Subgroups — C₁₉×#20370 #18026329

Order 4864.

908 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#2873	C₂	1	3	yes	maximal
2	C₁₉×#1747	C₂	1	8	yes	maximal, complemented
3	C₂×C₁₉×#338	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₉×H(3,ℤ/4)	C₂	1	3	yes	maximal
5	C₂²×M₁₆×C₁₉	C₂²	1	3	yes	—
6	C₂²×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
7	C₂×C₁₉×#117	C₂²	1	3	yes	—
8	C₁₉×#405	C₄	1	12	yes	complemented
9	C₁₉×#405	C₂²	1	12	yes	—
10	C₄×M₁₆×C₁₉	C₄	1	12	yes	complemented
11	C₁₉×#338	C₂²	1	4	yes	—
12	C₁₉×H(3,ℤ/4)	C₂²	1	12	yes	—
13	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
14	C₂×M₁₆×C₁₉	—	2	6	no	—
15	C₂×M₁₆×C₁₉	—	2	6	no	—
16	C₂×M₁₆×C₁₉	C₂×C₄	1	6	yes	—
17	C₂×M₁₆×C₁₉	C₂×C₄	1	6	yes	—
18	C₂×M₁₆×C₁₉	C₂×C₄	1	6	yes	—
19	C₂×M₁₆×C₁₉	C₂³	1	6	yes	—
20	C₂²×C₈×C₁₉	C₂×C₄	1	6	yes	—
21	C₂²×C₈×C₁₉	C₂³	1	2	yes	—
22	C₁₉×#117	—	2	3	no	—
23	C₁₉×#117	—	2	3	no	—
24	C₁₉×#117	C₂×C₄	1	6	yes	—
25	C₁₉×#117	C₂³	1	6	yes	—
26	C₂×C₄⋊C₄×C₁₉	—	2	3	no	—
27	C₂×C₂²⋊C₄×C₁₉	—	2	3	no	—
28	C₂×C₄²×C₁₉	C₂×C₄	1	6	yes	—
29	C₁₉×#97	C₂×C₄	1	24	yes	complemented
30	C₄×C₈×C₁₉	C₂×C₄	1	24	yes	complemented
31	C₂⁴×C₁₉	#41	1	1	yes	characteristic
32	C₂²×C₄×C₁₉	—	2	6	no	—
33	C₂²×C₄×C₁₉	C₂²×C₄	1	3	yes	—
34	C₂²×C₄×C₁₉	C₂²×C₄	1	3	yes	—
35	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
36	C₂²×C₄×C₁₉	#41	1	3	yes	—
37	C₂²×C₄×C₁₉	#41	1	3	yes	—
38	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
39	M₁₆×C₁₉	—	2	12	no	—
40	M₁₆×C₁₉	—	2	12	no	—
41	M₁₆×C₁₉	C₄²	1	12	yes	—
42	M₁₆×C₁₉	C₄²	1	4	yes	—
43	C₂×C₈×C₁₉	—	2	6	no	—
44	C₂×C₈×C₁₉	—	2	6	no	—
45	C₂×C₈×C₁₉	C₂²×C₄	1	12	yes	—
46	C₂×C₈×C₁₉	C₂²×C₄	1	12	yes	—
47	C₄⋊C₄×C₁₉	—	2	6	no	—
48	C₄⋊C₄×C₁₉	—	2	6	no	—
49	C₂²⋊C₄×C₁₉	—	2	6	no	—
50	C₂²⋊C₄×C₁₉	—	2	6	no	—
51	C₄²×C₁₉	—	2	6	no	—
52	C₄²×C₁₉	C₂²×C₄	1	12	yes	—
53	#20370	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
54	C₂³×C₁₉	—	8	1	no	—
55	C₂³×C₁₉	#117	1	3	yes	—
56	C₂³×C₁₉	#117	1	3	yes	—
57	C₂³×C₁₉	C₂×#41	1	1	yes	characteristic
58	C₂×C₄×C₁₉	—	2	12	no	—
59	C₂×C₄×C₁₉	—	8	6	no	—
60	C₂×C₄×C₁₉	C₂×C₄²	1	3	yes	—
61	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
62	C₂×C₄×C₁₉	#117	1	3	yes	—
63	C₂×C₄×C₁₉	#117	1	3	yes	—
64	C₂×C₄×C₁₉	#117	1	6	yes	—
65	C₂×C₄×C₁₉	#117	1	3	yes	—
66	C₂×C₄×C₁₉	#117	1	3	yes	—
67	C₂×C₄×C₁₉	C₂³×C₄	1	3	yes	—
68	C₂×C₄×C₁₉	C₂×#41	1	3	yes	—
69	C₈×C₁₉	—	2	12	no	—
70	C₈×C₁₉	C₂×C₄²	1	8	yes	—
71	#2873	C₂×C₁₉	1	3	yes	—
72	#1747	C₂×C₁₉	1	8	yes	complemented
73	C₂×#338	C₂×C₁₉	1	1	yes	characteristic
74	C₂×H(3,ℤ/4)	C₂×C₁₉	1	3	yes	—
75	C₂²×C₁₉	—	4	3	no	—
76	C₂²×C₁₉	—	4	3	no	—
77	C₂²×C₁₉	—	4	1	no	—
78	C₂²×C₁₉	#376	1	1	yes	characteristic
79	C₂²×C₁₉	#376	1	3	yes	—
80	C₂²×C₁₉	C₂×#117	1	3	yes	—
81	C₄×C₁₉	—	4	12	no	—
82	C₄×C₁₉	#376	1	1	yes	characteristic
83	C₄×C₁₉	#376	1	3	yes	—
84	C₄×C₁₉	C₂²×C₄²	1	1	yes	characteristic
85	C₄×C₁₉	C₂×#117	1	3	yes	—
86	C₂²×M₁₆	C₂²×C₁₉	1	3	yes	—
87	C₂²×M₁₆	C₂²×C₁₉	1	1	yes	characteristic
88	C₂×#117	C₂²×C₁₉	1	3	yes	—
89	#405	C₄×C₁₉	1	12	yes	complemented
90	#405	C₂²×C₁₉	1	12	yes	—
91	C₄×M₁₆	C₄×C₁₉	1	12	yes	complemented
92	#338	C₂²×C₁₉	1	4	yes	—
93	H(3,ℤ/4)	C₂²×C₁₉	1	12	yes	—
94	C₂×C₁₉	—	2	3	no	—
95	C₂×C₁₉	—	2	3	no	—
96	C₂×C₁₉	—	2	1	no	—
97	C₂×C₁₉	C₂×#376	1	1	yes	characteristic
98	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
99	C₂×M₁₆	—	2	6	no	—
100	C₂×M₁₆	—	2	6	no	—
101	C₂×M₁₆	C₂×C₄×C₁₉	1	6	yes	—
102	C₂×M₁₆	C₂×C₄×C₁₉	1	6	yes	—
103	C₂×M₁₆	C₂×C₄×C₁₉	1	6	yes	—
104	C₂×M₁₆	C₂³×C₁₉	1	6	yes	—
105	C₂²×C₈	C₂×C₄×C₁₉	1	6	yes	—
106	C₂²×C₈	C₂³×C₁₉	1	2	yes	—
107	#117	—	2	3	no	—
108	#117	—	2	3	no	—
109	#117	C₂×C₄×C₁₉	1	6	yes	—
110	#117	C₂³×C₁₉	1	6	yes	—
111	C₂×C₄⋊C₄	—	2	3	no	—
112	C₂×C₂²⋊C₄	—	2	3	no	—
113	C₂×C₄²	C₂×C₄×C₁₉	1	6	yes	—
114	#97	C₂×C₄×C₁₉	1	24	yes	complemented
115	C₄×C₈	C₂×C₄×C₁₉	1	24	yes	complemented
116	C₁₉	#20370	1	1	yes	characteristic, direct product factor, Sylow
117	C₂⁴	#41×C₁₉	1	1	yes	characteristic
118	C₂²×C₄	—	2	6	no	—
119	C₂²×C₄	C₂²×C₄×C₁₉	1	3	yes	—
120	C₂²×C₄	C₂²×C₄×C₁₉	1	3	yes	—
121	C₂²×C₄	#41×C₁₉	1	1	yes	characteristic
122	C₂²×C₄	#41×C₁₉	1	3	yes	—
123	C₂²×C₄	#41×C₁₉	1	3	yes	—
124	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
125	M₁₆	—	2	12	no	—
126	M₁₆	—	2	12	no	—
127	M₁₆	C₄²×C₁₉	1	12	yes	—
128	M₁₆	C₄²×C₁₉	1	4	yes	—
129	C₂×C₈	—	2	6	no	—
130	C₂×C₈	—	2	6	no	—
131	C₂×C₈	C₂²×C₄×C₁₉	1	12	yes	—
132	C₂×C₈	C₂²×C₄×C₁₉	1	12	yes	—
133	C₄⋊C₄	—	2	6	no	—
134	C₄⋊C₄	—	2	6	no	—
135	C₂²⋊C₄	—	2	6	no	—
136	C₂²⋊C₄	—	2	6	no	—
137	C₄²	—	2	6	no	—
138	C₄²	C₂²×C₄×C₁₉	1	12	yes	—
139	C₂³	—	8	1	no	—
140	C₂³	C₁₉×#117	1	3	yes	—
141	C₂³	C₁₉×#117	1	3	yes	—
142	C₂³	C₂×#41×C₁₉	1	1	yes	characteristic
143	C₂×C₄	—	2	12	no	—
144	C₂×C₄	—	8	6	no	—
145	C₂×C₄	C₂×C₄²×C₁₉	1	3	yes	—
146	C₂×C₄	C₂×C₄²×C₁₉	1	1	yes	characteristic
147	C₂×C₄	C₁₉×#117	1	3	yes	—
148	C₂×C₄	C₁₉×#117	1	3	yes	—
149	C₂×C₄	C₁₉×#117	1	3	yes	—
150	C₂×C₄	C₁₉×#117	1	3	yes	—
151	C₂×C₄	C₁₉×#117	1	6	yes	—
152	C₂×C₄	C₂³×C₄×C₁₉	1	3	yes	—
153	C₂×C₄	C₂×#41×C₁₉	1	3	yes	—
154	C₈	—	2	12	no	—
155	C₈	C₂×C₄²×C₁₉	1	8	yes	—
156	C₂²	—	4	3	no	—
157	C₂²	—	4	3	no	—
158	C₂²	—	4	1	no	—
159	C₂²	C₁₉×#376	1	1	yes	characteristic
160	C₂²	C₁₉×#376	1	3	yes	—
161	C₂²	C₂×C₁₉×#117	1	3	yes	—
162	C₄	—	4	12	no	—
163	C₄	C₁₉×#376	1	3	yes	—
164	C₄	C₁₉×#376	1	1	yes	characteristic
165	C₄	C₂²×C₄²×C₁₉	1	1	yes	characteristic
166	C₄	C₂×C₁₉×#117	1	3	yes	—
167	C₂	—	2	3	no	—
168	C₂	—	2	3	no	—
169	C₂	—	2	1	no	—
170	C₂	C₂×C₁₉×#376	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₉	C₂²×C₄²
commutator	C₄	C₂²×C₄²×C₁₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
socle	C₂×C₁₉	C₂×#376
prime radical	C₂⁴×C₁₉	#41
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₉×#1747	C₂	semidirect
8. C₁₉×#405	C₄	semidirect
10. C₄×M₁₆×C₁₉	C₄	semidirect
29. C₁₉×#97	C₂×C₄	semidirect
30. C₄×C₈×C₁₉	C₂×C₄	semidirect
53. #20370	C₁₉	direct (N × H)
72. #1747	C₂×C₁₉	semidirect
89. #405	C₄×C₁₉	semidirect
91. C₄×M₁₆	C₄×C₁₉	semidirect
114. #97	C₂×C₄×C₁₉	semidirect
115. C₄×C₈	C₂×C₄×C₁₉	semidirect
116. C₁₉	#20370	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×#2873	2
C₁₉×#1747	2
C₂×C₁₉×#338	2
C₂×C₁₉×H(3,ℤ/4)	2
#20370	19

Subgroups — C19×#20370 #18026329

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#20370 #18026329