Subgroups — C₁₉×#32885 #18038844

Order 4864.

3650 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₉×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₉×#449	C₂	1	3	yes	maximal, complemented
3	C₂×C₁₉×#448	C₂	1	3	yes	maximal, complemented
4	C₁₉×#2615	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#1788	C₂	1	3	yes	maximal, complemented
6	C₁₉×#1787	C₂	1	3	yes	maximal, complemented
7	C₁₉×#1780	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₉×2₋¹⁺⁴	—	2	12	no	—
9	C₂×C₁₉×2₋¹⁺⁴	—	2	12	no	—
10	C₂²×#41×C₁₉	—	2	3	no	—
11	C₂²×#41×C₁₉	C₂²	1	3	yes	—
12	C₂²×#41×C₁₉	C₂²	1	1	yes	characteristic
13	C₂³×Q₈×C₁₉	—	2	1	no	—
14	C₂³×Q₈×C₁₉	C₂²	1	3	yes	—
15	C₂²×SD₁₆×C₁₉	C₂²	1	3	yes	—
16	C₂²×D₈×C₁₉	C₂²	1	1	yes	characteristic
17	C₂×C₁₉×#121	C₂²	1	3	yes	—
18	C₁₉×#449	—	2	12	no	—
19	C₁₉×#448	—	2	12	no	—
20	C₂×C₁₉×#103	C₂²	1	6	yes	—
21	C₂×C₁₉×#103	C₂²	1	3	yes	—
22	C₂×C₁₉×#102	C₂²	1	3	yes	—
23	C₂×C₁₉×C₂²⋊C₈	C₂²	1	3	yes	—
24	C₂×C₁₉×C₂²⋊C₈	C₂²	1	3	yes	—
25	C₁₉×#393	—	2	1	no	—
26	C₁₉×#392	—	2	3	no	—
27	C₁₉×#380	C₂²	1	3	yes	—
28	C₁₉×2₋¹⁺⁴	—	2	32	no	—
29	C₁₉×2₋¹⁺⁴	—	2	96	no	—
30	C₂×#41×C₁₉	—	2	48	no	—
31	C₂×#41×C₁₉	—	2	24	no	—
32	C₂×#41×C₁₉	—	2	12	no	—
33	C₂×#41×C₁₉	—	2	12	no	—
34	C₂×#41×C₁₉	—	2	6	no	—
35	C₂×#41×C₁₉	—	2	6	no	—
36	C₂×#41×C₁₉	—	2	6	no	—
37	C₂×#41×C₁₉	D₄	1	12	yes	complemented
38	C₂²×Q₈×C₁₉	—	2	24	no	—
39	C₂²×Q₈×C₁₉	—	2	12	no	—
40	C₂²×Q₈×C₁₉	—	2	12	no	—
41	C₂²×Q₈×C₁₉	—	2	3	no	—
42	C₂²×Q₈×C₁₉	—	2	6	no	—
43	C₂²×Q₈×C₁₉	D₄	1	1	yes	characteristic
44	C₂²×Q₈×C₁₉	D₄	1	12	yes	complemented
45	C₂²×Q₈×C₁₉	C₂³	1	3	yes	—
46	C₂²×D₄×C₁₉	—	2	3	no	—
47	C₂²×D₄×C₁₉	—	2	3	no	—
48	C₂²×D₄×C₁₉	D₄	1	3	yes	—
49	C₂²×D₄×C₁₉	C₂³	1	1	yes	characteristic
50	C₂²×D₄×C₁₉	C₂³	1	1	yes	characteristic
51	C₂³×C₄×C₁₉	—	2	3	no	—
52	C₂³×C₄×C₁₉	—	2	3	no	—
53	C₂³×C₄×C₁₉	C₂³	1	3	yes	—
54	C₂×SD₁₆×C₁₉	—	2	12	no	—
55	C₂×SD₁₆×C₁₉	—	2	6	no	—
56	C₂×D₈×C₁₉	—	2	6	no	—
57	C₂²×C₈×C₁₉	C₂³	1	3	yes	—
58	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
59	C₁₉×#121	—	2	6	no	—
60	C₁₉×#121	—	2	6	no	—
61	C₂×C₄⋊C₄×C₁₉	C₂³	1	3	yes	—
62	C₂×C₂²⋊C₄×C₁₉	—	2	3	no	—
63	C₂×C₂²⋊C₄×C₁₉	—	2	3	no	—
64	C₁₉×#103	—	2	12	no	—
65	C₁₉×#103	—	2	6	no	—
66	C₁₉×#102	—	2	6	no	—
67	C₁₉×C₂²⋊C₈	—	2	6	no	—
68	C₁₉×C₂²⋊C₈	—	2	6	no	—
69	C₁₉×#95	—	2	3	no	—
70	C₂⁴×C₁₉	—	2	1	no	—
71	C₂⁴×C₁₉	—	2	1	no	—
72	C₂⁴×C₁₉	C₂×D₄	1	3	yes	—
73	#41×C₁₉	—	2	192	no	—
74	#41×C₁₉	—	2	48	no	—
75	#41×C₁₉	—	2	24	no	—
76	#41×C₁₉	—	2	8	no	—
77	#41×C₁₉	—	2	24	no	—
78	#41×C₁₉	—	2	24	no	—
79	C₂×Q₈×C₁₉	—	2	24	no	—
80	C₂×Q₈×C₁₉	—	2	64	no	—
81	C₂×Q₈×C₁₉	—	2	24	no	—
82	C₂×Q₈×C₁₉	—	2	24	no	—
83	C₂×Q₈×C₁₉	—	2	48	no	—
84	C₂×Q₈×C₁₉	—	2	6	no	—
85	C₂×Q₈×C₁₉	—	2	12	no	—
86	C₂×Q₈×C₁₉	—	2	3	no	—
87	C₂×Q₈×C₁₉	—	2	6	no	—
88	C₂×Q₈×C₁₉	C₂×D₄	1	6	yes	—
89	C₂×Q₈×C₁₉	C₂×D₄	1	12	yes	—
90	C₂×D₄×C₁₉	—	2	24	no	—
91	C₂×D₄×C₁₉	—	2	12	no	—
92	C₂×D₄×C₁₉	—	2	12	no	—
93	C₂×D₄×C₁₉	—	2	6	no	—
94	C₂×D₄×C₁₉	—	2	3	no	—
95	C₂×D₄×C₁₉	—	2	3	no	—
96	C₂×D₄×C₁₉	—	2	3	no	—
97	C₂×D₄×C₁₉	—	4	6	no	—
98	C₂×D₄×C₁₉	—	8	6	no	—
99	C₂×D₄×C₁₉	C₂×D₄	1	6	yes	—
100	C₂²×C₄×C₁₉	—	2	24	no	—
101	C₂²×C₄×C₁₉	—	2	24	no	—
102	C₂²×C₄×C₁₉	—	2	12	no	—
103	C₂²×C₄×C₁₉	—	2	3	no	—
104	C₂²×C₄×C₁₉	—	2	12	no	—
105	C₂²×C₄×C₁₉	—	2	6	no	—
106	C₂²×C₄×C₁₉	—	2	6	no	—
107	C₂²×C₄×C₁₉	—	2	3	no	—
108	C₂²×C₄×C₁₉	C₂×D₄	1	3	yes	—
109	C₂²×C₄×C₁₉	C₂×D₄	1	6	yes	—
110	C₂²×C₄×C₁₉	C₂×D₄	1	6	yes	—
111	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
112	SD₁₆×C₁₉	—	4	12	no	—
113	D₈×C₁₉	—	4	4	no	—
114	C₂×C₈×C₁₉	—	2	3	no	—
115	C₂×C₈×C₁₉	—	2	3	no	—
116	C₂×C₈×C₁₉	—	2	6	no	—
117	C₄⋊C₄×C₁₉	—	2	3	no	—
118	C₄⋊C₄×C₁₉	—	2	3	no	—
119	C₂²⋊C₄×C₁₉	—	4	6	no	—
120	C₂²⋊C₄×C₁₉	—	8	3	no	—
121	#32885	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₂³×C₁₉	—	2	6	no	—
123	C₂³×C₁₉	—	2	6	no	—
124	C₂³×C₁₉	—	2	3	no	—
125	C₂³×C₁₉	—	2	6	no	—
126	C₂³×C₁₉	—	2	3	no	—
127	C₂³×C₁₉	—	4	3	no	—
128	C₂³×C₁₉	—	4	1	no	—
129	C₂³×C₁₉	—	4	3	no	—
130	C₂³×C₁₉	—	4	3	no	—
131	C₂³×C₁₉	—	4	1	no	—
132	C₂³×C₁₉	#120	1	6	yes	—
133	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
134	Q₈×C₁₉	—	2	48	no	—
135	Q₈×C₁₉	—	2	8	no	—
136	Q₈×C₁₉	—	2	12	no	—
137	Q₈×C₁₉	—	2	6	no	—
138	Q₈×C₁₉	#120	1	12	yes	—
139	D₄×C₁₉	—	2	48	no	—
140	D₄×C₁₉	—	2	24	no	—
141	D₄×C₁₉	—	2	6	no	—
142	D₄×C₁₉	—	4	1	no	—
143	D₄×C₁₉	—	4	3	no	—
144	D₄×C₁₉	—	8	12	no	—
145	D₄×C₁₉	#120	1	4	yes	—
146	C₂×C₄×C₁₉	—	2	48	no	—
147	C₂×C₄×C₁₉	—	2	48	no	—
148	C₂×C₄×C₁₉	—	2	12	no	—
149	C₂×C₄×C₁₉	—	2	6	no	—
150	C₂×C₄×C₁₉	—	2	6	no	—
151	C₂×C₄×C₁₉	—	2	6	no	—
152	C₂×C₄×C₁₉	—	2	12	no	—
153	C₂×C₄×C₁₉	—	2	6	no	—
154	C₂×C₄×C₁₉	—	4	3	no	—
155	C₂×C₄×C₁₉	—	4	6	no	—
156	C₂×C₄×C₁₉	#120	1	6	yes	—
157	C₂×C₄×C₁₉	C₂²×D₄	1	3	yes	—
158	C₂×C₄×C₁₉	C₂²×D₄	1	3	yes	—
159	C₈×C₁₉	—	4	1	no	—
160	C₈×C₁₉	—	4	3	no	—
161	C₂²×2₋¹⁺⁴	C₂×C₁₉	1	1	yes	characteristic, complemented
162	C₂×#449	C₂×C₁₉	1	3	yes	complemented
163	C₂×#448	C₂×C₁₉	1	3	yes	complemented
164	#2615	C₂×C₁₉	1	1	yes	characteristic, complemented
165	#1788	C₂×C₁₉	1	3	yes	complemented
166	#1787	C₂×C₁₉	1	3	yes	complemented
167	#1780	C₂×C₁₉	1	1	yes	characteristic, complemented
168	C₂²×C₁₉	—	2	6	no	—
169	C₂²×C₁₉	—	2	4	no	—
170	C₂²×C₁₉	—	2	6	no	—
171	C₂²×C₁₉	—	2	6	no	—
172	C₂²×C₁₉	—	4	6	no	—
173	C₂²×C₁₉	—	4	6	no	—
174	C₂²×C₁₉	—	4	6	no	—
175	C₂²×C₁₉	—	4	6	no	—
176	C₂²×C₁₉	—	8	1	no	—
177	C₂²×C₁₉	—	8	3	no	—
178	C₂²×C₁₉	—	8	3	no	—
179	C₂²×C₁₉	C₂×#120	1	3	yes	—
180	C₂²×C₁₉	#577	1	3	yes	—
181	C₂²×C₁₉	#578	1	1	yes	characteristic
182	C₄×C₁₉	—	2	12	no	—
183	C₄×C₁₉	—	2	6	no	—
184	C₄×C₁₉	—	8	3	no	—
185	C₄×C₁₉	C₂×#120	1	3	yes	—
186	C₄×C₁₉	C₂×#120	1	1	yes	characteristic
187	C₂×2₋¹⁺⁴	—	2	12	no	—
188	C₂×2₋¹⁺⁴	—	2	12	no	—
189	C₂²×#41	—	2	3	no	—
190	C₂²×#41	C₂²×C₁₉	1	3	yes	—
191	C₂²×#41	C₂²×C₁₉	1	1	yes	characteristic
192	C₂³×Q₈	—	2	1	no	—
193	C₂³×Q₈	C₂²×C₁₉	1	3	yes	—
194	C₂²×SD₁₆	C₂²×C₁₉	1	3	yes	—
195	C₂²×D₈	C₂²×C₁₉	1	1	yes	characteristic
196	C₂×#121	C₂²×C₁₉	1	3	yes	—
197	#449	—	2	12	no	—
198	#448	—	2	12	no	—
199	C₂×#103	C₂²×C₁₉	1	6	yes	—
200	C₂×#103	C₂²×C₁₉	1	3	yes	—
201	C₂×#102	C₂²×C₁₉	1	3	yes	—
202	C₂×C₂²⋊C₈	C₂²×C₁₉	1	3	yes	—
203	C₂×C₂²⋊C₈	C₂²×C₁₉	1	3	yes	—
204	#393	—	2	1	no	—
205	#392	—	2	3	no	—
206	#380	C₂²×C₁₉	1	3	yes	—
207	C₂×C₁₉	—	4	6	no	—
208	C₂×C₁₉	—	4	4	no	—
209	C₂×C₁₉	—	16	1	no	—
210	C₂×C₁₉	#2846	1	1	yes	characteristic
211	C₂×C₁₉	#3010	1	3	yes	—
212	C₂×C₁₉	#3013	1	3	yes	—
213	2₋¹⁺⁴	—	2	32	no	—
214	2₋¹⁺⁴	—	2	96	no	—
215	C₂×#41	—	2	48	no	—
216	C₂×#41	—	2	24	no	—
217	C₂×#41	—	2	12	no	—
218	C₂×#41	—	2	12	no	—
219	C₂×#41	—	2	6	no	—
220	C₂×#41	—	2	6	no	—
221	C₂×#41	—	2	6	no	—
222	C₂×#41	D₄×C₁₉	1	12	yes	complemented
223	C₂²×Q₈	—	2	24	no	—
224	C₂²×Q₈	—	2	12	no	—
225	C₂²×Q₈	—	2	12	no	—
226	C₂²×Q₈	—	2	3	no	—
227	C₂²×Q₈	—	2	6	no	—
228	C₂²×Q₈	D₄×C₁₉	1	1	yes	characteristic
229	C₂²×Q₈	D₄×C₁₉	1	12	yes	complemented
230	C₂²×Q₈	C₂³×C₁₉	1	3	yes	—
231	C₂²×D₄	—	2	3	no	—
232	C₂²×D₄	—	2	3	no	—
233	C₂²×D₄	D₄×C₁₉	1	3	yes	—
234	C₂²×D₄	C₂³×C₁₉	1	1	yes	characteristic
235	C₂²×D₄	C₂³×C₁₉	1	1	yes	characteristic
236	C₂³×C₄	—	2	3	no	—
237	C₂³×C₄	—	2	3	no	—
238	C₂³×C₄	C₂³×C₁₉	1	3	yes	—
239	C₂×SD₁₆	—	2	12	no	—
240	C₂×SD₁₆	—	2	6	no	—
241	C₂×D₈	—	2	6	no	—
242	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
243	C₂²×C₈	C₂³×C₁₉	1	3	yes	—
244	#121	—	2	6	no	—
245	#121	—	2	6	no	—
246	C₂×C₄⋊C₄	C₂³×C₁₉	1	3	yes	—
247	C₂×C₂²⋊C₄	—	2	3	no	—
248	C₂×C₂²⋊C₄	—	2	3	no	—
249	#103	—	2	12	no	—
250	#103	—	2	6	no	—
251	#102	—	2	6	no	—
252	C₂²⋊C₈	—	2	6	no	—
253	C₂²⋊C₈	—	2	6	no	—
254	#95	—	2	3	no	—
255	C₁₉	#32885	1	1	yes	characteristic, direct product factor, Sylow
256	C₂⁴	—	2	1	no	—
257	C₂⁴	—	2	1	no	—
258	C₂⁴	C₂×D₄×C₁₉	1	3	yes	—
259	#41	—	2	192	no	—
260	#41	—	2	48	no	—
261	#41	—	2	24	no	—
262	#41	—	2	8	no	—
263	#41	—	2	24	no	—
264	#41	—	2	24	no	—
265	C₂×Q₈	—	2	24	no	—
266	C₂×Q₈	—	2	64	no	—
267	C₂×Q₈	—	2	24	no	—
268	C₂×Q₈	—	2	24	no	—
269	C₂×Q₈	—	2	48	no	—
270	C₂×Q₈	—	2	6	no	—
271	C₂×Q₈	—	2	12	no	—
272	C₂×Q₈	—	2	3	no	—
273	C₂×Q₈	—	2	6	no	—
274	C₂×Q₈	C₂×D₄×C₁₉	1	6	yes	—
275	C₂×Q₈	C₂×D₄×C₁₉	1	12	yes	—
276	C₂×D₄	—	2	24	no	—
277	C₂×D₄	—	2	12	no	—
278	C₂×D₄	—	2	12	no	—
279	C₂×D₄	—	2	6	no	—
280	C₂×D₄	—	2	3	no	—
281	C₂×D₄	—	2	3	no	—
282	C₂×D₄	—	2	3	no	—
283	C₂×D₄	—	4	6	no	—
284	C₂×D₄	—	8	6	no	—
285	C₂×D₄	C₂×D₄×C₁₉	1	6	yes	—
286	C₂²×C₄	—	2	24	no	—
287	C₂²×C₄	—	2	24	no	—
288	C₂²×C₄	—	2	12	no	—
289	C₂²×C₄	—	2	3	no	—
290	C₂²×C₄	—	2	12	no	—
291	C₂²×C₄	—	2	6	no	—
292	C₂²×C₄	—	2	6	no	—
293	C₂²×C₄	—	2	3	no	—
294	C₂²×C₄	C₂×D₄×C₁₉	1	3	yes	—
295	C₂²×C₄	C₂×D₄×C₁₉	1	6	yes	—
296	C₂²×C₄	C₂×D₄×C₁₉	1	6	yes	—
297	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
298	SD₁₆	—	4	12	no	—
299	D₈	—	4	4	no	—
300	C₂×C₈	—	2	3	no	—
301	C₂×C₈	—	2	3	no	—
302	C₂×C₈	—	2	6	no	—
303	C₄⋊C₄	—	2	3	no	—
304	C₄⋊C₄	—	2	3	no	—
305	C₂²⋊C₄	—	4	6	no	—
306	C₂²⋊C₄	—	8	3	no	—
307	C₂³	—	2	6	no	—
308	C₂³	—	2	6	no	—
309	C₂³	—	2	3	no	—
310	C₂³	—	2	6	no	—
311	C₂³	—	2	3	no	—
312	C₂³	—	4	3	no	—
313	C₂³	—	4	1	no	—
314	C₂³	—	4	3	no	—
315	C₂³	—	4	3	no	—
316	C₂³	—	4	1	no	—
317	C₂³	C₁₉×#120	1	6	yes	—
318	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
319	Q₈	—	2	48	no	—
320	Q₈	—	2	8	no	—
321	Q₈	—	2	12	no	—
322	Q₈	—	2	6	no	—
323	Q₈	C₁₉×#120	1	12	yes	—
324	D₄	—	2	48	no	—
325	D₄	—	2	24	no	—
326	D₄	—	2	6	no	—
327	D₄	—	4	1	no	—
328	D₄	—	4	3	no	—
329	D₄	—	8	12	no	—
330	D₄	C₁₉×#120	1	4	yes	—
331	C₂×C₄	—	2	48	no	—
332	C₂×C₄	—	2	48	no	—
333	C₂×C₄	—	2	12	no	—
334	C₂×C₄	—	2	6	no	—
335	C₂×C₄	—	2	6	no	—
336	C₂×C₄	—	2	6	no	—
337	C₂×C₄	—	2	12	no	—
338	C₂×C₄	—	2	6	no	—
339	C₂×C₄	—	4	3	no	—
340	C₂×C₄	—	4	6	no	—
341	C₂×C₄	C₁₉×#120	1	6	yes	—
342	C₂×C₄	C₂²×D₄×C₁₉	1	3	yes	—
343	C₂×C₄	C₂²×D₄×C₁₉	1	3	yes	—
344	C₈	—	4	1	no	—
345	C₈	—	4	3	no	—
346	C₂²	—	2	6	no	—
347	C₂²	—	2	6	no	—
348	C₂²	—	2	6	no	—
349	C₂²	—	2	4	no	—
350	C₂²	—	4	6	no	—
351	C₂²	—	4	6	no	—
352	C₂²	—	4	6	no	—
353	C₂²	—	4	6	no	—
354	C₂²	—	8	3	no	—
355	C₂²	—	8	3	no	—
356	C₂²	—	8	1	no	—
357	C₂²	C₂×C₁₉×#120	1	3	yes	—
358	C₂²	C₁₉×#577	1	3	yes	—
359	C₂²	C₁₉×#578	1	1	yes	characteristic
360	C₄	—	2	12	no	—
361	C₄	—	2	6	no	—
362	C₄	—	8	3	no	—
363	C₄	C₂×C₁₉×#120	1	3	yes	—
364	C₄	C₂×C₁₉×#120	1	1	yes	characteristic
365	C₂	—	4	6	no	—
366	C₂	—	4	4	no	—
367	C₂	—	16	1	no	—
368	C₂	C₁₉×#2846	1	1	yes	characteristic
369	C₂	C₁₉×#3010	1	3	yes	—
370	C₂	C₁₉×#3013	1	3	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ C₂²×#41×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₁₉×2₋¹⁺⁴	C₂	semidirect
2. C₂×C₁₉×#449	C₂	semidirect
3. C₂×C₁₉×#448	C₂	semidirect
4. C₁₉×#2615	C₂	semidirect
5. C₁₉×#1788	C₂	semidirect
6. C₁₉×#1787	C₂	semidirect
7. C₁₉×#1780	C₂	semidirect
37. C₂×#41×C₁₉	D₄	semidirect
44. C₂²×Q₈×C₁₉	D₄	semidirect
121. #32885	C₁₉	direct (N × H)
161. C₂²×2₋¹⁺⁴	C₂×C₁₉	semidirect
162. C₂×#449	C₂×C₁₉	semidirect
163. C₂×#448	C₂×C₁₉	semidirect
164. #2615	C₂×C₁₉	semidirect
165. #1788	C₂×C₁₉	semidirect
166. #1787	C₂×C₁₉	semidirect
167. #1780	C₂×C₁₉	semidirect
222. C₂×#41	D₄×C₁₉	semidirect
229. C₂²×Q₈	D₄×C₁₉	semidirect
255. C₁₉	#32885	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×C₁₉×2₋¹⁺⁴	2
C₂×C₁₉×#449	2
C₂×C₁₉×#448	2
C₁₉×#2615	2
C₁₉×#1788	2
C₁₉×#1787	2
C₁₉×#1780	2
#32885	19

Subgroups — C19×#32885 #18038844

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#32885 #18038844