Subgroups — C₁₉×#11155 #18017114

Order 4864.

1018 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₉×#100	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2309	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#2309	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1733	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₉×H(3,ℤ/4)	C₂	1	2	yes	maximal, complemented
6	C₂³×D₄×C₁₉	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₁₉	C₂²	1	2	yes	—
8	C₂×C₁₉×#117	C₂²	1	1	yes	characteristic
9	C₂×C₁₉×#117	C₂²	1	1	yes	characteristic
10	C₂×C₁₉×#117	C₂²	1	2	yes	—
11	C₂×C₁₉×#100	—	2	2	no	—
12	C₂×C₁₉×#100	—	2	4	no	—
13	C₁₉×#390	—	2	2	no	—
14	C₁₉×#390	—	2	4	no	—
15	C₁₉×#390	C₄	1	4	yes	complemented
16	C₁₉×#379	—	2	2	no	—
17	C₁₉×#379	—	2	2	no	—
18	C₁₉×#339	—	2	4	no	—
19	C₁₉×H(3,ℤ/4)	—	2	4	no	—
20	C₂⁵×C₁₉	D₄	1	2	yes	—
21	C₂²×D₄×C₁₉	—	2	2	no	—
22	C₂²×D₄×C₁₉	—	2	2	no	—
23	C₂²×D₄×C₁₉	—	2	2	no	—
24	C₂²×D₄×C₁₉	—	2	2	no	—
25	C₂²×D₄×C₁₉	—	2	2	no	—
26	C₂²×D₄×C₁₉	—	2	2	no	—
27	C₂²×D₄×C₁₉	C₂×C₄	1	2	yes	—
28	C₂²×D₄×C₁₉	D₄	1	2	yes	—
29	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
30	C₂×M₁₆×C₁₉	—	2	2	no	—
31	C₂×M₁₆×C₁₉	—	2	2	no	—
32	C₂×M₁₆×C₁₉	—	2	2	no	—
33	C₂×M₁₆×C₁₉	—	2	2	no	—
34	C₂×M₁₆×C₁₉	—	2	2	no	—
35	C₂×M₁₆×C₁₉	D₄	1	4	yes	—
36	C₂²×C₈×C₁₉	—	2	2	no	—
37	C₁₉×#117	—	2	2	no	—
38	C₁₉×#117	—	2	2	no	—
39	C₁₉×#117	—	2	2	no	—
40	C₁₉×#117	—	2	2	no	—
41	C₁₉×#117	—	2	2	no	—
42	C₁₉×#117	—	2	2	no	—
43	C₁₉×#117	—	2	2	no	—
44	C₁₉×#117	—	2	2	no	—
45	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
46	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
47	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
48	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
49	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
50	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
51	C₂×C₂²⋊C₄×C₁₉	—	2	8	no	—
52	C₂×C₂²⋊C₄×C₁₉	—	2	8	no	—
53	C₂×C₄²×C₁₉	—	2	1	no	—
54	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
55	C₂×C₄²×C₁₉	D₄	1	4	yes	—
56	C₁₉×#100	—	2	8	no	—
57	C₂⁴×C₁₉	—	2	2	no	—
58	C₂⁴×C₁₉	—	2	4	no	—
59	C₂⁴×C₁₉	—	2	4	no	—
60	C₂⁴×C₁₉	—	2	2	no	—
61	C₂⁴×C₁₉	—	8	4	no	—
62	C₂⁴×C₁₉	C₂²⋊C₄	1	4	yes	—
63	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
64	C₂×D₄×C₁₉	—	2	2	no	—
65	C₂×D₄×C₁₉	—	2	2	no	—
66	C₂×D₄×C₁₉	—	2	4	no	—
67	C₂×D₄×C₁₉	—	2	4	no	—
68	C₂×D₄×C₁₉	—	2	2	no	—
69	C₂×D₄×C₁₉	—	2	4	no	—
70	C₂×D₄×C₁₉	—	2	4	no	—
71	C₂×D₄×C₁₉	—	2	2	no	—
72	C₂×D₄×C₁₉	—	4	2	no	—
73	C₂×D₄×C₁₉	—	4	2	no	—
74	C₂×D₄×C₁₉	—	4	2	no	—
75	C₂×D₄×C₁₉	—	4	2	no	—
76	C₂×D₄×C₁₉	—	4	2	no	—
77	C₂×D₄×C₁₉	—	4	2	no	—
78	C₂×D₄×C₁₉	—	4	2	no	—
79	C₂×D₄×C₁₉	—	4	2	no	—
80	C₂²×C₄×C₁₉	—	2	2	no	—
81	C₂²×C₄×C₁₉	—	2	2	no	—
82	C₂²×C₄×C₁₉	—	2	4	no	—
83	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₉	#41	1	2	yes	—
88	C₂²×C₄×C₁₉	#41	1	2	yes	—
89	C₂²×C₄×C₁₉	#41	1	2	yes	—
90	C₂²×C₄×C₁₉	#41	1	2	yes	—
91	M₁₆×C₁₉	—	2	4	no	—
92	M₁₆×C₁₉	—	2	4	no	—
93	M₁₆×C₁₉	—	4	2	no	—
94	M₁₆×C₁₉	—	4	2	no	—
95	C₂×C₈×C₁₉	—	2	4	no	—
96	C₂×C₈×C₁₉	—	4	2	no	—
97	C₂×C₈×C₁₉	—	4	2	no	—
98	C₄⋊C₄×C₁₉	—	2	4	no	—
99	C₄⋊C₄×C₁₉	—	4	1	no	—
100	C₄⋊C₄×C₁₉	—	4	2	no	—
101	C₄⋊C₄×C₁₉	—	4	1	no	—
102	C₄⋊C₄×C₁₉	—	4	2	no	—
103	C₂²⋊C₄×C₁₉	—	4	1	no	—
104	C₂²⋊C₄×C₁₉	—	4	2	no	—
105	C₂²⋊C₄×C₁₉	—	4	2	no	—
106	C₂²⋊C₄×C₁₉	—	4	1	no	—
107	C₂²⋊C₄×C₁₉	—	4	2	no	—
108	C₂²⋊C₄×C₁₉	—	8	8	no	—
109	C₂²⋊C₄×C₁₉	—	8	8	no	—
110	C₄²×C₁₉	—	2	4	no	—
111	C₄²×C₁₉	—	2	4	no	—
112	C₄²×C₁₉	—	2	4	no	—
113	C₄²×C₁₉	—	4	2	no	—
114	#11155	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂³×C₁₉	—	2	4	no	—
116	C₂³×C₁₉	—	2	4	no	—
117	C₂³×C₁₉	—	2	4	no	—
118	C₂³×C₁₉	—	4	2	no	—
119	C₂³×C₁₉	—	4	8	no	—
120	C₂³×C₁₉	—	4	8	no	—
121	C₂³×C₁₉	—	4	4	no	—
122	C₂³×C₁₉	—	4	4	no	—
123	C₂³×C₁₉	—	8	4	no	—
124	C₂³×C₁₉	—	8	8	no	—
125	C₂³×C₁₉	—	8	8	no	—
126	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂³×C₁₉	#120	1	1	yes	characteristic
128	C₂³×C₁₉	#121	1	2	yes	—
129	C₂³×C₁₉	#123	1	1	yes	characteristic
130	C₂³×C₁₉	#123	1	2	yes	—
131	D₄×C₁₉	—	4	4	no	—
132	D₄×C₁₉	—	4	4	no	—
133	D₄×C₁₉	—	4	4	no	—
134	D₄×C₁₉	—	4	4	no	—
135	C₂×C₄×C₁₉	—	4	4	no	—
136	C₂×C₄×C₁₉	—	4	8	no	—
137	C₂×C₄×C₁₉	—	4	4	no	—
138	C₂×C₄×C₁₉	—	4	2	no	—
139	C₂×C₄×C₁₉	—	4	2	no	—
140	C₂×C₄×C₁₉	—	4	4	no	—
141	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
142	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
143	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
144	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
145	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
146	C₂×C₄×C₁₉	#121	1	2	yes	—
147	C₂×C₄×C₁₉	#121	1	2	yes	—
148	C₂×C₄×C₁₉	#121	1	2	yes	—
149	C₂×C₄×C₁₉	#123	1	2	yes	—
150	C₂×C₄×C₁₉	#123	1	2	yes	—
151	C₂×C₄×C₁₉	#123	1	2	yes	—
152	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
153	C₂×C₄×C₁₉	#124	1	2	yes	—
154	C₂×C₄×C₁₉	#124	1	2	yes	—
155	C₂×C₄×C₁₉	#126	1	2	yes	—
156	C₂×C₄×C₁₉	#126	1	2	yes	—
157	C₈×C₁₉	—	4	4	no	—
158	C₂²×#100	C₂×C₁₉	1	1	yes	characteristic
159	#2309	C₂×C₁₉	1	1	yes	characteristic
160	#2309	C₂×C₁₉	1	1	yes	characteristic
161	#1733	C₂×C₁₉	1	2	yes	complemented
162	C₂×H(3,ℤ/4)	C₂×C₁₉	1	2	yes	complemented
163	C₂²×C₁₉	—	2	2	no	—
164	C₂²×C₁₉	—	2	2	no	—
165	C₂²×C₁₉	—	2	2	no	—
166	C₂²×C₁₉	—	4	2	no	—
167	C₂²×C₁₉	—	4	2	no	—
168	C₂²×C₁₉	—	4	8	no	—
169	C₂²×C₁₉	—	4	8	no	—
170	C₂²×C₁₉	—	4	4	no	—
171	C₂²×C₁₉	—	8	4	no	—
172	C₂²×C₁₉	—	8	8	no	—
173	C₂²×C₁₉	—	8	8	no	—
174	C₂²×C₁₉	#381	1	1	yes	characteristic
175	C₂²×C₁₉	#386	1	1	yes	characteristic
176	C₂²×C₁₉	#386	1	1	yes	characteristic
177	C₂²×C₁₉	#394	1	2	yes	—
178	C₂²×C₁₉	#397	1	2	yes	—
179	C₄×C₁₉	—	8	2	no	—
180	C₄×C₁₉	—	8	4	no	—
181	C₄×C₁₉	—	8	2	no	—
182	C₄×C₁₉	#388	1	2	yes	—
183	C₄×C₁₉	#388	1	2	yes	—
184	C₄×C₁₉	#394	1	2	yes	—
185	C₄×C₁₉	#397	1	2	yes	—
186	C₂³×D₄	C₂²×C₁₉	1	1	yes	characteristic
187	C₂²×M₁₆	C₂²×C₁₉	1	2	yes	—
188	C₂×#117	C₂²×C₁₉	1	1	yes	characteristic
189	C₂×#117	C₂²×C₁₉	1	1	yes	characteristic
190	C₂×#117	C₂²×C₁₉	1	2	yes	—
191	C₂×#100	—	2	2	no	—
192	C₂×#100	—	2	4	no	—
193	#390	—	2	2	no	—
194	#390	—	2	4	no	—
195	#390	C₄×C₁₉	1	4	yes	complemented
196	#379	—	2	2	no	—
197	#379	—	2	2	no	—
198	#339	—	2	4	no	—
199	H(3,ℤ/4)	—	2	4	no	—
200	C₂×C₁₉	—	2	1	no	—
201	C₂×C₁₉	—	2	2	no	—
202	C₂×C₁₉	—	2	1	no	—
203	C₂×C₁₉	—	2	2	no	—
204	C₂×C₁₉	—	8	4	no	—
205	C₂×C₁₉	#1440	1	1	yes	characteristic
206	C₂×C₁₉	#1911	1	2	yes	—
207	C₂⁵	D₄×C₁₉	1	2	yes	—
208	C₂²×D₄	—	2	2	no	—
209	C₂²×D₄	—	2	2	no	—
210	C₂²×D₄	—	2	2	no	—
211	C₂²×D₄	—	2	2	no	—
212	C₂²×D₄	—	2	2	no	—
213	C₂²×D₄	—	2	2	no	—
214	C₂²×D₄	C₂×C₄×C₁₉	1	2	yes	—
215	C₂²×D₄	D₄×C₁₉	1	2	yes	—
216	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
217	C₂×M₁₆	—	2	2	no	—
218	C₂×M₁₆	—	2	2	no	—
219	C₂×M₁₆	—	2	2	no	—
220	C₂×M₁₆	—	2	2	no	—
221	C₂×M₁₆	—	2	2	no	—
222	C₂×M₁₆	D₄×C₁₉	1	4	yes	—
223	C₂²×C₈	—	2	2	no	—
224	#117	—	2	2	no	—
225	#117	—	2	2	no	—
226	#117	—	2	2	no	—
227	#117	—	2	2	no	—
228	#117	—	2	2	no	—
229	#117	—	2	2	no	—
230	#117	—	2	2	no	—
231	#117	—	2	2	no	—
232	C₂×C₄⋊C₄	—	2	1	no	—
233	C₂×C₄⋊C₄	—	2	2	no	—
234	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	2	yes	—
235	C₂×C₂²⋊C₄	—	2	1	no	—
236	C₂×C₂²⋊C₄	—	2	1	no	—
237	C₂×C₂²⋊C₄	—	2	2	no	—
238	C₂×C₂²⋊C₄	—	2	8	no	—
239	C₂×C₂²⋊C₄	—	2	8	no	—
240	C₂×C₄²	—	2	1	no	—
241	C₂×C₄²	C₂×C₄×C₁₉	1	2	yes	—
242	C₂×C₄²	D₄×C₁₉	1	4	yes	—
243	#100	—	2	8	no	—
244	C₁₉	#11155	1	1	yes	characteristic, direct product factor, Sylow
245	C₂⁴	—	2	2	no	—
246	C₂⁴	—	2	4	no	—
247	C₂⁴	—	2	4	no	—
248	C₂⁴	—	2	2	no	—
249	C₂⁴	—	8	4	no	—
250	C₂⁴	C₂²⋊C₄×C₁₉	1	4	yes	—
251	C₂⁴	C₂×D₄×C₁₉	1	1	yes	characteristic
252	C₂×D₄	—	2	2	no	—
253	C₂×D₄	—	2	2	no	—
254	C₂×D₄	—	2	4	no	—
255	C₂×D₄	—	2	4	no	—
256	C₂×D₄	—	2	2	no	—
257	C₂×D₄	—	2	4	no	—
258	C₂×D₄	—	2	4	no	—
259	C₂×D₄	—	2	2	no	—
260	C₂×D₄	—	4	2	no	—
261	C₂×D₄	—	4	2	no	—
262	C₂×D₄	—	4	2	no	—
263	C₂×D₄	—	4	2	no	—
264	C₂×D₄	—	4	2	no	—
265	C₂×D₄	—	4	2	no	—
266	C₂×D₄	—	4	2	no	—
267	C₂×D₄	—	4	2	no	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	—	2	2	no	—
270	C₂²×C₄	—	2	4	no	—
271	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
272	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
273	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
274	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
275	C₂²×C₄	#41×C₁₉	1	2	yes	—
276	C₂²×C₄	#41×C₁₉	1	2	yes	—
277	C₂²×C₄	#41×C₁₉	1	2	yes	—
278	C₂²×C₄	#41×C₁₉	1	2	yes	—
279	M₁₆	—	2	4	no	—
280	M₁₆	—	2	4	no	—
281	M₁₆	—	4	2	no	—
282	M₁₆	—	4	2	no	—
283	C₂×C₈	—	2	4	no	—
284	C₂×C₈	—	4	2	no	—
285	C₂×C₈	—	4	2	no	—
286	C₄⋊C₄	—	2	4	no	—
287	C₄⋊C₄	—	4	1	no	—
288	C₄⋊C₄	—	4	2	no	—
289	C₄⋊C₄	—	4	1	no	—
290	C₄⋊C₄	—	4	2	no	—
291	C₂²⋊C₄	—	4	1	no	—
292	C₂²⋊C₄	—	4	2	no	—
293	C₂²⋊C₄	—	4	2	no	—
294	C₂²⋊C₄	—	4	1	no	—
295	C₂²⋊C₄	—	4	2	no	—
296	C₂²⋊C₄	—	8	8	no	—
297	C₂²⋊C₄	—	8	8	no	—
298	C₄²	—	2	4	no	—
299	C₄²	—	2	4	no	—
300	C₄²	—	2	4	no	—
301	C₄²	—	4	2	no	—
302	C₂³	—	2	4	no	—
303	C₂³	—	2	4	no	—
304	C₂³	—	2	4	no	—
305	C₂³	—	4	2	no	—
306	C₂³	—	4	8	no	—
307	C₂³	—	4	8	no	—
308	C₂³	—	4	4	no	—
309	C₂³	—	4	4	no	—
310	C₂³	—	8	4	no	—
311	C₂³	—	8	8	no	—
312	C₂³	—	8	8	no	—
313	C₂³	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
314	C₂³	C₁₉×#120	1	1	yes	characteristic
315	C₂³	C₁₉×#121	1	2	yes	—
316	C₂³	C₁₉×#123	1	1	yes	characteristic
317	C₂³	C₁₉×#123	1	2	yes	—
318	D₄	—	4	4	no	—
319	D₄	—	4	4	no	—
320	D₄	—	4	4	no	—
321	D₄	—	4	4	no	—
322	C₂×C₄	—	4	4	no	—
323	C₂×C₄	—	4	8	no	—
324	C₂×C₄	—	4	4	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	4	no	—
328	C₂×C₄	C₁₉×#117	1	1	yes	characteristic
329	C₂×C₄	C₁₉×#117	1	1	yes	characteristic
330	C₂×C₄	C₄×D₄×C₁₉	1	2	yes	—
331	C₂×C₄	C₄×D₄×C₁₉	1	2	yes	—
332	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
333	C₂×C₄	C₁₉×#121	1	2	yes	—
334	C₂×C₄	C₁₉×#121	1	2	yes	—
335	C₂×C₄	C₁₉×#121	1	2	yes	—
336	C₂×C₄	C₁₉×#123	1	2	yes	—
337	C₂×C₄	C₁₉×#123	1	2	yes	—
338	C₂×C₄	C₁₉×#123	1	2	yes	—
339	C₂×C₄	C₁₉×#123	1	1	yes	characteristic
340	C₂×C₄	C₁₉×#124	1	2	yes	—
341	C₂×C₄	C₁₉×#124	1	2	yes	—
342	C₂×C₄	C₁₉×#126	1	2	yes	—
343	C₂×C₄	C₁₉×#126	1	2	yes	—
344	C₈	—	4	4	no	—
345	C₂²	—	2	2	no	—
346	C₂²	—	2	2	no	—
347	C₂²	—	2	2	no	—
348	C₂²	—	4	2	no	—
349	C₂²	—	4	2	no	—
350	C₂²	—	4	8	no	—
351	C₂²	—	4	8	no	—
352	C₂²	—	4	4	no	—
353	C₂²	—	8	4	no	—
354	C₂²	—	8	8	no	—
355	C₂²	—	8	8	no	—
356	C₂²	C₁₉×#381	1	1	yes	characteristic
357	C₂²	C₁₉×#386	1	1	yes	characteristic
358	C₂²	C₁₉×#386	1	1	yes	characteristic
359	C₂²	C₁₉×#394	1	2	yes	—
360	C₂²	C₁₉×#397	1	2	yes	—
361	C₄	—	8	2	no	—
362	C₄	—	8	4	no	—
363	C₄	—	8	2	no	—
364	C₄	C₁₉×#388	1	2	yes	—
365	C₄	C₁₉×#388	1	2	yes	—
366	C₄	C₁₉×#394	1	2	yes	—
367	C₄	C₁₉×#397	1	2	yes	—
368	C₂	—	2	1	no	—
369	C₂	—	2	2	no	—
370	C₂	—	2	1	no	—
371	C₂	—	2	2	no	—
372	C₂	—	8	4	no	—
373	C₂	C₁₉×#1440	1	1	yes	characteristic
374	C₂	C₁₉×#1911	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₉	#386
commutator	C₂²×C₄	C₂²×C₄×C₁₉
Frattini, cosocle	C₂³×C₄	C₂³×C₁₉
prime radical	C₂³×D₄×C₁₉	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂³×C₄×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₉×#1733	C₂	semidirect
5. C₂×C₁₉×H(3,ℤ/4)	C₂	semidirect
15. C₁₉×#390	C₄	semidirect
114. #11155	C₁₉	direct (N × H)
161. #1733	C₂×C₁₉	semidirect
162. C₂×H(3,ℤ/4)	C₂×C₁₉	semidirect
195. #390	C₄×C₁₉	semidirect
244. C₁₉	#11155	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₁₉×#100	2
C₁₉×#2309	2
C₁₉×#1733	2
C₂×C₁₉×H(3,ℤ/4)	2
#11155	19

Subgroups — C19×#11155 #18017114

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#11155 #18017114