Subgroups — #11926331

Order 1216.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#120	C₂	1	1	yes	maximal, characteristic
2	#10597291	C₂	1	1	yes	maximal, characteristic, complemented
3	#10597290	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁴×C₁₉	C₄	1	1	yes	characteristic, complemented
5	C₂×D₄×C₁₉	—	2	1	no	—
6	C₂×D₄×C₁₉	C₂²	1	1	yes	characteristic
7	C₂²⋊C₄×C₁₉	—	2	1	no	—
8	C₂²⋊C₄×C₁₉	C₄	1	1	yes	characteristic, complemented
9	C₂²⋊Dic₁₉	—	2	1	no	—
10	#64520	—	2	1	no	—
11	C₂³×C₁₉	—	2	1	no	—
12	C₂³×C₁₉	—	4	2	no	—
13	C₂³×C₁₉	—	4	1	no	—
14	C₂³×C₁₉	C₂×C₄	1	1	yes	characteristic
15	C₂³×C₁₉	D₄	1	1	yes	characteristic
16	D₄×C₁₉	—	4	1	no	—
17	D₄×C₁₉	—	4	2	no	—
18	C₂×C₄×C₁₉	—	2	1	no	—
19	C₂×C₄×C₁₉	D₄	1	1	yes	characteristic
20	C₂×Dic₁₉	—	4	1	no	—
21	C₁₉⋊C₈	—	4	1	no	—
22	C₂²×C₁₉	—	2	2	no	—
23	C₂²×C₁₉	—	2	1	no	—
24	C₂²×C₁₉	—	2	1	no	—
25	C₂²×C₁₉	—	4	1	no	—
26	C₂²×C₁₉	—	4	1	no	—
27	C₂²×C₁₉	—	4	1	no	—
28	C₂²×C₁₉	—	4	2	no	—
29	C₂²×C₁₉	—	4	2	no	—
30	C₂²×C₁₉	—	4	1	no	—
31	C₂²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
32	C₄×C₁₉	—	2	1	no	—
33	C₄×C₁₉	—	4	1	no	—
34	Dic₁₉	—	8	1	no	—
35	AΣL(1,𝔽₁₆)	—	19	1	no	maximal, Sylow
36	C₂×C₁₉	—	2	1	no	—
37	C₂×C₁₉	—	4	1	no	—
38	C₂×C₁₉	—	4	2	no	—
39	C₂×C₁₉	—	4	1	no	—
40	C₂×C₁₉	#99	1	1	yes	characteristic
41	#120	D₁₉	1	1	yes	characteristic
42	#100	—	19	1	no	—
43	#99	—	19	1	no	—
44	C₁₉	AΣL(1,𝔽₁₆)	1	1	yes	characteristic, complemented, Sylow
45	C₂⁴	Dic₁₉	1	1	yes	characteristic, complemented
46	C₂×D₄	—	2	1	no	—
47	C₂×D₄	C₂×D₁₉	1	1	yes	characteristic
48	M₁₆	—	38	1	no	—
49	C₂²⋊C₄	—	2	1	no	—
50	C₂²⋊C₄	—	38	1	no	—
51	C₂²⋊C₄	Dic₁₉	1	1	yes	characteristic, complemented
52	C₂³	—	2	1	no	—
53	C₂³	—	4	2	no	—
54	C₂³	—	4	1	no	—
55	C₂³	C₂×Dic₁₉	1	1	yes	characteristic
56	C₂³	C₂²⋊D₁₉	1	1	yes	characteristic
57	D₄	—	4	1	no	—
58	D₄	—	4	2	no	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	76	1	no	—
61	C₂×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
62	C₈	—	76	1	no	—
63	C₂²	—	2	2	no	—
64	C₂²	—	2	1	no	—
65	C₂²	—	2	1	no	—
66	C₂²	—	4	1	no	—
67	C₂²	—	4	1	no	—
68	C₂²	—	4	1	no	—
69	C₂²	—	4	2	no	—
70	C₂²	—	4	2	no	—
71	C₂²	—	4	1	no	—
72	C₂²	C₂²⋊Dic₁₉	1	1	yes	characteristic
73	C₄	—	2	1	no	—
74	C₄	—	4	1	no	—
75	C₄	—	152	1	no	—
76	C₂	—	2	1	no	—
77	C₂	—	4	1	no	—
78	C₂	—	4	2	no	—
79	C₂	—	4	1	no	—
80	C₂	#10597291	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#10597291
commutator	C₂³×C₁₉	C₂×C₄
Frattini	C₂×D₄	C₂×D₁₉
Fitting, prime radical	C₁₉×#120	C₂
socle	C₂×C₁₉	#99
cosocle	C₂×D₄×C₁₉	C₂²
hypercenter	#120	D₁₉
hypocenter	C₁₉	AΣL(1,𝔽₁₆)
LCS	G ▷ C₂³×C₁₉ ▷ C₂²×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×D₄ ◁ #120

Complements

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

N	H ≅ G/N	type
2. #10597291	C₂	semidirect
3. #10597290	C₂	semidirect
4. C₂⁴×C₁₉	C₄	semidirect
8. C₂²⋊C₄×C₁₉	C₄	semidirect
44. C₁₉	AΣL(1,𝔽₁₆)	semidirect
45. C₂⁴	Dic₁₉	semidirect
51. C₂²⋊C₄	Dic₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₁₉×#120	2
#10597291	2
#10597290	2
AΣL(1,𝔽₁₆)	19