Subgroups — C₁₉×#21780 #18027739

Order 4864.

1236 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3486	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2889	C₂	1	2	yes	maximal, complemented
3	C₁₉×#1915	C₂	1	4	yes	maximal, complemented
4	C₁₉×#1891	C₂	1	4	yes	maximal, complemented
5	C₁₉×#1830	C₂	1	2	yes	maximal, complemented
6	C₁₉×#1784	C₂	1	2	yes	maximal, complemented
7	C₂×C₁₉×2₋¹⁺⁴	C₂²	1	1	yes	characteristic
8	C₂³×Q₈×C₁₉	C₂²	1	1	yes	characteristic
9	C₁₉×#568	C₂²	1	4	yes	—
10	D₄×Q₈×C₁₉	—	2	4	no	—
11	C₁₉×#548	—	2	4	no	—
12	C₁₉×#536	C₄	1	4	yes	—
13	C₁₉×#536	C₂²	1	2	yes	—
14	C₁₉×#525	C₄	1	4	yes	—
15	C₁₉×#525	C₂²	1	2	yes	—
16	C₁₉×#519	C₂²	1	1	yes	characteristic
17	C₁₉×#413	—	2	4	no	—
18	C₁₉×#413	C₂²	1	4	yes	—
19	C₂×C₁₉×#101	—	2	4	no	—
20	C₂×C₁₉×#101	C₂²	1	4	yes	—
21	C₂×C₁₉×#101	C₂²	1	4	yes	—
22	C₁₉×#407	C₂²	1	4	yes	—
23	C₁₉×#344	C₂²	1	8	yes	—
24	C₁₉×2₋¹⁺⁴	—	4	4	no	—
25	C₂×#41×C₁₉	C₂×C₄	1	4	yes	—
26	C₂×#41×C₁₉	D₄	1	4	yes	—
27	C₂×#41×C₁₉	C₂³	1	2	yes	—
28	C₂²×Q₈×C₁₉	—	2	4	no	—
29	C₂²×Q₈×C₁₉	—	4	4	no	—
30	C₂²×Q₈×C₁₉	C₂×C₄	1	2	yes	—
31	C₂²×Q₈×C₁₉	D₄	1	4	yes	—
32	C₂²×Q₈×C₁₉	C₂³	1	1	yes	characteristic
33	C₂²×Q₈×C₁₉	C₂³	1	1	yes	characteristic
34	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
35	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
36	C₁₉×#131	—	4	8	no	—
37	C₂×M₁₆×C₁₉	—	2	8	no	—
38	C₂×M₁₆×C₁₉	C₂³	1	8	yes	—
39	C₁₉×C₄⋊Q₈	—	2	4	no	—
40	C₁₉×C₄⋊Q₈	—	2	2	no	—
41	C₁₉×#124	—	2	4	no	—
42	C₁₉×#124	—	2	2	no	—
43	C₁₉×#123	C₂×C₄	1	8	yes	—
44	C₁₉×#123	C₂×C₄	1	4	yes	—
45	C₁₉×C₂²⋊Q₈	—	2	4	no	—
46	C₁₉×C₂²⋊Q₈	—	2	2	no	—
47	C₁₉×C₂²⋊Q₈	—	2	8	no	—
48	C₁₉×C₂²⋊Q₈	—	2	4	no	—
49	C₁₉×#121	—	2	4	no	—
50	C₁₉×#121	—	2	2	no	—
51	C₁₉×#120	C₂×C₄	1	4	yes	—
52	C₄×Q₈×C₁₉	—	2	4	no	—
53	C₄×D₄×C₁₉	—	2	8	no	—
54	C₄×D₄×C₁₉	—	2	4	no	—
55	C₁₉×#117	C₂×C₄	1	4	yes	—
56	C₁₉×#117	C₂³	1	2	yes	—
57	C₁₉×#101	—	2	4	no	—
58	C₁₉×#101	—	2	8	no	—
59	C₁₉×#101	—	4	4	no	—
60	C₁₉×#100	—	2	4	no	—
61	C₁₉×C₂²⋊C₈	—	4	4	no	—
62	C₂⁴×C₁₉	C₂²×C₄	1	1	yes	characteristic
63	#41×C₁₉	—	2	8	no	—
64	#41×C₁₉	—	4	8	no	—
65	#41×C₁₉	—	4	8	no	—
66	C₂×Q₈×C₁₉	—	2	4	no	—
67	C₂×Q₈×C₁₉	—	2	2	no	—
68	C₂×Q₈×C₁₉	—	2	4	no	—
69	C₂×Q₈×C₁₉	—	2	4	no	—
70	C₂×Q₈×C₁₉	—	4	4	no	—
71	C₂×Q₈×C₁₉	—	4	8	no	—
72	C₂×Q₈×C₁₉	—	4	16	no	—
73	C₂×Q₈×C₁₉	C₂²⋊C₄	1	4	yes	—
74	C₂×Q₈×C₁₉	C₂×D₄	1	4	yes	—
75	C₂×Q₈×C₁₉	C₂×D₄	1	2	yes	—
76	C₂×D₄×C₁₉	—	2	4	no	—
77	C₂×D₄×C₁₉	C₂²⋊C₄	1	4	yes	—
78	C₂×D₄×C₁₉	C₂²×C₄	1	4	yes	—
79	C₂×D₄×C₁₉	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₁₉	—	2	4	no	—
81	C₂²×C₄×C₁₉	—	2	2	no	—
82	C₂²×C₄×C₁₉	—	2	4	no	—
83	C₂²×C₄×C₁₉	—	4	4	no	—
84	C₂²×C₄×C₁₉	C₂²⋊C₄	1	8	yes	—
85	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
86	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
88	C₂²×C₄×C₁₉	C₂×D₄	1	4	yes	—
89	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
90	M₁₆×C₁₉	—	2	8	no	—
91	M₁₆×C₁₉	—	2	8	no	—
92	M₁₆×C₁₉	—	4	8	no	—
93	C₂×C₈×C₁₉	—	2	8	no	—
94	C₂×C₈×C₁₉	—	4	8	no	—
95	C₄⋊C₄×C₁₉	—	2	4	no	—
96	C₄⋊C₄×C₁₉	—	2	8	no	—
97	C₄⋊C₄×C₁₉	—	2	4	no	—
98	C₄⋊C₄×C₁₉	—	2	2	no	—
99	C₂²⋊C₄×C₁₉	—	2	8	no	—
100	C₂²⋊C₄×C₁₉	—	2	4	no	—
101	C₂²⋊C₄×C₁₉	—	2	4	no	—
102	C₂²⋊C₄×C₁₉	C₂²×C₄	1	4	yes	—
103	C₄²×C₁₉	—	2	4	no	—
104	C₄²×C₁₉	—	2	2	no	—
105	#21780	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
106	C₂³×C₁₉	—	2	1	no	—
107	C₂³×C₁₉	—	2	2	no	—
108	C₂³×C₁₉	—	4	2	no	—
109	C₂³×C₁₉	C₂×C₂²⋊C₄	1	4	yes	—
110	C₂³×C₁₉	C₂³×C₄	1	1	yes	characteristic
111	Q₈×C₁₉	—	2	8	no	—
112	Q₈×C₁₉	—	2	4	no	—
113	Q₈×C₁₉	—	4	4	no	—
114	Q₈×C₁₉	—	4	8	no	—
115	Q₈×C₁₉	—	4	4	no	—
116	D₄×C₁₉	—	2	8	no	—
117	D₄×C₁₉	—	2	4	no	—
118	D₄×C₁₉	—	4	4	no	—
119	D₄×C₁₉	—	4	8	no	—
120	C₂×C₄×C₁₉	—	2	4	no	—
121	C₂×C₄×C₁₉	—	2	8	no	—
122	C₂×C₄×C₁₉	—	2	4	no	—
123	C₂×C₄×C₁₉	—	2	2	no	—
124	C₂×C₄×C₁₉	—	2	1	no	—
125	C₂×C₄×C₁₉	—	2	2	no	—
126	C₂×C₄×C₁₉	—	2	4	no	—
127	C₂×C₄×C₁₉	—	4	8	no	—
128	C₂×C₄×C₁₉	—	4	8	no	—
129	C₂×C₄×C₁₉	—	4	4	no	—
130	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	4	yes	—
131	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	4	yes	—
132	C₂×C₄×C₁₉	C₂²×D₄	1	2	yes	—
133	C₂×C₄×C₁₉	H(5,𝔽₂)	1	4	yes	—
134	C₈×C₁₉	—	4	8	no	—
135	#3486	C₂×C₁₉	1	1	yes	characteristic
136	#2889	C₂×C₁₉	1	2	yes	complemented
137	#1915	C₂×C₁₉	1	4	yes	complemented
138	#1891	C₂×C₁₉	1	4	yes	complemented
139	#1830	C₂×C₁₉	1	2	yes	complemented
140	#1784	C₂×C₁₉	1	2	yes	complemented
141	C₂²×C₁₉	—	2	2	no	—
142	C₂²×C₁₉	—	2	4	no	—
143	C₂²×C₁₉	—	2	4	no	—
144	C₂²×C₁₉	—	4	1	no	—
145	C₂²×C₁₉	—	4	4	no	—
146	C₂²×C₁₉	—	4	1	no	—
147	C₂²×C₁₉	—	4	2	no	—
148	C₂²×C₁₉	—	4	1	no	—
149	C₂²×C₁₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
150	C₂²×C₁₉	#518	1	2	yes	—
151	C₄×C₁₉	—	2	4	no	—
152	C₄×C₁₉	—	2	4	no	—
153	C₄×C₁₉	—	4	4	no	—
154	C₄×C₁₉	—	4	1	no	—
155	C₄×C₁₉	—	4	2	no	—
156	C₄×C₁₉	#534	1	4	yes	—
157	C₂×2₋¹⁺⁴	C₂²×C₁₉	1	1	yes	characteristic
158	C₂³×Q₈	C₂²×C₁₉	1	1	yes	characteristic
159	#568	C₂²×C₁₉	1	4	yes	—
160	D₄×Q₈	—	2	4	no	—
161	#548	—	2	4	no	—
162	#536	C₄×C₁₉	1	4	yes	—
163	#536	C₂²×C₁₉	1	2	yes	—
164	#525	C₄×C₁₉	1	4	yes	—
165	#525	C₂²×C₁₉	1	2	yes	—
166	#519	C₂²×C₁₉	1	1	yes	characteristic
167	#413	—	2	4	no	—
168	#413	C₂²×C₁₉	1	4	yes	—
169	C₂×#101	—	2	4	no	—
170	C₂×#101	C₂²×C₁₉	1	4	yes	—
171	C₂×#101	C₂²×C₁₉	1	4	yes	—
172	#407	C₂²×C₁₉	1	4	yes	—
173	#344	C₂²×C₁₉	1	8	yes	—
174	C₂×C₁₉	—	2	1	no	—
175	C₂×C₁₉	—	2	2	no	—
176	C₂×C₁₉	—	4	2	no	—
177	C₂×C₁₉	—	4	4	no	—
178	C₂×C₁₉	#2308	1	1	yes	characteristic
179	2₋¹⁺⁴	—	4	4	no	—
180	C₂×#41	C₂×C₄×C₁₉	1	4	yes	—
181	C₂×#41	D₄×C₁₉	1	4	yes	—
182	C₂×#41	C₂³×C₁₉	1	2	yes	—
183	C₂²×Q₈	—	2	4	no	—
184	C₂²×Q₈	—	4	4	no	—
185	C₂²×Q₈	C₂×C₄×C₁₉	1	2	yes	—
186	C₂²×Q₈	D₄×C₁₉	1	4	yes	—
187	C₂²×Q₈	C₂³×C₁₉	1	1	yes	characteristic
188	C₂²×Q₈	C₂³×C₁₉	1	1	yes	characteristic
189	C₂³×C₄	C₂×C₄×C₁₉	1	2	yes	—
190	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
191	#131	—	4	8	no	—
192	C₂×M₁₆	—	2	8	no	—
193	C₂×M₁₆	C₂³×C₁₉	1	8	yes	—
194	C₄⋊Q₈	—	2	4	no	—
195	C₄⋊Q₈	—	2	2	no	—
196	#124	—	2	4	no	—
197	#124	—	2	2	no	—
198	#123	C₂×C₄×C₁₉	1	8	yes	—
199	#123	C₂×C₄×C₁₉	1	4	yes	—
200	C₂²⋊Q₈	—	2	4	no	—
201	C₂²⋊Q₈	—	2	2	no	—
202	C₂²⋊Q₈	—	2	8	no	—
203	C₂²⋊Q₈	—	2	4	no	—
204	#121	—	2	4	no	—
205	#121	—	2	2	no	—
206	#120	C₂×C₄×C₁₉	1	4	yes	—
207	C₄×Q₈	—	2	4	no	—
208	C₄×D₄	—	2	8	no	—
209	C₄×D₄	—	2	4	no	—
210	#117	C₂×C₄×C₁₉	1	4	yes	—
211	#117	C₂³×C₁₉	1	2	yes	—
212	#101	—	2	4	no	—
213	#101	—	2	8	no	—
214	#101	—	4	4	no	—
215	#100	—	2	4	no	—
216	C₂²⋊C₈	—	4	4	no	—
217	C₁₉	#21780	1	1	yes	characteristic, direct product factor, Sylow
218	C₂⁴	C₂²×C₄×C₁₉	1	1	yes	characteristic
219	#41	—	2	8	no	—
220	#41	—	4	8	no	—
221	#41	—	4	8	no	—
222	C₂×Q₈	—	2	4	no	—
223	C₂×Q₈	—	2	2	no	—
224	C₂×Q₈	—	2	4	no	—
225	C₂×Q₈	—	2	4	no	—
226	C₂×Q₈	—	4	4	no	—
227	C₂×Q₈	—	4	8	no	—
228	C₂×Q₈	—	4	16	no	—
229	C₂×Q₈	C₂²⋊C₄×C₁₉	1	4	yes	—
230	C₂×Q₈	C₂×D₄×C₁₉	1	4	yes	—
231	C₂×Q₈	C₂×D₄×C₁₉	1	2	yes	—
232	C₂×D₄	—	2	4	no	—
233	C₂×D₄	C₂²⋊C₄×C₁₉	1	4	yes	—
234	C₂×D₄	C₂²×C₄×C₁₉	1	4	yes	—
235	C₂×D₄	C₂×D₄×C₁₉	1	2	yes	—
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	4	no	—
239	C₂²×C₄	—	4	4	no	—
240	C₂²×C₄	C₂²⋊C₄×C₁₉	1	8	yes	—
241	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
242	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
243	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
244	C₂²×C₄	C₂×D₄×C₁₉	1	4	yes	—
245	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
246	M₁₆	—	2	8	no	—
247	M₁₆	—	2	8	no	—
248	M₁₆	—	4	8	no	—
249	C₂×C₈	—	2	8	no	—
250	C₂×C₈	—	4	8	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	2	8	no	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₂²⋊C₄	—	2	8	no	—
256	C₂²⋊C₄	—	2	4	no	—
257	C₂²⋊C₄	—	2	4	no	—
258	C₂²⋊C₄	C₂²×C₄×C₁₉	1	4	yes	—
259	C₄²	—	2	4	no	—
260	C₄²	—	2	2	no	—
261	C₂³	—	2	1	no	—
262	C₂³	—	2	2	no	—
263	C₂³	—	4	2	no	—
264	C₂³	C₂×C₂²⋊C₄×C₁₉	1	4	yes	—
265	C₂³	C₂³×C₄×C₁₉	1	1	yes	characteristic
266	Q₈	—	2	8	no	—
267	Q₈	—	2	4	no	—
268	Q₈	—	4	4	no	—
269	Q₈	—	4	8	no	—
270	Q₈	—	4	4	no	—
271	D₄	—	2	8	no	—
272	D₄	—	2	4	no	—
273	D₄	—	4	4	no	—
274	D₄	—	4	8	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	8	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	4	8	no	—
283	C₂×C₄	—	4	8	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	4	yes	—
286	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	4	yes	—
287	C₂×C₄	C₂²×D₄×C₁₉	1	2	yes	—
288	C₂×C₄	C₁₉×H(5,𝔽₂)	1	4	yes	—
289	C₈	—	4	8	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	4	no	—
292	C₂²	—	2	4	no	—
293	C₂²	—	4	1	no	—
294	C₂²	—	4	4	no	—
295	C₂²	—	4	1	no	—
296	C₂²	—	4	2	no	—
297	C₂²	—	4	1	no	—
298	C₂²	C₂²×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
299	C₂²	C₁₉×#518	1	2	yes	—
300	C₄	—	2	4	no	—
301	C₄	—	2	4	no	—
302	C₄	—	4	4	no	—
303	C₄	—	4	1	no	—
304	C₄	—	4	2	no	—
305	C₄	C₁₉×#534	1	4	yes	—
306	C₂	—	2	1	no	—
307	C₂	—	2	2	no	—
308	C₂	—	4	2	no	—
309	C₂	—	4	4	no	—
310	C₂	C₁₉×#2308	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₉	#2308
commutator	C₂³	C₂³×C₄×C₁₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
prime radical	C₁₉×#3486	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₉×#2889	C₂	semidirect
3. C₁₉×#1915	C₂	semidirect
4. C₁₉×#1891	C₂	semidirect
5. C₁₉×#1830	C₂	semidirect
6. C₁₉×#1784	C₂	semidirect
105. #21780	C₁₉	direct (N × H)
136. #2889	C₂×C₁₉	semidirect
137. #1915	C₂×C₁₉	semidirect
138. #1891	C₂×C₁₉	semidirect
139. #1830	C₂×C₁₉	semidirect
140. #1784	C₂×C₁₉	semidirect
217. C₁₉	#21780	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×#3486	2
C₁₉×#2889	2
C₁₉×#1915	2
C₁₉×#1891	2
C₁₉×#1830	2
C₁₉×#1784	2
#21780	19

Subgroups — C19×#21780 #18027739

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#21780 #18027739