Subgroups — C₁₉×#17754 #18023713

Order 4864.

1234 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#532	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×#432	C₂	1	2	yes	maximal, complemented
3	C₁₉×#1558	C₂	1	2	yes	maximal, complemented
4	C₁₉×#1555	C₂	1	2	yes	maximal, complemented
5	C₁₉×#1551	C₂	1	8	yes	maximal, complemented
6	C₂²×#41×C₁₉	C₄	1	2	yes	—
7	C₁₉×#532	—	2	2	no	—
8	C₁₉×#532	—	2	2	no	—
9	C₁₉×#532	C₄	1	4	yes	—
10	C₁₉×#532	C₂²	1	4	yes	—
11	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₁₉×#127	C₂²	1	1	yes	characteristic
13	C₂×C₁₉×#124	C₄	1	2	yes	—
14	C₂×C₁₉×#121	—	2	2	no	—
15	C₂×C₄×D₄×C₁₉	—	2	2	no	—
16	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
17	C₁₉×#432	—	2	2	no	—
18	C₁₉×#432	—	2	2	no	—
19	C₁₉×#432	C₂²	1	4	yes	—
20	C₁₉×#432	C₂²	1	4	yes	complemented
21	C₂×C₁₉×C₄⋊C₈	C₂²	1	4	yes	complemented
22	C₂×C₁₉×C₂²⋊C₈	—	2	2	no	—
23	C₂×C₁₉×#97	—	2	2	no	—
24	C₁₉×#333	C₂²	1	8	yes	complemented
25	C₁₉×#331	C₂²	1	8	yes	complemented
26	C₂×#41×C₁₉	—	2	4	no	—
27	C₂×#41×C₁₉	—	4	8	no	—
28	C₂×#41×C₁₉	C₂×C₄	1	8	yes	—
29	C₂²×Q₈×C₁₉	C₂×C₄	1	2	yes	—
30	C₂²×D₄×C₁₉	—	2	2	no	—
31	C₂²×D₄×C₁₉	—	2	2	no	—
32	C₂²×D₄×C₁₉	C₂×C₄	1	2	yes	—
33	C₂³×C₄×C₁₉	—	2	2	no	—
34	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄×C₁₉	D₄	1	2	yes	—
36	C₂²×C₈×C₁₉	—	2	4	no	—
37	C₁₉×C₄⋊Q₈	—	2	2	no	—
38	C₁₉×C₄⋊Q₈	C₂×C₄	1	2	yes	—
39	C₁₉×C₄⋊Q₈	C₂³	1	2	yes	—
40	C₁₉×#127	—	2	2	no	—
41	C₁₉×#127	C₂×C₄	1	2	yes	—
42	C₁₉×#127	C₂³	1	2	yes	—
43	C₁₉×#124	—	2	2	no	—
44	C₁₉×#124	—	2	2	no	—
45	C₁₉×#124	C₂×C₄	1	8	yes	—
46	C₁₉×#121	—	2	4	no	—
47	C₁₉×#121	—	2	4	no	—
48	C₁₉×#121	—	2	4	no	—
49	C₁₉×#121	—	2	4	no	—
50	C₄×D₄×C₁₉	—	2	4	no	—
51	C₄×D₄×C₁₉	—	2	4	no	—
52	C₄×D₄×C₁₉	—	2	4	no	—
53	C₄×D₄×C₁₉	—	2	4	no	—
54	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
55	C₂×C₂²⋊C₄×C₁₉	—	2	4	no	—
56	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
58	C₂×C₄²×C₁₉	D₄	1	2	yes	—
59	C₂×C₄²×C₁₉	D₄	1	2	yes	—
60	C₂×C₄²×C₁₉	D₄	1	2	yes	—
61	C₂×C₄²×C₁₉	C₂³	1	2	yes	—
62	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
63	C₁₉×C₄⋊C₈	—	2	4	no	—
64	C₁₉×C₄⋊C₈	C₂³	1	8	yes	—
65	C₁₉×C₂²⋊C₈	—	2	4	no	—
66	C₁₉×C₂²⋊C₈	—	2	4	no	—
67	C₁₉×#97	—	2	4	no	—
68	C₁₉×#97	—	2	4	no	—
69	C₂⁴×C₁₉	—	2	2	no	—
70	C₂⁴×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
71	#41×C₁₉	—	4	16	no	—
72	#41×C₁₉	—	4	16	no	—
73	C₂×Q₈×C₁₉	—	2	2	no	—
74	C₂×Q₈×C₁₉	—	4	4	no	—
75	C₂×Q₈×C₁₉	C₂²×C₄	1	4	yes	—
76	C₂×D₄×C₁₉	—	2	8	no	—
77	C₂×D₄×C₁₉	—	2	4	no	—
78	C₂×D₄×C₁₉	—	2	2	no	—
79	C₂×D₄×C₁₉	—	2	4	no	—
80	C₂×D₄×C₁₉	—	4	4	no	—
81	C₂×D₄×C₁₉	—	4	8	no	—
82	C₂×D₄×C₁₉	—	4	8	no	—
83	C₂×D₄×C₁₉	C₂²×C₄	1	4	yes	—
84	C₂²×C₄×C₁₉	—	2	4	no	—
85	C₂²×C₄×C₁₉	—	2	2	no	—
86	C₂²×C₄×C₁₉	—	2	8	no	—
87	C₂²×C₄×C₁₉	—	4	4	no	—
88	C₂²×C₄×C₁₉	—	4	8	no	—
89	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
90	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
91	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
92	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
93	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
94	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
96	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
97	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
98	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
99	C₂×C₈×C₁₉	—	2	8	no	—
100	C₂×C₈×C₁₉	—	4	8	no	—
101	C₄⋊C₄×C₁₉	—	2	4	no	—
102	C₄⋊C₄×C₁₉	—	2	4	no	—
103	C₂²⋊C₄×C₁₉	—	2	8	no	—
104	C₂²⋊C₄×C₁₉	—	2	8	no	—
105	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
106	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
107	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
108	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
109	C₄²×C₁₉	C₂²×C₄	1	2	yes	—
110	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₁₉	C₂×D₄	1	2	yes	—
112	C₄²×C₁₉	C₂×D₄	1	2	yes	—
113	C₄²×C₁₉	C₂×D₄	1	2	yes	—
114	C₄²×C₁₉	C₂⁴	1	1	yes	characteristic
115	#17754	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	C₂³×C₁₉	—	2	4	no	—
117	C₂³×C₁₉	—	4	2	no	—
118	C₂³×C₁₉	—	4	8	no	—
119	C₂³×C₁₉	—	4	4	no	—
120	C₂³×C₁₉	#100	1	4	yes	—
121	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂³×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
123	Q₈×C₁₉	—	4	4	no	—
124	Q₈×C₁₉	—	4	4	no	—
125	D₄×C₁₉	—	4	4	no	—
126	D₄×C₁₉	—	4	8	no	—
127	D₄×C₁₉	—	4	4	no	—
128	D₄×C₁₉	—	4	16	no	—
129	D₄×C₁₉	—	4	8	no	—
130	C₂×C₄×C₁₉	—	2	2	no	—
131	C₂×C₄×C₁₉	—	2	4	no	—
132	C₂×C₄×C₁₉	—	2	2	no	—
133	C₂×C₄×C₁₉	—	2	4	no	—
134	C₂×C₄×C₁₉	—	2	4	no	—
135	C₂×C₄×C₁₉	—	2	4	no	—
136	C₂×C₄×C₁₉	—	2	4	no	—
137	C₂×C₄×C₁₉	—	2	4	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	16	no	—
142	C₂×C₄×C₁₉	—	4	4	no	—
143	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
144	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
145	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
146	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
147	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
149	C₂×C₄×C₁₉	C₂²×D₄	1	2	yes	—
150	C₈×C₁₉	—	4	8	no	—
151	C₂×#532	C₂×C₁₉	1	1	yes	characteristic
152	C₂×#432	C₂×C₁₉	1	2	yes	complemented
153	#1558	C₂×C₁₉	1	2	yes	complemented
154	#1555	C₂×C₁₉	1	2	yes	complemented
155	#1551	C₂×C₁₉	1	8	yes	complemented
156	C₂²×C₁₉	—	2	2	no	—
157	C₂²×C₁₉	—	2	2	no	—
158	C₂²×C₁₉	—	2	2	no	—
159	C₂²×C₁₉	—	4	4	no	—
160	C₂²×C₁₉	—	4	4	no	—
161	C₂²×C₁₉	—	4	8	no	—
162	C₂²×C₁₉	—	8	8	no	—
163	C₂²×C₁₉	C₂×#100	1	2	yes	—
164	C₂²×C₁₉	C₂×#100	1	4	yes	—
165	C₂²×C₁₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
166	C₄×C₁₉	—	2	2	no	—
167	C₄×C₁₉	—	2	1	no	—
168	C₄×C₁₉	—	2	1	no	—
169	C₄×C₁₉	—	2	2	no	—
170	C₄×C₁₉	—	2	2	no	—
171	C₄×C₁₉	—	2	2	no	—
172	C₄×C₁₉	—	4	4	no	—
173	C₄×C₁₉	#575	1	4	yes	—
174	C₂²×#41	C₄×C₁₉	1	2	yes	—
175	#532	—	2	2	no	—
176	#532	—	2	2	no	—
177	#532	C₄×C₁₉	1	4	yes	—
178	#532	C₂²×C₁₉	1	4	yes	—
179	C₂×C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
180	C₂×#127	C₂²×C₁₉	1	1	yes	characteristic
181	C₂×#124	C₄×C₁₉	1	2	yes	—
182	C₂×#121	—	2	2	no	—
183	C₂×C₄×D₄	—	2	2	no	—
184	C₂²×C₄²	C₂²×C₁₉	1	1	yes	characteristic
185	#432	—	2	2	no	—
186	#432	—	2	2	no	—
187	#432	C₂²×C₁₉	1	4	yes	—
188	#432	C₂²×C₁₉	1	4	yes	complemented
189	C₂×C₄⋊C₈	C₂²×C₁₉	1	4	yes	complemented
190	C₂×C₂²⋊C₈	—	2	2	no	—
191	C₂×#97	—	2	2	no	—
192	#333	C₂²×C₁₉	1	8	yes	complemented
193	#331	C₂²×C₁₉	1	8	yes	complemented
194	C₂×C₁₉	—	2	2	no	—
195	C₂×C₁₉	—	2	4	no	—
196	C₂×C₁₉	—	8	4	no	—
197	C₂×C₁₉	C₂²×#100	1	1	yes	characteristic
198	C₂×C₁₉	#2896	1	2	yes	—
199	C₂×#41	—	2	4	no	—
200	C₂×#41	—	4	8	no	—
201	C₂×#41	C₂×C₄×C₁₉	1	8	yes	—
202	C₂²×Q₈	C₂×C₄×C₁₉	1	2	yes	—
203	C₂²×D₄	—	2	2	no	—
204	C₂²×D₄	—	2	2	no	—
205	C₂²×D₄	C₂×C₄×C₁₉	1	2	yes	—
206	C₂³×C₄	—	2	2	no	—
207	C₂³×C₄	C₂×C₄×C₁₉	1	1	yes	characteristic
208	C₂³×C₄	D₄×C₁₉	1	2	yes	—
209	C₂²×C₈	—	2	4	no	—
210	C₄⋊Q₈	—	2	2	no	—
211	C₄⋊Q₈	C₂×C₄×C₁₉	1	2	yes	—
212	C₄⋊Q₈	C₂³×C₁₉	1	2	yes	—
213	#127	—	2	2	no	—
214	#127	C₂×C₄×C₁₉	1	2	yes	—
215	#127	C₂³×C₁₉	1	2	yes	—
216	#124	—	2	2	no	—
217	#124	—	2	2	no	—
218	#124	C₂×C₄×C₁₉	1	8	yes	—
219	#121	—	2	4	no	—
220	#121	—	2	4	no	—
221	#121	—	2	4	no	—
222	#121	—	2	4	no	—
223	C₄×D₄	—	2	4	no	—
224	C₄×D₄	—	2	4	no	—
225	C₄×D₄	—	2	4	no	—
226	C₄×D₄	—	2	4	no	—
227	C₂×C₄⋊C₄	—	2	2	no	—
228	C₂×C₂²⋊C₄	—	2	4	no	—
229	C₂×C₄²	C₂×C₄×C₁₉	1	1	yes	characteristic
230	C₂×C₄²	C₂×C₄×C₁₉	1	2	yes	—
231	C₂×C₄²	D₄×C₁₉	1	2	yes	—
232	C₂×C₄²	D₄×C₁₉	1	2	yes	—
233	C₂×C₄²	D₄×C₁₉	1	2	yes	—
234	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
235	C₂×C₄²	C₂³×C₁₉	1	2	yes	—
236	C₄⋊C₈	—	2	4	no	—
237	C₄⋊C₈	C₂³×C₁₉	1	8	yes	—
238	C₂²⋊C₈	—	2	4	no	—
239	C₂²⋊C₈	—	2	4	no	—
240	#97	—	2	4	no	—
241	#97	—	2	4	no	—
242	C₁₉	#17754	1	1	yes	characteristic, direct product factor, Sylow
243	C₂⁴	—	2	2	no	—
244	C₂⁴	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
245	#41	—	4	16	no	—
246	#41	—	4	16	no	—
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	4	4	no	—
249	C₂×Q₈	C₂²×C₄×C₁₉	1	4	yes	—
250	C₂×D₄	—	2	8	no	—
251	C₂×D₄	—	2	4	no	—
252	C₂×D₄	—	2	2	no	—
253	C₂×D₄	—	2	4	no	—
254	C₂×D₄	—	4	4	no	—
255	C₂×D₄	—	4	8	no	—
256	C₂×D₄	—	4	8	no	—
257	C₂×D₄	C₂²×C₄×C₁₉	1	4	yes	—
258	C₂²×C₄	—	2	4	no	—
259	C₂²×C₄	—	2	2	no	—
260	C₂²×C₄	—	2	8	no	—
261	C₂²×C₄	—	4	4	no	—
262	C₂²×C₄	—	4	8	no	—
263	C₂²×C₄	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
264	C₂²×C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
265	C₂²×C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
266	C₂²×C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
267	C₂²×C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
268	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
269	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
270	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
271	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
272	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
273	C₂×C₈	—	2	8	no	—
274	C₂×C₈	—	4	8	no	—
275	C₄⋊C₄	—	2	4	no	—
276	C₄⋊C₄	—	2	4	no	—
277	C₂²⋊C₄	—	2	8	no	—
278	C₂²⋊C₄	—	2	8	no	—
279	C₄²	C₂²⋊C₄×C₁₉	1	2	yes	—
280	C₄²	C₂²⋊C₄×C₁₉	1	2	yes	—
281	C₄²	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
282	C₄²	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
283	C₄²	C₂²×C₄×C₁₉	1	2	yes	—
284	C₄²	C₂²×C₄×C₁₉	1	1	yes	characteristic
285	C₄²	C₂×D₄×C₁₉	1	2	yes	—
286	C₄²	C₂×D₄×C₁₉	1	2	yes	—
287	C₄²	C₂×D₄×C₁₉	1	2	yes	—
288	C₄²	C₂⁴×C₁₉	1	1	yes	characteristic
289	C₂³	—	2	4	no	—
290	C₂³	—	4	2	no	—
291	C₂³	—	4	8	no	—
292	C₂³	—	4	4	no	—
293	C₂³	C₁₉×#100	1	4	yes	—
294	C₂³	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
295	C₂³	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
296	Q₈	—	4	4	no	—
297	Q₈	—	4	4	no	—
298	D₄	—	4	4	no	—
299	D₄	—	4	4	no	—
300	D₄	—	4	8	no	—
301	D₄	—	4	16	no	—
302	D₄	—	4	8	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	4	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	4	no	—
307	C₂×C₄	—	2	4	no	—
308	C₂×C₄	—	2	4	no	—
309	C₂×C₄	—	2	4	no	—
310	C₂×C₄	—	2	4	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	16	no	—
315	C₂×C₄	—	4	4	no	—
316	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
317	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
318	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
319	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
320	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
321	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
322	C₂×C₄	C₂²×D₄×C₁₉	1	2	yes	—
323	C₈	—	4	8	no	—
324	C₂²	—	2	2	no	—
325	C₂²	—	2	2	no	—
326	C₂²	—	2	2	no	—
327	C₂²	—	4	4	no	—
328	C₂²	—	4	4	no	—
329	C₂²	—	4	8	no	—
330	C₂²	—	8	8	no	—
331	C₂²	C₂×C₁₉×#100	1	2	yes	—
332	C₂²	C₂×C₁₉×#100	1	4	yes	—
333	C₂²	C₂²×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
334	C₄	—	2	2	no	—
335	C₄	—	2	1	no	—
336	C₄	—	2	1	no	—
337	C₄	—	2	2	no	—
338	C₄	—	2	2	no	—
339	C₄	—	2	2	no	—
340	C₄	—	4	4	no	—
341	C₄	C₁₉×#575	1	4	yes	—
342	C₂	—	2	2	no	—
343	C₂	—	2	4	no	—
344	C₂	—	8	4	no	—
345	C₂	C₂²×C₁₉×#100	1	1	yes	characteristic
346	C₂	C₁₉×#2896	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₁₉
Frattini, cosocle	C₄²	C₂⁴×C₁₉
prime radical	C₂×C₁₉×#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂²×C₄²×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₉×#432	C₂	semidirect
3. C₁₉×#1558	C₂	semidirect
4. C₁₉×#1555	C₂	semidirect
5. C₁₉×#1551	C₂	semidirect
20. C₁₉×#432	C₂²	semidirect
21. C₂×C₁₉×C₄⋊C₈	C₂²	semidirect
24. C₁₉×#333	C₂²	semidirect
25. C₁₉×#331	C₂²	semidirect
115. #17754	C₁₉	direct (N × H)
152. C₂×#432	C₂×C₁₉	semidirect
153. #1558	C₂×C₁₉	semidirect
154. #1555	C₂×C₁₉	semidirect
155. #1551	C₂×C₁₉	semidirect
188. #432	C₂²×C₁₉	semidirect
189. C₂×C₄⋊C₈	C₂²×C₁₉	semidirect
192. #333	C₂²×C₁₉	semidirect
193. #331	C₂²×C₁₉	semidirect
242. C₁₉	#17754	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×#532	2
C₂×C₁₉×#432	2
C₁₉×#1558	2
C₁₉×#1555	2
C₁₉×#1551	2
#17754	19

Subgroups — C19×#17754 #18023713

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#17754 #18023713