Subgroups — C₂×C₁₉×#1657 #18024941

Order 4864.

1010 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#563	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₉×#496	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₉×#490	C₂	1	2	yes	maximal, complemented
4	C₁₉×#1657	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₁₉×#330	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₉×#329	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×SD₁₆×C₁₉	—	2	1	no	—
8	C₂²×D₈×C₁₉	—	2	1	no	—
9	C₁₉×#563	—	2	4	no	—
10	C₁₉×#563	—	2	2	no	—
11	C₁₉×#563	C₂²	1	4	yes	complemented
12	C₂×C₁₉×C₄⋊Q₈	—	2	1	no	—
13	C₂×C₁₉×#127	C₂²	1	1	yes	characteristic
14	C₂×C₁₉×#126	—	2	2	no	—
15	C₂×C₁₉×#125	C₂²	1	2	yes	—
16	C₂×C₁₉×#124	C₂²	1	1	yes	characteristic
17	C₂×C₁₉×#123	—	2	1	no	—
18	C₂×C₁₉×C₂²⋊Q₈	—	2	2	no	—
19	C₁₉×#496	—	2	2	no	—
20	C₁₉×#496	C₂²	1	4	yes	—
21	C₁₉×#490	—	2	4	no	—
22	C₁₉×#490	C₂²	1	8	yes	complemented
23	C₂×C₁₉×#102	—	2	2	no	—
24	C₂×C₁₉×#102	—	2	2	no	—
25	C₂×C₁₉×#97	C₂²	1	2	yes	—
26	C₂×C₁₉×#97	C₂²	1	1	yes	characteristic
27	C₁₉×#330	C₂²	1	8	yes	complemented
28	C₁₉×#329	C₂²	1	4	yes	complemented
29	C₂²×Q₈×C₁₉	—	2	1	no	—
30	C₂²×Q₈×C₁₉	D₄	1	1	yes	characteristic
31	C₂²×D₄×C₁₉	—	2	2	no	—
32	C₂²×D₄×C₁₉	—	2	1	no	—
33	C₂²×D₄×C₁₉	D₄	1	1	yes	characteristic
34	C₂³×C₄×C₁₉	—	2	1	no	—
35	C₂×SD₁₆×C₁₉	—	2	4	no	—
36	C₂×SD₁₆×C₁₉	—	4	4	no	—
37	C₂×D₈×C₁₉	—	2	4	no	—
38	C₂×D₈×C₁₉	—	4	4	no	—
39	C₂²×C₈×C₁₉	—	2	1	no	—
40	C₂²×C₈×C₁₉	—	2	2	no	—
41	C₁₉×C₄⋊Q₈	—	2	4	no	—
42	C₁₉×C₄⋊Q₈	—	2	4	no	—
43	C₁₉×#127	—	2	2	no	—
44	C₁₉×#127	—	2	1	no	—
45	C₁₉×#127	C₂³	1	2	yes	—
46	C₁₉×#126	—	2	8	no	—
47	C₁₉×#126	—	2	8	no	—
48	C₁₉×#125	—	2	2	no	—
49	C₁₉×#125	—	2	2	no	—
50	C₁₉×#125	D₄	1	4	yes	—
51	C₁₉×#125	C₂³	1	4	yes	—
52	C₁₉×#124	—	2	2	no	—
53	C₁₉×#124	D₄	1	2	yes	—
54	C₁₉×#124	C₂³	1	2	yes	—
55	C₁₉×#123	—	2	4	no	—
56	C₁₉×#123	—	2	4	no	—
57	C₁₉×C₂²⋊Q₈	—	2	8	no	—
58	C₁₉×C₂²⋊Q₈	—	2	8	no	—
59	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
60	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
61	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
62	C₂×C₄⋊C₄×C₁₉	D₄	1	2	yes	—
63	C₂×C₄⋊C₄×C₁₉	D₄	1	2	yes	—
64	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
65	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
66	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
67	C₂×C₄²×C₁₉	—	2	1	no	—
68	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
69	C₁₉×#102	—	2	8	no	—
70	C₁₉×#102	—	2	8	no	—
71	C₁₉×#97	—	2	2	no	—
72	C₁₉×#97	—	2	1	no	—
73	C₁₉×#97	C₂³	1	4	yes	—
74	C₁₉×#97	C₂³	1	2	yes	—
75	C₂⁴×C₁₉	—	2	1	no	—
76	C₂⁴×C₁₉	—	4	1	no	—
77	C₂×Q₈×C₁₉	—	2	4	no	—
78	C₂×Q₈×C₁₉	—	2	1	no	—
79	C₂×Q₈×C₁₉	—	4	2	no	—
80	C₂×Q₈×C₁₉	—	8	2	no	—
81	C₂×Q₈×C₁₉	C₂×D₄	1	2	yes	—
82	C₂×D₄×C₁₉	—	2	1	no	—
83	C₂×D₄×C₁₉	—	2	2	no	—
84	C₂×D₄×C₁₉	—	2	4	no	—
85	C₂×D₄×C₁₉	—	4	2	no	—
86	C₂×D₄×C₁₉	—	4	2	no	—
87	C₂×D₄×C₁₉	—	4	2	no	—
88	C₂×D₄×C₁₉	—	4	8	no	—
89	C₂×D₄×C₁₉	—	4	2	no	—
90	C₂×D₄×C₁₉	—	4	1	no	—
91	C₂×D₄×C₁₉	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₁₉	—	2	2	no	—
93	C₂²×C₄×C₁₉	—	2	1	no	—
94	C₂²×C₄×C₁₉	—	2	2	no	—
95	C₂²×C₄×C₁₉	—	2	4	no	—
96	C₂²×C₄×C₁₉	—	8	2	no	—
97	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
98	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
99	SD₁₆×C₁₉	—	4	8	no	—
100	D₈×C₁₉	—	4	8	no	—
101	C₂×C₈×C₁₉	—	2	4	no	—
102	C₂×C₈×C₁₉	—	2	2	no	—
103	C₂×C₈×C₁₉	—	2	4	no	—
104	C₂×C₈×C₁₉	—	4	4	no	—
105	C₄⋊C₄×C₁₉	—	2	8	no	—
106	C₄⋊C₄×C₁₉	—	2	4	no	—
107	C₄⋊C₄×C₁₉	—	2	4	no	—
108	C₄⋊C₄×C₁₉	—	2	2	no	—
109	C₄⋊C₄×C₁₉	—	2	2	no	—
110	C₄⋊C₄×C₁₉	—	2	4	no	—
111	C₄⋊C₄×C₁₉	—	2	4	no	—
112	C₄⋊C₄×C₁₉	C₂×D₄	1	4	yes	—
113	C₄⋊C₄×C₁₉	C₂×D₄	1	4	yes	—
114	C₂²⋊C₄×C₁₉	—	2	8	no	—
115	C₂²⋊C₄×C₁₉	—	2	4	no	—
116	C₂²⋊C₄×C₁₉	—	2	4	no	—
117	C₂²⋊C₄×C₁₉	—	2	4	no	—
118	C₄²×C₁₉	—	2	4	no	—
119	C₄²×C₁₉	C₂×D₄	1	2	yes	—
120	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₁₉	C₂⁴	1	1	yes	characteristic
122	C₂×#1657	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₂³×C₁₉	—	2	2	no	—
124	C₂³×C₁₉	—	4	2	no	—
125	C₂³×C₁₉	—	4	4	no	—
126	C₂³×C₁₉	—	4	2	no	—
127	C₂³×C₁₉	—	8	2	no	—
128	C₂³×C₁₉	—	8	4	no	—
129	C₂³×C₁₉	#120	1	1	yes	characteristic
130	Q₈×C₁₉	—	4	2	no	—
131	Q₈×C₁₉	—	8	1	no	—
132	Q₈×C₁₉	—	8	4	no	—
133	D₄×C₁₉	—	4	2	no	—
134	D₄×C₁₉	—	4	2	no	—
135	D₄×C₁₉	—	4	2	no	—
136	D₄×C₁₉	—	4	8	no	—
137	D₄×C₁₉	—	8	1	no	—
138	D₄×C₁₉	—	8	2	no	—
139	D₄×C₁₉	—	8	4	no	—
140	C₂×C₄×C₁₉	—	2	2	no	—
141	C₂×C₄×C₁₉	—	2	4	no	—
142	C₂×C₄×C₁₉	—	2	4	no	—
143	C₂×C₄×C₁₉	—	2	2	no	—
144	C₂×C₄×C₁₉	—	2	4	no	—
145	C₂×C₄×C₁₉	—	4	4	no	—
146	C₂×C₄×C₁₉	—	4	2	no	—
147	C₂×C₄×C₁₉	—	4	4	no	—
148	C₂×C₄×C₁₉	—	8	4	no	—
149	C₂×C₄×C₁₉	#120	1	2	yes	—
150	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
151	C₂×C₄×C₁₉	C₂²×D₄	1	2	yes	—
152	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
153	C₈×C₁₉	—	2	4	no	—
154	C₈×C₁₉	—	4	4	no	—
155	C₂×#563	C₂×C₁₉	1	1	yes	characteristic, complemented
156	C₂×#496	C₂×C₁₉	1	1	yes	characteristic
157	C₂×#490	C₂×C₁₉	1	2	yes	complemented
158	#1657	C₂×C₁₉	1	8	yes	direct product factor
159	C₂×#330	C₂×C₁₉	1	2	yes	complemented
160	C₂×#329	C₂×C₁₉	1	1	yes	characteristic, complemented
161	C₂²×C₁₉	—	4	2	no	—
162	C₂²×C₁₉	—	4	4	no	—
163	C₂²×C₁₉	—	8	4	no	—
164	C₂²×C₁₉	—	8	2	no	—
165	C₂²×C₁₉	—	8	4	no	—
166	C₂²×C₁₉	—	16	4	no	—
167	C₂²×C₁₉	#453	1	2	yes	—
168	C₂²×C₁₉	#454	1	4	yes	—
169	C₂²×C₁₉	C₂×#120	1	1	yes	characteristic
170	C₄×C₁₉	—	2	1	no	—
171	C₄×C₁₉	—	2	1	no	—
172	C₄×C₁₉	—	2	2	no	—
173	C₄×C₁₉	—	2	2	no	—
174	C₄×C₁₉	—	4	4	no	—
175	C₄×C₁₉	—	4	2	no	—
176	C₄×C₁₉	—	4	4	no	—
177	C₂²×SD₁₆	—	2	1	no	—
178	C₂²×D₈	—	2	1	no	—
179	#563	—	2	4	no	—
180	#563	—	2	2	no	—
181	#563	C₂²×C₁₉	1	4	yes	complemented
182	C₂×C₄⋊Q₈	—	2	1	no	—
183	C₂×#127	C₂²×C₁₉	1	1	yes	characteristic
184	C₂×#126	—	2	2	no	—
185	C₂×#125	C₂²×C₁₉	1	2	yes	—
186	C₂×#124	C₂²×C₁₉	1	1	yes	characteristic
187	C₂×#123	—	2	1	no	—
188	C₂×C₂²⋊Q₈	—	2	2	no	—
189	#496	—	2	2	no	—
190	#496	C₂²×C₁₉	1	4	yes	—
191	#490	—	2	4	no	—
192	#490	C₂²×C₁₉	1	8	yes	complemented
193	C₂×#102	—	2	2	no	—
194	C₂×#102	—	2	2	no	—
195	C₂×#97	C₂²×C₁₉	1	2	yes	—
196	C₂×#97	C₂²×C₁₉	1	1	yes	characteristic
197	#330	C₂²×C₁₉	1	8	yes	complemented
198	#329	C₂²×C₁₉	1	4	yes	complemented
199	C₂×C₁₉	—	8	2	no	—
200	C₂×C₁₉	—	16	2	no	—
201	C₂×C₁₉	#1657	1	4	yes	direct product factor
202	C₂×C₁₉	C₂×#453	1	1	yes	characteristic
203	C₂×C₁₉	C₂×#454	1	2	yes	—
204	C₂²×Q₈	—	2	1	no	—
205	C₂²×Q₈	D₄×C₁₉	1	1	yes	characteristic
206	C₂²×D₄	—	2	1	no	—
207	C₂²×D₄	—	2	2	no	—
208	C₂²×D₄	D₄×C₁₉	1	1	yes	characteristic
209	C₂³×C₄	—	2	1	no	—
210	C₂×SD₁₆	—	2	4	no	—
211	C₂×SD₁₆	—	4	4	no	—
212	C₂×D₈	—	2	4	no	—
213	C₂×D₈	—	4	4	no	—
214	C₂²×C₈	—	2	1	no	—
215	C₂²×C₈	—	2	2	no	—
216	C₄⋊Q₈	—	2	4	no	—
217	C₄⋊Q₈	—	2	4	no	—
218	#127	—	2	2	no	—
219	#127	—	2	1	no	—
220	#127	C₂³×C₁₉	1	2	yes	—
221	#126	—	2	8	no	—
222	#126	—	2	8	no	—
223	#125	—	2	2	no	—
224	#125	—	2	2	no	—
225	#125	D₄×C₁₉	1	4	yes	—
226	#125	C₂³×C₁₉	1	4	yes	—
227	#124	—	2	2	no	—
228	#124	D₄×C₁₉	1	2	yes	—
229	#124	C₂³×C₁₉	1	2	yes	—
230	#123	—	2	4	no	—
231	#123	—	2	4	no	—
232	C₂²⋊Q₈	—	2	8	no	—
233	C₂²⋊Q₈	—	2	8	no	—
234	C₂×C₄⋊C₄	—	2	2	no	—
235	C₂×C₄⋊C₄	—	2	2	no	—
236	C₂×C₄⋊C₄	—	2	2	no	—
237	C₂×C₄⋊C₄	D₄×C₁₉	1	2	yes	—
238	C₂×C₄⋊C₄	D₄×C₁₉	1	2	yes	—
239	C₂×C₂²⋊C₄	—	2	2	no	—
240	C₂×C₂²⋊C₄	—	2	1	no	—
241	C₂×C₂²⋊C₄	—	2	2	no	—
242	C₂×C₄²	—	2	1	no	—
243	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
244	#102	—	2	8	no	—
245	#102	—	2	8	no	—
246	#97	—	2	2	no	—
247	#97	—	2	1	no	—
248	#97	C₂³×C₁₉	1	4	yes	—
249	#97	C₂³×C₁₉	1	2	yes	—
250	C₁₉	C₂×#1657	1	1	yes	characteristic, direct product factor, Sylow
251	C₂⁴	—	2	1	no	—
252	C₂⁴	—	4	1	no	—
253	C₂×Q₈	—	2	4	no	—
254	C₂×Q₈	—	2	1	no	—
255	C₂×Q₈	—	4	2	no	—
256	C₂×Q₈	—	8	2	no	—
257	C₂×Q₈	C₂×D₄×C₁₉	1	2	yes	—
258	C₂×D₄	—	2	1	no	—
259	C₂×D₄	—	2	4	no	—
260	C₂×D₄	—	2	2	no	—
261	C₂×D₄	—	4	2	no	—
262	C₂×D₄	—	4	2	no	—
263	C₂×D₄	—	4	2	no	—
264	C₂×D₄	—	4	8	no	—
265	C₂×D₄	—	4	2	no	—
266	C₂×D₄	—	4	1	no	—
267	C₂×D₄	C₂×D₄×C₁₉	1	2	yes	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	—	2	1	no	—
270	C₂²×C₄	—	2	2	no	—
271	C₂²×C₄	—	2	4	no	—
272	C₂²×C₄	—	8	2	no	—
273	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
274	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
275	SD₁₆	—	4	8	no	—
276	D₈	—	4	8	no	—
277	C₂×C₈	—	2	4	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	2	4	no	—
280	C₂×C₈	—	4	4	no	—
281	C₄⋊C₄	—	2	8	no	—
282	C₄⋊C₄	—	2	4	no	—
283	C₄⋊C₄	—	2	4	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	2	4	no	—
287	C₄⋊C₄	—	2	4	no	—
288	C₄⋊C₄	C₂×D₄×C₁₉	1	4	yes	—
289	C₄⋊C₄	C₂×D₄×C₁₉	1	4	yes	—
290	C₂²⋊C₄	—	2	8	no	—
291	C₂²⋊C₄	—	2	4	no	—
292	C₂²⋊C₄	—	2	4	no	—
293	C₂²⋊C₄	—	2	4	no	—
294	C₄²	—	2	4	no	—
295	C₄²	C₂×D₄×C₁₉	1	2	yes	—
296	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
297	C₄²	C₂⁴×C₁₉	1	1	yes	characteristic
298	C₂³	—	2	2	no	—
299	C₂³	—	4	2	no	—
300	C₂³	—	4	4	no	—
301	C₂³	—	4	2	no	—
302	C₂³	—	8	2	no	—
303	C₂³	—	8	4	no	—
304	C₂³	C₁₉×#120	1	1	yes	characteristic
305	Q₈	—	4	2	no	—
306	Q₈	—	8	1	no	—
307	Q₈	—	8	4	no	—
308	D₄	—	4	2	no	—
309	D₄	—	4	2	no	—
310	D₄	—	4	2	no	—
311	D₄	—	4	8	no	—
312	D₄	—	8	1	no	—
313	D₄	—	8	2	no	—
314	D₄	—	8	4	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	4	no	—
317	C₂×C₄	—	2	4	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	4	no	—
320	C₂×C₄	—	4	4	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	4	no	—
323	C₂×C₄	—	8	4	no	—
324	C₂×C₄	C₁₉×#120	1	2	yes	—
325	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
326	C₂×C₄	C₂²×D₄×C₁₉	1	2	yes	—
327	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
328	C₈	—	2	4	no	—
329	C₈	—	4	4	no	—
330	C₂²	—	4	2	no	—
331	C₂²	—	4	4	no	—
332	C₂²	—	8	4	no	—
333	C₂²	—	8	2	no	—
334	C₂²	—	8	4	no	—
335	C₂²	—	16	4	no	—
336	C₂²	C₁₉×#453	1	2	yes	—
337	C₂²	C₁₉×#454	1	4	yes	—
338	C₂²	C₂×C₁₉×#120	1	1	yes	characteristic
339	C₄	—	2	1	no	—
340	C₄	—	2	1	no	—
341	C₄	—	2	2	no	—
342	C₄	—	2	2	no	—
343	C₄	—	4	4	no	—
344	C₄	—	4	2	no	—
345	C₄	—	4	4	no	—
346	C₂	—	8	2	no	—
347	C₂	—	16	2	no	—
348	C₂	C₁₉×#1657	1	4	yes	direct product factor
349	C₂	C₂×C₁₉×#453	1	1	yes	characteristic
350	C₂	C₂×C₁₉×#454	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₉	#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₉
prime radical	C₂×C₁₉×#496	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₉×#563	C₂	semidirect
3. C₂×C₁₉×#490	C₂	semidirect
4. C₁₉×#1657	C₂	direct (N × H)
5. C₂×C₁₉×#330	C₂	semidirect
6. C₂×C₁₉×#329	C₂	semidirect
11. C₁₉×#563	C₂²	semidirect
22. C₁₉×#490	C₂²	semidirect
27. C₁₉×#330	C₂²	semidirect
28. C₁₉×#329	C₂²	semidirect
122. C₂×#1657	C₁₉	direct (N × H)
155. C₂×#563	C₂×C₁₉	semidirect
157. C₂×#490	C₂×C₁₉	semidirect
158. #1657	C₂×C₁₉	direct (N × H)
159. C₂×#330	C₂×C₁₉	semidirect
160. C₂×#329	C₂×C₁₉	semidirect
181. #563	C₂²×C₁₉	semidirect
192. #490	C₂²×C₁₉	semidirect
197. #330	C₂²×C₁₉	semidirect
198. #329	C₂²×C₁₉	semidirect
201. C₂×C₁₉	#1657	direct (N × H)
250. C₁₉	C₂×#1657	direct (N × H)
348. C₂	C₁₉×#1657	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×#563	2
C₂×C₁₉×#496	2
C₂×C₁₉×#490	2
C₁₉×#1657	2
C₂×C₁₉×#330	2
C₂×C₁₉×#329	2
C₂×#1657	19

Subgroups — C2×C19×#1657 #18024941

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₉×#1657 #18024941