Subgroups — #11926311

Order 1216.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊C₄×C₁₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊Dic₁₉	C₂	1	2	yes	maximal
3	C₂⁴×C₁₉	C₂²	1	1	yes	characteristic
4	C₂²×C₄×C₁₉	C₄	1	2	yes	complemented
5	C₂²×Dic₁₉	—	2	2	no	—
6	C₂²⋊C₄×C₁₉	—	2	2	no	—
7	C₂²⋊Dic₁₉	—	2	2	no	—
8	C₂²⋊Dic₁₉	C₄	1	4	yes	complemented
9	C₂³×C₁₉	—	4	2	no	—
10	C₂³×C₁₉	C₂×C₄	1	2	yes	—
11	C₂³×C₁₉	C₂×C₄	1	1	yes	characteristic
12	C₂³×C₁₉	D₄	1	1	yes	characteristic
13	C₂³×C₁₉	D₄	1	2	yes	—
14	C₂³×C₁₉	Q₈	1	1	yes	characteristic
15	C₂×C₄×C₁₉	—	2	2	no	—
16	C₂×C₄×C₁₉	—	4	2	no	—
17	C₂×Dic₁₉	—	2	4	no	—
18	C₂×Dic₁₉	—	4	4	no	—
19	C₂²×C₁₉	—	2	4	no	—
20	C₂²×C₁₉	—	2	2	no	—
21	C₂²×C₁₉	—	4	2	no	—
22	C₂²×C₁₉	—	4	1	no	—
23	C₂²×C₁₉	—	4	1	no	—
24	C₂²×C₁₉	C₄²	1	1	yes	characteristic
25	C₂²×C₁₉	C₂²⋊C₄	1	2	yes	—
26	C₂²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
27	C₂²×C₁₉	C₄⋊C₄	1	2	yes	—
28	C₂²×C₁₉	C₄⋊C₄	1	1	yes	characteristic
29	C₄×C₁₉	—	4	2	no	—
30	Dic₁₉	—	4	4	no	—
31	#342	—	19	1	no	maximal, Sylow
32	C₂×C₁₉	—	2	2	no	—
33	C₂×C₁₉	—	2	1	no	—
34	C₂×C₁₉	—	2	2	no	—
35	C₂×C₁₉	—	2	1	no	—
36	C₂×C₁₉	#95	1	1	yes	characteristic
37	C₂×C₁₉	#99	1	2	yes	—
38	C₂×C₂²⋊C₄	—	19	2	no	—
39	C₂×C₂²⋊C₄	D₁₉	1	1	yes	characteristic
40	C₁₉	#342	1	1	yes	characteristic, complemented, Sylow
41	C₂⁴	C₂×D₁₉	1	1	yes	characteristic
42	C₂²×C₄	—	38	2	no	—
43	C₂²×C₄	Dic₁₉	1	2	yes	complemented
44	C₂²⋊C₄	—	2	2	no	—
45	C₂²⋊C₄	—	19	4	no	—
46	C₂²⋊C₄	—	38	2	no	—
47	C₂³	—	4	2	no	—
48	C₂³	Dic₃₈	1	1	yes	characteristic
49	C₂³	C₄×D₁₉	1	2	yes	—
50	C₂³	D₇₆	1	1	yes	characteristic
51	C₂³	C₂×Dic₁₉	1	1	yes	characteristic
52	C₂³	C₂²⋊D₁₉	1	2	yes	—
53	C₂×C₄	—	2	2	no	—
54	C₂×C₄	—	4	2	no	—
55	C₂×C₄	—	38	4	no	—
56	C₂×C₄	—	76	4	no	—
57	C₂²	—	2	2	no	—
58	C₂²	—	2	4	no	—
59	C₂²	—	4	1	no	—
60	C₂²	—	4	2	no	—
61	C₂²	—	4	1	no	—
62	C₂²	C₄×Dic₁₉	1	1	yes	characteristic
63	C₂²	Dic₁₉⋊C₄	1	2	yes	—
64	C₂²	C₄⋊Dic₁₉	1	1	yes	characteristic
65	C₂²	(C₂×D₁₉)⋊C₄	1	2	yes	—
66	C₂²	C₂²⋊Dic₁₉	1	1	yes	characteristic
67	C₄	—	4	2	no	—
68	C₄	—	76	4	no	—
69	C₂	—	2	2	no	—
70	C₂	—	2	1	no	—
71	C₂	—	2	2	no	—
72	C₂	—	2	1	no	—
73	C₂	C₁₉⋊#95	1	1	yes	characteristic
74	C₂	#10597291	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²⋊Dic₁₉
commutator	C₂²×C₁₉	C₄²
Frattini	C₂⁴	C₂×D₁₉
Fitting	C₂×C₂²⋊C₄×C₁₉	C₂
socle	C₂²×C₁₉	C₂²⋊C₄
prime radical, cosocle	C₂⁴×C₁₉	C₂²
hypercenter	C₂×C₂²⋊C₄	D₁₉
hypocenter	C₁₉	#342
LCS	G ▷ C₂²×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
4. C₂²×C₄×C₁₉	C₄	semidirect
8. C₂²⋊Dic₁₉	C₄	semidirect
40. C₁₉	#342	semidirect
43. 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
C₂×C₂²⋊C₄×C₁₉	2
C₂×C₂²⋊Dic₁₉	2
#342	19