Subgroups — C₁₉×#32766 #18038725

Order 4864.

836 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#2621	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₉×#2619	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#2074	C₂	1	2	yes	maximal, complemented
4	C₁₉×#2073	C₂	1	2	yes	maximal, complemented
5	C₁₉×#2014	C₂	1	2	yes	maximal, complemented
6	C₁₉×#2013	C₂	1	2	yes	maximal, complemented
7	C₁₉×#2000	C₂	1	4	yes	maximal, complemented
8	C₁₉×#1818	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×SD₁₆×C₁₉	C₂²	1	4	yes	—
10	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₁₉×#127	C₂²	1	1	yes	characteristic
12	C₂×C₁₉×C₂²⋊Q₈	C₂²	1	2	yes	—
13	C₂×C₁₉×#121	C₂²	1	2	yes	—
14	C₂×C₁₉×#117	C₂²	1	1	yes	characteristic
15	C₂×C₁₉×#103	C₂²	1	4	yes	—
16	C₂×C₁₉×#102	C₂²	1	4	yes	—
17	C₂×C₁₉×C₂²⋊C₈	C₂²	1	2	yes	—
18	C₁₉×#398	—	2	2	no	—
19	C₁₉×#397	—	2	2	no	—
20	C₁₉×#394	—	2	2	no	—
21	C₁₉×#392	—	2	2	no	—
22	C₁₉×#391	C₂²	1	2	yes	—
23	C₁₉×#390	C₂²	1	2	yes	—
24	C₁₉×#382	C₂²	1	1	yes	characteristic
25	C₁₉×#382	C₂²	1	1	yes	characteristic
26	C₁₉×#340	C₂²	1	4	yes	—
27	C₁₉×#336	C₂²	1	2	yes	—
28	C₁₉×#336	C₂²	1	2	yes	—
29	C₂²×Q₈×C₁₉	C₂³	1	2	yes	—
30	C₂²×D₄×C₁₉	—	2	2	no	—
31	C₂²×D₄×C₁₉	—	2	2	no	—
32	C₂²×D₄×C₁₉	C₂³	1	2	yes	—
33	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
34	C₂×SD₁₆×C₁₉	—	4	4	no	—
35	C₂×SD₁₆×C₁₉	—	4	4	no	—
36	C₂×SD₁₆×C₁₉	—	4	4	no	—
37	C₂²×C₈×C₁₉	C₂³	1	4	yes	—
38	C₁₉×C₄⋊Q₈	—	4	2	no	—
39	C₁₉×#127	—	4	2	no	—
40	C₁₉×C₂²⋊Q₈	—	4	4	no	—
41	C₁₉×#121	—	4	4	no	—
42	C₁₉×#117	—	4	2	no	—
43	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
44	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
45	C₂×C₄⋊C₄×C₁₉	D₄	1	2	yes	—
46	C₂×C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
48	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
49	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
50	C₂×C₂²⋊C₄×C₁₉	—	2	4	no	—
51	C₂×C₂²⋊C₄×C₁₉	D₄	1	2	yes	—
52	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
54	C₁₉×#103	—	4	4	no	—
55	C₁₉×#102	—	4	4	no	—
56	C₁₉×C₂²⋊C₈	—	4	2	no	—
57	C₁₉×#95	—	2	2	no	—
58	C₁₉×#95	—	2	2	no	—
59	C₁₉×#95	—	2	4	no	—
60	C₂⁴×C₁₉	—	2	2	no	—
61	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
62	C₂×Q₈×C₁₉	—	2	4	no	—
63	C₂×Q₈×C₁₉	—	2	4	no	—
64	C₂×Q₈×C₁₉	—	2	2	no	—
65	C₂×Q₈×C₁₉	—	2	2	no	—
66	C₂×D₄×C₁₉	—	2	4	no	—
67	C₂×D₄×C₁₉	—	2	4	no	—
68	C₂×D₄×C₁₉	—	2	2	no	—
69	C₂×D₄×C₁₉	—	2	2	no	—
70	C₂×D₄×C₁₉	—	4	4	no	—
71	C₂×D₄×C₁₉	—	4	4	no	—
72	C₂×D₄×C₁₉	—	8	4	no	—
73	C₂×D₄×C₁₉	—	8	4	no	—
74	C₂²×C₄×C₁₉	—	2	2	no	—
75	C₂²×C₄×C₁₉	—	2	2	no	—
76	C₂²×C₄×C₁₉	—	2	2	no	—
77	C₂²×C₄×C₁₉	—	2	2	no	—
78	C₂²×C₄×C₁₉	—	2	2	no	—
79	C₂²×C₄×C₁₉	—	2	2	no	—
80	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
84	SD₁₆×C₁₉	—	4	16	no	—
85	C₂×C₈×C₁₉	—	2	4	no	—
86	C₂×C₈×C₁₉	—	2	4	no	—
87	C₂×C₈×C₁₉	—	2	4	no	—
88	C₄⋊C₄×C₁₉	—	2	4	no	—
89	C₄⋊C₄×C₁₉	—	2	4	no	—
90	C₄⋊C₄×C₁₉	—	4	2	no	—
91	C₄⋊C₄×C₁₉	—	4	4	no	—
92	C₄⋊C₄×C₁₉	—	4	4	no	—
93	C₂²⋊C₄×C₁₉	—	4	2	no	—
94	C₂²⋊C₄×C₁₉	—	4	4	no	—
95	C₂²⋊C₄×C₁₉	—	4	4	no	—
96	C₂²⋊C₄×C₁₉	—	8	4	no	—
97	C₄²×C₁₉	—	2	2	no	—
98	C₄²×C₁₉	—	2	2	no	—
99	#32766	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂³×C₁₉	—	2	1	no	—
101	C₂³×C₁₉	—	2	1	no	—
102	C₂³×C₁₉	—	2	1	no	—
103	C₂³×C₁₉	—	2	4	no	—
104	C₂³×C₁₉	—	4	2	no	—
105	C₂³×C₁₉	—	4	2	no	—
106	C₂³×C₁₉	—	4	2	no	—
107	C₂³×C₁₉	—	4	4	no	—
108	C₂³×C₁₉	—	4	4	no	—
109	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
110	Q₈×C₁₉	—	4	4	no	—
111	Q₈×C₁₉	—	4	4	no	—
112	D₄×C₁₉	—	4	4	no	—
113	D₄×C₁₉	—	4	4	no	—
114	D₄×C₁₉	—	8	8	no	—
115	D₄×C₁₉	—	8	8	no	—
116	C₂×C₄×C₁₉	—	2	2	no	—
117	C₂×C₄×C₁₉	—	2	2	no	—
118	C₂×C₄×C₁₉	—	2	1	no	—
119	C₂×C₄×C₁₉	—	2	1	no	—
120	C₂×C₄×C₁₉	—	2	1	no	—
121	C₂×C₄×C₁₉	—	2	4	no	—
122	C₂×C₄×C₁₉	—	2	4	no	—
123	C₂×C₄×C₁₉	—	4	4	no	—
124	C₂×C₄×C₁₉	—	4	2	no	—
125	C₂×C₄×C₁₉	—	4	2	no	—
126	C₂×C₄×C₁₉	—	4	2	no	—
127	C₂×C₄×C₁₉	—	4	2	no	—
128	C₂×C₄×C₁₉	—	4	2	no	—
129	C₂×C₄×C₁₉	—	4	4	no	—
130	C₂×C₄×C₁₉	—	4	4	no	—
131	C₂×C₄×C₁₉	H(5,𝔽₂)	1	2	yes	—
132	C₂×C₄×C₁₉	H(5,𝔽₂)	1	2	yes	—
133	C₂×C₄×C₁₉	H(5,𝔽₂)	1	2	yes	—
134	C₈×C₁₉	—	4	4	no	—
135	#2621	C₂×C₁₉	1	1	yes	characteristic, complemented
136	#2619	C₂×C₁₉	1	1	yes	characteristic
137	#2074	C₂×C₁₉	1	2	yes	complemented
138	#2073	C₂×C₁₉	1	2	yes	complemented
139	#2014	C₂×C₁₉	1	2	yes	complemented
140	#2013	C₂×C₁₉	1	2	yes	complemented
141	#2000	C₂×C₁₉	1	4	yes	complemented
142	#1818	C₂×C₁₉	1	1	yes	characteristic, complemented
143	C₂²×C₁₉	—	4	2	no	—
144	C₂²×C₁₉	—	4	2	no	—
145	C₂²×C₁₉	—	4	2	no	—
146	C₂²×C₁₉	—	4	1	no	—
147	C₂²×C₁₉	—	8	4	no	—
148	C₂²×C₁₉	—	8	4	no	—
149	C₂²×C₁₉	—	8	2	no	—
150	C₂²×C₁₉	—	8	2	no	—
151	C₂²×C₁₉	—	8	2	no	—
152	C₂²×C₁₉	#534	1	1	yes	characteristic
153	C₂²×C₁₉	#535	1	1	yes	characteristic
154	C₂²×C₁₉	#535	1	1	yes	characteristic
155	C₂²×C₁₉	#577	1	4	yes	—
156	C₄×C₁₉	—	4	2	no	—
157	C₄×C₁₉	—	4	2	no	—
158	C₄×C₁₉	—	4	1	no	—
159	C₄×C₁₉	—	8	2	no	—
160	C₄×C₁₉	—	8	2	no	—
161	C₄×C₁₉	—	8	2	no	—
162	C₄×C₁₉	#560	1	4	yes	—
163	C₂²×SD₁₆	C₂²×C₁₉	1	4	yes	—
164	C₂×C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
165	C₂×#127	C₂²×C₁₉	1	1	yes	characteristic
166	C₂×C₂²⋊Q₈	C₂²×C₁₉	1	2	yes	—
167	C₂×#121	C₂²×C₁₉	1	2	yes	—
168	C₂×#117	C₂²×C₁₉	1	1	yes	characteristic
169	C₂×#103	C₂²×C₁₉	1	4	yes	—
170	C₂×#102	C₂²×C₁₉	1	4	yes	—
171	C₂×C₂²⋊C₈	C₂²×C₁₉	1	2	yes	—
172	#398	—	2	2	no	—
173	#397	—	2	2	no	—
174	#394	—	2	2	no	—
175	#392	—	2	2	no	—
176	#391	C₂²×C₁₉	1	2	yes	—
177	#390	C₂²×C₁₉	1	2	yes	—
178	#382	C₂²×C₁₉	1	1	yes	characteristic
179	#382	C₂²×C₁₉	1	1	yes	characteristic
180	#340	C₂²×C₁₉	1	4	yes	—
181	#336	C₂²×C₁₉	1	2	yes	—
182	#336	C₂²×C₁₉	1	2	yes	—
183	C₂×C₁₉	—	8	1	no	—
184	C₂×C₁₉	—	16	2	no	—
185	C₂×C₁₉	#2815	1	1	yes	characteristic
186	C₂×C₁₉	#3200	1	2	yes	—
187	C₂×C₁₉	#3221	1	2	yes	—
188	C₂×C₁₉	#3222	1	2	yes	—
189	C₂²×Q₈	C₂³×C₁₉	1	2	yes	—
190	C₂²×D₄	—	2	2	no	—
191	C₂²×D₄	—	2	2	no	—
192	C₂²×D₄	C₂³×C₁₉	1	2	yes	—
193	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
194	C₂×SD₁₆	—	4	4	no	—
195	C₂×SD₁₆	—	4	4	no	—
196	C₂×SD₁₆	—	4	4	no	—
197	C₂²×C₈	C₂³×C₁₉	1	4	yes	—
198	C₄⋊Q₈	—	4	2	no	—
199	#127	—	4	2	no	—
200	C₂²⋊Q₈	—	4	4	no	—
201	#121	—	4	4	no	—
202	#117	—	4	2	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	D₄×C₁₉	1	2	yes	—
206	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
207	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
208	C₂×C₂²⋊C₄	—	2	2	no	—
209	C₂×C₂²⋊C₄	—	2	2	no	—
210	C₂×C₂²⋊C₄	—	2	4	no	—
211	C₂×C₂²⋊C₄	D₄×C₁₉	1	2	yes	—
212	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
213	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
214	#103	—	4	4	no	—
215	#102	—	4	4	no	—
216	C₂²⋊C₈	—	4	2	no	—
217	#95	—	2	2	no	—
218	#95	—	2	2	no	—
219	#95	—	2	4	no	—
220	C₁₉	#32766	1	1	yes	characteristic, direct product factor, Sylow
221	C₂⁴	—	2	2	no	—
222	C₂⁴	C₂×D₄×C₁₉	1	1	yes	characteristic
223	C₂×Q₈	—	2	2	no	—
224	C₂×Q₈	—	2	4	no	—
225	C₂×Q₈	—	2	4	no	—
226	C₂×Q₈	—	2	2	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	2	4	no	—
229	C₂×D₄	—	2	4	no	—
230	C₂×D₄	—	2	2	no	—
231	C₂×D₄	—	4	4	no	—
232	C₂×D₄	—	4	4	no	—
233	C₂×D₄	—	8	4	no	—
234	C₂×D₄	—	8	4	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
242	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
243	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
244	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
245	SD₁₆	—	4	16	no	—
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	2	4	no	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄⋊C₄	—	4	4	no	—
253	C₄⋊C₄	—	4	4	no	—
254	C₂²⋊C₄	—	4	2	no	—
255	C₂²⋊C₄	—	4	4	no	—
256	C₂²⋊C₄	—	4	4	no	—
257	C₂²⋊C₄	—	8	4	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	2	2	no	—
260	C₂³	—	2	1	no	—
261	C₂³	—	2	1	no	—
262	C₂³	—	2	1	no	—
263	C₂³	—	2	4	no	—
264	C₂³	—	4	2	no	—
265	C₂³	—	4	2	no	—
266	C₂³	—	4	2	no	—
267	C₂³	—	4	4	no	—
268	C₂³	—	4	4	no	—
269	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
270	Q₈	—	4	4	no	—
271	Q₈	—	4	4	no	—
272	D₄	—	4	4	no	—
273	D₄	—	4	4	no	—
274	D₄	—	8	8	no	—
275	D₄	—	8	8	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	4	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	C₁₉×H(5,𝔽₂)	1	2	yes	—
292	C₂×C₄	C₁₉×H(5,𝔽₂)	1	2	yes	—
293	C₂×C₄	C₁₉×H(5,𝔽₂)	1	2	yes	—
294	C₈	—	4	4	no	—
295	C₂²	—	4	2	no	—
296	C₂²	—	4	2	no	—
297	C₂²	—	4	2	no	—
298	C₂²	—	4	1	no	—
299	C₂²	—	8	4	no	—
300	C₂²	—	8	4	no	—
301	C₂²	—	8	2	no	—
302	C₂²	—	8	2	no	—
303	C₂²	—	8	2	no	—
304	C₂²	C₁₉×#534	1	1	yes	characteristic
305	C₂²	C₁₉×#535	1	1	yes	characteristic
306	C₂²	C₁₉×#535	1	1	yes	characteristic
307	C₂²	C₁₉×#577	1	4	yes	—
308	C₄	—	4	2	no	—
309	C₄	—	4	2	no	—
310	C₄	—	4	1	no	—
311	C₄	—	8	2	no	—
312	C₄	—	8	2	no	—
313	C₄	—	8	2	no	—
314	C₄	C₁₉×#560	1	4	yes	—
315	C₂	—	8	1	no	—
316	C₂	—	16	2	no	—
317	C₂	C₁₉×#2815	1	1	yes	characteristic
318	C₂	C₁₉×#3200	1	2	yes	—
319	C₂	C₁₉×#3221	1	2	yes	—
320	C₂	C₁₉×#3222	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
prime radical	C₁₉×#2619	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ C₂×C₁₉×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₉×#2621	C₂	semidirect
3. C₁₉×#2074	C₂	semidirect
4. C₁₉×#2073	C₂	semidirect
5. C₁₉×#2014	C₂	semidirect
6. C₁₉×#2013	C₂	semidirect
7. C₁₉×#2000	C₂	semidirect
8. C₁₉×#1818	C₂	semidirect
99. #32766	C₁₉	direct (N × H)
135. #2621	C₂×C₁₉	semidirect
137. #2074	C₂×C₁₉	semidirect
138. #2073	C₂×C₁₉	semidirect
139. #2014	C₂×C₁₉	semidirect
140. #2013	C₂×C₁₉	semidirect
141. #2000	C₂×C₁₉	semidirect
142. #1818	C₂×C₁₉	semidirect
220. C₁₉	#32766	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×#2621	2
C₁₉×#2619	2
C₁₉×#2074	2
C₁₉×#2073	2
C₁₉×#2014	2
C₁₉×#2013	2
C₁₉×#2000	2
C₁₉×#1818	2
#32766	19

Subgroups — C19×#32766 #18038725

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#32766 #18038725