
51	#131	—	57	1	no	Sylow
52	C₂×C₃×C₄	C₂×D₁₉	1	1	yes	characteristic
53	#67	—	19	2	no	—
54	D₁₂	—	19	1	no	—
55	C₄×D₃	—	19	2	no	—
56	Dic₆	—	19	1	no	—
57	C₃×C₈	—	19	1	no	—
58	C₃×C₈	—	19	1	no	—
59	C₃⋊C₈	C₂×D₁₉	1	2	yes	—
60	C₁₉	#901	1	1	yes	characteristic, complemented, Sylow
61	#41	—	57	1	no	—
62	M₁₆	—	3	1	no	—
63	M₁₆	—	57	2	no	—
64	C₂×C₈	—	19	1	no	—
65	C₂×C₈	—	57	2	no	—
66	C₂²×C₃	C₄×D₁₉	1	1	yes	characteristic
67	C₂×D₃	—	19	2	no	—
68	C₃×C₄	C₄×D₁₉	1	1	yes	characteristic
69	C₃×C₄	C₂²×D₁₉	1	1	yes	characteristic
70	Dic₃	—	19	2	no	—
71	Q₈	—	57	1	no	—
72	D₄	—	57	2	no	—
73	D₄	—	57	1	no	—
74	C₂×C₄	—	57	2	no	—
75	C₂×C₄	D₃×D₁₉	1	1	yes	characteristic
76	C₈	—	3	2	no	—
77	C₈	—	19	1	no	—
78	C₈	—	19	1	no	—
79	C₂×C₃	—	2	1	no	—
80	C₂×C₃	C₂×C₄×D₁₉	1	1	yes	characteristic
81	D₃	—	38	2	no	—
82	C₂²	—	57	2	no	—
83	C₂²	D₅₇⋊C₄	1	1	yes	characteristic
84	C₄	—	57	2	no	—
85	C₄	D₅₇⋊C₄	1	1	yes	characteristic
86	C₄	C₂×D₃×D₁₉	1	1	yes	characteristic
87	C₃	#10597353	1	1	yes	characteristic, complemented, Sylow
88	C₂	—	2	1	no	—
89	C₂	—	114	2	no	—
90	C₂	C₂×D₅₇⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂×D₃×D₁₉
commutator, socle	C₂×C₃×C₁₉	C₂²×C₄
Fitting	C₂×C₃×C₄×C₁₉	C₂²
prime radical	#11744985	C₂
cosocle	C₃×C₄×C₁₉	C₂³
hypercenter	C₂×C₄	D₃×D₁₉
hypocenter	C₃×C₁₉	#131
LCS	G ▷ C₂×C₃×C₁₉ ▷ C₃×C₁₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
2. C₁₉×#208	C₂	semidirect
3. C₂×C₃×C₁₉⋊C₈	C₂	semidirect
4. #11744817	C₂	semidirect
5. D₅₇⋊C₈	C₂	semidirect
17. C₂×C₁₉⋊C₈	D₃	semidirect
42. C₃×C₁₉	#131	semidirect
45. #208	D₁₉	semidirect
60. C₁₉	#901	semidirect
87. C₃	#10597353	semidirect

Maximal subgroups

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

subgroup	index
#11744985	2
C₁₉×#208	2
C₂×C₃×C₁₉⋊C₈	2
#11744817	2
D₅₇⋊C₈	2
#10597353	3
#901	19