Subgroups — C₁₉×#13641 #18019600

Order 4864.

180 conjugacy classes of subgroups.

The subgroup lattice has 140 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#2108	C₂	1	1	yes	maximal, characteristic
2	C₁₉×C₄⋊C₃₂	C₂	1	2	yes	maximal, complemented
3	C₁₉×C₂²⋊C₃₂	C₂	1	2	yes	maximal
4	C₁₉×#1397	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×C₁₉×C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₉×M₃₂	C₄	1	1	yes	characteristic
7	C₂²×C₁₆×C₁₉	C₂²	1	1	yes	characteristic
8	C₁₉×#405	C₄	1	1	yes	characteristic
9	C₂×C₁₉×C₃₂	C₂²	1	2	yes	—
10	C₂×C₁₉×C₃₂	C₂²	1	2	yes	—
11	C₁₉×#346	C₄	1	1	yes	characteristic
12	C₁₉×#346	C₄	1	1	yes	characteristic
13	C₄×C₁₆×C₁₉	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₁₉	C₂²	1	1	yes	characteristic
15	C₂×M₁₆×C₁₉	C₈	1	1	yes	characteristic
16	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
17	C₁₉×#117	C₈	1	1	yes	characteristic
18	C₁₉×M₃₂	—	2	2	no	—
19	C₂×C₁₆×C₁₉	—	2	2	no	—
20	C₂×C₁₆×C₁₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₆×C₁₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆×C₁₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆×C₁₉	C₂³	1	1	yes	characteristic
24	C₁₉×#97	C₈	1	1	yes	characteristic
25	C₁₉×#97	C₈	1	1	yes	characteristic
26	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
28	C₁₉×C₃₂	—	2	2	no	—
29	C₁₉×C₃₂	—	2	2	no	—
30	C₂²×C₄×C₁₉	C₂×C₈	1	1	yes	characteristic
31	M₁₆×C₁₉	—	2	2	no	—
32	C₂×C₈×C₁₉	—	2	1	no	—
33	C₂×C₈×C₁₉	—	2	1	no	—
34	C₂×C₈×C₁₉	C₂×C₈	1	1	yes	characteristic
35	C₂×C₈×C₁₉	C₂×C₈	1	1	yes	characteristic
36	C₂×C₈×C₁₉	C₂×C₈	1	1	yes	characteristic
37	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
38	C₄⋊C₄×C₁₉	C₁₆	1	2	yes	—
39	C₂²⋊C₄×C₁₉	C₁₆	1	2	yes	—
40	C₄²×C₁₉	C₂×C₈	1	1	yes	characteristic
41	C₄²×C₁₉	C₂×C₈	1	1	yes	characteristic
42	C₁₆×C₁₉	—	2	1	no	—
43	C₁₆×C₁₉	—	2	1	no	—
44	C₁₆×C₁₉	M₁₆	1	2	yes	—
45	C₁₆×C₁₉	#41	1	2	yes	—
46	#13641	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
47	C₂³×C₁₉	C₂×C₁₆	1	1	yes	characteristic
48	C₂×C₄×C₁₉	—	2	1	no	—
49	C₂×C₄×C₁₉	—	2	1	no	—
50	C₂×C₄×C₁₉	C₂×C₁₆	1	2	yes	—
51	C₂×C₄×C₁₉	C₂×C₁₆	1	2	yes	—
52	C₂×C₄×C₁₉	C₂×C₁₆	1	1	yes	characteristic
53	C₂×C₄×C₁₉	C₂²×C₈	1	1	yes	characteristic
54	C₈×C₁₉	—	2	1	no	—
55	C₈×C₁₉	—	2	1	no	—
56	C₈×C₁₉	M₃₂	1	2	yes	—
57	C₈×C₁₉	#117	1	1	yes	characteristic
58	C₈×C₁₉	C₂×M₁₆	1	1	yes	characteristic
59	#2108	C₂×C₁₉	1	1	yes	characteristic
60	C₄⋊C₃₂	C₂×C₁₉	1	2	yes	complemented
61	C₂²⋊C₃₂	C₂×C₁₉	1	2	yes	—
62	#1397	C₂×C₁₉	1	1	yes	characteristic, complemented
63	C₄×C₃₂	C₂×C₁₉	1	1	yes	characteristic, complemented
64	C₂²×C₁₉	—	2	1	no	—
65	C₂²×C₁₉	—	2	2	no	—
66	C₂²×C₁₉	C₂²×C₁₆	1	1	yes	characteristic
67	C₄×C₁₉	—	2	1	no	—
68	C₄×C₁₉	—	2	2	no	—
69	C₄×C₁₉	—	2	2	no	—
70	C₄×C₁₉	#431	1	1	yes	characteristic
71	C₄×C₁₉	C₂×M₃₂	1	1	yes	characteristic
72	C₂×M₃₂	C₄×C₁₉	1	1	yes	characteristic
73	C₂²×C₁₆	C₂²×C₁₉	1	1	yes	characteristic
74	#405	C₄×C₁₉	1	1	yes	characteristic
75	C₂×C₃₂	C₂²×C₁₉	1	2	yes	—
76	C₂×C₃₂	C₂²×C₁₉	1	2	yes	—
77	#346	C₄×C₁₉	1	1	yes	characteristic
78	#346	C₄×C₁₉	1	1	yes	characteristic
79	C₄×C₁₆	C₂²×C₁₉	1	1	yes	characteristic
80	C₄×C₁₆	C₂²×C₁₉	1	1	yes	characteristic
81	C₂×C₁₉	—	4	1	no	—
82	C₂×C₁₉	#2162	1	1	yes	characteristic
83	C₂×C₁₉	#2258	1	2	yes	—
84	C₂×M₁₆	C₈×C₁₉	1	1	yes	characteristic
85	C₂²×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
86	#117	C₈×C₁₉	1	1	yes	characteristic
87	M₃₂	—	2	2	no	—
88	C₂×C₁₆	—	2	2	no	—
89	C₂×C₁₆	C₂×C₄×C₁₉	1	1	yes	characteristic
90	C₂×C₁₆	C₂×C₄×C₁₉	1	1	yes	characteristic
91	C₂×C₁₆	C₂×C₄×C₁₉	1	1	yes	characteristic
92	C₂×C₁₆	C₂³×C₁₉	1	1	yes	characteristic
93	#97	C₈×C₁₉	1	1	yes	characteristic
94	#97	C₈×C₁₉	1	1	yes	characteristic
95	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
96	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
97	C₃₂	—	2	2	no	—
98	C₃₂	—	2	2	no	—
99	C₁₉	#13641	1	1	yes	characteristic, direct product factor, Sylow
100	C₂²×C₄	C₂×C₈×C₁₉	1	1	yes	characteristic
101	M₁₆	—	2	2	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	C₂×C₈×C₁₉	1	1	yes	characteristic
105	C₂×C₈	C₂×C₈×C₁₉	1	1	yes	characteristic
106	C₂×C₈	C₂×C₈×C₁₉	1	1	yes	characteristic
107	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
108	C₄⋊C₄	C₁₆×C₁₉	1	2	yes	—
109	C₂²⋊C₄	C₁₆×C₁₉	1	2	yes	—
110	C₄²	C₂×C₈×C₁₉	1	1	yes	characteristic
111	C₄²	C₂×C₈×C₁₉	1	1	yes	characteristic
112	C₁₆	—	2	1	no	—
113	C₁₆	—	2	1	no	—
114	C₁₆	M₁₆×C₁₉	1	2	yes	—
115	C₁₆	#41×C₁₉	1	2	yes	—
116	C₂³	C₂×C₁₆×C₁₉	1	1	yes	characteristic
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	C₂×C₁₆×C₁₉	1	2	yes	—
120	C₂×C₄	C₂×C₁₆×C₁₉	1	2	yes	—
121	C₂×C₄	C₂×C₁₆×C₁₉	1	1	yes	characteristic
122	C₂×C₄	C₂²×C₈×C₁₉	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	—	2	1	no	—
125	C₈	C₁₉×M₃₂	1	2	yes	—
126	C₈	C₁₉×#117	1	1	yes	characteristic
127	C₈	C₂×M₁₆×C₁₉	1	1	yes	characteristic
128	C₂²	—	2	1	no	—
129	C₂²	—	2	2	no	—
130	C₂²	C₂²×C₁₆×C₁₉	1	1	yes	characteristic
131	C₄	—	2	1	no	—
132	C₄	—	2	2	no	—
133	C₄	—	2	2	no	—
134	C₄	C₁₉×#431	1	1	yes	characteristic
135	C₄	C₂×C₁₉×M₃₂	1	1	yes	characteristic
136	C₂	—	4	1	no	—
137	C₂	C₁₉×#2162	1	1	yes	characteristic
138	C₂	C₁₉×#2258	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₆×C₁₉	C₂³
commutator	C₂²	C₂²×C₁₆×C₁₉
Frattini, cosocle	C₂×C₁₆	C₂³×C₁₉
socle	C₂²×C₁₉	C₂²×C₁₆
prime radical	C₂³×C₁₉	C₂×C₁₆
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₁₆×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₉×C₄⋊C₃₂	C₂	semidirect
4. C₁₉×#1397	C₂	semidirect
5. C₄×C₁₉×C₃₂	C₂	semidirect
46. #13641	C₁₉	direct (N × H)
60. C₄⋊C₃₂	C₂×C₁₉	semidirect
62. #1397	C₂×C₁₉	semidirect
63. C₄×C₃₂	C₂×C₁₉	semidirect
99. C₁₉	#13641	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₉×#2108	2
C₁₉×C₄⋊C₃₂	2
C₁₉×C₂²⋊C₃₂	2
C₁₉×#1397	2
C₄×C₁₉×C₃₂	2
#13641	19

Subgroups — C19×#13641 #18019600

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#13641 #18019600