Subgroups — C₁₉×#32692 #18038651

Order 4864.

610 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×#2584	C₂	1	2	yes	maximal
2	C₁₉×#2088	C₂	1	2	yes	maximal
3	C₁₉×#2059	C₂	1	2	yes	maximal
4	C₁₉×#2059	C₂	1	2	yes	maximal
5	C₁₉×#1985	C₂	1	1	yes	maximal, characteristic
6	C₁₉×#1944	C₂	1	1	yes	maximal, characteristic
7	C₁₉×#1944	C₂	1	1	yes	maximal, characteristic
8	C₁₉×#1920	C₂	1	2	yes	maximal
9	C₁₉×#1919	C₂	1	2	yes	maximal
10	C₂×C₁₉×#125	C₂²	1	2	yes	—
11	C₂×C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
12	C₂×C₁₉×#107	C₂²	1	1	yes	characteristic
13	C₂×C₁₉×#107	C₂²	1	1	yes	characteristic
14	C₂×C₁₉×#106	C₂²	1	2	yes	—
15	C₂×C₁₉×#106	C₂²	1	1	yes	characteristic
16	C₂×C₁₉×#106	C₂²	1	1	yes	characteristic
17	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₁₉×#97	C₂²	1	1	yes	characteristic
19	C₁₉×#398	—	2	2	no	—
20	C₁₉×#395	—	2	2	no	—
21	C₁₉×#389	—	2	2	no	—
22	C₁₉×#389	—	2	2	no	—
23	C₁₉×#389	C₂²	1	2	yes	—
24	C₁₉×#389	C₂²	1	2	yes	—
25	C₁₉×#389	C₂²	1	2	yes	—
26	C₁₉×#389	C₂²	1	2	yes	—
27	C₄×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
28	C₁₉×#340	C₂²	1	2	yes	—
29	C₁₉×#340	C₂²	1	2	yes	—
30	C₁₉×#340	C₂²	1	2	yes	—
31	C₁₉×#340	C₂²	1	2	yes	—
32	C₁₉×#340	C₂²	1	2	yes	—
33	C₁₉×#336	C₂²	1	2	yes	—
34	C₁₉×#336	C₂²	1	2	yes	—
35	C₂²×Q₈×C₁₉	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₉	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
39	C₁₉×#125	—	4	2	no	—
40	C₁₉×#125	—	4	2	no	—
41	C₄×Q₈×C₁₉	—	4	1	no	—
42	C₄×Q₈×C₁₉	—	4	1	no	—
43	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
44	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
45	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
46	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
47	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
48	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
49	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₁₉	D₄	1	2	yes	—
51	C₂×C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
52	C₂×C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
53	C₂×C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
54	C₂×C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
55	C₂×C₄²×C₁₉	—	2	2	no	—
56	C₂×C₄²×C₁₉	C₂³	1	2	yes	—
57	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
58	C₁₉×#107	—	4	1	no	—
59	C₁₉×#107	—	4	1	no	—
60	C₁₉×#106	—	4	2	no	—
61	C₁₉×#106	Q₈	1	4	yes	complemented
62	C₁₉×#106	Q₈	1	4	yes	complemented
63	C₁₉×C₄⋊C₈	—	4	1	no	—
64	C₁₉×#97	—	4	1	no	—
65	C₁₉×#95	—	2	2	no	—
66	C₁₉×#95	—	2	2	no	—
67	C₁₉×#95	—	2	2	no	—
68	C₁₉×#95	—	2	2	no	—
69	C₁₉×#95	—	2	2	no	—
70	C₂×Q₈×C₁₉	—	4	1	no	—
71	C₂×Q₈×C₁₉	—	4	2	no	—
72	C₂²×C₄×C₁₉	—	2	2	no	—
73	C₂²×C₄×C₁₉	—	2	2	no	—
74	C₂²×C₄×C₁₉	—	2	2	no	—
75	C₂²×C₄×C₁₉	—	2	2	no	—
76	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
81	C₂×C₈×C₁₉	—	2	1	no	—
82	C₂×C₈×C₁₉	—	2	1	no	—
83	C₂×C₈×C₁₉	—	2	1	no	—
84	C₂×C₈×C₁₉	—	2	1	no	—
85	C₂×C₈×C₁₉	—	2	2	no	—
86	C₂×C₈×C₁₉	—	2	2	no	—
87	C₂×C₈×C₁₉	—	2	2	no	—
88	C₂×C₈×C₁₉	C₂×Q₈	1	2	yes	—
89	C₂×C₈×C₁₉	C₂×Q₈	1	2	yes	—
90	C₄⋊C₄×C₁₉	—	2	2	no	—
91	C₄⋊C₄×C₁₉	—	2	2	no	—
92	C₄⋊C₄×C₁₉	—	2	2	no	—
93	C₄⋊C₄×C₁₉	—	2	2	no	—
94	C₄⋊C₄×C₁₉	—	2	2	no	—
95	C₄⋊C₄×C₁₉	—	2	2	no	—
96	C₄⋊C₄×C₁₉	—	4	1	no	—
97	C₄⋊C₄×C₁₉	—	4	2	no	—
98	C₄⋊C₄×C₁₉	—	4	4	no	—
99	C₄⋊C₄×C₁₉	—	4	4	no	—
100	C₄⋊C₄×C₁₉	—	4	4	no	—
101	C₄⋊C₄×C₁₉	—	4	4	no	—
102	C₄⋊C₄×C₁₉	—	8	2	no	—
103	C₄⋊C₄×C₁₉	—	8	2	no	—
104	C₄⋊C₄×C₁₉	C₂×Q₈	1	4	yes	—
105	C₄⋊C₄×C₁₉	C₂×Q₈	1	4	yes	—
106	C₄²×C₁₉	—	2	2	no	—
107	C₄²×C₁₉	—	2	2	no	—
108	C₄²×C₁₉	—	2	1	no	—
109	C₄²×C₁₉	—	2	1	no	—
110	C₄²×C₁₉	—	8	2	no	—
111	#32692	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
112	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
113	Q₈×C₁₉	—	4	4	no	—
114	C₂×C₄×C₁₉	—	2	2	no	—
115	C₂×C₄×C₁₉	—	2	2	no	—
116	C₂×C₄×C₁₉	—	2	1	no	—
117	C₂×C₄×C₁₉	—	2	1	no	—
118	C₂×C₄×C₁₉	—	2	4	no	—
119	C₂×C₄×C₁₉	—	2	4	no	—
120	C₂×C₄×C₁₉	—	2	2	no	—
121	C₂×C₄×C₁₉	—	2	2	no	—
122	C₂×C₄×C₁₉	—	2	2	no	—
123	C₂×C₄×C₁₉	—	2	2	no	—
124	C₂×C₄×C₁₉	—	2	2	no	—
125	C₂×C₄×C₁₉	—	2	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	2	no	—
130	C₂×C₄×C₁₉	—	4	2	no	—
131	C₂×C₄×C₁₉	—	4	2	no	—
132	C₂×C₄×C₁₉	—	4	2	no	—
133	C₂×C₄×C₁₉	—	4	2	no	—
134	C₂×C₄×C₁₉	—	4	2	no	—
135	C₂×C₄×C₁₉	—	4	2	no	—
136	C₂×C₄×C₁₉	C₂²×Q₈	1	1	yes	characteristic
137	C₂×C₄×C₁₉	C₂²×Q₈	1	1	yes	characteristic
138	C₂×C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
139	C₂×C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
140	C₂×C₄×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
141	C₂×C₄×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
142	C₈×C₁₉	—	2	2	no	—
143	C₈×C₁₉	—	2	2	no	—
144	C₈×C₁₉	—	4	2	no	—
145	#2584	C₂×C₁₉	1	2	yes	—
146	#2088	C₂×C₁₉	1	2	yes	—
147	#2059	C₂×C₁₉	1	2	yes	—
148	#2059	C₂×C₁₉	1	2	yes	—
149	#1985	C₂×C₁₉	1	1	yes	characteristic
150	#1944	C₂×C₁₉	1	1	yes	characteristic
151	#1944	C₂×C₁₉	1	1	yes	characteristic
152	#1920	C₂×C₁₉	1	2	yes	—
153	#1919	C₂×C₁₉	1	2	yes	—
154	C₂²×C₁₉	#534	1	1	yes	characteristic
155	C₂²×C₁₉	D₄×Q₈	1	1	yes	characteristic
156	C₂²×C₁₉	D₄×Q₈	1	1	yes	characteristic
157	C₂²×C₁₉	#575	1	2	yes	—
158	C₂²×C₁₉	#577	1	2	yes	—
159	C₄×C₁₉	—	4	4	no	—
160	C₄×C₁₉	—	4	4	no	—
161	C₄×C₁₉	—	4	2	no	—
162	C₄×C₁₉	—	4	1	no	—
163	C₄×C₁₉	—	4	2	no	—
164	C₄×C₁₉	—	4	1	no	—
165	C₄×C₁₉	—	8	2	no	—
166	C₄×C₁₉	—	8	2	no	—
167	C₄×C₁₉	#544	1	1	yes	characteristic
168	C₄×C₁₉	#544	1	1	yes	characteristic
169	C₄×C₁₉	Q₈²	1	1	yes	characteristic
170	C₄×C₁₉	#563	1	1	yes	characteristic
171	C₂×#125	C₂²×C₁₉	1	2	yes	—
172	C₂×C₄×Q₈	C₂²×C₁₉	1	1	yes	characteristic
173	C₂×#107	C₂²×C₁₉	1	1	yes	characteristic
174	C₂×#107	C₂²×C₁₉	1	1	yes	characteristic
175	C₂×#106	C₂²×C₁₉	1	2	yes	—
176	C₂×#106	C₂²×C₁₉	1	1	yes	characteristic
177	C₂×#106	C₂²×C₁₉	1	1	yes	characteristic
178	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
179	C₂×#97	C₂²×C₁₉	1	1	yes	characteristic
180	#398	—	2	2	no	—
181	#395	—	2	2	no	—
182	#389	—	2	2	no	—
183	#389	—	2	2	no	—
184	#389	C₂²×C₁₉	1	2	yes	—
185	#389	C₂²×C₁₉	1	2	yes	—
186	#389	C₂²×C₁₉	1	2	yes	—
187	#389	C₂²×C₁₉	1	2	yes	—
188	C₄×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
189	#340	C₂²×C₁₉	1	2	yes	—
190	#340	C₂²×C₁₉	1	2	yes	—
191	#340	C₂²×C₁₉	1	2	yes	—
192	#340	C₂²×C₁₉	1	2	yes	—
193	#340	C₂²×C₁₉	1	2	yes	—
194	#336	C₂²×C₁₉	1	2	yes	—
195	#336	C₂²×C₁₉	1	2	yes	—
196	C₂×C₁₉	#2734	1	1	yes	characteristic
197	C₂×C₁₉	#3197	1	1	yes	characteristic
198	C₂×C₁₉	#3204	1	1	yes	characteristic
199	C₂×C₁₉	#3388	1	2	yes	—
200	C₂×C₁₉	#3388	1	2	yes	—
201	C₂²×Q₈	D₄×C₁₉	1	1	yes	characteristic
202	C₂²×C₈	C₂³×C₁₉	1	2	yes	—
203	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
204	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
205	#125	—	4	2	no	—
206	#125	—	4	2	no	—
207	C₄×Q₈	—	4	1	no	—
208	C₄×Q₈	—	4	1	no	—
209	C₂×C₄⋊C₄	—	2	2	no	—
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₄⋊C₄	—	2	2	no	—
212	C₂×C₄⋊C₄	—	2	2	no	—
213	C₂×C₄⋊C₄	—	2	2	no	—
214	C₂×C₄⋊C₄	—	2	2	no	—
215	C₂×C₄⋊C₄	D₄×C₁₉	1	1	yes	characteristic
216	C₂×C₄⋊C₄	D₄×C₁₉	1	2	yes	—
217	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
218	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
219	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
220	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
221	C₂×C₄²	—	2	2	no	—
222	C₂×C₄²	C₂³×C₁₉	1	2	yes	—
223	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
224	#107	—	4	1	no	—
225	#107	—	4	1	no	—
226	#106	—	4	2	no	—
227	#106	Q₈×C₁₉	1	4	yes	complemented
228	#106	Q₈×C₁₉	1	4	yes	complemented
229	C₄⋊C₈	—	4	1	no	—
230	#97	—	4	1	no	—
231	#95	—	2	2	no	—
232	#95	—	2	2	no	—
233	#95	—	2	2	no	—
234	#95	—	2	2	no	—
235	#95	—	2	2	no	—
236	C₁₉	#32692	1	1	yes	characteristic, direct product factor, Sylow
237	C₂×Q₈	—	4	1	no	—
238	C₂×Q₈	—	4	2	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
245	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
246	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
247	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	C₂×Q₈×C₁₉	1	2	yes	—
256	C₂×C₈	C₂×Q₈×C₁₉	1	2	yes	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	4	1	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	—	4	4	no	—
266	C₄⋊C₄	—	4	4	no	—
267	C₄⋊C₄	—	4	4	no	—
268	C₄⋊C₄	—	4	4	no	—
269	C₄⋊C₄	—	8	2	no	—
270	C₄⋊C₄	—	8	2	no	—
271	C₄⋊C₄	C₂×Q₈×C₁₉	1	4	yes	—
272	C₄⋊C₄	C₂×Q₈×C₁₉	1	4	yes	—
273	C₄²	—	2	1	no	—
274	C₄²	—	2	1	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	2	2	no	—
277	C₄²	—	8	2	no	—
278	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
279	Q₈	—	4	4	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	C₂²×Q₈×C₁₉	1	1	yes	characteristic
303	C₂×C₄	C₂²×Q₈×C₁₉	1	1	yes	characteristic
304	C₂×C₄	C₁₉×H(5,𝔽₂)	1	1	yes	characteristic
305	C₂×C₄	C₁₉×H(5,𝔽₂)	1	1	yes	characteristic
306	C₂×C₄	C₁₉×2₋¹⁺⁴	1	1	yes	characteristic
307	C₂×C₄	C₁₉×2₋¹⁺⁴	1	1	yes	characteristic
308	C₈	—	2	2	no	—
309	C₈	—	2	2	no	—
310	C₈	—	4	2	no	—
311	C₂²	C₁₉×#534	1	1	yes	characteristic
312	C₂²	D₄×Q₈×C₁₉	1	1	yes	characteristic
313	C₂²	D₄×Q₈×C₁₉	1	1	yes	characteristic
314	C₂²	C₁₉×#575	1	2	yes	—
315	C₂²	C₁₉×#577	1	2	yes	—
316	C₄	—	4	2	no	—
317	C₄	—	4	1	no	—
318	C₄	—	4	2	no	—
319	C₄	—	4	1	no	—
320	C₄	—	4	4	no	—
321	C₄	—	4	4	no	—
322	C₄	—	8	2	no	—
323	C₄	—	8	2	no	—
324	C₄	C₁₉×#544	1	1	yes	characteristic
325	C₄	C₁₉×#544	1	1	yes	characteristic
326	C₄	Q₈²×C₁₉	1	1	yes	characteristic
327	C₄	C₁₉×#563	1	1	yes	characteristic
328	C₂	C₁₉×#2734	1	1	yes	characteristic
329	C₂	C₁₉×#3197	1	1	yes	characteristic
330	C₂	C₁₉×#3204	1	1	yes	characteristic
331	C₂	C₁₉×#3388	1	2	yes	—
332	C₂	C₁₉×#3388	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ C₂×C₄×Q₈×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
61. C₁₉×#106	Q₈	semidirect
62. C₁₉×#106	Q₈	semidirect
111. #32692	C₁₉	direct (N × H)
227. #106	Q₈×C₁₉	semidirect
228. #106	Q₈×C₁₉	semidirect
236. C₁₉	#32692	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×#2584	2
C₁₉×#2088	2
C₁₉×#2059	2
C₁₉×#1985	2
C₁₉×#1944	2
C₁₉×#1920	2
C₁₉×#1919	2
#32692	19

Subgroups — C19×#32692 #18038651

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#32692 #18038651