Subgroups — #11926404

Order 1216.

234 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×C₁₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊Dic₁₉	C₂	1	2	yes	maximal
3	C₂×C₄⋊Dic₁₉	C₂	1	2	yes	maximal, complemented
4	C₁₉⋊#95	C₂	1	2	yes	maximal, complemented
5	C₂⁴×C₁₉	C₂²	1	1	yes	characteristic
6	C₂²×C₄×C₁₉	—	2	4	no	—
7	C₂²×C₄×C₁₉	C₄	1	4	yes	complemented
8	C₂²×C₄×C₁₉	C₂²	1	1	yes	characteristic
9	C₂²×C₄×C₁₉	C₂²	1	1	yes	characteristic
10	C₂²×Dic₁₉	C₂²	1	4	yes	—
11	C₂²⋊Dic₁₉	—	2	4	no	—
12	C₄⋊Dic₁₉	—	2	4	no	—
13	C₂³×C₁₉	—	2	2	no	—
14	C₂³×C₁₉	—	2	2	no	—
15	C₂³×C₁₉	C₂×C₄	1	2	yes	—
16	C₂³×C₁₉	D₄	1	2	yes	—
17	C₂³×C₁₉	Q₈	1	2	yes	—
18	C₂³×C₁₉	C₂³	1	1	yes	characteristic
19	C₂×C₄×C₁₉	—	2	2	no	—
20	C₂×C₄×C₁₉	—	2	8	no	—
21	C₂×C₄×C₁₉	C₂×C₄	1	2	yes	—
22	C₂×C₄×C₁₉	C₂×C₄	1	2	yes	—
23	C₂×C₄×C₁₉	D₄	1	4	yes	—
24	C₂×Dic₁₉	—	2	4	no	—
25	C₂×Dic₁₉	—	2	4	no	—
26	C₂×Dic₁₉	—	2	4	no	—
27	C₂²×C₁₉	—	2	2	no	—
28	C₂²×C₁₉	—	2	2	no	—
29	C₂²×C₁₉	—	2	4	no	—
30	C₂²×C₁₉	—	2	4	no	—
31	C₂²×C₁₉	C₄⋊C₄	1	4	yes	complemented
32	C₂²×C₁₉	C₂²×C₄	1	1	yes	characteristic
33	C₂²×C₁₉	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₁₉	C₂×D₄	1	2	yes	—
35	C₂²×C₁₉	C₂×Q₈	1	1	yes	characteristic
36	C₂²×C₁₉	#41	1	2	yes	—
37	C₄×C₁₉	—	2	2	no	—
38	C₄×C₁₉	C₂²⋊C₄	1	4	yes	complemented
39	Dic₁₉	—	4	4	no	—
40	#380	—	19	1	no	maximal, Sylow
41	C₂×C₁₉	—	2	4	no	—
42	C₂×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₁₉	#117	1	1	yes	characteristic
45	C₂×C₁₉	#121	1	2	yes	—
46	C₂×C₁₉	C₂²⋊Q₈	1	2	yes	—
47	C₂³×C₄	D₁₉	1	1	yes	characteristic
48	C₂×C₄⋊C₄	—	19	2	no	—
49	C₂×C₂²⋊C₄	—	19	2	no	—
50	#95	—	19	2	no	—
51	C₁₉	#380	1	1	yes	characteristic, complemented, Sylow
52	C₂⁴	C₂×D₁₉	1	1	yes	characteristic
53	C₂²×C₄	—	2	4	no	—
54	C₂²×C₄	—	19	4	no	—
55	C₂²×C₄	Dic₁₉	1	4	yes	complemented
56	C₂²×C₄	C₂×D₁₉	1	1	yes	characteristic
57	C₂²×C₄	C₂×D₁₉	1	1	yes	characteristic
58	C₄⋊C₄	—	38	4	no	—
59	C₂²⋊C₄	—	38	4	no	—
60	C₂³	—	2	2	no	—
61	C₂³	—	2	2	no	—
62	C₂³	Dic₃₈	1	2	yes	—
63	C₂³	D₇₆	1	2	yes	—
64	C₂³	C₂×Dic₁₉	1	2	yes	—
65	C₂³	C₂²×D₁₉	1	1	yes	characteristic
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	—	2	8	no	—
68	C₂×C₄	—	38	4	no	—
69	C₂×C₄	—	38	4	no	—
70	C₂×C₄	—	38	4	no	—
71	C₂×C₄	C₂×Dic₁₉	1	2	yes	—
72	C₂×C₄	C₂×Dic₁₉	1	2	yes	—
73	C₂×C₄	C₂²⋊D₁₉	1	4	yes	—
74	C₂²	—	2	4	no	—
75	C₂²	—	2	2	no	—
76	C₂²	—	2	4	no	—
77	C₂²	—	2	2	no	—
78	C₂²	C₄⋊Dic₁₉	1	4	yes	complemented
79	C₂²	C₂×Dic₃₈	1	1	yes	characteristic
80	C₂²	C₂×D₇₆	1	1	yes	characteristic
81	C₂²	#64541	1	2	yes	—
82	C₂²	C₂²×Dic₁₉	1	1	yes	characteristic
83	C₂²	C₂×C₂²⋊D₁₉	1	2	yes	—
84	C₄	—	2	2	no	—
85	C₄	—	76	4	no	—
86	C₄	C₂²⋊Dic₁₉	1	4	yes	complemented
87	C₂	—	2	4	no	—
88	C₂	#10597370	1	2	yes	—
89	C₂	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
90	C₂	#10597372	1	1	yes	characteristic
91	C₂	#10597376	1	2	yes	—
92	C₂	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₁₉
commutator	C₂²×C₁₉	C₂²×C₄
Fitting	C₂³×C₄×C₁₉	C₂
socle, cosocle	C₂³×C₁₉	C₂³
prime radical	C₂⁴×C₁₉	C₂²
hypercenter	C₂³×C₄	D₁₉
hypocenter	C₁₉	#380
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
3. C₂×C₄⋊Dic₁₉	C₂	semidirect
4. C₁₉⋊#95	C₂	semidirect
7. C₂²×C₄×C₁₉	C₄	semidirect
31. C₂²×C₁₉	C₄⋊C₄	semidirect
38. C₄×C₁₉	C₂²⋊C₄	semidirect
51. C₁₉	#380	semidirect
55. C₂²×C₄	Dic₁₉	semidirect
78. C₂²	C₄⋊Dic₁₉	semidirect
86. C₄	C₂²⋊Dic₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₄×C₁₉	2
C₂×C₂²⋊Dic₁₉	2
C₂×C₄⋊Dic₁₉	2
C₁₉⋊#95	2
#380	19