Subgroups — C₁₉×#33954 #18039913

Order 4864.

450 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3383	C₂	1	1	yes	maximal, characteristic
2	Q₈×Q₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#2255	C₂	1	2	yes	maximal
4	C₁₉×#2254	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#2229	C₂	1	2	yes	maximal
6	C₁₉×#2227	C₂	1	1	yes	maximal, characteristic
7	C₁₉×#2225	C₂	1	2	yes	maximal
8	C₁₉×#2225	C₂	1	1	yes	maximal, characteristic
9	C₁₉×#2183	C₂	1	1	yes	maximal, characteristic
10	C₁₉×#2176	C₂	1	2	yes	maximal
11	C₄×C₁₉×Q₃₂	C₂	1	1	yes	maximal, characteristic
12	Q₈²×C₁₉	—	2	1	no	—
13	C₁₉×#557	—	2	1	no	—
14	C₂×C₁₉×Q₃₂	C₂²	1	1	yes	characteristic
15	C₁₉×#500	C₂²	1	1	yes	characteristic
16	C₁₉×#500	C₂²	1	2	yes	—
17	C₁₉×#500	C₂²	1	1	yes	characteristic
18	C₁₉×#499	C₂²	1	2	yes	—
19	C₁₉×#479	—	2	2	no	—
20	C₁₉×#477	—	2	1	no	—
21	C₁₉×#477	—	2	2	no	—
22	C₁₉×#475	—	2	1	no	—
23	C₈×Q₈×C₁₉	C₂²	1	1	yes	characteristic
24	C₄×Q₁₆×C₁₉	C₂²	1	1	yes	characteristic
25	C₄×Q₁₆×C₁₉	C₂²	1	1	yes	characteristic
26	C₄×Q₁₆×C₁₉	C₂²	1	2	yes	—
27	C₄×Q₁₆×C₁₉	C₂²	1	2	yes	—
28	C₁₉×#367	C₂²	1	2	yes	—
29	C₁₉×#367	C₂²	1	2	yes	—
30	C₁₉×#367	C₂²	1	2	yes	—
31	C₁₉×#366	C₂²	1	1	yes	characteristic
32	C₁₉×#366	C₂²	1	2	yes	—
33	C₁₉×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
34	C₁₉×C₄⋊C₁₆	C₂²	1	2	yes	—
35	C₁₉×#358	C₂²	1	1	yes	characteristic
36	C₁₉×#358	C₂²	1	1	yes	characteristic
37	C₁₉×#358	C₂²	1	2	yes	—
38	C₁₉×#358	C₂²	1	2	yes	—
39	C₁₉×#346	C₂²	1	2	yes	—
40	C₄×C₁₆×C₁₉	C₂²	1	1	yes	characteristic
41	C₂×Q₁₆×C₁₉	C₂³	1	1	yes	characteristic
42	C₂×Q₁₆×C₁₉	C₂³	1	1	yes	characteristic
43	C₁₉×C₄⋊Q₈	—	2	1	no	—
44	C₁₉×C₄⋊Q₈	—	2	2	no	—
45	C₁₉×C₄⋊Q₈	—	2	1	no	—
46	C₁₉×C₄⋊Q₈	—	4	1	no	—
47	C₁₉×C₄⋊Q₈	—	4	1	no	—
48	C₁₉×C₄⋊Q₈	—	4	2	no	—
49	C₁₉×#125	—	2	2	no	—
50	C₁₉×#125	—	4	2	no	—
51	C₄×Q₈×C₁₉	—	2	1	no	—
52	C₄×Q₈×C₁₉	—	2	2	no	—
53	C₄×Q₈×C₁₉	—	2	1	no	—
54	C₄×Q₈×C₁₉	—	2	2	no	—
55	C₄×Q₈×C₁₉	—	4	1	no	—
56	C₄×Q₈×C₁₉	—	4	1	no	—
57	C₄×Q₈×C₁₉	D₄	1	1	yes	characteristic
58	C₁₉×Q₃₂	Q₈	1	4	yes	complemented
59	C₂×C₁₆×C₁₉	C₂³	1	1	yes	characteristic
60	C₂×C₁₆×C₁₉	C₂³	1	2	yes	—
61	C₂×C₁₆×C₁₉	C₂³	1	1	yes	characteristic
62	C₁₉×#107	—	2	2	no	—
63	C₁₉×#107	—	2	1	no	—
64	C₁₉×#107	—	2	1	no	—
65	C₁₉×#107	C₂³	1	1	yes	characteristic
66	C₁₉×#107	C₂³	1	1	yes	characteristic
67	C₁₉×#107	C₂³	1	2	yes	—
68	C₁₉×#107	C₂³	1	2	yes	—
69	C₁₉×#106	—	2	2	no	—
70	C₁₉×C₄⋊C₈	D₄	1	1	yes	characteristic
71	C₁₉×C₄⋊C₈	D₄	1	2	yes	—
72	C₁₉×#103	—	2	2	no	—
73	C₁₉×#103	—	2	2	no	—
74	C₁₉×#103	—	2	1	no	—
75	C₁₉×#103	—	2	1	no	—
76	C₄×C₈×C₁₉	C₂³	1	1	yes	characteristic
77	C₄×C₈×C₁₉	C₂³	1	2	yes	—
78	C₂×Q₈×C₁₉	—	2	1	no	—
79	C₂×Q₈×C₁₉	—	2	1	no	—
80	C₂×Q₈×C₁₉	—	4	1	no	—
81	C₂×Q₈×C₁₉	—	4	2	no	—
82	C₂×Q₈×C₁₉	—	4	1	no	—
83	C₂×Q₈×C₁₉	D₈	1	1	yes	characteristic
84	Q₁₆×C₁₉	—	2	1	no	—
85	Q₁₆×C₁₉	—	2	1	no	—
86	Q₁₆×C₁₉	C₂×Q₈	1	2	yes	—
87	Q₁₆×C₁₉	C₂×Q₈	1	2	yes	—
88	C₂×C₈×C₁₉	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₁₉	C₂⁴	1	1	yes	characteristic
91	C₄⋊C₄×C₁₉	—	2	1	no	—
92	C₄⋊C₄×C₁₉	—	2	2	no	—
93	C₄⋊C₄×C₁₉	—	2	2	no	—
94	C₄⋊C₄×C₁₉	—	2	1	no	—
95	C₄⋊C₄×C₁₉	—	4	1	no	—
96	C₄⋊C₄×C₁₉	—	4	2	no	—
97	C₄⋊C₄×C₁₉	—	4	2	no	—
98	C₄⋊C₄×C₁₉	—	4	1	no	—
99	C₄⋊C₄×C₁₉	—	4	2	no	—
100	C₄⋊C₄×C₁₉	—	4	2	no	—
101	C₄⋊C₄×C₁₉	—	4	1	no	—
102	C₄⋊C₄×C₁₉	—	4	2	no	—
103	C₄⋊C₄×C₁₉	—	4	1	no	—
104	C₄⋊C₄×C₁₉	D₈	1	1	yes	characteristic
105	C₄⋊C₄×C₁₉	D₈	1	2	yes	—
106	C₄²×C₁₉	—	4	2	no	—
107	C₄²×C₁₉	—	4	1	no	—
108	C₄²×C₁₉	—	4	2	no	—
109	C₄²×C₁₉	—	4	1	no	—
110	C₄²×C₁₉	C₂×D₄	1	2	yes	—
111	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
112	C₁₆×C₁₉	—	2	1	no	—
113	C₁₆×C₁₉	—	2	2	no	—
114	C₁₆×C₁₉	C₂×Q₈	1	2	yes	—
115	#33954	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	Q₈×C₁₉	—	2	2	no	—
117	Q₈×C₁₉	—	2	2	no	—
118	Q₈×C₁₉	—	2	2	no	—
119	Q₈×C₁₉	—	4	1	no	—
120	Q₈×C₁₉	—	4	1	no	—
121	Q₈×C₁₉	—	8	2	no	—
122	Q₈×C₁₉	—	16	2	no	—
123	Q₈×C₁₉	—	16	1	no	—
124	C₂×C₄×C₁₉	—	4	1	no	—
125	C₂×C₄×C₁₉	—	4	1	no	—
126	C₂×C₄×C₁₉	—	4	1	no	—
127	C₂×C₄×C₁₉	—	4	2	no	—
128	C₂×C₄×C₁₉	—	4	2	no	—
129	C₂×C₄×C₁₉	—	4	1	no	—
130	C₂×C₄×C₁₉	C₂×D₈	1	2	yes	—
131	C₂×C₄×C₁₉	C₂×D₈	1	2	yes	—
132	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
135	C₈×C₁₉	—	2	2	no	—
136	C₈×C₁₉	—	2	1	no	—
137	C₈×C₁₉	C₂²×Q₈	1	1	yes	characteristic
138	C₈×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
139	#3383	C₂×C₁₉	1	1	yes	characteristic
140	Q₈×Q₁₆	C₂×C₁₉	1	1	yes	characteristic
141	#2255	C₂×C₁₉	1	2	yes	—
142	#2254	C₂×C₁₉	1	1	yes	characteristic
143	#2229	C₂×C₁₉	1	2	yes	—
144	#2227	C₂×C₁₉	1	1	yes	characteristic
145	#2225	C₂×C₁₉	1	2	yes	—
146	#2225	C₂×C₁₉	1	1	yes	characteristic
147	#2183	C₂×C₁₉	1	1	yes	characteristic
148	#2176	C₂×C₁₉	1	2	yes	—
149	C₄×Q₃₂	C₂×C₁₉	1	1	yes	characteristic
150	C₂²×C₁₉	C₂²×D₈	1	1	yes	characteristic
151	C₄×C₁₉	—	2	2	no	—
152	C₄×C₁₉	—	2	2	no	—
153	C₄×C₁₉	—	2	1	no	—
154	C₄×C₁₉	—	4	2	no	—
155	C₄×C₁₉	—	4	2	no	—
156	C₄×C₁₉	—	8	2	no	—
157	C₄×C₁₉	—	8	1	no	—
158	C₄×C₁₉	—	8	2	no	—
159	C₄×C₁₉	—	8	1	no	—
160	C₄×C₁₉	#510	1	2	yes	—
161	C₄×C₁₉	D₄×Q₈	1	1	yes	characteristic
162	C₄×C₁₉	#578	1	1	yes	characteristic
163	Q₈²	—	2	1	no	—
164	#557	—	2	1	no	—
165	C₂×Q₃₂	C₂²×C₁₉	1	1	yes	characteristic
166	#500	C₂²×C₁₉	1	2	yes	—
167	#500	C₂²×C₁₉	1	1	yes	characteristic
168	#500	C₂²×C₁₉	1	1	yes	characteristic
169	#499	C₂²×C₁₉	1	2	yes	—
170	#479	—	2	2	no	—
171	#477	—	2	1	no	—
172	#477	—	2	2	no	—
173	#475	—	2	1	no	—
174	C₈×Q₈	C₂²×C₁₉	1	1	yes	characteristic
175	C₄×Q₁₆	C₂²×C₁₉	1	1	yes	characteristic
176	C₄×Q₁₆	C₂²×C₁₉	1	2	yes	—
177	C₄×Q₁₆	C₂²×C₁₉	1	1	yes	characteristic
178	C₄×Q₁₆	C₂²×C₁₉	1	2	yes	—
179	#367	C₂²×C₁₉	1	2	yes	—
180	#367	C₂²×C₁₉	1	2	yes	—
181	#367	C₂²×C₁₉	1	2	yes	—
182	#366	C₂²×C₁₉	1	1	yes	characteristic
183	#366	C₂²×C₁₉	1	2	yes	—
184	C₄⋊C₁₆	C₂²×C₁₉	1	1	yes	characteristic
185	C₄⋊C₁₆	C₂²×C₁₉	1	2	yes	—
186	#358	C₂²×C₁₉	1	1	yes	characteristic
187	#358	C₂²×C₁₉	1	1	yes	characteristic
188	#358	C₂²×C₁₉	1	2	yes	—
189	#358	C₂²×C₁₉	1	2	yes	—
190	#346	C₂²×C₁₉	1	2	yes	—
191	C₄×C₁₆	C₂²×C₁₉	1	1	yes	characteristic
192	C₂×C₁₉	Q₈×D₈	1	1	yes	characteristic
193	C₂×C₁₉	C₂×#510	1	1	yes	characteristic
194	C₂×C₁₉	#3416	1	1	yes	characteristic
195	C₂×Q₁₆	C₂³×C₁₉	1	1	yes	characteristic
196	C₂×Q₁₆	C₂³×C₁₉	1	1	yes	characteristic
197	C₄⋊Q₈	—	2	1	no	—
198	C₄⋊Q₈	—	2	2	no	—
199	C₄⋊Q₈	—	2	1	no	—
200	C₄⋊Q₈	—	4	1	no	—
201	C₄⋊Q₈	—	4	1	no	—
202	C₄⋊Q₈	—	4	2	no	—
203	#125	—	2	2	no	—
204	#125	—	4	2	no	—
205	C₄×Q₈	—	2	1	no	—
206	C₄×Q₈	—	2	2	no	—
207	C₄×Q₈	—	2	2	no	—
208	C₄×Q₈	—	2	1	no	—
209	C₄×Q₈	—	4	1	no	—
210	C₄×Q₈	—	4	1	no	—
211	C₄×Q₈	D₄×C₁₉	1	1	yes	characteristic
212	Q₃₂	Q₈×C₁₉	1	4	yes	complemented
213	C₂×C₁₆	C₂³×C₁₉	1	1	yes	characteristic
214	C₂×C₁₆	C₂³×C₁₉	1	2	yes	—
215	C₂×C₁₆	C₂³×C₁₉	1	1	yes	characteristic
216	#107	—	2	2	no	—
217	#107	—	2	1	no	—
218	#107	—	2	1	no	—
219	#107	C₂³×C₁₉	1	1	yes	characteristic
220	#107	C₂³×C₁₉	1	1	yes	characteristic
221	#107	C₂³×C₁₉	1	2	yes	—
222	#107	C₂³×C₁₉	1	2	yes	—
223	#106	—	2	2	no	—
224	C₄⋊C₈	D₄×C₁₉	1	1	yes	characteristic
225	C₄⋊C₈	D₄×C₁₉	1	2	yes	—
226	#103	—	2	2	no	—
227	#103	—	2	2	no	—
228	#103	—	2	1	no	—
229	#103	—	2	1	no	—
230	C₄×C₈	C₂³×C₁₉	1	1	yes	characteristic
231	C₄×C₈	C₂³×C₁₉	1	2	yes	—
232	C₁₉	#33954	1	1	yes	characteristic, direct product factor, Sylow
233	C₂×Q₈	—	2	1	no	—
234	C₂×Q₈	—	2	1	no	—
235	C₂×Q₈	—	4	1	no	—
236	C₂×Q₈	—	4	2	no	—
237	C₂×Q₈	—	4	1	no	—
238	C₂×Q₈	D₈×C₁₉	1	1	yes	characteristic
239	Q₁₆	—	2	1	no	—
240	Q₁₆	—	2	1	no	—
241	Q₁₆	C₂×Q₈×C₁₉	1	2	yes	—
242	Q₁₆	C₂×Q₈×C₁₉	1	2	yes	—
243	C₂×C₈	C₂×D₄×C₁₉	1	2	yes	—
244	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
245	C₂×C₈	C₂⁴×C₁₉	1	1	yes	characteristic
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	—	4	1	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	—	4	1	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	—	4	1	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	4	1	no	—
259	C₄⋊C₄	D₈×C₁₉	1	1	yes	characteristic
260	C₄⋊C₄	D₈×C₁₉	1	2	yes	—
261	C₄²	—	4	2	no	—
262	C₄²	—	4	1	no	—
263	C₄²	—	4	2	no	—
264	C₄²	—	4	1	no	—
265	C₄²	C₂×D₄×C₁₉	1	2	yes	—
266	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
267	C₁₆	—	2	1	no	—
268	C₁₆	—	2	2	no	—
269	C₁₆	C₂×Q₈×C₁₉	1	2	yes	—
270	Q₈	—	2	2	no	—
271	Q₈	—	2	2	no	—
272	Q₈	—	2	2	no	—
273	Q₈	—	4	1	no	—
274	Q₈	—	4	1	no	—
275	Q₈	—	8	2	no	—
276	Q₈	—	16	2	no	—
277	Q₈	—	16	1	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	1	no	—
284	C₂×C₄	C₂×D₈×C₁₉	1	2	yes	—
285	C₂×C₄	C₂×D₈×C₁₉	1	2	yes	—
286	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
287	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
288	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
289	C₈	—	2	2	no	—
290	C₈	—	2	1	no	—
291	C₈	C₂²×Q₈×C₁₉	1	1	yes	characteristic
292	C₈	C₁₉×2₋¹⁺⁴	1	1	yes	characteristic
293	C₂²	C₂²×D₈×C₁₉	1	1	yes	characteristic
294	C₄	—	2	2	no	—
295	C₄	—	2	2	no	—
296	C₄	—	2	1	no	—
297	C₄	—	4	2	no	—
298	C₄	—	4	2	no	—
299	C₄	—	8	2	no	—
300	C₄	—	8	2	no	—
301	C₄	—	8	1	no	—
302	C₄	—	8	1	no	—
303	C₄	C₁₉×#510	1	2	yes	—
304	C₄	D₄×Q₈×C₁₉	1	1	yes	characteristic
305	C₄	C₁₉×#578	1	1	yes	characteristic
306	C₂	Q₈×D₈×C₁₉	1	1	yes	characteristic
307	C₂	C₂×C₁₉×#510	1	1	yes	characteristic
308	C₂	C₁₉×#3416	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₁₉	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₉
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₄×Q₈×C₁₉ ◁ C₈×Q₈×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
58. C₁₉×Q₃₂	Q₈	semidirect
115. #33954	C₁₉	direct (N × H)
212. Q₃₂	Q₈×C₁₉	semidirect
232. C₁₉	#33954	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×#3383	2
Q₈×Q₁₆×C₁₉	2
C₁₉×#2255	2
C₁₉×#2254	2
C₁₉×#2229	2
C₁₉×#2227	2
C₁₉×#2225	2
C₁₉×#2183	2
C₁₉×#2176	2
C₄×C₁₉×Q₃₂	2
#33954	19

Subgroups — C19×#33954 #18039913

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#33954 #18039913