Subgroups — C₁₉×#12011 #18017970

Order 4864.

220 conjugacy classes of subgroups.

The subgroup lattice has 136 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#364	C₂	1	2	yes	maximal
2	C₄×C₁₉×M₃₂	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1373	C₂	1	4	yes	maximal, complemented
4	C₂×C₁₉×M₃₂	—	2	2	no	—
5	C₂×C₁₉×M₃₂	C₄	1	1	yes	characteristic
6	C₂×C₁₉×M₃₂	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
8	C₁₉×#364	—	2	2	no	—
9	C₁₉×#364	C₂²	1	4	yes	—
10	C₁₉×#346	C₄	1	2	yes	—
11	C₄×C₁₆×C₁₉	C₂²	1	2	yes	—
12	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
14	C₂×C₄²×C₁₉	Q₈	1	1	yes	characteristic
15	C₁₉×M₃₂	—	2	2	no	—
16	C₁₉×M₃₂	—	2	4	no	—
17	C₁₉×M₃₂	—	4	2	no	—
18	C₁₉×M₃₂	C₈	1	4	yes	—
19	C₂×C₁₆×C₁₉	—	4	2	no	—
20	C₂×C₁₆×C₁₉	C₂×C₄	1	2	yes	—
21	C₂×C₁₆×C₁₉	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
24	C₄×C₈×C₁₉	Q₈	1	1	yes	characteristic
25	C₄×C₈×C₁₉	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₁₉	—	2	1	no	—
27	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₈×C₁₉	C₂×C₈	1	2	yes	—
31	C₂×C₈×C₁₉	M₁₆	1	2	yes	—
32	C₂×C₈×C₁₉	D₈	1	2	yes	—
33	C₂×C₈×C₁₉	Q₁₆	1	2	yes	—
34	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
36	C₄²×C₁₉	—	2	1	no	—
37	C₄²×C₁₉	C₄⋊C₄	1	1	yes	characteristic
38	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
39	C₁₆×C₁₉	—	2	2	no	—
40	C₁₆×C₁₉	—	4	4	no	—
41	C₁₆×C₁₉	C₂×C₈	1	4	yes	—
42	#12011	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
43	C₂³×C₁₉	#108	1	1	yes	characteristic
44	C₂×C₄×C₁₉	—	2	1	no	—
45	C₂×C₄×C₁₉	—	2	1	no	—
46	C₂×C₄×C₁₉	—	4	2	no	—
47	C₂×C₄×C₁₉	C₄⋊C₈	1	2	yes	—
48	C₂×C₄×C₁₉	#107	1	2	yes	—
49	C₂×C₄×C₁₉	#108	1	1	yes	characteristic
50	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₈×C₁₉	C₄⋊C₈	1	2	yes	—
52	C₈×C₁₉	#107	1	2	yes	—
53	C₈×C₁₉	C₂²×C₈	1	1	yes	characteristic
54	C₈×C₁₉	C₂×M₁₆	1	1	yes	characteristic
55	C₈×C₁₉	C₂×D₈	1	1	yes	characteristic
56	C₈×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
57	C₂×#364	C₂×C₁₉	1	2	yes	—
58	C₄×M₃₂	C₂×C₁₉	1	1	yes	characteristic
59	#1373	C₂×C₁₉	1	4	yes	complemented
60	C₂²×C₁₉	—	4	1	no	—
61	C₂²×C₁₉	#335	1	2	yes	—
62	C₂²×C₁₉	C₂×#108	1	1	yes	characteristic
63	C₄×C₁₉	—	2	2	no	—
64	C₄×C₁₉	—	4	1	no	—
65	C₄×C₁₉	#335	1	2	yes	—
66	C₄×C₁₉	C₂×C₄⋊C₈	1	1	yes	characteristic
67	C₄×C₁₉	C₂×#107	1	1	yes	characteristic
68	C₂×M₃₂	—	2	2	no	—
69	C₂×M₃₂	C₄×C₁₉	1	1	yes	characteristic
70	C₂×M₃₂	C₄×C₁₉	1	1	yes	characteristic
71	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
72	#364	—	2	2	no	—
73	#364	C₂²×C₁₉	1	4	yes	—
74	#346	C₄×C₁₉	1	2	yes	—
75	C₄×C₁₆	C₂²×C₁₉	1	2	yes	—
76	C₂×C₁₉	—	2	1	no	—
77	C₂×C₁₉	—	2	2	no	—
78	C₂×C₁₉	C₂×#335	1	1	yes	characteristic
79	C₂²×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
80	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
81	C₂×C₄²	Q₈×C₁₉	1	1	yes	characteristic
82	M₃₂	—	2	2	no	—
83	M₃₂	—	2	4	no	—
84	M₃₂	—	4	2	no	—
85	M₃₂	C₈×C₁₉	1	4	yes	—
86	C₂×C₁₆	—	4	2	no	—
87	C₂×C₁₆	C₂×C₄×C₁₉	1	2	yes	—
88	C₂×C₁₆	C₂×C₄×C₁₉	1	2	yes	—
89	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
90	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
91	C₄×C₈	Q₈×C₁₉	1	1	yes	characteristic
92	C₄×C₈	C₂³×C₁₉	1	1	yes	characteristic
93	C₁₉	#12011	1	1	yes	characteristic, direct product factor, Sylow
94	C₂²×C₄	—	2	1	no	—
95	C₂²×C₄	C₄⋊C₄×C₁₉	1	1	yes	characteristic
96	C₂×C₈	C₄⋊C₄×C₁₉	1	1	yes	characteristic
97	C₂×C₈	C₄⋊C₄×C₁₉	1	1	yes	characteristic
98	C₂×C₈	C₂×C₈×C₁₉	1	2	yes	—
99	C₂×C₈	M₁₆×C₁₉	1	2	yes	—
100	C₂×C₈	D₈×C₁₉	1	2	yes	—
101	C₂×C₈	Q₁₆×C₁₉	1	2	yes	—
102	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
103	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
104	C₄²	—	2	1	no	—
105	C₄²	C₄⋊C₄×C₁₉	1	1	yes	characteristic
106	C₄²	C₂×Q₈×C₁₉	1	1	yes	characteristic
107	C₁₆	—	2	2	no	—
108	C₁₆	—	4	4	no	—
109	C₁₆	C₂×C₈×C₁₉	1	4	yes	—
110	C₂³	C₁₉×#108	1	1	yes	characteristic
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	C₁₉×C₄⋊C₈	1	2	yes	—
115	C₂×C₄	C₁₉×#107	1	2	yes	—
116	C₂×C₄	C₁₉×#108	1	1	yes	characteristic
117	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	1	yes	characteristic
118	C₈	C₁₉×C₄⋊C₈	1	2	yes	—
119	C₈	C₁₉×#107	1	2	yes	—
120	C₈	C₂²×C₈×C₁₉	1	1	yes	characteristic
121	C₈	C₂×M₁₆×C₁₉	1	1	yes	characteristic
122	C₈	C₂×D₈×C₁₉	1	1	yes	characteristic
123	C₈	C₂×Q₁₆×C₁₉	1	1	yes	characteristic
124	C₂²	—	4	1	no	—
125	C₂²	C₁₉×#335	1	2	yes	—
126	C₂²	C₂×C₁₉×#108	1	1	yes	characteristic
127	C₄	—	2	2	no	—
128	C₄	—	4	1	no	—
129	C₄	C₁₉×#335	1	2	yes	—
130	C₄	C₂×C₁₉×C₄⋊C₈	1	1	yes	characteristic
131	C₄	C₂×C₁₉×#107	1	1	yes	characteristic
132	C₂	—	2	1	no	—
133	C₂	—	2	2	no	—
134	C₂	C₂×C₁₉×#335	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₁₉	C₂×D₈
commutator	C₈	C₂²×C₈×C₁₉
Frattini, cosocle	C₄×C₈	C₂³×C₁₉
socle	C₂×C₁₉	C₂×#335
prime radical	C₂³×C₁₉	#108
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₉ ◁ C₂²×C₈×C₁₉ ◁ C₂×C₄×C₈×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₉×#1373	C₂	semidirect
42. #12011	C₁₉	direct (N × H)
59. #1373	C₂×C₁₉	semidirect
93. C₁₉	#12011	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₉×#364	2
C₄×C₁₉×M₃₂	2
C₁₉×#1373	2
#12011	19

Subgroups — C19×#12011 #18017970

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#12011 #18017970