Subgroups — C₁₉×#17755 #18023714

Order 4864.

954 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#533	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×#432	C₂	1	2	yes	maximal
3	C₁₉×#1556	C₂	1	4	yes	maximal, complemented
4	C₁₉×#1552	C₂	1	8	yes	maximal, complemented
5	C₁₉×#533	—	2	2	no	—
6	C₁₉×#533	—	2	2	no	—
7	C₁₉×#533	C₄	1	4	yes	—
8	C₁₉×#533	C₂²	1	4	yes	—
9	C₂×C₁₉×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₂×C₁₉×#125	C₄	1	2	yes	—
11	C₂×C₁₉×C₂²⋊Q₈	—	2	2	no	—
12	C₂×C₄×Q₈×C₁₉	—	2	2	no	—
13	C₂×C₁₉×#117	C₄	1	2	yes	—
14	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
15	C₁₉×#432	—	2	2	no	—
16	C₁₉×#432	—	2	2	no	—
17	C₁₉×#432	C₂²	1	4	yes	—
18	C₁₉×#432	C₂²	1	4	yes	—
19	C₂×C₁₉×C₄⋊C₈	C₂²	1	4	yes	—
20	C₂×C₁₉×C₂²⋊C₈	—	2	2	no	—
21	C₂×C₁₉×#97	—	2	2	no	—
22	C₁₉×#332	C₂²	1	16	yes	complemented
23	C₂²×Q₈×C₁₉	—	2	2	no	—
24	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₁₉	D₄	1	2	yes	—
26	C₂²×C₈×C₁₉	—	2	4	no	—
27	C₁₉×C₄⋊Q₈	—	2	2	no	—
28	C₁₉×C₄⋊Q₈	—	2	2	no	—
29	C₁₉×C₄⋊Q₈	C₂×C₄	1	4	yes	—
30	C₁₉×C₄⋊Q₈	C₂³	1	4	yes	—
31	C₁₉×#125	—	2	2	no	—
32	C₁₉×#125	—	2	2	no	—
33	C₁₉×#125	C₂×C₄	1	8	yes	—
34	C₁₉×C₂²⋊Q₈	—	2	4	no	—
35	C₁₉×C₂²⋊Q₈	—	2	4	no	—
36	C₁₉×C₂²⋊Q₈	—	2	4	no	—
37	C₁₉×C₂²⋊Q₈	—	2	4	no	—
38	C₄×Q₈×C₁₉	—	2	4	no	—
39	C₄×Q₈×C₁₉	—	2	4	no	—
40	C₄×Q₈×C₁₉	—	2	4	no	—
41	C₄×Q₈×C₁₉	—	2	4	no	—
42	C₁₉×#117	—	2	4	no	—
43	C₁₉×#117	C₂×C₄	1	8	yes	—
44	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
45	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
46	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	4	yes	—
47	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
48	C₂×C₄²×C₁₉	—	2	2	no	—
49	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₁₉	D₄	1	2	yes	—
52	C₂×C₄²×C₁₉	D₄	1	2	yes	—
53	C₂×C₄²×C₁₉	D₄	1	2	yes	—
54	C₂×C₄²×C₁₉	C₂³	1	2	yes	—
55	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
56	C₁₉×C₄⋊C₈	—	2	4	no	—
57	C₁₉×C₄⋊C₈	C₂³	1	8	yes	—
58	C₁₉×C₂²⋊C₈	—	2	4	no	—
59	C₁₉×C₂²⋊C₈	—	2	4	no	—
60	C₁₉×#97	—	2	4	no	—
61	C₁₉×#97	—	2	4	no	—
62	C₂⁴×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×Q₈×C₁₉	—	2	4	no	—
64	C₂×Q₈×C₁₉	—	2	4	no	—
65	C₂×Q₈×C₁₉	—	4	8	no	—
66	C₂²×C₄×C₁₉	—	2	4	no	—
67	C₂²×C₄×C₁₉	—	2	4	no	—
68	C₂²×C₄×C₁₉	—	4	4	no	—
69	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
71	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
72	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
73	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
74	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
79	C₂×C₈×C₁₉	—	2	8	no	—
80	C₂×C₈×C₁₉	—	4	8	no	—
81	C₄⋊C₄×C₁₉	—	2	8	no	—
82	C₄⋊C₄×C₁₉	—	2	8	no	—
83	C₄⋊C₄×C₁₉	—	2	4	no	—
84	C₄⋊C₄×C₁₉	—	2	4	no	—
85	C₄⋊C₄×C₁₉	—	2	4	no	—
86	C₄⋊C₄×C₁₉	C₂²×C₄	1	8	yes	—
87	C₂²⋊C₄×C₁₉	—	2	8	no	—
88	C₄²×C₁₉	—	2	8	no	—
89	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
90	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
91	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
92	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
93	C₄²×C₁₉	C₂²×C₄	1	2	yes	—
94	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₁₉	C₂×D₄	1	2	yes	—
96	C₄²×C₁₉	C₂×D₄	1	2	yes	—
97	C₄²×C₁₉	C₂×D₄	1	2	yes	—
98	C₄²×C₁₉	C₂⁴	1	1	yes	characteristic
99	#17755	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂³×C₁₉	—	4	2	no	—
101	C₂³×C₁₉	#101	1	4	yes	—
102	C₂³×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
103	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	Q₈×C₁₉	—	4	8	no	—
105	Q₈×C₁₉	—	4	8	no	—
106	C₂×C₄×C₁₉	—	2	2	no	—
107	C₂×C₄×C₁₉	—	2	4	no	—
108	C₂×C₄×C₁₉	—	2	2	no	—
109	C₂×C₄×C₁₉	—	2	4	no	—
110	C₂×C₄×C₁₉	—	2	4	no	—
111	C₂×C₄×C₁₉	—	2	4	no	—
112	C₂×C₄×C₁₉	—	2	4	no	—
113	C₂×C₄×C₁₉	—	2	8	no	—
114	C₂×C₄×C₁₉	—	4	2	no	—
115	C₂×C₄×C₁₉	—	4	2	no	—
116	C₂×C₄×C₁₉	—	4	2	no	—
117	C₂×C₄×C₁₉	—	4	8	no	—
118	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
119	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
120	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
121	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
122	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₁₉	C₂²×D₄	1	2	yes	—
125	C₈×C₁₉	—	4	8	no	—
126	C₂×#533	C₂×C₁₉	1	1	yes	characteristic
127	C₂×#432	C₂×C₁₉	1	2	yes	—
128	#1556	C₂×C₁₉	1	4	yes	complemented
129	#1552	C₂×C₁₉	1	8	yes	complemented
130	C₂²×C₁₉	—	2	2	no	—
131	C₂²×C₁₉	—	2	2	no	—
132	C₂²×C₁₉	—	2	2	no	—
133	C₂²×C₁₉	—	4	4	no	—
134	C₂²×C₁₉	C₂×#101	1	4	yes	—
135	C₂²×C₁₉	C₂×#101	1	2	yes	—
136	C₂²×C₁₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
137	C₄×C₁₉	—	2	2	no	—
138	C₄×C₁₉	—	2	1	no	—
139	C₄×C₁₉	—	2	1	no	—
140	C₄×C₁₉	—	2	2	no	—
141	C₄×C₁₉	—	2	2	no	—
142	C₄×C₁₉	—	2	2	no	—
143	C₄×C₁₉	—	4	8	no	—
144	C₄×C₁₉	#575	1	4	yes	—
145	#533	—	2	2	no	—
146	#533	—	2	2	no	—
147	#533	C₄×C₁₉	1	4	yes	—
148	#533	C₂²×C₁₉	1	4	yes	—
149	C₂×C₄⋊Q₈	C₂²×C₁₉	1	2	yes	—
150	C₂×#125	C₄×C₁₉	1	2	yes	—
151	C₂×C₂²⋊Q₈	—	2	2	no	—
152	C₂×C₄×Q₈	—	2	2	no	—
153	C₂×#117	C₄×C₁₉	1	2	yes	—
154	C₂²×C₄²	C₂²×C₁₉	1	1	yes	characteristic
155	#432	—	2	2	no	—
156	#432	—	2	2	no	—
157	#432	C₂²×C₁₉	1	4	yes	—
158	#432	C₂²×C₁₉	1	4	yes	—
159	C₂×C₄⋊C₈	C₂²×C₁₉	1	4	yes	—
160	C₂×C₂²⋊C₈	—	2	2	no	—
161	C₂×#97	—	2	2	no	—
162	#332	C₂²×C₁₉	1	16	yes	complemented
163	C₂×C₁₉	—	2	2	no	—
164	C₂×C₁₉	—	2	4	no	—
165	C₂×C₁₉	C₂²×#101	1	1	yes	characteristic
166	C₂×C₁₉	#2896	1	2	yes	—
167	C₂²×Q₈	—	2	2	no	—
168	C₂³×C₄	C₂×C₄×C₁₉	1	1	yes	characteristic
169	C₂³×C₄	D₄×C₁₉	1	2	yes	—
170	C₂²×C₈	—	2	4	no	—
171	C₄⋊Q₈	—	2	2	no	—
172	C₄⋊Q₈	—	2	2	no	—
173	C₄⋊Q₈	C₂×C₄×C₁₉	1	4	yes	—
174	C₄⋊Q₈	C₂³×C₁₉	1	4	yes	—
175	#125	—	2	2	no	—
176	#125	—	2	2	no	—
177	#125	C₂×C₄×C₁₉	1	8	yes	—
178	C₂²⋊Q₈	—	2	4	no	—
179	C₂²⋊Q₈	—	2	4	no	—
180	C₂²⋊Q₈	—	2	4	no	—
181	C₂²⋊Q₈	—	2	4	no	—
182	C₄×Q₈	—	2	4	no	—
183	C₄×Q₈	—	2	4	no	—
184	C₄×Q₈	—	2	4	no	—
185	C₄×Q₈	—	2	4	no	—
186	#117	—	2	4	no	—
187	#117	C₂×C₄×C₁₉	1	8	yes	—
188	C₂×C₄⋊C₄	—	2	4	no	—
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	4	yes	—
191	C₂×C₂²⋊C₄	—	2	2	no	—
192	C₂×C₄²	—	2	2	no	—
193	C₂×C₄²	C₂×C₄×C₁₉	1	1	yes	characteristic
194	C₂×C₄²	C₂×C₄×C₁₉	1	2	yes	—
195	C₂×C₄²	D₄×C₁₉	1	2	yes	—
196	C₂×C₄²	D₄×C₁₉	1	2	yes	—
197	C₂×C₄²	D₄×C₁₉	1	2	yes	—
198	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
199	C₂×C₄²	C₂³×C₁₉	1	2	yes	—
200	C₄⋊C₈	—	2	4	no	—
201	C₄⋊C₈	C₂³×C₁₉	1	8	yes	—
202	C₂²⋊C₈	—	2	4	no	—
203	C₂²⋊C₈	—	2	4	no	—
204	#97	—	2	4	no	—
205	#97	—	2	4	no	—
206	C₁₉	#17755	1	1	yes	characteristic, direct product factor, Sylow
207	C₂⁴	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
208	C₂×Q₈	—	2	4	no	—
209	C₂×Q₈	—	2	4	no	—
210	C₂×Q₈	—	4	8	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	4	4	no	—
214	C₂²×C₄	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
215	C₂²×C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
216	C₂²×C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
217	C₂²×C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
218	C₂²×C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
219	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
220	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
221	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
222	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
223	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
224	C₂×C₈	—	2	8	no	—
225	C₂×C₈	—	4	8	no	—
226	C₄⋊C₄	—	2	8	no	—
227	C₄⋊C₄	—	2	8	no	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	C₂²×C₄×C₁₉	1	8	yes	—
232	C₂²⋊C₄	—	2	8	no	—
233	C₄²	—	2	8	no	—
234	C₄²	C₂²⋊C₄×C₁₉	1	2	yes	—
235	C₄²	C₂²⋊C₄×C₁₉	1	2	yes	—
236	C₄²	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
237	C₄²	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
238	C₄²	C₂²×C₄×C₁₉	1	2	yes	—
239	C₄²	C₂²×C₄×C₁₉	1	1	yes	characteristic
240	C₄²	C₂×D₄×C₁₉	1	2	yes	—
241	C₄²	C₂×D₄×C₁₉	1	2	yes	—
242	C₄²	C₂×D₄×C₁₉	1	2	yes	—
243	C₄²	C₂⁴×C₁₉	1	1	yes	characteristic
244	C₂³	—	4	2	no	—
245	C₂³	C₁₉×#101	1	4	yes	—
246	C₂³	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
247	C₂³	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
248	Q₈	—	4	8	no	—
249	Q₈	—	4	8	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	8	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	8	no	—
262	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
263	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
264	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
265	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
266	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
267	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
268	C₂×C₄	C₂²×D₄×C₁₉	1	2	yes	—
269	C₈	—	4	8	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	4	4	no	—
274	C₂²	C₂×C₁₉×#101	1	4	yes	—
275	C₂²	C₂×C₁₉×#101	1	2	yes	—
276	C₂²	C₂²×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
277	C₄	—	2	2	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	2	2	no	—
281	C₄	—	2	2	no	—
282	C₄	—	2	2	no	—
283	C₄	—	4	8	no	—
284	C₄	C₁₉×#575	1	4	yes	—
285	C₂	—	2	2	no	—
286	C₂	—	2	4	no	—
287	C₂	C₂²×C₁₉×#101	1	1	yes	characteristic
288	C₂	C₁₉×#2896	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₁₉
Frattini, cosocle	C₄²	C₂⁴×C₁₉
prime radical	C₂⁴×C₁₉	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂²×C₄²×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₉×#1556	C₂	semidirect
4. C₁₉×#1552	C₂	semidirect
22. C₁₉×#332	C₂²	semidirect
99. #17755	C₁₉	direct (N × H)
128. #1556	C₂×C₁₉	semidirect
129. #1552	C₂×C₁₉	semidirect
162. #332	C₂²×C₁₉	semidirect
206. C₁₉	#17755	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₁₉×#533	2
C₂×C₁₉×#432	2
C₁₉×#1556	2
C₁₉×#1552	2
#17755	19

Subgroups — C19×#17755 #18023714

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#17755 #18023714