Subgroups — C₁₉×#33649 #18039608

Order 4864.

580 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#467	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×#427	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#2237	C₂	1	4	yes	maximal, complemented
4	C₁₉×#2236	C₂	1	2	yes	maximal, complemented
5	C₁₉×#2235	C₂	1	2	yes	maximal, complemented
6	C₁₉×#2159	C₂	1	2	yes	maximal, complemented
7	C₁₉×#2141	C₂	1	2	yes	maximal, complemented
8	C₁₉×#2112	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₁₆×C₁₉	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
11	C₂×C₁₉×C₂²⋊Q₈	—	2	1	no	—
12	C₂×C₁₉×#117	—	2	1	no	—
13	C₂×C₁₉×M₃₂	C₂²	1	2	yes	—
14	C₁₉×#467	—	2	1	no	—
15	C₁₉×#467	—	2	1	no	—
16	C₁₉×#467	C₂²	1	2	yes	—
17	C₁₉×#467	C₂²	1	2	yes	—
18	C₁₉×#427	—	2	1	no	—
19	C₁₉×#427	—	2	1	no	—
20	C₁₉×#427	C₂²	1	2	yes	—
21	C₁₉×#427	C₂²	1	2	yes	—
22	C₂×C₁₉×#107	C₂²	1	1	yes	characteristic
23	C₂×C₁₉×#107	C₂²	1	1	yes	characteristic
24	C₂×C₁₉×#107	C₂²	1	1	yes	characteristic
25	C₂×C₁₉×#106	—	2	1	no	—
26	C₂×C₁₉×#103	—	2	1	no	—
27	C₁₉×#367	C₂²	1	4	yes	complemented
28	C₁₉×#366	C₂²	1	4	yes	complemented
29	C₁₉×#358	C₂²	1	4	yes	complemented
30	C₁₉×#358	C₂²	1	4	yes	complemented
31	C₁₉×C₂²⋊C₁₆	C₂²	1	2	yes	—
32	C₁₉×C₂²⋊C₁₆	C₂²	1	2	yes	—
33	C₂²×Q₈×C₁₉	—	2	1	no	—
34	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
35	C₂×Q₁₆×C₁₉	—	2	1	no	—
36	C₂×Q₁₆×C₁₉	—	2	4	no	—
37	C₂×Q₁₆×C₁₉	C₂³	1	2	yes	—
38	C₂²×C₈×C₁₉	—	2	4	no	—
39	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
45	C₁₉×C₂²⋊Q₈	—	2	2	no	—
46	C₁₉×C₂²⋊Q₈	—	2	2	no	—
47	C₁₉×C₂²⋊Q₈	—	4	1	no	—
48	C₁₉×C₂²⋊Q₈	—	4	1	no	—
49	C₁₉×#117	—	2	2	no	—
50	C₁₉×#117	—	2	2	no	—
51	C₁₉×#117	—	4	1	no	—
52	C₁₉×#117	—	4	1	no	—
53	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
54	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
55	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
56	C₂×C₄⋊C₄×C₁₉	—	4	1	no	—
57	C₂×C₂²⋊C₄×C₁₉	—	4	1	no	—
58	C₂×C₂²⋊C₄×C₁₉	—	4	1	no	—
59	C₂×C₄²×C₁₉	—	4	1	no	—
60	C₁₉×M₃₂	—	2	4	no	—
61	C₂×C₁₆×C₁₉	C₂³	1	4	yes	—
62	C₁₉×#107	—	2	1	no	—
63	C₁₉×#107	—	2	1	no	—
64	C₁₉×#107	—	2	1	no	—
65	C₁₉×#107	C₂³	1	2	yes	—
66	C₁₉×#107	C₂³	1	2	yes	—
67	C₁₉×#107	C₂³	1	2	yes	—
68	C₁₉×#106	—	2	2	no	—
69	C₁₉×#106	—	2	2	no	—
70	C₁₉×#103	—	2	2	no	—
71	C₁₉×#103	—	2	2	no	—
72	C₂⁴×C₁₉	D₈	1	1	yes	characteristic
73	C₂×Q₈×C₁₉	—	2	2	no	—
74	C₂×Q₈×C₁₉	—	4	1	no	—
75	C₂×Q₈×C₁₉	—	4	2	no	—
76	C₂×Q₈×C₁₉	—	4	2	no	—
77	C₂²×C₄×C₁₉	—	2	2	no	—
78	C₂²×C₄×C₁₉	—	2	2	no	—
79	C₂²×C₄×C₁₉	—	4	1	no	—
80	C₂²×C₄×C₁₉	—	4	1	no	—
81	C₂²×C₄×C₁₉	—	4	1	no	—
82	C₂²×C₄×C₁₉	—	4	1	no	—
83	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
84	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
85	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
86	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
89	Q₁₆×C₁₉	—	2	4	no	—
90	Q₁₆×C₁₉	—	4	2	no	—
91	C₂×C₈×C₁₉	—	2	2	no	—
92	C₂×C₈×C₁₉	—	2	2	no	—
93	C₂×C₈×C₁₉	—	2	2	no	—
94	C₂×C₈×C₁₉	—	2	2	no	—
95	C₂×C₈×C₁₉	—	2	2	no	—
96	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₁₉	#41	1	4	yes	—
100	C₂×C₈×C₁₉	C₂⁴	1	1	yes	characteristic
101	C₄⋊C₄×C₁₉	—	2	2	no	—
102	C₄⋊C₄×C₁₉	—	2	2	no	—
103	C₄⋊C₄×C₁₉	—	2	2	no	—
104	C₄⋊C₄×C₁₉	—	4	1	no	—
105	C₄⋊C₄×C₁₉	—	4	1	no	—
106	C₄⋊C₄×C₁₉	—	4	1	no	—
107	C₄⋊C₄×C₁₉	—	4	2	no	—
108	C₄⋊C₄×C₁₉	—	4	2	no	—
109	C₂²⋊C₄×C₁₉	—	4	2	no	—
110	C₂²⋊C₄×C₁₉	—	4	2	no	—
111	C₂²⋊C₄×C₁₉	—	4	2	no	—
112	C₂²⋊C₄×C₁₉	—	4	2	no	—
113	C₄²×C₁₉	—	4	2	no	—
114	C₄²×C₁₉	—	4	2	no	—
115	C₁₆×C₁₉	—	2	4	no	—
116	#33649	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₂³×C₁₉	—	2	2	no	—
118	C₂³×C₁₉	—	2	2	no	—
119	C₂³×C₁₉	—	4	1	no	—
120	C₂³×C₁₉	C₂×D₈	1	1	yes	characteristic
121	C₂³×C₁₉	C₂×D₈	1	1	yes	characteristic
122	C₂³×C₁₉	C₂×D₈	1	1	yes	characteristic
123	Q₈×C₁₉	—	4	2	no	—
124	Q₈×C₁₉	—	4	2	no	—
125	Q₈×C₁₉	—	8	2	no	—
126	C₂×C₄×C₁₉	—	2	2	no	—
127	C₂×C₄×C₁₉	—	2	1	no	—
128	C₂×C₄×C₁₉	—	2	1	no	—
129	C₂×C₄×C₁₉	—	2	1	no	—
130	C₂×C₄×C₁₉	—	2	1	no	—
131	C₂×C₄×C₁₉	—	2	2	no	—
132	C₂×C₄×C₁₉	—	2	2	no	—
133	C₂×C₄×C₁₉	—	4	2	no	—
134	C₂×C₄×C₁₉	—	4	2	no	—
135	C₂×C₄×C₁₉	—	4	2	no	—
136	C₂×C₄×C₁₉	—	4	2	no	—
137	C₂×C₄×C₁₉	—	8	2	no	—
138	C₂×C₄×C₁₉	—	8	2	no	—
139	C₂×C₄×C₁₉	—	8	2	no	—
140	C₂×C₄×C₁₉	—	8	2	no	—
141	C₂×C₄×C₁₉	#123	1	2	yes	—
142	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₁₉	#137	1	2	yes	—
146	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
147	C₈×C₁₉	—	2	1	no	—
148	C₈×C₁₉	—	2	1	no	—
149	C₈×C₁₉	#123	1	2	yes	—
150	C₈×C₁₉	C₂×#41	1	2	yes	—
151	C₂×#467	C₂×C₁₉	1	1	yes	characteristic
152	C₂×#427	C₂×C₁₉	1	1	yes	characteristic
153	#2237	C₂×C₁₉	1	4	yes	complemented
154	#2236	C₂×C₁₉	1	2	yes	complemented
155	#2235	C₂×C₁₉	1	2	yes	complemented
156	#2159	C₂×C₁₉	1	2	yes	complemented
157	#2141	C₂×C₁₉	1	2	yes	complemented
158	#2112	C₂×C₁₉	1	1	yes	characteristic
159	C₂²×C₁₉	—	2	1	no	—
160	C₂²×C₁₉	—	2	1	no	—
161	C₂²×C₁₉	—	2	1	no	—
162	C₂²×C₁₉	—	2	1	no	—
163	C₂²×C₁₉	—	2	1	no	—
164	C₂²×C₁₉	—	2	1	no	—
165	C₂²×C₁₉	—	2	1	no	—
166	C₂²×C₁₉	—	4	4	no	—
167	C₂²×C₁₉	#480	1	2	yes	—
168	C₂²×C₁₉	#510	1	2	yes	—
169	C₂²×C₁₉	C₂²×D₈	1	1	yes	characteristic
170	C₄×C₁₉	—	2	1	no	—
171	C₄×C₁₉	—	2	1	no	—
172	C₄×C₁₉	—	8	2	no	—
173	C₄×C₁₉	—	8	2	no	—
174	C₄×C₁₉	—	8	2	no	—
175	C₄×C₁₉	—	8	2	no	—
176	C₄×C₁₉	#480	1	2	yes	—
177	C₄×C₁₉	C₂×#123	1	1	yes	characteristic
178	C₄×C₁₉	C₂×#137	1	1	yes	characteristic
179	C₂²×Q₁₆	C₂²×C₁₉	1	1	yes	characteristic
180	C₂³×C₈	C₂²×C₁₉	1	1	yes	characteristic
181	C₂×C₂²⋊Q₈	—	2	1	no	—
182	C₂×#117	—	2	1	no	—
183	C₂×M₃₂	C₂²×C₁₉	1	2	yes	—
184	#467	—	2	1	no	—
185	#467	—	2	1	no	—
186	#467	C₂²×C₁₉	1	2	yes	—
187	#467	C₂²×C₁₉	1	2	yes	—
188	#427	—	2	1	no	—
189	#427	—	2	1	no	—
190	#427	C₂²×C₁₉	1	2	yes	—
191	#427	C₂²×C₁₉	1	2	yes	—
192	C₂×#107	C₂²×C₁₉	1	1	yes	characteristic
193	C₂×#107	C₂²×C₁₉	1	1	yes	characteristic
194	C₂×#107	C₂²×C₁₉	1	1	yes	characteristic
195	C₂×#106	—	2	1	no	—
196	C₂×#103	—	2	1	no	—
197	#367	C₂²×C₁₉	1	4	yes	complemented
198	#366	C₂²×C₁₉	1	4	yes	complemented
199	#358	C₂²×C₁₉	1	4	yes	complemented
200	#358	C₂²×C₁₉	1	4	yes	complemented
201	C₂²⋊C₁₆	C₂²×C₁₉	1	2	yes	—
202	C₂²⋊C₁₆	C₂²×C₁₉	1	2	yes	—
203	C₂×C₁₉	—	2	2	no	—
204	C₂×C₁₉	—	2	2	no	—
205	C₂×C₁₉	—	4	1	no	—
206	C₂×C₁₉	C₂×#480	1	1	yes	characteristic
207	C₂×C₁₉	C₂×#510	1	1	yes	characteristic
208	C₂×C₁₉	#3414	1	1	yes	characteristic
209	C₂²×Q₈	—	2	1	no	—
210	C₂³×C₄	D₄×C₁₉	1	1	yes	characteristic
211	C₂×Q₁₆	—	2	1	no	—
212	C₂×Q₁₆	—	2	4	no	—
213	C₂×Q₁₆	C₂³×C₁₉	1	2	yes	—
214	C₂²×C₈	—	2	4	no	—
215	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
216	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
217	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
218	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
219	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
220	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
221	C₂²⋊Q₈	—	2	2	no	—
222	C₂²⋊Q₈	—	2	2	no	—
223	C₂²⋊Q₈	—	4	1	no	—
224	C₂²⋊Q₈	—	4	1	no	—
225	#117	—	2	2	no	—
226	#117	—	2	2	no	—
227	#117	—	4	1	no	—
228	#117	—	4	1	no	—
229	C₂×C₄⋊C₄	—	2	1	no	—
230	C₂×C₄⋊C₄	—	2	1	no	—
231	C₂×C₄⋊C₄	—	2	1	no	—
232	C₂×C₄⋊C₄	—	4	1	no	—
233	C₂×C₂²⋊C₄	—	4	1	no	—
234	C₂×C₂²⋊C₄	—	4	1	no	—
235	C₂×C₄²	—	4	1	no	—
236	M₃₂	—	2	4	no	—
237	C₂×C₁₆	C₂³×C₁₉	1	4	yes	—
238	#107	—	2	1	no	—
239	#107	—	2	1	no	—
240	#107	—	2	1	no	—
241	#107	C₂³×C₁₉	1	2	yes	—
242	#107	C₂³×C₁₉	1	2	yes	—
243	#107	C₂³×C₁₉	1	2	yes	—
244	#106	—	2	2	no	—
245	#106	—	2	2	no	—
246	#103	—	2	2	no	—
247	#103	—	2	2	no	—
248	C₁₉	#33649	1	1	yes	characteristic, direct product factor, Sylow
249	C₂⁴	D₈×C₁₉	1	1	yes	characteristic
250	C₂×Q₈	—	2	2	no	—
251	C₂×Q₈	—	4	1	no	—
252	C₂×Q₈	—	4	2	no	—
253	C₂×Q₈	—	4	2	no	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	—	2	2	no	—
256	C₂²×C₄	—	4	1	no	—
257	C₂²×C₄	—	4	1	no	—
258	C₂²×C₄	—	4	1	no	—
259	C₂²×C₄	—	4	1	no	—
260	C₂²×C₄	D₈×C₁₉	1	1	yes	characteristic
261	C₂²×C₄	D₈×C₁₉	1	1	yes	characteristic
262	C₂²×C₄	D₈×C₁₉	1	1	yes	characteristic
263	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
264	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
265	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
266	Q₁₆	—	2	4	no	—
267	Q₁₆	—	4	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
274	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
275	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
276	C₂×C₈	#41×C₁₉	1	4	yes	—
277	C₂×C₈	C₂⁴×C₁₉	1	1	yes	characteristic
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₄⋊C₄	—	4	1	no	—
283	C₄⋊C₄	—	4	1	no	—
284	C₄⋊C₄	—	4	2	no	—
285	C₄⋊C₄	—	4	2	no	—
286	C₂²⋊C₄	—	4	2	no	—
287	C₂²⋊C₄	—	4	2	no	—
288	C₂²⋊C₄	—	4	2	no	—
289	C₂²⋊C₄	—	4	2	no	—
290	C₄²	—	4	2	no	—
291	C₄²	—	4	2	no	—
292	C₁₆	—	2	4	no	—
293	C₂³	—	2	2	no	—
294	C₂³	—	2	2	no	—
295	C₂³	—	4	1	no	—
296	C₂³	C₂×D₈×C₁₉	1	1	yes	characteristic
297	C₂³	C₂×D₈×C₁₉	1	1	yes	characteristic
298	C₂³	C₂×D₈×C₁₉	1	1	yes	characteristic
299	Q₈	—	4	2	no	—
300	Q₈	—	4	2	no	—
301	Q₈	—	8	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	8	2	no	—
314	C₂×C₄	—	8	2	no	—
315	C₂×C₄	—	8	2	no	—
316	C₂×C₄	—	8	2	no	—
317	C₂×C₄	C₁₉×#123	1	2	yes	—
318	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
319	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
320	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
321	C₂×C₄	C₁₉×#137	1	2	yes	—
322	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
323	C₈	—	2	1	no	—
324	C₈	—	2	1	no	—
325	C₈	C₁₉×#123	1	2	yes	—
326	C₈	C₂×#41×C₁₉	1	2	yes	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	4	4	no	—
335	C₂²	C₁₉×#480	1	2	yes	—
336	C₂²	C₁₉×#510	1	2	yes	—
337	C₂²	C₂²×D₈×C₁₉	1	1	yes	characteristic
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	8	2	no	—
341	C₄	—	8	2	no	—
342	C₄	—	8	2	no	—
343	C₄	—	8	2	no	—
344	C₄	C₁₉×#480	1	2	yes	—
345	C₄	C₂×C₁₉×#123	1	1	yes	characteristic
346	C₄	C₂×C₁₉×#137	1	1	yes	characteristic
347	C₂	—	2	2	no	—
348	C₂	—	2	2	no	—
349	C₂	—	4	1	no	—
350	C₂	C₂×C₁₉×#480	1	1	yes	characteristic
351	C₂	C₂×C₁₉×#510	1	1	yes	characteristic
352	C₂	C₁₉×#3414	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₉
prime radical	C₂⁴×C₁₉	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂³×C₄×C₁₉ ◁ C₂³×C₈×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₉×#2237	C₂	semidirect
4. C₁₉×#2236	C₂	semidirect
5. C₁₉×#2235	C₂	semidirect
6. C₁₉×#2159	C₂	semidirect
7. C₁₉×#2141	C₂	semidirect
27. C₁₉×#367	C₂²	semidirect
28. C₁₉×#366	C₂²	semidirect
29. C₁₉×#358	C₂²	semidirect
30. C₁₉×#358	C₂²	semidirect
116. #33649	C₁₉	direct (N × H)
153. #2237	C₂×C₁₉	semidirect
154. #2236	C₂×C₁₉	semidirect
155. #2235	C₂×C₁₉	semidirect
156. #2159	C₂×C₁₉	semidirect
157. #2141	C₂×C₁₉	semidirect
197. #367	C₂²×C₁₉	semidirect
198. #366	C₂²×C₁₉	semidirect
199. #358	C₂²×C₁₉	semidirect
200. #358	C₂²×C₁₉	semidirect
248. C₁₉	#33649	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×#467	2
C₂×C₁₉×#427	2
C₁₉×#2237	2
C₁₉×#2236	2
C₁₉×#2235	2
C₁₉×#2159	2
C₁₉×#2141	2
C₁₉×#2112	2
#33649	19

Subgroups — C19×#33649 #18039608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#33649 #18039608