Subgroups — C₂×C₁₉×#1704 #18025113

Order 4864.

618 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#487	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×#477	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₉×#475	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₉×#446	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1704	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₁₉×#334	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₉×#333	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₉×#332	C₂	1	1	yes	maximal, characteristic
9	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
12	C₁₉×#487	—	2	2	no	—
13	C₁₉×#487	C₂²	1	4	yes	—
14	C₁₉×#477	—	2	2	no	—
15	C₁₉×#477	C₂²	1	4	yes	—
16	C₁₉×#475	—	2	2	no	—
17	C₁₉×#475	C₂²	1	4	yes	—
18	C₁₉×#446	—	2	2	no	—
19	C₁₉×#446	C₂²	1	4	yes	—
20	C₂×C₁₉×#107	—	2	1	no	—
21	C₂×C₁₉×#106	—	2	1	no	—
22	C₂×C₁₉×C₄⋊C₈	—	2	1	no	—
23	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₁₉×#103	—	2	1	no	—
27	C₂×C₁₉×#103	—	2	1	no	—
28	C₂×C₁₉×#103	—	2	1	no	—
29	C₂×C₁₉×#103	—	2	1	no	—
30	C₂×C₁₉×#97	—	2	1	no	—
31	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
32	C₁₉×#334	C₂²	1	4	yes	—
33	C₁₉×#333	C₂²	1	4	yes	—
34	C₁₉×#332	C₂²	1	4	yes	—
35	C₂²×Q₈×C₁₉	—	2	1	no	—
36	C₂²×Q₈×C₁₉	—	2	1	no	—
37	C₂²×Q₈×C₁₉	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₉	—	2	1	no	—
39	C₂²×C₈×C₁₉	—	2	1	no	—
40	C₂²×C₈×C₁₉	—	2	1	no	—
41	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
43	C₁₉×C₄⋊Q₈	—	2	1	no	—
44	C₁₉×C₄⋊Q₈	—	2	1	no	—
45	C₁₉×C₄⋊Q₈	—	2	1	no	—
46	C₁₉×C₄⋊Q₈	—	2	1	no	—
47	C₁₉×C₄⋊Q₈	—	2	1	no	—
48	C₁₉×C₄⋊Q₈	—	2	1	no	—
49	C₁₉×C₄⋊Q₈	C₂³	1	2	yes	—
50	C₁₉×C₄⋊Q₈	C₂³	1	2	yes	—
51	C₄×Q₈×C₁₉	—	2	1	no	—
52	C₄×Q₈×C₁₉	—	2	1	no	—
53	C₄×Q₈×C₁₉	D₄	1	2	yes	—
54	C₄×Q₈×C₁₉	C₂³	1	2	yes	—
55	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
56	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
57	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
58	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
59	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
60	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
61	C₂×C₄²×C₁₉	—	2	1	no	—
62	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
63	C₁₉×#107	—	2	4	no	—
64	C₁₉×#106	—	2	4	no	—
65	C₁₉×C₄⋊C₈	—	2	4	no	—
66	C₁₉×C₄⋊C₈	—	2	1	no	—
67	C₁₉×C₄⋊C₈	—	2	1	no	—
68	C₁₉×C₄⋊C₈	—	2	1	no	—
69	C₁₉×C₄⋊C₈	C₂³	1	2	yes	—
70	C₁₉×C₄⋊C₈	C₂³	1	2	yes	—
71	C₁₉×C₄⋊C₈	C₂³	1	2	yes	—
72	C₁₉×#103	—	2	4	no	—
73	C₁₉×#103	—	2	4	no	—
74	C₁₉×#103	—	2	4	no	—
75	C₁₉×#103	—	2	4	no	—
76	C₁₉×#97	—	2	4	no	—
77	C₄×C₈×C₁₉	D₄	1	2	yes	—
78	C₄×C₈×C₁₉	C₂³	1	2	yes	—
79	C₂×Q₈×C₁₉	—	2	1	no	—
80	C₂×Q₈×C₁₉	—	2	2	no	—
81	C₂×Q₈×C₁₉	—	2	2	no	—
82	C₂×Q₈×C₁₉	—	2	2	no	—
83	C₂×Q₈×C₁₉	—	2	2	no	—
84	C₂×Q₈×C₁₉	—	4	1	no	—
85	C₂×Q₈×C₁₉	—	4	1	no	—
86	C₂×Q₈×C₁₉	—	4	2	no	—
87	C₂×Q₈×C₁₉	—	4	2	no	—
88	C₂×Q₈×C₁₉	—	8	2	no	—
89	C₂×Q₈×C₁₉	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₁₉	—	2	1	no	—
91	C₂²×C₄×C₁₉	—	2	1	no	—
92	C₂²×C₄×C₁₉	—	4	1	no	—
93	C₂²×C₄×C₁₉	—	4	1	no	—
94	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
97	C₂×C₈×C₁₉	—	2	2	no	—
98	C₂×C₈×C₁₉	—	2	2	no	—
99	C₂×C₈×C₁₉	—	2	2	no	—
100	C₂×C₈×C₁₉	—	2	2	no	—
101	C₂×C₈×C₁₉	—	2	2	no	—
102	C₂×C₈×C₁₉	—	4	2	no	—
103	C₂×C₈×C₁₉	—	4	2	no	—
104	C₂×C₈×C₁₉	—	4	2	no	—
105	C₂×C₈×C₁₉	C₂×D₄	1	2	yes	—
106	C₂×C₈×C₁₉	C₂×D₄	1	2	yes	—
107	C₄⋊C₄×C₁₉	—	2	2	no	—
108	C₄⋊C₄×C₁₉	—	2	1	no	—
109	C₄⋊C₄×C₁₉	—	2	2	no	—
110	C₄⋊C₄×C₁₉	—	2	2	no	—
111	C₄⋊C₄×C₁₉	—	2	2	no	—
112	C₄⋊C₄×C₁₉	—	2	2	no	—
113	C₄⋊C₄×C₁₉	—	2	2	no	—
114	C₄⋊C₄×C₁₉	—	4	1	no	—
115	C₄⋊C₄×C₁₉	—	4	1	no	—
116	C₄⋊C₄×C₁₉	—	4	1	no	—
117	C₄⋊C₄×C₁₉	—	4	1	no	—
118	C₄⋊C₄×C₁₉	C₂×D₄	1	2	yes	—
119	C₄²×C₁₉	—	2	2	no	—
120	C₄²×C₁₉	—	2	2	no	—
121	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
122	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₁₉	#41	1	1	yes	characteristic
124	C₄²×C₁₉	C₂⁴	1	1	yes	characteristic
125	C₂×#1704	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
126	C₂³×C₁₉	#121	1	1	yes	characteristic
127	Q₈×C₁₉	—	2	2	no	—
128	Q₈×C₁₉	—	2	2	no	—
129	Q₈×C₁₉	—	4	2	no	—
130	Q₈×C₁₉	—	4	2	no	—
131	Q₈×C₁₉	—	4	2	no	—
132	Q₈×C₁₉	—	8	2	no	—
133	Q₈×C₁₉	—	8	2	no	—
134	Q₈×C₁₉	—	8	2	no	—
135	C₂×C₄×C₁₉	—	2	2	no	—
136	C₂×C₄×C₁₉	—	2	2	no	—
137	C₂×C₄×C₁₉	—	2	2	no	—
138	C₂×C₄×C₁₉	—	4	2	no	—
139	C₂×C₄×C₁₉	—	4	2	no	—
140	C₂×C₄×C₁₉	—	4	2	no	—
141	C₂×C₄×C₁₉	—	4	2	no	—
142	C₂×C₄×C₁₉	—	8	2	no	—
143	C₂×C₄×C₁₉	—	8	2	no	—
144	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
145	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
146	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
147	C₂×C₄×C₁₉	#135	1	2	yes	—
148	C₂×C₄×C₁₉	#137	1	2	yes	—
149	C₂×C₄×C₁₉	#137	1	2	yes	—
150	C₂×C₄×C₁₉	#137	1	2	yes	—
151	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₁₉	C₂×#41	1	1	yes	characteristic
154	C₈×C₁₉	—	2	2	no	—
155	C₈×C₁₉	—	2	2	no	—
156	C₈×C₁₉	—	4	2	no	—
157	C₈×C₁₉	—	4	2	no	—
158	C₈×C₁₉	—	4	2	no	—
159	C₂×#487	C₂×C₁₉	1	1	yes	characteristic
160	C₂×#477	C₂×C₁₉	1	1	yes	characteristic
161	C₂×#475	C₂×C₁₉	1	1	yes	characteristic
162	C₂×#446	C₂×C₁₉	1	1	yes	characteristic
163	#1704	C₂×C₁₉	1	8	yes	direct product factor
164	C₂×#334	C₂×C₁₉	1	1	yes	characteristic
165	C₂×#333	C₂×C₁₉	1	1	yes	characteristic
166	C₂×#332	C₂×C₁₉	1	1	yes	characteristic
167	C₂²×C₁₉	#464	1	2	yes	—
168	C₂²×C₁₉	#470	1	2	yes	—
169	C₂²×C₁₉	#471	1	2	yes	—
170	C₂²×C₁₉	C₂×#121	1	1	yes	characteristic
171	C₄×C₁₉	—	2	1	no	—
172	C₄×C₁₉	—	2	1	no	—
173	C₄×C₁₉	—	4	2	no	—
174	C₄×C₁₉	—	4	2	no	—
175	C₄×C₁₉	—	8	2	no	—
176	C₄×C₁₉	—	8	2	no	—
177	C₄×C₁₉	#464	1	2	yes	—
178	C₄×C₁₉	#470	1	2	yes	—
179	C₄×C₁₉	C₂×#135	1	1	yes	characteristic
180	C₄×C₁₉	C₂×#137	1	1	yes	characteristic
181	C₄×C₁₉	C₂×#137	1	1	yes	characteristic
182	C₄×C₁₉	C₂×#137	1	1	yes	characteristic
183	C₂×C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
184	C₂×C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
185	C₂×C₄×Q₈	C₂²×C₁₉	1	1	yes	characteristic
186	#487	—	2	2	no	—
187	#487	C₂²×C₁₉	1	4	yes	—
188	#477	—	2	2	no	—
189	#477	C₂²×C₁₉	1	4	yes	—
190	#475	—	2	2	no	—
191	#475	C₂²×C₁₉	1	4	yes	—
192	#446	—	2	2	no	—
193	#446	C₂²×C₁₉	1	4	yes	—
194	C₂×#107	—	2	1	no	—
195	C₂×#106	—	2	1	no	—
196	C₂×C₄⋊C₈	—	2	1	no	—
197	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
198	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
199	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
200	C₂×#103	—	2	1	no	—
201	C₂×#103	—	2	1	no	—
202	C₂×#103	—	2	1	no	—
203	C₂×#103	—	2	1	no	—
204	C₂×#97	—	2	1	no	—
205	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
206	#334	C₂²×C₁₉	1	4	yes	—
207	#333	C₂²×C₁₉	1	4	yes	—
208	#332	C₂²×C₁₉	1	4	yes	—
209	C₂×C₁₉	#1704	1	4	yes	direct product factor
210	C₂×C₁₉	C₂×#464	1	1	yes	characteristic
211	C₂×C₁₉	C₂×#470	1	1	yes	characteristic
212	C₂×C₁₉	C₂×#471	1	1	yes	characteristic
213	C₂²×Q₈	—	2	1	no	—
214	C₂²×Q₈	—	2	1	no	—
215	C₂²×Q₈	D₄×C₁₉	1	1	yes	characteristic
216	C₂²×C₈	—	2	1	no	—
217	C₂²×C₈	—	2	1	no	—
218	C₂²×C₈	—	2	1	no	—
219	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
220	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
221	C₄⋊Q₈	—	2	1	no	—
222	C₄⋊Q₈	—	2	1	no	—
223	C₄⋊Q₈	—	2	1	no	—
224	C₄⋊Q₈	—	2	1	no	—
225	C₄⋊Q₈	—	2	1	no	—
226	C₄⋊Q₈	—	2	1	no	—
227	C₄⋊Q₈	C₂³×C₁₉	1	2	yes	—
228	C₄⋊Q₈	C₂³×C₁₉	1	2	yes	—
229	C₄×Q₈	—	2	1	no	—
230	C₄×Q₈	—	2	1	no	—
231	C₄×Q₈	D₄×C₁₉	1	2	yes	—
232	C₄×Q₈	C₂³×C₁₉	1	2	yes	—
233	C₂×C₄⋊C₄	—	2	1	no	—
234	C₂×C₄⋊C₄	—	2	1	no	—
235	C₂×C₄⋊C₄	—	2	1	no	—
236	C₂×C₄⋊C₄	—	2	1	no	—
237	C₂×C₄⋊C₄	—	2	1	no	—
238	C₂×C₄⋊C₄	D₄×C₁₉	1	1	yes	characteristic
239	C₂×C₄²	—	2	1	no	—
240	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
241	#107	—	2	4	no	—
242	#106	—	2	4	no	—
243	C₄⋊C₈	—	2	4	no	—
244	C₄⋊C₈	—	2	1	no	—
245	C₄⋊C₈	—	2	1	no	—
246	C₄⋊C₈	—	2	1	no	—
247	C₄⋊C₈	C₂³×C₁₉	1	2	yes	—
248	C₄⋊C₈	C₂³×C₁₉	1	2	yes	—
249	C₄⋊C₈	C₂³×C₁₉	1	2	yes	—
250	#103	—	2	4	no	—
251	#103	—	2	4	no	—
252	#103	—	2	4	no	—
253	#103	—	2	4	no	—
254	#97	—	2	4	no	—
255	C₄×C₈	D₄×C₁₉	1	2	yes	—
256	C₄×C₈	C₂³×C₁₉	1	2	yes	—
257	C₁₉	C₂×#1704	1	1	yes	characteristic, direct product factor, Sylow
258	C₂×Q₈	—	2	1	no	—
259	C₂×Q₈	—	2	2	no	—
260	C₂×Q₈	—	2	2	no	—
261	C₂×Q₈	—	2	2	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	4	1	no	—
264	C₂×Q₈	—	4	1	no	—
265	C₂×Q₈	—	4	2	no	—
266	C₂×Q₈	—	4	2	no	—
267	C₂×Q₈	—	8	2	no	—
268	C₂×Q₈	C₂×D₄×C₁₉	1	2	yes	—
269	C₂²×C₄	—	2	1	no	—
270	C₂²×C₄	—	2	1	no	—
271	C₂²×C₄	—	4	1	no	—
272	C₂²×C₄	—	4	1	no	—
273	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
274	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
275	C₂²×C₄	#41×C₁₉	1	1	yes	characteristic
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	4	2	no	—
282	C₂×C₈	—	4	2	no	—
283	C₂×C₈	—	4	2	no	—
284	C₂×C₈	C₂×D₄×C₁₉	1	2	yes	—
285	C₂×C₈	C₂×D₄×C₁₉	1	2	yes	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	2	1	no	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	4	1	no	—
294	C₄⋊C₄	—	4	1	no	—
295	C₄⋊C₄	—	4	1	no	—
296	C₄⋊C₄	—	4	1	no	—
297	C₄⋊C₄	C₂×D₄×C₁₉	1	2	yes	—
298	C₄²	—	2	2	no	—
299	C₄²	—	2	2	no	—
300	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
301	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
302	C₄²	#41×C₁₉	1	1	yes	characteristic
303	C₄²	C₂⁴×C₁₉	1	1	yes	characteristic
304	C₂³	C₁₉×#121	1	1	yes	characteristic
305	Q₈	—	2	2	no	—
306	Q₈	—	2	2	no	—
307	Q₈	—	4	2	no	—
308	Q₈	—	4	2	no	—
309	Q₈	—	4	2	no	—
310	Q₈	—	8	2	no	—
311	Q₈	—	8	2	no	—
312	Q₈	—	8	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	8	2	no	—
321	C₂×C₄	—	8	2	no	—
322	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
323	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
324	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
325	C₂×C₄	C₁₉×#135	1	2	yes	—
326	C₂×C₄	C₁₉×#137	1	2	yes	—
327	C₂×C₄	C₁₉×#137	1	2	yes	—
328	C₂×C₄	C₁₉×#137	1	2	yes	—
329	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
330	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
331	C₂×C₄	C₂×#41×C₁₉	1	1	yes	characteristic
332	C₈	—	2	2	no	—
333	C₈	—	2	2	no	—
334	C₈	—	4	2	no	—
335	C₈	—	4	2	no	—
336	C₈	—	4	2	no	—
337	C₂²	C₁₉×#464	1	2	yes	—
338	C₂²	C₁₉×#470	1	2	yes	—
339	C₂²	C₁₉×#471	1	2	yes	—
340	C₂²	C₂×C₁₉×#121	1	1	yes	characteristic
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	4	2	no	—
344	C₄	—	4	2	no	—
345	C₄	—	8	2	no	—
346	C₄	—	8	2	no	—
347	C₄	C₁₉×#464	1	2	yes	—
348	C₄	C₁₉×#470	1	2	yes	—
349	C₄	C₂×C₁₉×#135	1	1	yes	characteristic
350	C₄	C₂×C₁₉×#137	1	1	yes	characteristic
351	C₄	C₂×C₁₉×#137	1	1	yes	characteristic
352	C₄	C₂×C₁₉×#137	1	1	yes	characteristic
353	C₂	C₁₉×#1704	1	4	yes	direct product factor
354	C₂	C₂×C₁₉×#464	1	1	yes	characteristic
355	C₂	C₂×C₁₉×#470	1	1	yes	characteristic
356	C₂	C₂×C₁₉×#471	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₉	#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₉
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
5. C₁₉×#1704	C₂	direct (N × H)
125. C₂×#1704	C₁₉	direct (N × H)
163. #1704	C₂×C₁₉	direct (N × H)
209. C₂×C₁₉	#1704	direct (N × H)
257. C₁₉	C₂×#1704	direct (N × H)
353. C₂	C₁₉×#1704	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×#487	2
C₂×C₁₉×#477	2
C₂×C₁₉×#475	2
C₂×C₁₉×#446	2
C₁₉×#1704	2
C₂×C₁₉×#334	2
C₂×C₁₉×#333	2
C₂×C₁₉×#332	2
C₂×#1704	19

Subgroups — C2×C19×#1704 #18025113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₉×#1704 #18025113