Subgroups — C₁₉×#33588 #18039547

Order 4864.

680 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#510	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₉×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₉×#3156	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#2950	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#2211	C₂	1	4	yes	maximal, complemented
6	C₁₉×#2209	C₂	1	4	yes	maximal, complemented
7	C₁₉×#2151	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₉×#2141	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₉×#2138	C₂	1	1	yes	maximal, characteristic
10	C₂×C₁₉×#137	—	2	1	no	—
11	C₂×C₁₉×#135	C₂²	1	1	yes	characteristic
12	C₂²×Q₁₆×C₁₉	C₂²	1	1	yes	characteristic
13	C₂²×Q₁₆×C₁₉	C₂²	1	1	yes	characteristic
14	C₁₉×#536	—	2	1	no	—
15	C₁₉×#519	—	2	1	no	—
16	C₁₉×#510	—	2	4	no	—
17	C₂×C₁₉×Q₃₂	—	2	4	no	—
18	C₂×C₁₉×Q₃₂	C₂²	1	2	yes	—
19	C₂×C₁₉×Q₃₂	C₂²	1	4	yes	—
20	C₂×C₁₉×SD₃₂	C₂²	1	2	yes	—
21	C₂×C₁₉×M₃₂	C₂²	1	1	yes	characteristic
22	C₂²×C₁₆×C₁₉	C₂²	1	1	yes	characteristic
23	C₁₉×#497	—	2	2	no	—
24	C₁₉×#495	C₂²	1	2	yes	—
25	C₁₉×#494	C₂²	1	2	yes	—
26	C₁₉×#441	—	2	2	no	—
27	C₄×Q₁₆×C₁₉	C₂²	1	4	yes	—
28	C₁₉×#427	C₂²	1	1	yes	characteristic
29	C₁₉×#419	—	2	1	no	—
30	C₁₉×#405	C₂²	1	1	yes	characteristic
31	C₁₉×C₄⋊C₁₆	C₂²	1	4	yes	—
32	C₁₉×#358	C₂²	1	2	yes	—
33	C₁₉×#358	C₂²	1	4	yes	—
34	C₁₉×#357	C₂²	1	2	yes	—
35	C₂×#41×C₁₉	—	2	1	no	—
36	C₂²×Q₈×C₁₉	—	2	1	no	—
37	C₂²×Q₈×C₁₉	—	2	1	no	—
38	C₁₉×#137	—	4	4	no	—
39	C₁₉×#135	—	2	2	no	—
40	C₁₉×#135	—	2	2	no	—
41	C₂×Q₁₆×C₁₉	—	2	1	no	—
42	C₂×Q₁₆×C₁₉	—	2	2	no	—
43	C₂×Q₁₆×C₁₉	—	2	2	no	—
44	C₂×Q₁₆×C₁₉	—	2	1	no	—
45	C₂×Q₁₆×C₁₉	—	2	2	no	—
46	C₂×Q₁₆×C₁₉	—	2	2	no	—
47	C₂×Q₁₆×C₁₉	D₄	1	4	yes	—
48	C₂×Q₁₆×C₁₉	C₂³	1	1	yes	characteristic
49	C₂×Q₁₆×C₁₉	C₂³	1	2	yes	—
50	C₂×Q₁₆×C₁₉	C₂³	1	2	yes	—
51	C₂×SD₁₆×C₁₉	—	2	2	no	—
52	C₂×SD₁₆×C₁₉	—	2	1	no	—
53	C₂×D₈×C₁₉	C₂³	1	1	yes	characteristic
54	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
56	C₁₉×C₄⋊Q₈	—	2	2	no	—
57	C₁₉×#124	—	2	2	no	—
58	C₁₉×#123	—	4	2	no	—
59	C₁₉×C₂²⋊Q₈	—	4	2	no	—
60	C₄×Q₈×C₁₉	—	2	4	no	—
61	C₄×Q₈×C₁₉	—	4	2	no	—
62	C₁₉×#117	—	2	1	no	—
63	C₁₉×#117	—	4	2	no	—
64	C₁₉×#117	D₄	1	1	yes	characteristic
65	C₁₉×Q₃₂	—	2	8	no	—
66	C₁₉×Q₃₂	—	2	4	no	—
67	C₁₉×SD₃₂	—	2	4	no	—
68	C₁₉×M₃₂	—	2	2	no	—
69	C₂×C₁₆×C₁₉	—	2	2	no	—
70	C₂×C₁₆×C₁₉	C₂³	1	2	yes	—
71	C₂×C₁₆×C₁₉	C₂³	1	2	yes	—
72	C₁₉×#107	C₂³	1	2	yes	—
73	C₁₉×#106	—	2	1	no	—
74	C₁₉×#103	—	2	4	no	—
75	C₁₉×#97	D₄	1	2	yes	—
76	C₄×C₈×C₁₉	C₂³	1	2	yes	—
77	#41×C₁₉	—	4	2	no	—
78	#41×C₁₉	—	4	2	no	—
79	C₂×Q₈×C₁₉	—	2	4	no	—
80	C₂×Q₈×C₁₉	—	2	2	no	—
81	C₂×Q₈×C₁₉	—	2	2	no	—
82	C₂×Q₈×C₁₉	—	2	1	no	—
83	C₂×Q₈×C₁₉	—	4	1	no	—
84	C₂×Q₈×C₁₉	—	4	1	no	—
85	C₂×Q₈×C₁₉	—	4	2	no	—
86	C₂×Q₈×C₁₉	—	4	2	no	—
87	C₂×Q₈×C₁₉	—	4	2	no	—
88	C₂×D₄×C₁₉	—	2	1	no	—
89	C₂×D₄×C₁₉	—	4	1	no	—
90	C₂²×C₄×C₁₉	—	4	1	no	—
91	C₂²×C₄×C₁₉	—	4	1	no	—
92	C₂²×C₄×C₁₉	—	4	1	no	—
93	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
94	Q₁₆×C₁₉	—	2	4	no	—
95	Q₁₆×C₁₉	—	2	2	no	—
96	Q₁₆×C₁₉	—	2	1	no	—
97	Q₁₆×C₁₉	—	2	2	no	—
98	Q₁₆×C₁₉	—	2	2	no	—
99	Q₁₆×C₁₉	—	2	1	no	—
100	Q₁₆×C₁₉	—	4	4	no	—
101	Q₁₆×C₁₉	—	4	2	no	—
102	Q₁₆×C₁₉	C₂×D₄	1	4	yes	—
103	SD₁₆×C₁₉	—	4	4	no	—
104	SD₁₆×C₁₉	—	4	2	no	—
105	D₈×C₁₉	—	2	1	no	—
106	D₈×C₁₉	—	2	1	no	—
107	M₁₆×C₁₉	—	2	2	no	—
108	C₂×C₈×C₁₉	C₂×D₄	1	2	yes	—
109	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₁₉	C₂×D₄	1	2	yes	—
111	C₂×C₈×C₁₉	#41	1	2	yes	—
112	C₂×C₈×C₁₉	C₂⁴	1	1	yes	characteristic
113	C₄⋊C₄×C₁₉	—	2	2	no	—
114	C₄⋊C₄×C₁₉	—	4	4	no	—
115	C₄⋊C₄×C₁₉	—	4	4	no	—
116	C₄⋊C₄×C₁₉	D₈	1	2	yes	—
117	C₂²⋊C₄×C₁₉	—	4	1	no	—
118	C₂²⋊C₄×C₁₉	—	4	4	no	—
119	C₂²⋊C₄×C₁₉	D₈	1	2	yes	—
120	C₄²×C₁₉	—	4	1	no	—
121	C₄²×C₁₉	—	4	4	no	—
122	C₄²×C₁₉	C₂×D₄	1	2	yes	—
123	C₁₆×C₁₉	—	2	2	no	—
124	C₁₆×C₁₉	—	2	2	no	—
125	#33588	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
126	C₂³×C₁₉	—	4	1	no	—
127	C₂³×C₁₉	C₂×D₈	1	1	yes	characteristic
128	Q₈×C₁₉	—	2	4	no	—
129	Q₈×C₁₉	—	4	2	no	—
130	Q₈×C₁₉	—	4	4	no	—
131	Q₈×C₁₉	—	4	1	no	—
132	Q₈×C₁₉	—	4	2	no	—
133	Q₈×C₁₉	—	4	2	no	—
134	Q₈×C₁₉	—	4	1	no	—
135	Q₈×C₁₉	—	8	2	no	—
136	D₄×C₁₉	—	4	1	no	—
137	D₄×C₁₉	—	4	1	no	—
138	D₄×C₁₉	—	8	2	no	—
139	C₂×C₄×C₁₉	—	4	2	no	—
140	C₂×C₄×C₁₉	—	4	4	no	—
141	C₂×C₄×C₁₉	—	4	2	no	—
142	C₂×C₄×C₁₉	—	4	2	no	—
143	C₂×C₄×C₁₉	—	4	1	no	—
144	C₂×C₄×C₁₉	—	8	2	no	—
145	C₂×C₄×C₁₉	—	8	2	no	—
146	C₂×C₄×C₁₉	#121	1	2	yes	—
147	C₂×C₄×C₁₉	C₂×D₈	1	4	yes	—
148	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
149	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
150	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
151	C₈×C₁₉	—	2	2	no	—
152	C₈×C₁₉	#121	1	2	yes	—
153	C₈×C₁₉	C₂²×D₄	1	1	yes	characteristic
154	C₈×C₁₉	C₂×#41	1	1	yes	characteristic
155	C₂×#510	C₂×C₁₉	1	1	yes	characteristic
156	C₂²×Q₃₂	C₂×C₁₉	1	1	yes	characteristic, complemented
157	#3156	C₂×C₁₉	1	1	yes	characteristic
158	#2950	C₂×C₁₉	1	1	yes	characteristic, complemented
159	#2211	C₂×C₁₉	1	4	yes	complemented
160	#2209	C₂×C₁₉	1	4	yes	complemented
161	#2151	C₂×C₁₉	1	1	yes	characteristic, complemented
162	#2141	C₂×C₁₉	1	1	yes	characteristic, complemented
163	#2138	C₂×C₁₉	1	1	yes	characteristic
164	C₂²×C₁₉	—	2	2	no	—
165	C₂²×C₁₉	—	8	2	no	—
166	C₂²×C₁₉	—	8	1	no	—
167	C₂²×C₁₉	#459	1	2	yes	—
168	C₂²×C₁₉	C₂²×D₈	1	1	yes	characteristic
169	C₄×C₁₉	—	2	4	no	—
170	C₄×C₁₉	—	4	4	no	—
171	C₄×C₁₉	—	8	2	no	—
172	C₄×C₁₉	—	8	1	no	—
173	C₄×C₁₉	—	8	2	no	—
174	C₄×C₁₉	#459	1	2	yes	—
175	C₄×C₁₉	C₂×#121	1	1	yes	characteristic
176	C₄×C₁₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
177	C₂×#137	—	2	1	no	—
178	C₂×#135	C₂²×C₁₉	1	1	yes	characteristic
179	C₂²×Q₁₆	C₂²×C₁₉	1	1	yes	characteristic
180	C₂²×Q₁₆	C₂²×C₁₉	1	1	yes	characteristic
181	#536	—	2	1	no	—
182	#519	—	2	1	no	—
183	#510	—	2	4	no	—
184	C₂×Q₃₂	—	2	4	no	—
185	C₂×Q₃₂	C₂²×C₁₉	1	4	yes	—
186	C₂×Q₃₂	C₂²×C₁₉	1	2	yes	—
187	C₂×SD₃₂	C₂²×C₁₉	1	2	yes	—
188	C₂×M₃₂	C₂²×C₁₉	1	1	yes	characteristic
189	C₂²×C₁₆	C₂²×C₁₉	1	1	yes	characteristic
190	#497	—	2	2	no	—
191	#495	C₂²×C₁₉	1	2	yes	—
192	#494	C₂²×C₁₉	1	2	yes	—
193	#441	—	2	2	no	—
194	C₄×Q₁₆	C₂²×C₁₉	1	4	yes	—
195	#427	C₂²×C₁₉	1	1	yes	characteristic
196	#419	—	2	1	no	—
197	#405	C₂²×C₁₉	1	1	yes	characteristic
198	C₄⋊C₁₆	C₂²×C₁₉	1	4	yes	—
199	#358	C₂²×C₁₉	1	2	yes	—
200	#358	C₂²×C₁₉	1	4	yes	—
201	#357	C₂²×C₁₉	1	2	yes	—
202	C₂×C₁₉	—	2	2	no	—
203	C₂×C₁₉	—	16	1	no	—
204	C₂×C₁₉	C₂×#459	1	1	yes	characteristic
205	C₂×C₁₉	#3417	1	2	yes	—
206	C₂×#41	—	2	1	no	—
207	C₂²×Q₈	—	2	1	no	—
208	C₂²×Q₈	—	2	1	no	—
209	#137	—	4	4	no	—
210	#135	—	2	2	no	—
211	#135	—	2	2	no	—
212	C₂×Q₁₆	—	2	1	no	—
213	C₂×Q₁₆	—	2	2	no	—
214	C₂×Q₁₆	—	2	1	no	—
215	C₂×Q₁₆	—	2	2	no	—
216	C₂×Q₁₆	—	2	2	no	—
217	C₂×Q₁₆	—	2	2	no	—
218	C₂×Q₁₆	D₄×C₁₉	1	4	yes	—
219	C₂×Q₁₆	C₂³×C₁₉	1	2	yes	—
220	C₂×Q₁₆	C₂³×C₁₉	1	1	yes	characteristic
221	C₂×Q₁₆	C₂³×C₁₉	1	2	yes	—
222	C₂×SD₁₆	—	2	1	no	—
223	C₂×SD₁₆	—	2	2	no	—
224	C₂×D₈	C₂³×C₁₉	1	1	yes	characteristic
225	C₂×M₁₆	D₄×C₁₉	1	1	yes	characteristic
226	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
227	C₄⋊Q₈	—	2	2	no	—
228	#124	—	2	2	no	—
229	#123	—	4	2	no	—
230	C₂²⋊Q₈	—	4	2	no	—
231	C₄×Q₈	—	2	4	no	—
232	C₄×Q₈	—	4	2	no	—
233	#117	—	2	1	no	—
234	#117	—	4	2	no	—
235	#117	D₄×C₁₉	1	1	yes	characteristic
236	Q₃₂	—	2	8	no	—
237	Q₃₂	—	2	4	no	—
238	SD₃₂	—	2	4	no	—
239	M₃₂	—	2	2	no	—
240	C₂×C₁₆	—	2	2	no	—
241	C₂×C₁₆	C₂³×C₁₉	1	2	yes	—
242	C₂×C₁₆	C₂³×C₁₉	1	2	yes	—
243	#107	C₂³×C₁₉	1	2	yes	—
244	#106	—	2	1	no	—
245	#103	—	2	4	no	—
246	#97	D₄×C₁₉	1	2	yes	—
247	C₄×C₈	C₂³×C₁₉	1	2	yes	—
248	C₁₉	#33588	1	1	yes	characteristic, direct product factor, Sylow
249	#41	—	4	2	no	—
250	#41	—	4	2	no	—
251	C₂×Q₈	—	2	4	no	—
252	C₂×Q₈	—	2	2	no	—
253	C₂×Q₈	—	2	2	no	—
254	C₂×Q₈	—	2	1	no	—
255	C₂×Q₈	—	4	1	no	—
256	C₂×Q₈	—	4	1	no	—
257	C₂×Q₈	—	4	2	no	—
258	C₂×Q₈	—	4	2	no	—
259	C₂×Q₈	—	4	2	no	—
260	C₂×D₄	—	2	1	no	—
261	C₂×D₄	—	4	1	no	—
262	C₂²×C₄	—	4	1	no	—
263	C₂²×C₄	—	4	1	no	—
264	C₂²×C₄	—	4	1	no	—
265	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
266	Q₁₆	—	2	4	no	—
267	Q₁₆	—	2	2	no	—
268	Q₁₆	—	2	1	no	—
269	Q₁₆	—	2	2	no	—
270	Q₁₆	—	2	2	no	—
271	Q₁₆	—	2	1	no	—
272	Q₁₆	—	4	4	no	—
273	Q₁₆	—	4	2	no	—
274	Q₁₆	C₂×D₄×C₁₉	1	4	yes	—
275	SD₁₆	—	4	4	no	—
276	SD₁₆	—	4	2	no	—
277	D₈	—	2	1	no	—
278	D₈	—	2	1	no	—
279	M₁₆	—	2	2	no	—
280	C₂×C₈	C₂×D₄×C₁₉	1	2	yes	—
281	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
282	C₂×C₈	C₂×D₄×C₁₉	1	2	yes	—
283	C₂×C₈	#41×C₁₉	1	2	yes	—
284	C₂×C₈	C₂⁴×C₁₉	1	1	yes	characteristic
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	4	4	no	—
287	C₄⋊C₄	—	4	4	no	—
288	C₄⋊C₄	D₈×C₁₉	1	2	yes	—
289	C₂²⋊C₄	—	4	1	no	—
290	C₂²⋊C₄	—	4	4	no	—
291	C₂²⋊C₄	D₈×C₁₉	1	2	yes	—
292	C₄²	—	4	1	no	—
293	C₄²	—	4	4	no	—
294	C₄²	C₂×D₄×C₁₉	1	2	yes	—
295	C₁₆	—	2	2	no	—
296	C₁₆	—	2	2	no	—
297	C₂³	—	4	1	no	—
298	C₂³	C₂×D₈×C₁₉	1	1	yes	characteristic
299	Q₈	—	2	4	no	—
300	Q₈	—	4	4	no	—
301	Q₈	—	4	2	no	—
302	Q₈	—	4	1	no	—
303	Q₈	—	4	2	no	—
304	Q₈	—	4	2	no	—
305	Q₈	—	4	1	no	—
306	Q₈	—	8	2	no	—
307	D₄	—	4	1	no	—
308	D₄	—	4	1	no	—
309	D₄	—	8	2	no	—
310	C₂×C₄	—	4	4	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	8	2	no	—
316	C₂×C₄	—	8	2	no	—
317	C₂×C₄	C₁₉×#121	1	2	yes	—
318	C₂×C₄	C₂×D₈×C₁₉	1	4	yes	—
319	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
320	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	2	yes	—
321	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
322	C₈	—	2	2	no	—
323	C₈	C₁₉×#121	1	2	yes	—
324	C₈	C₂²×D₄×C₁₉	1	1	yes	characteristic
325	C₈	C₂×#41×C₁₉	1	1	yes	characteristic
326	C₂²	—	2	2	no	—
327	C₂²	—	8	1	no	—
328	C₂²	—	8	2	no	—
329	C₂²	C₁₉×#459	1	2	yes	—
330	C₂²	C₂²×D₈×C₁₉	1	1	yes	characteristic
331	C₄	—	2	4	no	—
332	C₄	—	4	4	no	—
333	C₄	—	8	1	no	—
334	C₄	—	8	2	no	—
335	C₄	—	8	2	no	—
336	C₄	C₁₉×#459	1	2	yes	—
337	C₄	C₂×C₁₉×#121	1	1	yes	characteristic
338	C₄	C₂×C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
339	C₂	—	2	2	no	—
340	C₂	—	16	1	no	—
341	C₂	C₂×C₁₉×#459	1	1	yes	characteristic
342	C₂	C₁₉×#3417	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₉
prime radical	C₂×C₁₉×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₁₉×#117 ◁ C₁₉×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₁₉×Q₃₂	C₂	semidirect
4. C₁₉×#2950	C₂	semidirect
5. C₁₉×#2211	C₂	semidirect
6. C₁₉×#2209	C₂	semidirect
7. C₁₉×#2151	C₂	semidirect
8. C₁₉×#2141	C₂	semidirect
125. #33588	C₁₉	direct (N × H)
156. C₂²×Q₃₂	C₂×C₁₉	semidirect
158. #2950	C₂×C₁₉	semidirect
159. #2211	C₂×C₁₉	semidirect
160. #2209	C₂×C₁₉	semidirect
161. #2151	C₂×C₁₉	semidirect
162. #2141	C₂×C₁₉	semidirect
248. C₁₉	#33588	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×#510	2
C₂²×C₁₉×Q₃₂	2
C₁₉×#3156	2
C₁₉×#2950	2
C₁₉×#2211	2
C₁₉×#2209	2
C₁₉×#2151	2
C₁₉×#2141	2
C₁₉×#2138	2
#33588	19

Subgroups — C19×#33588 #18039547

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#33588 #18039547