Subgroups — C₁₉×#19293 #18025252

Order 4864.

680 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#494	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₉×#3079	C₂	1	2	yes	maximal
3	C₁₉×#2965	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1690	C₂	1	4	yes	maximal, complemented
5	C₁₉×#1689	C₂	1	4	yes	maximal, complemented
6	C₁₉×#1566	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1549	C₂	1	2	yes	maximal, complemented
8	C₂²×Q₁₆×C₁₉	—	2	2	no	—
9	C₂×C₁₉×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₄×#41×C₁₉	C₂²	1	1	yes	characteristic
11	C₁₉×#494	—	2	1	no	—
12	C₁₉×#494	—	2	1	no	—
13	C₁₉×#494	C₂²	1	4	yes	—
14	C₁₉×#477	—	2	2	no	—
15	C₁₉×#476	—	2	2	no	—
16	C₁₉×#475	C₂²	1	4	yes	—
17	C₁₉×#474	C₂²	1	4	yes	—
18	C₁₉×#446	C₂²	1	2	yes	—
19	C₁₉×#436	C₂²	1	2	yes	—
20	C₁₉×#435	—	2	2	no	—
21	C₁₉×#431	—	2	1	no	—
22	C₁₉×#431	C₂²	1	1	yes	characteristic
23	C₁₉×#428	—	2	2	no	—
24	C₁₉×#423	C₂²	1	2	yes	—
25	C₁₉×#417	—	2	2	no	—
26	C₄×M₁₆×C₁₉	—	2	1	no	—
27	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
28	C₁₉×#334	C₂²	1	4	yes	—
29	C₁₉×#332	C₂²	1	8	yes	—
30	C₂×#41×C₁₉	D₄	1	1	yes	characteristic
31	C₂²×Q₈×C₁₉	—	2	2	no	—
32	C₂×Q₁₆×C₁₉	—	2	4	no	—
33	C₂×Q₁₆×C₁₉	—	2	4	no	—
34	C₂×Q₁₆×C₁₉	—	4	8	no	—
35	C₂×M₁₆×C₁₉	—	2	2	no	—
36	C₂×M₁₆×C₁₉	—	2	2	no	—
37	C₂²×C₈×C₁₉	D₄	1	2	yes	—
38	C₁₉×C₄⋊Q₈	—	2	2	no	—
39	C₁₉×C₄⋊Q₈	—	2	2	no	—
40	C₁₉×C₄⋊Q₈	—	2	2	no	—
41	C₁₉×C₄⋊Q₈	C₂³	1	4	yes	—
42	C₄×Q₈×C₁₉	D₄	1	1	yes	characteristic
43	C₄×Q₈×C₁₉	C₂³	1	1	yes	characteristic
44	C₄×D₄×C₁₉	—	2	1	no	—
45	C₄×D₄×C₁₉	—	2	1	no	—
46	C₄×D₄×C₁₉	D₄	1	1	yes	characteristic
47	C₄×D₄×C₁₉	C₂³	1	1	yes	characteristic
48	C₁₉×#117	—	2	1	no	—
49	C₁₉×#117	D₄	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
51	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
52	C₂×C₄²×C₁₉	—	2	1	no	—
53	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
54	C₁₉×#107	—	2	4	no	—
55	C₁₉×#106	—	2	4	no	—
56	C₁₉×C₄⋊C₈	—	2	2	no	—
57	C₁₉×C₄⋊C₈	—	2	2	no	—
58	C₁₉×C₄⋊C₈	C₂³	1	4	yes	—
59	C₁₉×C₄⋊C₈	C₂³	1	2	yes	—
60	C₁₉×#103	—	2	4	no	—
61	C₁₉×#102	—	2	4	no	—
62	C₁₉×C₂²⋊C₈	—	2	1	no	—
63	C₁₉×C₂²⋊C₈	—	2	2	no	—
64	C₁₉×#97	—	2	2	no	—
65	C₄×C₈×C₁₉	—	2	1	no	—
66	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
67	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
68	C₄×C₈×C₁₉	C₂³	1	2	yes	—
69	#41×C₁₉	—	2	4	no	—
70	C₂×Q₈×C₁₉	—	2	4	no	—
71	C₂×Q₈×C₁₉	—	4	8	no	—
72	C₂×Q₈×C₁₉	—	4	2	no	—
73	C₂×Q₈×C₁₉	C₂×D₄	1	1	yes	characteristic
74	C₂×D₄×C₁₉	—	2	1	no	—
75	C₂×D₄×C₁₉	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₉	—	2	1	no	—
77	C₂²×C₄×C₁₉	—	2	1	no	—
78	C₂²×C₄×C₁₉	—	2	1	no	—
79	C₂²×C₄×C₁₉	—	4	2	no	—
80	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
83	Q₁₆×C₁₉	—	4	8	no	—
84	Q₁₆×C₁₉	—	4	8	no	—
85	M₁₆×C₁₉	—	4	4	no	—
86	M₁₆×C₁₉	—	4	4	no	—
87	C₂×C₈×C₁₉	—	2	4	no	—
88	C₂×C₈×C₁₉	—	2	2	no	—
89	C₂×C₈×C₁₉	—	2	4	no	—
90	C₂×C₈×C₁₉	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₁₉	C₂×D₄	1	2	yes	—
92	C₄⋊C₄×C₁₉	—	2	1	no	—
93	C₄⋊C₄×C₁₉	—	2	1	no	—
94	C₄⋊C₄×C₁₉	—	2	4	no	—
95	C₄⋊C₄×C₁₉	—	2	4	no	—
96	C₄⋊C₄×C₁₉	—	4	2	no	—
97	C₄⋊C₄×C₁₉	—	4	2	no	—
98	C₄⋊C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
99	C₄⋊C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
100	C₂²⋊C₄×C₁₉	—	2	1	no	—
101	C₂²⋊C₄×C₁₉	—	2	1	no	—
102	C₂²⋊C₄×C₁₉	—	2	1	no	—
103	C₄²×C₁₉	—	2	1	no	—
104	C₄²×C₁₉	—	2	1	no	—
105	C₄²×C₁₉	—	2	1	no	—
106	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₁₉	#41	1	1	yes	characteristic
109	C₄²×C₁₉	C₂⁴	1	1	yes	characteristic
110	#19293	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₂³×C₁₉	—	2	1	no	—
112	C₂³×C₁₉	#121	1	1	yes	characteristic
113	Q₈×C₁₉	—	2	2	no	—
114	Q₈×C₁₉	—	4	8	no	—
115	Q₈×C₁₉	—	8	4	no	—
116	D₄×C₁₉	—	2	2	no	—
117	D₄×C₁₉	—	4	2	no	—
118	C₂×C₄×C₁₉	—	2	2	no	—
119	C₂×C₄×C₁₉	—	2	2	no	—
120	C₂×C₄×C₁₉	—	2	1	no	—
121	C₂×C₄×C₁₉	—	2	1	no	—
122	C₂×C₄×C₁₉	—	2	1	no	—
123	C₂×C₄×C₁₉	—	4	2	no	—
124	C₂×C₄×C₁₉	—	4	2	no	—
125	C₂×C₄×C₁₉	—	4	2	no	—
126	C₂×C₄×C₁₉	—	4	4	no	—
127	C₂×C₄×C₁₉	—	8	4	no	—
128	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
129	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
130	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
131	C₂×C₄×C₁₉	#137	1	4	yes	—
132	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₉	C₂×#41	1	1	yes	characteristic
135	C₈×C₁₉	—	2	2	no	—
136	C₈×C₁₉	—	4	2	no	—
137	C₈×C₁₉	—	4	4	no	—
138	C₈×C₁₉	AGL(1,ℤ/8)	1	4	yes	—
139	C₂×#494	C₂×C₁₉	1	1	yes	characteristic, complemented
140	#3079	C₂×C₁₉	1	2	yes	—
141	#2965	C₂×C₁₉	1	1	yes	characteristic
142	#1690	C₂×C₁₉	1	4	yes	complemented
143	#1689	C₂×C₁₉	1	4	yes	complemented
144	#1566	C₂×C₁₉	1	1	yes	characteristic, complemented
145	#1549	C₂×C₁₉	1	2	yes	complemented
146	C₂²×C₁₉	—	2	1	no	—
147	C₂²×C₁₉	—	2	1	no	—
148	C₂²×C₁₉	—	4	1	no	—
149	C₂²×C₁₉	—	4	1	no	—
150	C₂²×C₁₉	—	4	1	no	—
151	C₂²×C₁₉	#470	1	2	yes	—
152	C₂²×C₁₉	C₂×#121	1	1	yes	characteristic
153	C₄×C₁₉	—	2	1	no	—
154	C₄×C₁₉	—	2	1	no	—
155	C₄×C₁₉	—	4	1	no	—
156	C₄×C₁₉	—	4	1	no	—
157	C₄×C₁₉	—	4	1	no	—
158	C₄×C₁₉	—	8	4	no	—
159	C₄×C₁₉	#470	1	2	yes	—
160	C₄×C₁₉	#473	1	2	yes	—
161	C₄×C₁₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
162	C₄×C₁₉	C₂×#137	1	2	yes	—
163	C₂²×Q₁₆	—	2	2	no	—
164	C₂×C₄⋊Q₈	C₂²×C₁₉	1	2	yes	—
165	C₄×#41	C₂²×C₁₉	1	1	yes	characteristic
166	#494	—	2	1	no	—
167	#494	—	2	1	no	—
168	#494	C₂²×C₁₉	1	4	yes	—
169	#477	—	2	2	no	—
170	#476	—	2	2	no	—
171	#475	C₂²×C₁₉	1	4	yes	—
172	#474	C₂²×C₁₉	1	4	yes	—
173	#446	C₂²×C₁₉	1	2	yes	—
174	#436	C₂²×C₁₉	1	2	yes	—
175	#435	—	2	2	no	—
176	#431	—	2	1	no	—
177	#431	C₂²×C₁₉	1	1	yes	characteristic
178	#428	—	2	2	no	—
179	#423	C₂²×C₁₉	1	2	yes	—
180	#417	—	2	2	no	—
181	C₄×M₁₆	—	2	1	no	—
182	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
183	#334	C₂²×C₁₉	1	4	yes	—
184	#332	C₂²×C₁₉	1	8	yes	—
185	C₂×C₁₉	—	2	2	no	—
186	C₂×C₁₉	—	8	1	no	—
187	C₂×C₁₉	#3038	1	1	yes	characteristic
188	C₂×C₁₉	C₂×#470	1	1	yes	characteristic
189	C₂×C₁₉	C₂×#473	1	1	yes	characteristic
190	C₂×#41	D₄×C₁₉	1	1	yes	characteristic
191	C₂²×Q₈	—	2	2	no	—
192	C₂×Q₁₆	—	2	4	no	—
193	C₂×Q₁₆	—	2	4	no	—
194	C₂×Q₁₆	—	4	8	no	—
195	C₂×M₁₆	—	2	2	no	—
196	C₂×M₁₆	—	2	2	no	—
197	C₂²×C₈	D₄×C₁₉	1	2	yes	—
198	C₄⋊Q₈	—	2	2	no	—
199	C₄⋊Q₈	—	2	2	no	—
200	C₄⋊Q₈	—	2	2	no	—
201	C₄⋊Q₈	C₂³×C₁₉	1	4	yes	—
202	C₄×Q₈	D₄×C₁₉	1	1	yes	characteristic
203	C₄×Q₈	C₂³×C₁₉	1	1	yes	characteristic
204	C₄×D₄	—	2	1	no	—
205	C₄×D₄	—	2	1	no	—
206	C₄×D₄	D₄×C₁₉	1	1	yes	characteristic
207	C₄×D₄	C₂³×C₁₉	1	1	yes	characteristic
208	#117	—	2	1	no	—
209	#117	D₄×C₁₉	1	1	yes	characteristic
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₄⋊C₄	—	2	2	no	—
212	C₂×C₄²	—	2	1	no	—
213	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
214	#107	—	2	4	no	—
215	#106	—	2	4	no	—
216	C₄⋊C₈	—	2	2	no	—
217	C₄⋊C₈	—	2	2	no	—
218	C₄⋊C₈	C₂³×C₁₉	1	2	yes	—
219	C₄⋊C₈	C₂³×C₁₉	1	4	yes	—
220	#103	—	2	4	no	—
221	#102	—	2	4	no	—
222	C₂²⋊C₈	—	2	1	no	—
223	C₂²⋊C₈	—	2	2	no	—
224	#97	—	2	2	no	—
225	C₄×C₈	—	2	1	no	—
226	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
227	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
228	C₄×C₈	C₂³×C₁₉	1	2	yes	—
229	C₁₉	#19293	1	1	yes	characteristic, direct product factor, Sylow
230	#41	—	2	4	no	—
231	C₂×Q₈	—	2	4	no	—
232	C₂×Q₈	—	4	8	no	—
233	C₂×Q₈	—	4	2	no	—
234	C₂×Q₈	C₂×D₄×C₁₉	1	1	yes	characteristic
235	C₂×D₄	—	2	1	no	—
236	C₂×D₄	C₂×D₄×C₁₉	1	1	yes	characteristic
237	C₂²×C₄	—	2	1	no	—
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	2	1	no	—
240	C₂²×C₄	—	4	2	no	—
241	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
242	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
243	C₂²×C₄	#41×C₁₉	1	1	yes	characteristic
244	Q₁₆	—	4	8	no	—
245	Q₁₆	—	4	8	no	—
246	M₁₆	—	4	4	no	—
247	M₁₆	—	4	4	no	—
248	C₂×C₈	—	2	4	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	4	no	—
251	C₂×C₈	C₂×D₄×C₁₉	1	2	yes	—
252	C₂×C₈	C₂×D₄×C₁₉	1	2	yes	—
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	2	4	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₄⋊C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
260	C₄⋊C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
261	C₂²⋊C₄	—	2	1	no	—
262	C₂²⋊C₄	—	2	1	no	—
263	C₂²⋊C₄	—	2	1	no	—
264	C₄²	—	2	1	no	—
265	C₄²	—	2	1	no	—
266	C₄²	—	2	1	no	—
267	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
268	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
269	C₄²	#41×C₁₉	1	1	yes	characteristic
270	C₄²	C₂⁴×C₁₉	1	1	yes	characteristic
271	C₂³	—	2	1	no	—
272	C₂³	C₁₉×#121	1	1	yes	characteristic
273	Q₈	—	2	2	no	—
274	Q₈	—	4	8	no	—
275	Q₈	—	8	4	no	—
276	D₄	—	2	2	no	—
277	D₄	—	4	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	—	8	4	no	—
288	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
289	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
290	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
291	C₂×C₄	C₁₉×#137	1	4	yes	—
292	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
293	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
294	C₂×C₄	C₂×#41×C₁₉	1	1	yes	characteristic
295	C₈	—	2	2	no	—
296	C₈	—	4	2	no	—
297	C₈	—	4	4	no	—
298	C₈	C₁₉×AGL(1,ℤ/8)	1	4	yes	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	4	1	no	—
302	C₂²	—	4	1	no	—
303	C₂²	—	4	1	no	—
304	C₂²	C₁₉×#470	1	2	yes	—
305	C₂²	C₂×C₁₉×#121	1	1	yes	characteristic
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	4	1	no	—
309	C₄	—	4	1	no	—
310	C₄	—	4	1	no	—
311	C₄	—	8	4	no	—
312	C₄	C₁₉×#470	1	2	yes	—
313	C₄	C₁₉×#473	1	2	yes	—
314	C₄	C₂×C₁₉×AGL(1,ℤ/8)	1	2	yes	—
315	C₄	C₂×C₁₉×#137	1	2	yes	—
316	C₂	—	2	2	no	—
317	C₂	—	8	1	no	—
318	C₂	C₁₉×#3038	1	1	yes	characteristic
319	C₂	C₂×C₁₉×#470	1	1	yes	characteristic
320	C₂	C₂×C₁₉×#473	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₉
prime radical	C₂×#41×C₁₉	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₉×#494	C₂	semidirect
4. C₁₉×#1690	C₂	semidirect
5. C₁₉×#1689	C₂	semidirect
6. C₁₉×#1566	C₂	semidirect
7. C₁₉×#1549	C₂	semidirect
110. #19293	C₁₉	direct (N × H)
139. C₂×#494	C₂×C₁₉	semidirect
142. #1690	C₂×C₁₉	semidirect
143. #1689	C₂×C₁₉	semidirect
144. #1566	C₂×C₁₉	semidirect
145. #1549	C₂×C₁₉	semidirect
229. C₁₉	#19293	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×#494	2
C₁₉×#3079	2
C₁₉×#2965	2
C₁₉×#1690	2
C₁₉×#1689	2
C₁₉×#1566	2
C₁₉×#1549	2
#19293	19

Subgroups — C19×#19293 #18025252

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#19293 #18025252