Subgroups — C₁₉×#13117 #18019076

Order 4864.

254 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#2155	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#1945	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₉×#108	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1389	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1388	C₂	1	1	yes	maximal, characteristic
6	C₁₉×#1385	C₂	1	1	yes	maximal, characteristic
7	C₁₉×#1376	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₉×M₃₂	C₂²	1	1	yes	characteristic
9	C₂×C₁₉×M₃₂	C₂²	1	1	yes	characteristic
10	C₂×C₁₉×#108	C₂²	1	1	yes	characteristic
11	C₂×C₁₉×#108	C₂²	1	1	yes	characteristic
12	C₂×C₁₉×#108	C₂²	1	1	yes	characteristic
13	C₁₉×#427	C₂²	1	1	yes	characteristic
14	C₁₉×#424	—	2	1	no	—
15	C₄×M₁₆×C₁₉	—	2	1	no	—
16	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
17	C₁₉×#365	C₄	1	4	yes	complemented
18	C₁₉×#341	—	2	1	no	—
19	C₁₉×#339	—	2	1	no	—
20	C₂×M₁₆×C₁₉	—	2	1	no	—
21	C₂×M₁₆×C₁₉	—	2	1	no	—
22	C₂×M₁₆×C₁₉	—	2	1	no	—
23	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
25	C₁₉×#117	—	2	1	no	—
26	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
27	C₁₉×M₃₂	—	2	1	no	—
28	C₁₉×M₃₂	—	2	1	no	—
29	C₁₉×M₃₂	—	2	1	no	—
30	C₁₉×M₃₂	C₂×C₄	1	2	yes	—
31	C₂×C₁₆×C₁₉	—	2	1	no	—
32	C₂×C₁₆×C₁₉	—	2	1	no	—
33	C₁₉×#108	—	2	1	no	—
34	C₁₉×#108	—	2	1	no	—
35	C₁₉×#108	—	2	1	no	—
36	C₁₉×#108	—	2	1	no	—
37	C₁₉×#108	C₂×C₄	1	2	yes	—
38	C₁₉×#108	Q₈	1	2	yes	—
39	C₁₉×#107	—	2	1	no	—
40	C₁₉×#106	C₂×C₄	1	2	yes	—
41	C₁₉×C₄⋊C₈	—	4	2	no	—
42	C₁₉×#97	—	4	1	no	—
43	C₄×C₈×C₁₉	—	2	1	no	—
44	C₄×C₈×C₁₉	—	2	1	no	—
45	C₄×C₈×C₁₉	—	4	1	no	—
46	C₂²×C₄×C₁₉	—	2	1	no	—
47	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
48	M₁₆×C₁₉	—	2	2	no	—
49	M₁₆×C₁₉	—	2	2	no	—
50	M₁₆×C₁₉	—	4	1	no	—
51	M₁₆×C₁₉	—	4	1	no	—
52	M₁₆×C₁₉	—	4	1	no	—
53	M₁₆×C₁₉	—	4	1	no	—
54	C₂×C₈×C₁₉	—	2	1	no	—
55	C₂×C₈×C₁₉	—	2	1	no	—
56	C₂×C₈×C₁₉	—	2	1	no	—
57	C₂×C₈×C₁₉	—	4	1	no	—
58	C₂×C₈×C₁₉	—	4	1	no	—
59	C₂×C₈×C₁₉	—	4	1	no	—
60	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₁₉	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
63	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
64	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
65	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
66	C₄⋊C₄×C₁₉	—	2	2	no	—
67	C₂²⋊C₄×C₁₉	—	4	1	no	—
68	C₄²×C₁₉	—	2	1	no	—
69	C₄²×C₁₉	—	4	1	no	—
70	C₄²×C₁₉	SD₁₆	1	1	yes	characteristic
71	C₄²×C₁₉	SD₁₆	1	1	yes	characteristic
72	C₁₆×C₁₉	—	2	2	no	—
73	C₁₆×C₁₉	—	4	1	no	—
74	#13117	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂³×C₁₉	#137	1	1	yes	characteristic
76	C₂×C₄×C₁₉	—	2	1	no	—
77	C₂×C₄×C₁₉	—	2	1	no	—
78	C₂×C₄×C₁₉	—	4	2	no	—
79	C₂×C₄×C₁₉	—	4	2	no	—
80	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
81	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
82	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
83	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
84	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
85	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
86	C₈×C₁₉	—	4	2	no	—
87	C₈×C₁₉	—	4	2	no	—
88	C₈×C₁₉	—	4	1	no	—
89	C₈×C₁₉	—	8	1	no	—
90	C₈×C₁₉	#117	1	1	yes	characteristic
91	C₈×C₁₉	C₄×Q₈	1	1	yes	characteristic
92	C₈×C₁₉	#125	1	1	yes	characteristic
93	C₈×C₁₉	#126	1	1	yes	characteristic
94	#2155	C₂×C₁₉	1	1	yes	characteristic
95	#1945	C₂×C₁₉	1	1	yes	characteristic
96	C₄×#108	C₂×C₁₉	1	1	yes	characteristic
97	#1389	C₂×C₁₉	1	1	yes	characteristic
98	#1388	C₂×C₁₉	1	1	yes	characteristic
99	#1385	C₂×C₁₉	1	1	yes	characteristic
100	#1376	C₂×C₁₉	1	1	yes	characteristic
101	C₂²×C₁₉	—	4	1	no	—
102	C₂²×C₁₉	#440	1	1	yes	characteristic
103	C₂²×C₁₉	#479	1	1	yes	characteristic
104	C₂²×C₁₉	#483	1	1	yes	characteristic
105	C₄×C₁₉	—	2	2	no	—
106	C₄×C₁₉	—	4	1	no	—
107	C₄×C₁₉	—	8	2	no	—
108	C₄×C₁₉	#387	1	1	yes	characteristic
109	C₄×C₁₉	C₄×SD₁₆	1	1	yes	characteristic
110	C₄×C₁₉	#476	1	1	yes	characteristic
111	C₄×C₁₉	#482	1	1	yes	characteristic
112	C₂×M₃₂	C₂²×C₁₉	1	1	yes	characteristic
113	C₂×M₃₂	C₂²×C₁₉	1	1	yes	characteristic
114	C₂×#108	C₂²×C₁₉	1	1	yes	characteristic
115	C₂×#108	C₂²×C₁₉	1	1	yes	characteristic
116	C₂×#108	C₂²×C₁₉	1	1	yes	characteristic
117	#427	C₂²×C₁₉	1	1	yes	characteristic
118	#424	—	2	1	no	—
119	C₄×M₁₆	—	2	1	no	—
120	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
121	#365	C₄×C₁₉	1	4	yes	complemented
122	#341	—	2	1	no	—
123	#339	—	2	1	no	—
124	C₂×C₁₉	—	2	1	no	—
125	C₂×C₁₉	—	2	1	no	—
126	C₂×C₁₉	—	2	1	no	—
127	C₂×C₁₉	#1920	1	1	yes	characteristic
128	C₂×M₁₆	—	2	1	no	—
129	C₂×M₁₆	—	2	1	no	—
130	C₂×M₁₆	—	2	1	no	—
131	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
132	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
133	#117	—	2	1	no	—
134	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
135	M₃₂	—	2	1	no	—
136	M₃₂	—	2	1	no	—
137	M₃₂	—	2	1	no	—
138	M₃₂	C₂×C₄×C₁₉	1	2	yes	—
139	C₂×C₁₆	—	2	1	no	—
140	C₂×C₁₆	—	2	1	no	—
141	#108	—	2	1	no	—
142	#108	—	2	1	no	—
143	#108	—	2	1	no	—
144	#108	—	2	1	no	—
145	#108	C₂×C₄×C₁₉	1	2	yes	—
146	#108	Q₈×C₁₉	1	2	yes	—
147	#107	—	2	1	no	—
148	#106	C₂×C₄×C₁₉	1	2	yes	—
149	C₄⋊C₈	—	4	2	no	—
150	#97	—	4	1	no	—
151	C₄×C₈	—	2	1	no	—
152	C₄×C₈	—	2	1	no	—
153	C₄×C₈	—	4	1	no	—
154	C₁₉	#13117	1	1	yes	characteristic, direct product factor, Sylow
155	C₂²×C₄	—	2	1	no	—
156	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
157	M₁₆	—	2	2	no	—
158	M₁₆	—	2	2	no	—
159	M₁₆	—	4	1	no	—
160	M₁₆	—	4	1	no	—
161	M₁₆	—	4	1	no	—
162	M₁₆	—	4	1	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	4	1	no	—
167	C₂×C₈	—	4	1	no	—
168	C₂×C₈	—	4	1	no	—
169	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
170	C₂×C₈	C₂×Q₈×C₁₉	1	1	yes	characteristic
171	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
172	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
173	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
174	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
175	C₄⋊C₄	—	2	2	no	—
176	C₂²⋊C₄	—	4	1	no	—
177	C₄²	—	2	1	no	—
178	C₄²	—	4	1	no	—
179	C₄²	SD₁₆×C₁₉	1	1	yes	characteristic
180	C₄²	SD₁₆×C₁₉	1	1	yes	characteristic
181	C₁₆	—	2	2	no	—
182	C₁₆	—	4	1	no	—
183	C₂³	C₁₉×#137	1	1	yes	characteristic
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
189	C₂×C₄	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
190	C₂×C₄	C₁₉×#123	1	1	yes	characteristic
191	C₂×C₄	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
192	C₂×C₄	C₁₉×#135	1	1	yes	characteristic
193	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
194	C₈	—	4	2	no	—
195	C₈	—	4	2	no	—
196	C₈	—	4	1	no	—
197	C₈	—	8	1	no	—
198	C₈	C₁₉×#117	1	1	yes	characteristic
199	C₈	C₄×Q₈×C₁₉	1	1	yes	characteristic
200	C₈	C₁₉×#125	1	1	yes	characteristic
201	C₈	C₁₉×#126	1	1	yes	characteristic
202	C₂²	—	4	1	no	—
203	C₂²	C₁₉×#440	1	1	yes	characteristic
204	C₂²	C₁₉×#479	1	1	yes	characteristic
205	C₂²	C₁₉×#483	1	1	yes	characteristic
206	C₄	—	2	2	no	—
207	C₄	—	4	1	no	—
208	C₄	—	8	2	no	—
209	C₄	C₁₉×#387	1	1	yes	characteristic
210	C₄	C₄×SD₁₆×C₁₉	1	1	yes	characteristic
211	C₄	C₁₉×#476	1	1	yes	characteristic
212	C₄	C₁₉×#482	1	1	yes	characteristic
213	C₂	—	2	1	no	—
214	C₂	—	2	1	no	—
215	C₂	—	2	1	no	—
216	C₂	C₁₉×#1920	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₉	#476
commutator	C₂×C₈	C₂²×C₄×C₁₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
socle	C₂×C₁₉	#1920
prime radical	C₂³×C₁₉	#137
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₂×C₄×C₈×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
17. C₁₉×#365	C₄	semidirect
74. #13117	C₁₉	direct (N × H)
121. #365	C₄×C₁₉	semidirect
154. C₁₉	#13117	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×#2155	2
C₁₉×#1945	2
C₄×C₁₉×#108	2
C₁₉×#1389	2
C₁₉×#1388	2
C₁₉×#1385	2
C₁₉×#1376	2
#13117	19

Subgroups — C19×#13117 #18019076

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#13117 #18019076