Subgroups — C₁₉×#13170 #18019129

Order 4864.

278 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#368	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2158	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₉×#346	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1945	C₂	1	2	yes	maximal
5	C₁₉×#1382	C₂	1	2	yes	maximal
6	C₂²×C₁₆×C₁₉	C₂²	1	1	yes	characteristic
7	C₂²×C₁₆×C₁₉	C₂²	1	1	yes	characteristic
8	C₂×C₁₉×#108	C₂²	1	2	yes	—
9	C₁₉×#427	C₂²	1	2	yes	—
10	C₁₉×#424	—	2	2	no	—
11	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
12	C₁₉×#368	—	2	2	no	—
13	C₁₉×#367	C₄	1	2	yes	complemented
14	C₁₉×#366	C₄	1	2	yes	complemented
15	C₁₉×#346	—	2	2	no	—
16	C₁₉×#339	—	2	2	no	—
17	C₂×M₁₆×C₁₉	—	2	2	no	—
18	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
20	C₁₉×#117	—	2	2	no	—
21	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
22	C₂×C₁₆×C₁₉	—	2	1	no	—
23	C₂×C₁₆×C₁₉	—	2	1	no	—
24	C₂×C₁₆×C₁₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₁₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₁₉	D₄	1	2	yes	—
27	C₂×C₁₆×C₁₉	Q₈	1	2	yes	—
28	C₂×C₁₆×C₁₉	Q₈	1	2	yes	—
29	C₁₉×#108	—	2	2	no	—
30	C₁₉×#108	—	2	2	no	—
31	C₁₉×#107	C₂×C₄	1	4	yes	—
32	C₁₉×#106	—	2	2	no	—
33	C₁₉×C₄⋊C₈	—	4	4	no	—
34	C₄×C₈×C₁₉	—	2	1	no	—
35	C₄×C₈×C₁₉	—	2	1	no	—
36	C₂²×C₄×C₁₉	—	2	1	no	—
37	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
38	M₁₆×C₁₉	—	4	2	no	—
39	M₁₆×C₁₉	—	4	2	no	—
40	C₂×C₈×C₁₉	—	2	1	no	—
41	C₂×C₈×C₁₉	—	2	1	no	—
42	C₂×C₈×C₁₉	—	4	2	no	—
43	C₂×C₈×C₁₉	D₈	1	2	yes	—
44	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₁₉	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₁₉	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
49	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
50	C₄⋊C₄×C₁₉	—	2	4	no	—
51	C₂²⋊C₄×C₁₉	—	4	2	no	—
52	C₄²×C₁₉	—	2	1	no	—
53	C₄²×C₁₉	—	4	2	no	—
54	C₄²×C₁₉	Q₁₆	1	2	yes	—
55	C₁₆×C₁₉	—	2	2	no	—
56	C₁₆×C₁₉	C₄⋊C₄	1	2	yes	—
57	C₁₆×C₁₉	C₄⋊C₄	1	2	yes	—
58	#13170	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂³×C₁₉	#135	1	1	yes	characteristic
60	C₂×C₄×C₁₉	—	2	2	no	—
61	C₂×C₄×C₁₉	—	4	2	no	—
62	C₂×C₄×C₁₉	—	4	4	no	—
63	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
64	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
65	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
68	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
69	C₈×C₁₉	—	2	2	no	—
70	C₈×C₁₉	—	8	2	no	—
71	C₈×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₈×C₁₉	C₄×Q₈	1	1	yes	characteristic
73	C₈×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
74	C₈×C₁₉	#125	1	1	yes	characteristic
75	C₂×#368	C₂×C₁₉	1	1	yes	characteristic
76	#2158	C₂×C₁₉	1	1	yes	characteristic
77	C₂×#346	C₂×C₁₉	1	1	yes	characteristic
78	#1945	C₂×C₁₉	1	2	yes	—
79	#1382	C₂×C₁₉	1	2	yes	—
80	C₂²×C₁₉	—	2	2	no	—
81	C₂²×C₁₉	C₄×D₈	1	1	yes	characteristic
82	C₂²×C₁₉	#467	1	1	yes	characteristic
83	C₂²×C₁₉	#499	1	1	yes	characteristic
84	C₄×C₁₉	—	4	2	no	—
85	C₄×C₁₉	—	8	4	no	—
86	C₄×C₁₉	#389	1	1	yes	characteristic
87	C₄×C₁₉	C₄×Q₁₆	1	1	yes	characteristic
88	C₄×C₁₉	#466	1	1	yes	characteristic
89	C₄×C₁₉	#500	1	1	yes	characteristic
90	C₂²×C₁₆	C₂²×C₁₉	1	1	yes	characteristic
91	C₂²×C₁₆	C₂²×C₁₉	1	1	yes	characteristic
92	C₂×#108	C₂²×C₁₉	1	2	yes	—
93	#427	C₂²×C₁₉	1	2	yes	—
94	#424	—	2	2	no	—
95	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
96	#368	—	2	2	no	—
97	#367	C₄×C₁₉	1	2	yes	complemented
98	#366	C₄×C₁₉	1	2	yes	complemented
99	#346	—	2	2	no	—
100	#339	—	2	2	no	—
101	C₂×C₁₉	—	2	1	no	—
102	C₂×C₁₉	—	2	1	no	—
103	C₂×C₁₉	#1942	1	1	yes	characteristic
104	C₂×C₁₉	#2179	1	2	yes	—
105	C₂×M₁₆	—	2	2	no	—
106	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
107	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
108	#117	—	2	2	no	—
109	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
110	C₂×C₁₆	—	2	1	no	—
111	C₂×C₁₆	—	2	1	no	—
112	C₂×C₁₆	C₂×C₄×C₁₉	1	1	yes	characteristic
113	C₂×C₁₆	C₂×C₄×C₁₉	1	1	yes	characteristic
114	C₂×C₁₆	D₄×C₁₉	1	2	yes	—
115	C₂×C₁₆	Q₈×C₁₉	1	2	yes	—
116	C₂×C₁₆	Q₈×C₁₉	1	2	yes	—
117	#108	—	2	2	no	—
118	#108	—	2	2	no	—
119	#107	C₂×C₄×C₁₉	1	4	yes	—
120	#106	—	2	2	no	—
121	C₄⋊C₈	—	4	4	no	—
122	C₄×C₈	—	2	1	no	—
123	C₄×C₈	—	2	1	no	—
124	C₁₉	#13170	1	1	yes	characteristic, direct product factor, Sylow
125	C₂²×C₄	—	2	1	no	—
126	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
127	M₁₆	—	4	2	no	—
128	M₁₆	—	4	2	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	4	2	no	—
132	C₂×C₈	D₈×C₁₉	1	2	yes	—
133	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
134	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
135	C₂×C₈	C₂×Q₈×C₁₉	1	1	yes	characteristic
136	C₂×C₈	C₂×Q₈×C₁₉	1	1	yes	characteristic
137	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
138	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
139	C₄⋊C₄	—	2	4	no	—
140	C₂²⋊C₄	—	4	2	no	—
141	C₄²	—	2	1	no	—
142	C₄²	—	4	2	no	—
143	C₄²	Q₁₆×C₁₉	1	2	yes	—
144	C₁₆	—	2	2	no	—
145	C₁₆	C₄⋊C₄×C₁₉	1	2	yes	—
146	C₁₆	C₄⋊C₄×C₁₉	1	2	yes	—
147	C₂³	C₁₉×#135	1	1	yes	characteristic
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	—	4	4	no	—
151	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
152	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
153	C₂×C₄	C₁₉×C₄⋊Q₈	1	1	yes	characteristic
154	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
155	C₂×C₄	C₂×Q₁₆×C₁₉	1	1	yes	characteristic
156	C₂×C₄	C₁₉×#135	1	1	yes	characteristic
157	C₈	—	2	2	no	—
158	C₈	—	8	2	no	—
159	C₈	C₂×C₄⋊C₄×C₁₉	1	1	yes	characteristic
160	C₈	C₄×Q₈×C₁₉	1	1	yes	characteristic
161	C₈	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
162	C₈	C₁₉×#125	1	1	yes	characteristic
163	C₂²	—	2	2	no	—
164	C₂²	C₄×D₈×C₁₉	1	1	yes	characteristic
165	C₂²	C₁₉×#467	1	1	yes	characteristic
166	C₂²	C₁₉×#499	1	1	yes	characteristic
167	C₄	—	4	2	no	—
168	C₄	—	8	4	no	—
169	C₄	C₁₉×#389	1	1	yes	characteristic
170	C₄	C₄×Q₁₆×C₁₉	1	1	yes	characteristic
171	C₄	C₁₉×#466	1	1	yes	characteristic
172	C₄	C₁₉×#500	1	1	yes	characteristic
173	C₂	—	2	1	no	—
174	C₂	—	2	1	no	—
175	C₂	C₁₉×#1942	1	1	yes	characteristic
176	C₂	C₁₉×#2179	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₉	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
socle	C₂²×C₁₉	C₄×D₈
prime radical	C₂³×C₁₉	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×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
13. C₁₉×#367	C₄	semidirect
14. C₁₉×#366	C₄	semidirect
58. #13170	C₁₉	direct (N × H)
97. #367	C₄×C₁₉	semidirect
98. #366	C₄×C₁₉	semidirect
124. C₁₉	#13170	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×#368	2
C₁₉×#2158	2
C₂×C₁₉×#346	2
C₁₉×#1945	2
C₁₉×#1382	2
#13170	19

Subgroups — C19×#13170 #18019129

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#13170 #18019129