Subgroups — C₁₉×#13106 #18019065

Order 4864.

272 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#2151	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2108	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1772	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1379	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1379	C₂	1	1	yes	maximal, characteristic
6	C₁₉×#1376	C₂	1	1	yes	maximal, characteristic
7	C₁₉×#1374	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	C₂²×C₁₆×C₁₉	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₁₉	C₂²	1	1	yes	characteristic
12	C₂×C₁₉×#108	C₄	1	1	yes	characteristic
13	C₁₉×#427	C₄	1	1	yes	characteristic
14	C₁₉×#405	C₂²	1	1	yes	characteristic
15	C₁₉×#405	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
17	C₁₉×C₄⋊C₁₆	—	2	1	no	—
18	C₁₉×C₄⋊C₁₆	—	2	1	no	—
19	C₁₉×#346	—	2	1	no	—
20	C₄×C₁₆×C₁₉	—	2	1	no	—
21	C₁₉×#339	C₄	1	1	yes	characteristic
22	C₁₉×#339	C₄	1	1	yes	characteristic
23	C₂×M₁₆×C₁₉	C₂×C₄	1	1	yes	characteristic
24	C₂×M₁₆×C₁₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
27	C₁₉×#117	C₂×C₄	1	1	yes	characteristic
28	C₁₉×#117	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
30	C₁₉×M₃₂	—	2	1	no	—
31	C₁₉×M₃₂	—	2	1	no	—
32	C₁₉×M₃₂	—	2	1	no	—
33	C₁₉×M₃₂	Q₈	1	2	yes	—
34	C₂×C₁₆×C₁₉	—	2	1	no	—
35	C₂×C₁₆×C₁₉	—	2	1	no	—
36	C₂×C₁₆×C₁₉	—	2	1	no	—
37	C₂×C₁₆×C₁₉	—	2	1	no	—
38	C₂×C₁₆×C₁₉	—	2	1	no	—
39	C₂×C₁₆×C₁₉	—	2	1	no	—
40	C₂×C₁₆×C₁₉	—	2	1	no	—
41	C₂×C₁₆×C₁₉	D₄	1	1	yes	characteristic
42	C₂×C₁₆×C₁₉	D₄	1	1	yes	characteristic
43	C₁₉×#108	—	2	2	no	—
44	C₁₉×#107	C₈	1	1	yes	characteristic
45	C₁₉×#107	C₈	1	1	yes	characteristic
46	C₁₉×#106	C₈	1	1	yes	characteristic
47	C₁₉×#106	C₈	1	1	yes	characteristic
48	C₁₉×#97	—	2	1	no	—
49	C₁₉×#97	—	2	1	no	—
50	C₄×C₈×C₁₉	—	2	1	no	—
51	C₄×C₈×C₁₉	—	2	1	no	—
52	C₄×C₈×C₁₉	—	2	1	no	—
53	C₄×C₈×C₁₉	—	2	1	no	—
54	C₂²×C₄×C₁₉	—	2	1	no	—
55	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
56	M₁₆×C₁₉	—	2	1	no	—
57	M₁₆×C₁₉	—	2	1	no	—
58	M₁₆×C₁₉	—	2	1	no	—
59	M₁₆×C₁₉	M₁₆	1	2	yes	—
60	C₂×C₈×C₁₉	—	2	1	no	—
61	C₂×C₈×C₁₉	—	2	1	no	—
62	C₂×C₈×C₁₉	—	2	1	no	—
63	C₂×C₈×C₁₉	—	2	1	no	—
64	C₂×C₈×C₁₉	C₂×C₈	1	1	yes	characteristic
65	C₂×C₈×C₁₉	C₂×C₈	1	1	yes	characteristic
66	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₁₉	C₂×Q₈	1	1	yes	characteristic
68	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
69	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
70	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
71	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
72	C₄⋊C₄×C₁₉	C₂×C₈	1	1	yes	characteristic
73	C₄⋊C₄×C₁₉	C₂×C₈	1	1	yes	characteristic
74	C₄⋊C₄×C₁₉	C₂×C₈	1	1	yes	characteristic
75	C₄⋊C₄×C₁₉	C₂×C₈	1	1	yes	characteristic
76	C₂²⋊C₄×C₁₉	—	2	1	no	—
77	C₂²⋊C₄×C₁₉	—	2	1	no	—
78	C₄²×C₁₉	—	2	1	no	—
79	C₄²×C₁₉	—	2	1	no	—
80	C₄²×C₁₉	—	2	1	no	—
81	C₄²×C₁₉	—	2	1	no	—
82	C₁₆×C₁₉	—	2	2	no	—
83	C₁₆×C₁₉	—	2	1	no	—
84	C₁₆×C₁₉	—	2	1	no	—
85	C₁₆×C₁₉	—	4	1	no	—
86	C₁₆×C₁₉	—	4	1	no	—
87	#13106	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂³×C₁₉	#131	1	1	yes	characteristic
89	C₂×C₄×C₁₉	—	2	2	no	—
90	C₂×C₄×C₁₉	—	2	1	no	—
91	C₂×C₄×C₁₉	—	2	1	no	—
92	C₂×C₄×C₁₉	—	2	1	no	—
93	C₂×C₄×C₁₉	—	2	1	no	—
94	C₂×C₄×C₁₉	—	4	1	no	—
95	C₂×C₄×C₁₉	—	4	1	no	—
96	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
97	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
98	C₂×C₄×C₁₉	C₄×Q₈	1	1	yes	characteristic
99	C₂×C₄×C₁₉	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
101	C₂×C₄×C₁₉	#131	1	1	yes	characteristic
102	C₈×C₁₉	—	2	1	no	—
103	C₈×C₁₉	—	2	1	no	—
104	C₈×C₁₉	—	2	2	no	—
105	C₈×C₁₉	—	4	1	no	—
106	C₈×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
107	C₈×C₁₉	#123	1	1	yes	characteristic
108	C₈×C₁₉	#125	1	1	yes	characteristic
109	C₈×C₁₉	#126	1	1	yes	characteristic
110	#2151	C₂×C₁₉	1	1	yes	characteristic
111	#2108	C₂×C₁₉	1	1	yes	characteristic
112	#1772	C₂×C₁₉	1	1	yes	characteristic
113	#1379	C₂×C₁₉	1	1	yes	characteristic
114	#1379	C₂×C₁₉	1	1	yes	characteristic
115	#1376	C₂×C₁₉	1	1	yes	characteristic
116	#1374	C₂×C₁₉	1	1	yes	characteristic
117	C₂²×C₁₉	—	2	1	no	—
118	C₂²×C₁₉	—	2	1	no	—
119	C₂²×C₁₉	#433	1	1	yes	characteristic
120	C₂²×C₁₉	C₈×D₄	1	1	yes	characteristic
121	C₂²×C₁₉	#446	1	1	yes	characteristic
122	C₄×C₁₉	—	4	2	no	—
123	C₄×C₁₉	—	4	1	no	—
124	C₄×C₁₉	—	4	1	no	—
125	C₄×C₁₉	—	4	1	no	—
126	C₄×C₁₉	—	4	1	no	—
127	C₄×C₁₉	#387	1	1	yes	characteristic
128	C₄×C₁₉	#431	1	1	yes	characteristic
129	C₄×C₁₉	#435	1	1	yes	characteristic
130	C₄×C₁₉	C₈×Q₈	1	1	yes	characteristic
131	C₂×M₃₂	C₂²×C₁₉	1	1	yes	characteristic
132	C₂×M₃₂	C₂²×C₁₉	1	1	yes	characteristic
133	C₂²×C₁₆	C₂²×C₁₉	1	1	yes	characteristic
134	C₂²×C₁₆	C₂²×C₁₉	1	1	yes	characteristic
135	C₂×#108	C₄×C₁₉	1	1	yes	characteristic
136	#427	C₄×C₁₉	1	1	yes	characteristic
137	#405	C₂²×C₁₉	1	1	yes	characteristic
138	#405	C₂²×C₁₉	1	1	yes	characteristic
139	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
140	C₄⋊C₁₆	—	2	1	no	—
141	C₄⋊C₁₆	—	2	1	no	—
142	#346	—	2	1	no	—
143	C₄×C₁₆	—	2	1	no	—
144	#339	C₄×C₁₉	1	1	yes	characteristic
145	#339	C₄×C₁₉	1	1	yes	characteristic
146	C₂×C₁₉	—	2	1	no	—
147	C₂×C₁₉	—	2	1	no	—
148	C₂×C₁₉	#1916	1	1	yes	characteristic
149	C₂×C₁₉	#2170	1	1	yes	characteristic
150	C₂×C₁₉	#2171	1	1	yes	characteristic
151	C₂×M₁₆	C₂×C₄×C₁₉	1	1	yes	characteristic
152	C₂×M₁₆	C₂×C₄×C₁₉	1	1	yes	characteristic
153	C₂²×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
154	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
155	#117	C₂×C₄×C₁₉	1	1	yes	characteristic
156	#117	C₂×C₄×C₁₉	1	1	yes	characteristic
157	C₂×C₄²	C₂×C₄×C₁₉	1	1	yes	characteristic
158	M₃₂	—	2	1	no	—
159	M₃₂	—	2	1	no	—
160	M₃₂	—	2	1	no	—
161	M₃₂	Q₈×C₁₉	1	2	yes	—
162	C₂×C₁₆	—	2	1	no	—
163	C₂×C₁₆	—	2	1	no	—
164	C₂×C₁₆	—	2	1	no	—
165	C₂×C₁₆	—	2	1	no	—
166	C₂×C₁₆	—	2	1	no	—
167	C₂×C₁₆	—	2	1	no	—
168	C₂×C₁₆	—	2	1	no	—
169	C₂×C₁₆	D₄×C₁₉	1	1	yes	characteristic
170	C₂×C₁₆	D₄×C₁₉	1	1	yes	characteristic
171	#108	—	2	2	no	—
172	#107	C₈×C₁₉	1	1	yes	characteristic
173	#107	C₈×C₁₉	1	1	yes	characteristic
174	#106	C₈×C₁₉	1	1	yes	characteristic
175	#106	C₈×C₁₉	1	1	yes	characteristic
176	#97	—	2	1	no	—
177	#97	—	2	1	no	—
178	C₄×C₈	—	2	1	no	—
179	C₄×C₈	—	2	1	no	—
180	C₄×C₈	—	2	1	no	—
181	C₄×C₈	—	2	1	no	—
182	C₁₉	#13106	1	1	yes	characteristic, direct product factor, Sylow
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
185	M₁₆	—	2	1	no	—
186	M₁₆	—	2	1	no	—
187	M₁₆	—	2	1	no	—
188	M₁₆	M₁₆×C₁₉	1	2	yes	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	2	1	no	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	C₂×C₈×C₁₉	1	1	yes	characteristic
194	C₂×C₈	C₂×C₈×C₁₉	1	1	yes	characteristic
195	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
196	C₂×C₈	C₂×Q₈×C₁₉	1	1	yes	characteristic
197	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
198	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
199	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
200	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
201	C₄⋊C₄	C₂×C₈×C₁₉	1	1	yes	characteristic
202	C₄⋊C₄	C₂×C₈×C₁₉	1	1	yes	characteristic
203	C₄⋊C₄	C₂×C₈×C₁₉	1	1	yes	characteristic
204	C₄⋊C₄	C₂×C₈×C₁₉	1	1	yes	characteristic
205	C₂²⋊C₄	—	2	1	no	—
206	C₂²⋊C₄	—	2	1	no	—
207	C₄²	—	2	1	no	—
208	C₄²	—	2	1	no	—
209	C₄²	—	2	1	no	—
210	C₄²	—	2	1	no	—
211	C₁₆	—	2	2	no	—
212	C₁₆	—	2	1	no	—
213	C₁₆	—	2	1	no	—
214	C₁₆	—	4	1	no	—
215	C₁₆	—	4	1	no	—
216	C₂³	C₁₉×#131	1	1	yes	characteristic
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	4	1	no	—
223	C₂×C₄	—	4	1	no	—
224	C₂×C₄	C₁₉×#117	1	1	yes	characteristic
225	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
226	C₂×C₄	C₄×Q₈×C₁₉	1	1	yes	characteristic
227	C₂×C₄	C₂²×C₈×C₁₉	1	1	yes	characteristic
228	C₂×C₄	C₂×M₁₆×C₁₉	1	1	yes	characteristic
229	C₂×C₄	C₁₉×#131	1	1	yes	characteristic
230	C₈	—	2	1	no	—
231	C₈	—	2	1	no	—
232	C₈	—	2	2	no	—
233	C₈	—	4	1	no	—
234	C₈	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
235	C₈	C₁₉×#123	1	1	yes	characteristic
236	C₈	C₁₉×#125	1	1	yes	characteristic
237	C₈	C₁₉×#126	1	1	yes	characteristic
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	C₁₉×#433	1	1	yes	characteristic
241	C₂²	C₈×D₄×C₁₉	1	1	yes	characteristic
242	C₂²	C₁₉×#446	1	1	yes	characteristic
243	C₄	—	4	2	no	—
244	C₄	—	4	1	no	—
245	C₄	—	4	1	no	—
246	C₄	—	4	1	no	—
247	C₄	—	4	1	no	—
248	C₄	C₁₉×#387	1	1	yes	characteristic
249	C₄	C₁₉×#431	1	1	yes	characteristic
250	C₄	C₁₉×#435	1	1	yes	characteristic
251	C₄	C₈×Q₈×C₁₉	1	1	yes	characteristic
252	C₂	—	2	1	no	—
253	C₂	—	2	1	no	—
254	C₂	C₁₉×#1916	1	1	yes	characteristic
255	C₂	C₁₉×#2170	1	1	yes	characteristic
256	C₂	C₁₉×#2171	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₁₉	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₁₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
socle	C₂²×C₁₉	C₈×D₄
prime radical	C₂³×C₁₉	#131
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₉ ◁ C₂²×C₁₆×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
87. #13106	C₁₉	direct (N × H)
182. C₁₉	#13106	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×#2151	2
C₁₉×#2108	2
C₁₉×#1772	2
C₁₉×#1379	2
C₁₉×#1376	2
C₁₉×#1374	2
#13106	19

Subgroups — C19×#13106 #18019065

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#13106 #18019065