Subgroups — C₁₉×#21710 #18027669

Order 4864.

1014 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3150	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×#444	C₂	1	2	yes	maximal, complemented
3	C₂×C₁₉×#443	C₂	1	2	yes	maximal, complemented
4	C₁₉×#2932	C₂	1	2	yes	maximal, complemented
5	C₁₉×#1975	C₂	1	2	yes	maximal, complemented
6	C₁₉×#1974	C₂	1	2	yes	maximal, complemented
7	C₁₉×#1962	C₂	1	2	yes	maximal, complemented
8	C₁₉×#1849	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₁₉×#97	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₁₉×#135	C₂²	1	2	yes	—
11	C₂×C₁₉×#135	C₂²	1	2	yes	—
12	C₂×C₁₉×#131	C₂²	1	4	yes	—
13	C₁₉×#532	C₂²	1	2	yes	—
14	C₁₉×#497	C₄	1	4	yes	complemented
15	C₁₉×#496	C₄	1	4	yes	complemented
16	C₁₉×#444	—	2	8	no	—
17	C₁₉×#443	—	2	8	no	—
18	C₁₉×#435	—	2	8	no	—
19	C₂×C₁₉×#108	C₂²	1	2	yes	—
20	C₂×C₁₉×#108	C₂²	1	2	yes	—
21	C₁₉×#423	C₂²	1	2	yes	—
22	C₂×C₁₉×#104	C₂²	1	4	yes	—
23	C₂×C₁₉×#104	C₂²	1	4	yes	—
24	C₁₉×#408	C₂²	1	4	yes	—
25	C₂×C₁₉×#97	C₂²	1	1	yes	characteristic
26	C₂×C₁₉×#97	C₂²	1	1	yes	characteristic
27	C₂×C₁₉×#97	C₂²	1	1	yes	characteristic
28	C₂×C₁₉×#97	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
31	C₄³×C₁₉	C₂²	1	1	yes	characteristic
32	C₂×#41×C₁₉	C₂³	1	4	yes	—
33	C₁₉×#135	—	2	4	no	—
34	C₁₉×#135	—	2	4	no	—
35	C₁₉×#135	—	2	4	no	—
36	C₁₉×#135	—	2	4	no	—
37	C₂×Q₁₆×C₁₉	C₂×C₄	1	2	yes	—
38	C₂×Q₁₆×C₁₉	C₂×C₄	1	2	yes	—
39	C₂×SD₁₆×C₁₉	C₂×C₄	1	4	yes	—
40	C₂×SD₁₆×C₁₉	C₂×C₄	1	4	yes	—
41	C₂×D₈×C₁₉	C₂×C₄	1	2	yes	—
42	C₂×D₈×C₁₉	C₂×C₄	1	2	yes	—
43	C₁₉×#131	—	2	8	no	—
44	C₁₉×#131	—	2	8	no	—
45	C₂×M₁₆×C₁₉	—	2	4	no	—
46	C₂×M₁₆×C₁₉	C₂³	1	4	yes	—
47	C₂²×C₈×C₁₉	—	2	4	no	—
48	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
52	C₁₉×C₄⋊Q₈	C₂×C₄	1	2	yes	—
53	C₁₉×#127	C₂×C₄	1	2	yes	—
54	C₁₉×#124	C₂×C₄	1	4	yes	—
55	C₁₉×#121	—	2	4	no	—
56	C₄×D₄×C₁₉	—	2	4	no	—
57	C₂×C₄²×C₁₉	—	2	2	no	—
58	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
61	C₁₉×#108	—	2	4	no	—
62	C₁₉×#108	—	2	4	no	—
63	C₁₉×C₄⋊C₈	—	2	4	no	—
64	C₁₉×#104	—	2	8	no	—
65	C₁₉×#104	—	2	8	no	—
66	C₁₉×C₂²⋊C₈	—	2	8	no	—
67	C₁₉×#97	—	2	2	no	—
68	C₁₉×#97	C₂×C₄	1	4	yes	—
69	C₁₉×#97	D₄	1	4	yes	—
70	C₁₉×#97	D₄	1	4	yes	complemented
71	C₄×C₈×C₁₉	—	2	2	no	—
72	C₄×C₈×C₁₉	D₄	1	4	yes	complemented
73	#41×C₁₉	—	2	4	no	—
74	#41×C₁₉	—	2	4	no	—
75	#41×C₁₉	—	2	4	no	—
76	#41×C₁₉	—	2	4	no	—
77	C₂×Q₈×C₁₉	C₂²×C₄	1	4	yes	—
78	C₂×D₄×C₁₉	—	2	4	no	—
79	C₂×D₄×C₁₉	—	2	4	no	—
80	C₂×D₄×C₁₉	C₂²×C₄	1	4	yes	—
81	C₂²×C₄×C₁₉	—	2	1	no	—
82	C₂²×C₄×C₁₉	—	2	1	no	—
83	C₂²×C₄×C₁₉	—	2	4	no	—
84	C₂²×C₄×C₁₉	#41	1	2	yes	—
85	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
86	Q₁₆×C₁₉	—	2	4	no	—
87	Q₁₆×C₁₉	—	2	4	no	—
88	SD₁₆×C₁₉	—	2	8	no	—
89	SD₁₆×C₁₉	—	2	8	no	—
90	D₈×C₁₉	—	2	4	no	—
91	D₈×C₁₉	—	2	4	no	—
92	M₁₆×C₁₉	—	2	4	no	—
93	M₁₆×C₁₉	—	2	4	no	—
94	M₁₆×C₁₉	—	4	8	no	—
95	C₂×C₈×C₁₉	—	2	1	no	—
96	C₂×C₈×C₁₉	—	2	1	no	—
97	C₂×C₈×C₁₉	—	2	4	no	—
98	C₂×C₈×C₁₉	—	4	8	no	—
99	C₂×C₈×C₁₉	C₂²×C₄	1	2	yes	—
100	C₂×C₈×C₁₉	C₂²×C₄	1	2	yes	—
101	C₂×C₈×C₁₉	C₂×D₄	1	2	yes	—
102	C₂×C₈×C₁₉	C₂×D₄	1	2	yes	—
103	C₂×C₈×C₁₉	C₂×D₄	1	2	yes	—
104	C₂×C₈×C₁₉	C₂×D₄	1	2	yes	—
105	C₂×C₈×C₁₉	C₂×D₄	1	2	yes	—
106	C₂×C₈×C₁₉	C₂×D₄	1	2	yes	—
107	C₂×C₈×C₁₉	#41	1	2	yes	—
108	C₂×C₈×C₁₉	#41	1	2	yes	—
109	C₄⋊C₄×C₁₉	—	2	4	no	—
110	C₂²⋊C₄×C₁₉	—	2	8	no	—
111	C₄²×C₁₉	—	2	1	no	—
112	C₄²×C₁₉	—	2	1	no	—
113	C₄²×C₁₉	—	4	4	no	—
114	C₄²×C₁₉	C₂²×C₄	1	2	yes	—
115	C₄²×C₁₉	C₂×D₄	1	2	yes	—
116	C₄²×C₁₉	C₂×D₄	1	2	yes	—
117	C₄²×C₁₉	C₂×D₄	1	2	yes	—
118	#21710	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	C₂³×C₁₉	—	2	4	no	—
120	C₂³×C₁₉	C₂×#41	1	1	yes	characteristic
121	Q₈×C₁₉	—	2	4	no	—
122	Q₈×C₁₉	—	2	4	no	—
123	D₄×C₁₉	—	2	4	no	—
124	D₄×C₁₉	—	2	4	no	—
125	D₄×C₁₉	—	4	4	no	—
126	D₄×C₁₉	—	4	4	no	—
127	D₄×C₁₉	—	8	4	no	—
128	C₂×C₄×C₁₉	—	2	4	no	—
129	C₂×C₄×C₁₉	—	2	2	no	—
130	C₂×C₄×C₁₉	—	2	4	no	—
131	C₂×C₄×C₁₉	—	2	2	no	—
132	C₂×C₄×C₁₉	—	4	2	no	—
133	C₂×C₄×C₁₉	—	4	2	no	—
134	C₂×C₄×C₁₉	—	4	4	no	—
135	C₂×C₄×C₁₉	—	4	4	no	—
136	C₂×C₄×C₁₉	C₄×D₄	1	4	yes	—
137	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₁₉	C₂×#41	1	1	yes	characteristic
142	C₂×C₄×C₁₉	C₂×#41	1	1	yes	characteristic
143	C₈×C₁₉	—	2	2	no	—
144	C₈×C₁₉	—	4	4	no	—
145	C₈×C₁₉	C₄×D₄	1	4	yes	—
146	C₈×C₁₉	C₄×D₄	1	4	yes	—
147	C₈×C₁₉	#127	1	4	yes	—
148	#3150	C₂×C₁₉	1	1	yes	characteristic
149	C₂×#444	C₂×C₁₉	1	2	yes	complemented
150	C₂×#443	C₂×C₁₉	1	2	yes	complemented
151	#2932	C₂×C₁₉	1	2	yes	complemented
152	#1975	C₂×C₁₉	1	2	yes	complemented
153	#1974	C₂×C₁₉	1	2	yes	complemented
154	#1962	C₂×C₁₉	1	2	yes	complemented
155	#1849	C₂×C₁₉	1	1	yes	characteristic, complemented
156	C₄×#97	C₂×C₁₉	1	1	yes	characteristic, complemented
157	C₂²×C₁₉	—	2	2	no	—
158	C₂²×C₁₉	—	4	4	no	—
159	C₂²×C₁₉	—	4	8	no	—
160	C₂²×C₁₉	C₂×C₄×D₄	1	1	yes	characteristic
161	C₂²×C₁₉	#532	1	1	yes	characteristic
162	C₂²×C₁₉	#532	1	1	yes	characteristic
163	C₄×C₁₉	—	2	4	no	—
164	C₄×C₁₉	—	2	4	no	—
165	C₄×C₁₉	—	4	2	no	—
166	C₄×C₁₉	—	4	4	no	—
167	C₄×C₁₉	C₂×C₄×D₄	1	1	yes	characteristic
168	C₄×C₁₉	C₂×C₄×D₄	1	1	yes	characteristic
169	C₄×C₁₉	C₂×#127	1	1	yes	characteristic
170	C₄×C₁₉	#532	1	1	yes	characteristic
171	C₂×#135	C₂²×C₁₉	1	2	yes	—
172	C₂×#135	C₂²×C₁₉	1	2	yes	—
173	C₂×#131	C₂²×C₁₉	1	4	yes	—
174	#532	C₂²×C₁₉	1	2	yes	—
175	#497	C₄×C₁₉	1	4	yes	complemented
176	#496	C₄×C₁₉	1	4	yes	complemented
177	#444	—	2	8	no	—
178	#443	—	2	8	no	—
179	#435	—	2	8	no	—
180	C₂×#108	C₂²×C₁₉	1	2	yes	—
181	C₂×#108	C₂²×C₁₉	1	2	yes	—
182	#423	C₂²×C₁₉	1	2	yes	—
183	C₂×#104	C₂²×C₁₉	1	4	yes	—
184	C₂×#104	C₂²×C₁₉	1	4	yes	—
185	#408	C₂²×C₁₉	1	4	yes	—
186	C₂×#97	C₂²×C₁₉	1	1	yes	characteristic
187	C₂×#97	C₂²×C₁₉	1	1	yes	characteristic
188	C₂×#97	C₂²×C₁₉	1	1	yes	characteristic
189	C₂×#97	C₂²×C₁₉	1	1	yes	characteristic
190	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
191	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
192	C₄³	C₂²×C₁₉	1	1	yes	characteristic
193	C₂×C₁₉	—	2	1	no	—
194	C₂×C₁₉	—	2	1	no	—
195	C₂×C₁₉	—	8	4	no	—
196	C₂×C₁₉	C₄×#127	1	1	yes	characteristic
197	C₂×C₁₉	#2956	1	2	yes	—
198	C₂×#41	C₂³×C₁₉	1	4	yes	—
199	#135	—	2	4	no	—
200	#135	—	2	4	no	—
201	#135	—	2	4	no	—
202	#135	—	2	4	no	—
203	C₂×Q₁₆	C₂×C₄×C₁₉	1	2	yes	—
204	C₂×Q₁₆	C₂×C₄×C₁₉	1	2	yes	—
205	C₂×SD₁₆	C₂×C₄×C₁₉	1	4	yes	—
206	C₂×SD₁₆	C₂×C₄×C₁₉	1	4	yes	—
207	C₂×D₈	C₂×C₄×C₁₉	1	2	yes	—
208	C₂×D₈	C₂×C₄×C₁₉	1	2	yes	—
209	#131	—	2	8	no	—
210	#131	—	2	8	no	—
211	C₂×M₁₆	—	2	4	no	—
212	C₂×M₁₆	C₂³×C₁₉	1	4	yes	—
213	C₂²×C₈	—	2	4	no	—
214	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
215	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
216	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
217	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
218	C₄⋊Q₈	C₂×C₄×C₁₉	1	2	yes	—
219	#127	C₂×C₄×C₁₉	1	2	yes	—
220	#124	C₂×C₄×C₁₉	1	4	yes	—
221	#121	—	2	4	no	—
222	C₄×D₄	—	2	4	no	—
223	C₂×C₄²	—	2	2	no	—
224	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
225	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
226	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
227	#108	—	2	4	no	—
228	#108	—	2	4	no	—
229	C₄⋊C₈	—	2	4	no	—
230	#104	—	2	8	no	—
231	#104	—	2	8	no	—
232	C₂²⋊C₈	—	2	8	no	—
233	#97	—	2	2	no	—
234	#97	C₂×C₄×C₁₉	1	4	yes	—
235	#97	D₄×C₁₉	1	4	yes	—
236	#97	D₄×C₁₉	1	4	yes	complemented
237	C₄×C₈	—	2	2	no	—
238	C₄×C₈	D₄×C₁₉	1	4	yes	complemented
239	C₁₉	#21710	1	1	yes	characteristic, direct product factor, Sylow
240	#41	—	2	4	no	—
241	#41	—	2	4	no	—
242	#41	—	2	4	no	—
243	#41	—	2	4	no	—
244	C₂×Q₈	C₂²×C₄×C₁₉	1	4	yes	—
245	C₂×D₄	—	2	4	no	—
246	C₂×D₄	—	2	4	no	—
247	C₂×D₄	C₂²×C₄×C₁₉	1	4	yes	—
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	—	2	4	no	—
251	C₂²×C₄	#41×C₁₉	1	2	yes	—
252	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
253	Q₁₆	—	2	4	no	—
254	Q₁₆	—	2	4	no	—
255	SD₁₆	—	2	8	no	—
256	SD₁₆	—	2	8	no	—
257	D₈	—	2	4	no	—
258	D₈	—	2	4	no	—
259	M₁₆	—	2	4	no	—
260	M₁₆	—	2	4	no	—
261	M₁₆	—	4	8	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	2	4	no	—
265	C₂×C₈	—	4	8	no	—
266	C₂×C₈	C₂²×C₄×C₁₉	1	2	yes	—
267	C₂×C₈	C₂²×C₄×C₁₉	1	2	yes	—
268	C₂×C₈	C₂×D₄×C₁₉	1	2	yes	—
269	C₂×C₈	C₂×D₄×C₁₉	1	2	yes	—
270	C₂×C₈	C₂×D₄×C₁₉	1	2	yes	—
271	C₂×C₈	C₂×D₄×C₁₉	1	2	yes	—
272	C₂×C₈	C₂×D₄×C₁₉	1	2	yes	—
273	C₂×C₈	C₂×D₄×C₁₉	1	2	yes	—
274	C₂×C₈	#41×C₁₉	1	2	yes	—
275	C₂×C₈	#41×C₁₉	1	2	yes	—
276	C₄⋊C₄	—	2	4	no	—
277	C₂²⋊C₄	—	2	8	no	—
278	C₄²	—	2	1	no	—
279	C₄²	—	2	1	no	—
280	C₄²	—	4	4	no	—
281	C₄²	C₂²×C₄×C₁₉	1	2	yes	—
282	C₄²	C₂×D₄×C₁₉	1	2	yes	—
283	C₄²	C₂×D₄×C₁₉	1	2	yes	—
284	C₄²	C₂×D₄×C₁₉	1	2	yes	—
285	C₂³	—	2	4	no	—
286	C₂³	C₂×#41×C₁₉	1	1	yes	characteristic
287	Q₈	—	2	4	no	—
288	Q₈	—	2	4	no	—
289	D₄	—	2	4	no	—
290	D₄	—	2	4	no	—
291	D₄	—	4	4	no	—
292	D₄	—	4	4	no	—
293	D₄	—	8	4	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	4	no	—
301	C₂×C₄	—	4	4	no	—
302	C₂×C₄	C₄×D₄×C₁₉	1	4	yes	—
303	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
304	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
305	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
306	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
307	C₂×C₄	C₂×#41×C₁₉	1	1	yes	characteristic
308	C₂×C₄	C₂×#41×C₁₉	1	1	yes	characteristic
309	C₈	—	2	2	no	—
310	C₈	—	4	4	no	—
311	C₈	C₄×D₄×C₁₉	1	4	yes	—
312	C₈	C₄×D₄×C₁₉	1	4	yes	—
313	C₈	C₁₉×#127	1	4	yes	—
314	C₂²	—	2	2	no	—
315	C₂²	—	4	8	no	—
316	C₂²	—	4	4	no	—
317	C₂²	C₂×C₄×D₄×C₁₉	1	1	yes	characteristic
318	C₂²	C₁₉×#532	1	1	yes	characteristic
319	C₂²	C₁₉×#532	1	1	yes	characteristic
320	C₄	—	2	4	no	—
321	C₄	—	2	4	no	—
322	C₄	—	4	2	no	—
323	C₄	—	4	4	no	—
324	C₄	C₂×C₄×D₄×C₁₉	1	1	yes	characteristic
325	C₄	C₂×C₄×D₄×C₁₉	1	1	yes	characteristic
326	C₄	C₂×C₁₉×#127	1	1	yes	characteristic
327	C₄	C₁₉×#532	1	1	yes	characteristic
328	C₂	—	2	1	no	—
329	C₂	—	2	1	no	—
330	C₂	—	8	4	no	—
331	C₂	C₄×C₁₉×#127	1	1	yes	characteristic
332	C₂	C₁₉×#2956	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₉	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
socle	C₂²×C₁₉	C₂×C₄×D₄
prime radical	C₁₉×#3150	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉ ◁ C₂×C₁₉×#97 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₉×#444	C₂	semidirect
3. C₂×C₁₉×#443	C₂	semidirect
4. C₁₉×#2932	C₂	semidirect
5. C₁₉×#1975	C₂	semidirect
6. C₁₉×#1974	C₂	semidirect
7. C₁₉×#1962	C₂	semidirect
8. C₁₉×#1849	C₂	semidirect
9. C₄×C₁₉×#97	C₂	semidirect
14. C₁₉×#497	C₄	semidirect
15. C₁₉×#496	C₄	semidirect
70. C₁₉×#97	D₄	semidirect
72. C₄×C₈×C₁₉	D₄	semidirect
118. #21710	C₁₉	direct (N × H)
149. C₂×#444	C₂×C₁₉	semidirect
150. C₂×#443	C₂×C₁₉	semidirect
151. #2932	C₂×C₁₉	semidirect
152. #1975	C₂×C₁₉	semidirect
153. #1974	C₂×C₁₉	semidirect
154. #1962	C₂×C₁₉	semidirect
155. #1849	C₂×C₁₉	semidirect
156. C₄×#97	C₂×C₁₉	semidirect
175. #497	C₄×C₁₉	semidirect
176. #496	C₄×C₁₉	semidirect
236. #97	D₄×C₁₉	semidirect
238. C₄×C₈	D₄×C₁₉	semidirect
239. C₁₉	#21710	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₁₉×#3150	2
C₂×C₁₉×#444	2
C₂×C₁₉×#443	2
C₁₉×#2932	2
C₁₉×#1975	2
C₁₉×#1974	2
C₁₉×#1962	2
C₁₉×#1849	2
C₄×C₁₉×#97	2
#21710	19

Subgroups — C19×#21710 #18027669

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#21710 #18027669