Subgroups — C₁₉×#32686 #18038645

Order 4864.

616 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	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×#1984	C₂	1	1	yes	maximal, characteristic
6	C₁₉×#1942	C₂	1	1	yes	maximal, characteristic
7	C₁₉×#1942	C₂	1	1	yes	maximal, characteristic
8	C₁₉×#1921	C₂	1	2	yes	maximal
9	C₁₉×#1918	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×#107	C₂²	1	1	yes	characteristic
15	C₂×C₁₉×#107	C₂²	1	1	yes	characteristic
16	C₂×C₁₉×#106	C₂²	1	2	yes	—
17	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₁₉	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×#107	Q₈	1	4	yes	complemented
61	C₁₉×#107	Q₈	1	4	yes	complemented
62	C₁₉×#106	—	4	2	no	—
63	C₁₉×C₄⋊C₈	—	4	1	no	—
64	C₄×C₈×C₁₉	—	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	—	2	2	no	—
71	C₂×Q₈×C₁₉	—	4	1	no	—
72	C₂×Q₈×C₁₉	—	4	1	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	—	2	2	no	—
77	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
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	1	no	—
86	C₂×C₈×C₁₉	—	2	2	no	—
87	C₂×C₈×C₁₉	—	2	2	no	—
88	C₂×C₈×C₁₉	—	2	2	no	—
89	C₂×C₈×C₁₉	C₂×Q₈	1	2	yes	—
90	C₂×C₈×C₁₉	C₂×Q₈	1	2	yes	—
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	—	2	2	no	—
97	C₄⋊C₄×C₁₉	—	4	1	no	—
98	C₄⋊C₄×C₁₉	—	4	2	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	—	4	4	no	—
103	C₄⋊C₄×C₁₉	—	8	2	no	—
104	C₄⋊C₄×C₁₉	—	8	2	no	—
105	C₄⋊C₄×C₁₉	C₂×Q₈	1	4	yes	—
106	C₄⋊C₄×C₁₉	C₂×Q₈	1	4	yes	—
107	C₄²×C₁₉	—	2	2	no	—
108	C₄²×C₁₉	—	2	2	no	—
109	C₄²×C₁₉	—	2	1	no	—
110	C₄²×C₁₉	—	2	1	no	—
111	C₄²×C₁₉	—	8	2	no	—
112	#32686	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
114	Q₈×C₁₉	—	4	4	no	—
115	C₂×C₄×C₁₉	—	2	1	no	—
116	C₂×C₄×C₁₉	—	2	1	no	—
117	C₂×C₄×C₁₉	—	2	2	no	—
118	C₂×C₄×C₁₉	—	2	2	no	—
119	C₂×C₄×C₁₉	—	2	1	no	—
120	C₂×C₄×C₁₉	—	2	4	no	—
121	C₂×C₄×C₁₉	—	2	4	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	1	no	—
126	C₂×C₄×C₁₉	—	2	2	no	—
127	C₂×C₄×C₁₉	—	2	1	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	—	4	2	no	—
137	C₂×C₄×C₁₉	—	4	2	no	—
138	C₂×C₄×C₁₉	#135	1	2	yes	—
139	C₂×C₄×C₁₉	C₂²×Q₈	1	1	yes	characteristic
140	C₂×C₄×C₁₉	C₂²×Q₈	1	1	yes	characteristic
141	C₂×C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
142	C₂×C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
143	C₂×C₄×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
144	C₂×C₄×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
145	C₈×C₁₉	—	2	2	no	—
146	C₈×C₁₉	—	2	2	no	—
147	C₈×C₁₉	—	4	2	no	—
148	#2584	C₂×C₁₉	1	2	yes	—
149	#2088	C₂×C₁₉	1	2	yes	—
150	#2059	C₂×C₁₉	1	2	yes	—
151	#2059	C₂×C₁₉	1	2	yes	—
152	#1984	C₂×C₁₉	1	1	yes	characteristic
153	#1942	C₂×C₁₉	1	1	yes	characteristic
154	#1942	C₂×C₁₉	1	1	yes	characteristic
155	#1921	C₂×C₁₉	1	2	yes	—
156	#1918	C₂×C₁₉	1	2	yes	—
157	C₂²×C₁₉	#534	1	1	yes	characteristic
158	C₂²×C₁₉	D₄×Q₈	1	1	yes	characteristic
159	C₂²×C₁₉	D₄×Q₈	1	1	yes	characteristic
160	C₂²×C₁₉	C₂×#135	1	1	yes	characteristic
161	C₂²×C₁₉	#575	1	1	yes	characteristic
162	C₂²×C₁₉	#576	1	1	yes	characteristic
163	C₂²×C₁₉	#578	1	1	yes	characteristic
164	C₄×C₁₉	—	2	2	no	—
165	C₄×C₁₉	—	4	4	no	—
166	C₄×C₁₉	—	4	4	no	—
167	C₄×C₁₉	—	4	2	no	—
168	C₄×C₁₉	—	4	2	no	—
169	C₄×C₁₉	—	4	1	no	—
170	C₄×C₁₉	—	8	2	no	—
171	C₄×C₁₉	—	8	2	no	—
172	C₄×C₁₉	#544	1	1	yes	characteristic
173	C₄×C₁₉	#544	1	1	yes	characteristic
174	C₄×C₁₉	Q₈²	1	1	yes	characteristic
175	C₄×C₁₉	#563	1	1	yes	characteristic
176	C₂×#125	C₂²×C₁₉	1	2	yes	—
177	C₂×C₄×Q₈	C₂²×C₁₉	1	1	yes	characteristic
178	C₂×#107	C₂²×C₁₉	1	1	yes	characteristic
179	C₂×#107	C₂²×C₁₉	1	1	yes	characteristic
180	C₂×#107	C₂²×C₁₉	1	1	yes	characteristic
181	C₂×#107	C₂²×C₁₉	1	1	yes	characteristic
182	C₂×#106	C₂²×C₁₉	1	2	yes	—
183	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
184	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
185	#398	—	2	2	no	—
186	#395	—	2	2	no	—
187	#389	—	2	2	no	—
188	#389	—	2	2	no	—
189	#389	C₂²×C₁₉	1	2	yes	—
190	#389	C₂²×C₁₉	1	2	yes	—
191	#389	C₂²×C₁₉	1	2	yes	—
192	#389	C₂²×C₁₉	1	2	yes	—
193	C₄×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
194	#340	C₂²×C₁₉	1	2	yes	—
195	#340	C₂²×C₁₉	1	2	yes	—
196	#340	C₂²×C₁₉	1	2	yes	—
197	#340	C₂²×C₁₉	1	2	yes	—
198	#340	C₂²×C₁₉	1	2	yes	—
199	#336	C₂²×C₁₉	1	2	yes	—
200	#336	C₂²×C₁₉	1	2	yes	—
201	C₂×C₁₉	#2734	1	1	yes	characteristic
202	C₂×C₁₉	#3191	1	1	yes	characteristic
203	C₂×C₁₉	#3203	1	1	yes	characteristic
204	C₂×C₁₉	#3380	1	1	yes	characteristic
205	C₂×C₁₉	#3383	1	1	yes	characteristic
206	C₂×C₁₉	#3389	1	1	yes	characteristic
207	C₂×C₁₉	#3390	1	1	yes	characteristic
208	C₂²×Q₈	D₄×C₁₉	1	1	yes	characteristic
209	C₂²×C₈	C₂³×C₁₉	1	2	yes	—
210	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
211	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
212	#125	—	4	2	no	—
213	#125	—	4	2	no	—
214	C₄×Q₈	—	4	1	no	—
215	C₄×Q₈	—	4	1	no	—
216	C₂×C₄⋊C₄	—	2	2	no	—
217	C₂×C₄⋊C₄	—	2	2	no	—
218	C₂×C₄⋊C₄	—	2	2	no	—
219	C₂×C₄⋊C₄	—	2	2	no	—
220	C₂×C₄⋊C₄	—	2	2	no	—
221	C₂×C₄⋊C₄	—	2	2	no	—
222	C₂×C₄⋊C₄	D₄×C₁₉	1	1	yes	characteristic
223	C₂×C₄⋊C₄	D₄×C₁₉	1	2	yes	—
224	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
225	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
226	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
227	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
228	C₂×C₄²	—	2	2	no	—
229	C₂×C₄²	C₂³×C₁₉	1	2	yes	—
230	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
231	#107	—	4	1	no	—
232	#107	—	4	1	no	—
233	#107	Q₈×C₁₉	1	4	yes	complemented
234	#107	Q₈×C₁₉	1	4	yes	complemented
235	#106	—	4	2	no	—
236	C₄⋊C₈	—	4	1	no	—
237	C₄×C₈	—	4	1	no	—
238	#95	—	2	2	no	—
239	#95	—	2	2	no	—
240	#95	—	2	2	no	—
241	#95	—	2	2	no	—
242	#95	—	2	2	no	—
243	C₁₉	#32686	1	1	yes	characteristic, direct product factor, Sylow
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	—	4	1	no	—
246	C₂×Q₈	—	4	1	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
252	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
253	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
254	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
255	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	C₂×Q₈×C₁₉	1	2	yes	—
264	C₂×C₈	C₂×Q₈×C₁₉	1	2	yes	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	4	1	no	—
272	C₄⋊C₄	—	4	2	no	—
273	C₄⋊C₄	—	4	4	no	—
274	C₄⋊C₄	—	4	4	no	—
275	C₄⋊C₄	—	4	4	no	—
276	C₄⋊C₄	—	4	4	no	—
277	C₄⋊C₄	—	8	2	no	—
278	C₄⋊C₄	—	8	2	no	—
279	C₄⋊C₄	C₂×Q₈×C₁₉	1	4	yes	—
280	C₄⋊C₄	C₂×Q₈×C₁₉	1	4	yes	—
281	C₄²	—	2	1	no	—
282	C₄²	—	2	1	no	—
283	C₄²	—	2	2	no	—
284	C₄²	—	2	2	no	—
285	C₄²	—	8	2	no	—
286	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
287	Q₈	—	4	4	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	C₁₉×#135	1	2	yes	—
312	C₂×C₄	C₂²×Q₈×C₁₉	1	1	yes	characteristic
313	C₂×C₄	C₂²×Q₈×C₁₉	1	1	yes	characteristic
314	C₂×C₄	C₁₉×H(5,𝔽₂)	1	1	yes	characteristic
315	C₂×C₄	C₁₉×H(5,𝔽₂)	1	1	yes	characteristic
316	C₂×C₄	C₁₉×2₋¹⁺⁴	1	1	yes	characteristic
317	C₂×C₄	C₁₉×2₋¹⁺⁴	1	1	yes	characteristic
318	C₈	—	2	2	no	—
319	C₈	—	2	2	no	—
320	C₈	—	4	2	no	—
321	C₂²	C₁₉×#534	1	1	yes	characteristic
322	C₂²	D₄×Q₈×C₁₉	1	1	yes	characteristic
323	C₂²	D₄×Q₈×C₁₉	1	1	yes	characteristic
324	C₂²	C₂×C₁₉×#135	1	1	yes	characteristic
325	C₂²	C₁₉×#575	1	1	yes	characteristic
326	C₂²	C₁₉×#576	1	1	yes	characteristic
327	C₂²	C₁₉×#578	1	1	yes	characteristic
328	C₄	—	2	2	no	—
329	C₄	—	4	2	no	—
330	C₄	—	4	2	no	—
331	C₄	—	4	1	no	—
332	C₄	—	4	4	no	—
333	C₄	—	4	4	no	—
334	C₄	—	8	2	no	—
335	C₄	—	8	2	no	—
336	C₄	C₁₉×#544	1	1	yes	characteristic
337	C₄	C₁₉×#544	1	1	yes	characteristic
338	C₄	Q₈²×C₁₉	1	1	yes	characteristic
339	C₄	C₁₉×#563	1	1	yes	characteristic
340	C₂	C₁₉×#2734	1	1	yes	characteristic
341	C₂	C₁₉×#3191	1	1	yes	characteristic
342	C₂	C₁₉×#3203	1	1	yes	characteristic
343	C₂	C₁₉×#3380	1	1	yes	characteristic
344	C₂	C₁₉×#3383	1	1	yes	characteristic
345	C₂	C₁₉×#3389	1	1	yes	characteristic
346	C₂	C₁₉×#3390	1	1	yes	characteristic

	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
60. C₁₉×#107	Q₈	semidirect
61. C₁₉×#107	Q₈	semidirect
112. #32686	C₁₉	direct (N × H)
233. #107	Q₈×C₁₉	semidirect
234. #107	Q₈×C₁₉	semidirect
243. C₁₉	#32686	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×#1984	2
C₁₉×#1942	2
C₁₉×#1921	2
C₁₉×#1918	2
#32686	19

Subgroups — C19×#32686 #18038645

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#32686 #18038645