Subgroups — C₁₉×#34197 #18040156

Order 4864.

2314 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	C₂×C₄×#41×C₁₉	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2357	C₂	1	6	yes	maximal, complemented
3	C₁₉×#2351	C₂	1	2	yes	maximal, complemented
4	C₁₉×#2346	C₂	1	6	yes	maximal, complemented
5	C₁₉×#2341	C₂	1	2	yes	maximal
6	C₁₉×#2319	C₂	1	6	yes	maximal, complemented
7	C₁₉×#2283	C₂	1	3	yes	maximal, complemented
8	C₁₉×#2283	C₂	1	3	yes	maximal, complemented
9	C₁₉×#2276	C₂	1	2	yes	maximal, complemented
10	C₂²×#41×C₁₉	C₂²	1	1	yes	characteristic
11	C₄×#41×C₁₉	C₄	1	16	yes	complemented
12	C₂×C₄×Q₈×C₁₉	C₂²	1	4	yes	—
13	C₂×C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₁₉	C₂²	1	3	yes	—
16	C₂×C₄×D₄×C₁₉	C₂²	1	3	yes	—
17	C₂×C₁₉×#117	C₂²	1	3	yes	—
18	C₂²×C₄²×C₁₉	C₂²	1	3	yes	—
19	C₁₉×#390	C₂²	1	12	yes	—
20	C₁₉×#389	C₂²	1	12	yes	—
21	C₁₉×#388	C₂²	1	24	yes	—
22	C₁₉×#383	C₂²	1	1	yes	characteristic
23	C₁₉×#383	C₂²	1	3	yes	—
24	C₁₉×#382	C₂²	1	6	yes	—
25	C₁₉×#382	C₂²	1	6	yes	—
26	C₁₉×#381	C₂²	1	6	yes	—
27	C₁₉×#381	C₂²	1	6	yes	—
28	C₁₉×#380	C₂²	1	6	yes	—
29	C₁₉×#380	C₂²	1	6	yes	—
30	C₄×C₄⋊C₄×C₁₉	C₂²	1	6	yes	—
31	C₄×C₄⋊C₄×C₁₉	C₂²	1	12	yes	—
32	C₄×C₄⋊C₄×C₁₉	C₂²	1	6	yes	—
33	C₄×C₂²⋊C₄×C₁₉	C₂²	1	6	yes	—
34	C₄×C₂²⋊C₄×C₁₉	C₂²	1	6	yes	—
35	C₁₉×#376	C₂²	1	6	yes	—
36	C₁₉×#376	C₂²	1	6	yes	—
37	C₂×#41×C₁₉	—	2	8	no	—
38	C₂×#41×C₁₉	C₂×C₄	1	8	yes	—
39	C₂²×Q₈×C₁₉	C₂³	1	1	yes	characteristic
40	C₂²×D₄×C₁₉	C₂³	1	3	yes	—
41	C₂³×C₄×C₁₉	C₂³	1	3	yes	—
42	C₂³×C₄×C₁₉	C₂³	1	3	yes	—
43	C₂³×C₄×C₁₉	C₂³	1	3	yes	—
44	C₄×Q₈×C₁₉	—	2	16	no	—
45	C₄×Q₈×C₁₉	C₂×C₄	1	8	yes	—
46	C₄×Q₈×C₁₉	C₂×C₄	1	8	yes	—
47	C₄×D₄×C₁₉	C₂×C₄	1	24	yes	—
48	C₄×D₄×C₁₉	C₂×C₄	1	24	yes	—
49	C₁₉×#117	C₂×C₄	1	24	yes	—
50	C₂×C₄⋊C₄×C₁₉	C₂³	1	12	yes	—
51	C₂×C₄⋊C₄×C₁₉	C₂³	1	12	yes	—
52	C₂×C₄⋊C₄×C₁₉	C₂³	1	3	yes	—
53	C₂×C₄⋊C₄×C₁₉	C₂³	1	3	yes	—
54	C₂×C₂²⋊C₄×C₁₉	C₂³	1	12	yes	—
55	C₂×C₂²⋊C₄×C₁₉	C₂³	1	12	yes	—
56	C₂×C₂²⋊C₄×C₁₉	C₂³	1	6	yes	—
57	C₂×C₄²×C₁₉	C₂×C₄	1	24	yes	—
58	C₂×C₄²×C₁₉	C₂³	1	12	yes	—
59	C₂×C₄²×C₁₉	C₂³	1	12	yes	—
60	C₂×C₄²×C₁₉	C₂³	1	3	yes	—
61	C₂×C₄²×C₁₉	C₂³	1	3	yes	—
62	C₂×C₄²×C₁₉	C₂³	1	3	yes	—
63	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
64	C₁₉×#95	C₂³	1	2	yes	—
65	C₁₉×#95	C₂³	1	6	yes	—
66	C₁₉×#95	C₂³	1	6	yes	—
67	C₁₉×#95	C₂³	1	12	yes	—
68	C₁₉×#95	C₂³	1	12	yes	—
69	C₁₉×#95	C₂³	1	2	yes	—
70	C₁₉×#95	C₂³	1	6	yes	—
71	C₁₉×#95	C₂³	1	2	yes	—
72	C₂⁴×C₁₉	C₂⁴	1	3	yes	—
73	#41×C₁₉	—	2	32	no	—
74	C₂×Q₈×C₁₉	C₂²×C₄	1	4	yes	—
75	C₂×Q₈×C₁₉	#41	1	8	yes	—
76	C₂×D₄×C₁₉	—	2	12	no	—
77	C₂×D₄×C₁₉	C₂²×C₄	1	12	yes	—
78	C₂²×C₄×C₁₉	—	2	12	no	—
79	C₂²×C₄×C₁₉	—	4	6	no	—
80	C₂²×C₄×C₁₉	—	4	6	no	—
81	C₂²×C₄×C₁₉	C₂²×C₄	1	12	yes	—
82	C₂²×C₄×C₁₉	C₂²×C₄	1	12	yes	—
83	C₂²×C₄×C₁₉	C₂²×C₄	1	12	yes	—
84	C₂²×C₄×C₁₉	C₂⁴	1	4	yes	—
85	C₂²×C₄×C₁₉	C₂⁴	1	12	yes	—
86	C₂²×C₄×C₁₉	C₂⁴	1	3	yes	—
87	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
88	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
89	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
90	C₂²×C₄×C₁₉	C₂⁴	1	3	yes	—
91	C₂²×C₄×C₁₉	C₂⁴	1	3	yes	—
92	C₄⋊C₄×C₁₉	—	2	24	no	—
93	C₄⋊C₄×C₁₉	—	4	12	no	—
94	C₄⋊C₄×C₁₉	C₂²×C₄	1	12	yes	—
95	C₄⋊C₄×C₁₉	C₂²×C₄	1	12	yes	—
96	C₂²⋊C₄×C₁₉	—	4	12	no	—
97	C₂²⋊C₄×C₁₉	—	4	12	no	—
98	C₂²⋊C₄×C₁₉	C₂²×C₄	1	24	yes	—
99	C₄²×C₁₉	—	2	24	no	—
100	C₄²×C₁₉	—	4	12	no	—
101	C₄²×C₁₉	C₂²×C₄	1	4	yes	—
102	C₄²×C₁₉	C₂²×C₄	1	12	yes	—
103	C₄²×C₁₉	C₂²×C₄	1	12	yes	—
104	C₄²×C₁₉	C₂²×C₄	1	12	yes	—
105	#34197	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
106	C₂³×C₁₉	—	2	6	no	—
107	C₂³×C₁₉	—	2	6	no	—
108	C₂³×C₁₉	—	2	6	no	—
109	C₂³×C₁₉	C₂³×C₄	1	6	yes	—
110	C₂³×C₁₉	C₂⁵	1	1	yes	characteristic
111	Q₈×C₁₉	#117	1	16	yes	—
112	D₄×C₁₉	—	2	24	no	—
113	C₂×C₄×C₁₉	—	2	2	no	—
114	C₂×C₄×C₁₉	—	2	6	no	—
115	C₂×C₄×C₁₉	—	2	6	no	—
116	C₂×C₄×C₁₉	—	2	2	no	—
117	C₂×C₄×C₁₉	—	2	2	no	—
118	C₂×C₄×C₁₉	—	2	24	no	—
119	C₂×C₄×C₁₉	—	2	4	no	—
120	C₂×C₄×C₁₉	—	2	12	no	—
121	C₂×C₄×C₁₉	—	2	12	no	—
122	C₂×C₄×C₁₉	—	2	4	no	—
123	C₂×C₄×C₁₉	—	2	12	no	—
124	C₂×C₄×C₁₉	—	2	4	no	—
125	C₂×C₄×C₁₉	—	4	12	no	—
126	C₂×C₄×C₁₉	—	4	12	no	—
127	C₂×C₄×C₁₉	C₂³×C₄	1	2	yes	—
128	C₂×C₄×C₁₉	C₂³×C₄	1	2	yes	—
129	C₂×C₄×C₁₉	C₂³×C₄	1	2	yes	—
130	C₂×C₄×C₁₉	C₂³×C₄	1	6	yes	—
131	C₂×C₄×C₁₉	C₂³×C₄	1	6	yes	—
132	C₂×C₄×C₁₉	C₂³×C₄	1	6	yes	—
133	C₂×C₄×C₁₉	C₂×#41	1	12	yes	—
134	C₂×C₄×#41	C₂×C₁₉	1	1	yes	characteristic
135	#2357	C₂×C₁₉	1	6	yes	complemented
136	#2351	C₂×C₁₉	1	2	yes	complemented
137	#2346	C₂×C₁₉	1	6	yes	complemented
138	#2341	C₂×C₁₉	1	2	yes	—
139	#2319	C₂×C₁₉	1	6	yes	complemented
140	#2283	C₂×C₁₉	1	3	yes	complemented
141	#2283	C₂×C₁₉	1	3	yes	complemented
142	#2276	C₂×C₁₉	1	2	yes	complemented
143	C₂²×C₁₉	—	2	6	no	—
144	C₂²×C₁₉	—	2	6	no	—
145	C₂²×C₁₉	—	2	6	no	—
146	C₂²×C₁₉	—	4	12	no	—
147	C₂²×C₁₉	C₂⁴×C₄	1	1	yes	characteristic
148	C₂²×C₁₉	C₂²×#41	1	2	yes	—
149	C₂²×C₁₉	#585	1	2	yes	—
150	C₂²×C₁₉	#585	1	2	yes	—
151	C₄×C₁₉	—	2	2	no	—
152	C₄×C₁₉	—	2	2	no	—
153	C₄×C₁₉	—	2	2	no	—
154	C₄×C₁₉	—	2	6	no	—
155	C₄×C₁₉	—	2	6	no	—
156	C₄×C₁₉	—	4	4	no	—
157	C₄×C₁₉	—	4	12	no	—
158	C₄×C₁₉	C₂×#117	1	12	yes	—
159	C₂²×#41	C₂²×C₁₉	1	1	yes	characteristic
160	C₄×#41	C₄×C₁₉	1	16	yes	complemented
161	C₂×C₄×Q₈	C₂²×C₁₉	1	4	yes	—
162	C₂×C₄×Q₈	C₂²×C₁₉	1	1	yes	characteristic
163	C₂×C₄×Q₈	C₂²×C₁₉	1	1	yes	characteristic
164	C₂×C₄×D₄	C₂²×C₁₉	1	3	yes	—
165	C₂×C₄×D₄	C₂²×C₁₉	1	3	yes	—
166	C₂×#117	C₂²×C₁₉	1	3	yes	—
167	C₂²×C₄²	C₂²×C₁₉	1	3	yes	—
168	#390	C₂²×C₁₉	1	12	yes	—
169	#389	C₂²×C₁₉	1	12	yes	—
170	#388	C₂²×C₁₉	1	24	yes	—
171	#383	C₂²×C₁₉	1	1	yes	characteristic
172	#383	C₂²×C₁₉	1	3	yes	—
173	#382	C₂²×C₁₉	1	6	yes	—
174	#382	C₂²×C₁₉	1	6	yes	—
175	#381	C₂²×C₁₉	1	6	yes	—
176	#381	C₂²×C₁₉	1	6	yes	—
177	#380	C₂²×C₁₉	1	6	yes	—
178	#380	C₂²×C₁₉	1	6	yes	—
179	C₄×C₄⋊C₄	C₂²×C₁₉	1	6	yes	—
180	C₄×C₄⋊C₄	C₂²×C₁₉	1	12	yes	—
181	C₄×C₄⋊C₄	C₂²×C₁₉	1	6	yes	—
182	C₄×C₂²⋊C₄	C₂²×C₁₉	1	6	yes	—
183	C₄×C₂²⋊C₄	C₂²×C₁₉	1	6	yes	—
184	#376	C₂²×C₁₉	1	6	yes	—
185	#376	C₂²×C₁₉	1	6	yes	—
186	C₂×C₁₉	—	4	6	no	—
187	C₂×C₁₉	C₂²×#117	1	1	yes	characteristic
188	C₂×C₁₉	#3428	1	1	yes	characteristic
189	C₂×C₁₉	#3428	1	1	yes	characteristic
190	C₂×C₁₉	#3524	1	4	yes	—
191	C₂×#41	—	2	8	no	—
192	C₂×#41	C₂×C₄×C₁₉	1	8	yes	—
193	C₂²×Q₈	C₂³×C₁₉	1	1	yes	characteristic
194	C₂²×D₄	C₂³×C₁₉	1	3	yes	—
195	C₂³×C₄	C₂³×C₁₉	1	3	yes	—
196	C₂³×C₄	C₂³×C₁₉	1	3	yes	—
197	C₂³×C₄	C₂³×C₁₉	1	3	yes	—
198	C₄×Q₈	—	2	16	no	—
199	C₄×Q₈	C₂×C₄×C₁₉	1	8	yes	—
200	C₄×Q₈	C₂×C₄×C₁₉	1	8	yes	—
201	C₄×D₄	C₂×C₄×C₁₉	1	24	yes	—
202	C₄×D₄	C₂×C₄×C₁₉	1	24	yes	—
203	#117	C₂×C₄×C₁₉	1	24	yes	—
204	C₂×C₄⋊C₄	C₂³×C₁₉	1	12	yes	—
205	C₂×C₄⋊C₄	C₂³×C₁₉	1	12	yes	—
206	C₂×C₄⋊C₄	C₂³×C₁₉	1	3	yes	—
207	C₂×C₄⋊C₄	C₂³×C₁₉	1	3	yes	—
208	C₂×C₂²⋊C₄	C₂³×C₁₉	1	12	yes	—
209	C₂×C₂²⋊C₄	C₂³×C₁₉	1	12	yes	—
210	C₂×C₂²⋊C₄	C₂³×C₁₉	1	6	yes	—
211	C₂×C₄²	C₂×C₄×C₁₉	1	24	yes	—
212	C₂×C₄²	C₂³×C₁₉	1	12	yes	—
213	C₂×C₄²	C₂³×C₁₉	1	12	yes	—
214	C₂×C₄²	C₂³×C₁₉	1	3	yes	—
215	C₂×C₄²	C₂³×C₁₉	1	3	yes	—
216	C₂×C₄²	C₂³×C₁₉	1	3	yes	—
217	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
218	#95	C₂³×C₁₉	1	2	yes	—
219	#95	C₂³×C₁₉	1	6	yes	—
220	#95	C₂³×C₁₉	1	6	yes	—
221	#95	C₂³×C₁₉	1	12	yes	—
222	#95	C₂³×C₁₉	1	12	yes	—
223	#95	C₂³×C₁₉	1	2	yes	—
224	#95	C₂³×C₁₉	1	6	yes	—
225	#95	C₂³×C₁₉	1	2	yes	—
226	C₁₉	#34197	1	1	yes	characteristic, direct product factor, Sylow
227	C₂⁴	C₂⁴×C₁₉	1	3	yes	—
228	#41	—	2	32	no	—
229	C₂×Q₈	C₂²×C₄×C₁₉	1	4	yes	—
230	C₂×Q₈	#41×C₁₉	1	8	yes	—
231	C₂×D₄	—	2	12	no	—
232	C₂×D₄	C₂²×C₄×C₁₉	1	12	yes	—
233	C₂²×C₄	—	2	12	no	—
234	C₂²×C₄	—	4	6	no	—
235	C₂²×C₄	—	4	6	no	—
236	C₂²×C₄	C₂²×C₄×C₁₉	1	12	yes	—
237	C₂²×C₄	C₂²×C₄×C₁₉	1	12	yes	—
238	C₂²×C₄	C₂²×C₄×C₁₉	1	12	yes	—
239	C₂²×C₄	C₂⁴×C₁₉	1	4	yes	—
240	C₂²×C₄	C₂⁴×C₁₉	1	12	yes	—
241	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
242	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
243	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
244	C₂²×C₄	C₂⁴×C₁₉	1	3	yes	—
245	C₂²×C₄	C₂⁴×C₁₉	1	3	yes	—
246	C₂²×C₄	C₂⁴×C₁₉	1	3	yes	—
247	C₄⋊C₄	—	2	24	no	—
248	C₄⋊C₄	—	4	12	no	—
249	C₄⋊C₄	C₂²×C₄×C₁₉	1	12	yes	—
250	C₄⋊C₄	C₂²×C₄×C₁₉	1	12	yes	—
251	C₂²⋊C₄	—	4	12	no	—
252	C₂²⋊C₄	—	4	12	no	—
253	C₂²⋊C₄	C₂²×C₄×C₁₉	1	24	yes	—
254	C₄²	—	2	24	no	—
255	C₄²	—	4	12	no	—
256	C₄²	C₂²×C₄×C₁₉	1	4	yes	—
257	C₄²	C₂²×C₄×C₁₉	1	12	yes	—
258	C₄²	C₂²×C₄×C₁₉	1	12	yes	—
259	C₄²	C₂²×C₄×C₁₉	1	12	yes	—
260	C₂³	—	2	6	no	—
261	C₂³	—	2	6	no	—
262	C₂³	—	2	6	no	—
263	C₂³	C₂³×C₄×C₁₉	1	6	yes	—
264	C₂³	C₂⁵×C₁₉	1	1	yes	characteristic
265	Q₈	C₁₉×#117	1	16	yes	—
266	D₄	—	2	24	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	6	no	—
269	C₂×C₄	—	2	6	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	24	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	12	no	—
275	C₂×C₄	—	2	12	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	12	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	4	12	no	—
280	C₂×C₄	—	4	12	no	—
281	C₂×C₄	C₂³×C₄×C₁₉	1	2	yes	—
282	C₂×C₄	C₂³×C₄×C₁₉	1	2	yes	—
283	C₂×C₄	C₂³×C₄×C₁₉	1	2	yes	—
284	C₂×C₄	C₂³×C₄×C₁₉	1	6	yes	—
285	C₂×C₄	C₂³×C₄×C₁₉	1	6	yes	—
286	C₂×C₄	C₂³×C₄×C₁₉	1	6	yes	—
287	C₂×C₄	C₂×#41×C₁₉	1	12	yes	—
288	C₂²	—	2	6	no	—
289	C₂²	—	2	6	no	—
290	C₂²	—	2	6	no	—
291	C₂²	—	4	12	no	—
292	C₂²	C₂⁴×C₄×C₁₉	1	1	yes	characteristic
293	C₂²	C₂²×#41×C₁₉	1	2	yes	—
294	C₂²	C₁₉×#585	1	2	yes	—
295	C₂²	C₁₉×#585	1	2	yes	—
296	C₄	—	2	2	no	—
297	C₄	—	2	2	no	—
298	C₄	—	2	6	no	—
299	C₄	—	2	6	no	—
300	C₄	—	2	2	no	—
301	C₄	—	4	4	no	—
302	C₄	—	4	12	no	—
303	C₄	C₂×C₁₉×#117	1	12	yes	—
304	C₂	—	4	6	no	—
305	C₂	C₂²×C₁₉×#117	1	1	yes	characteristic
306	C₂	C₁₉×#3428	1	1	yes	characteristic
307	C₂	C₁₉×#3428	1	1	yes	characteristic
308	C₂	C₁₉×#3524	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₉	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₁₉
Frattini, cosocle	C₂³	C₂⁵×C₁₉
prime radical	C₂²×#41×C₁₉	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₉×#2357	C₂	semidirect
3. C₁₉×#2351	C₂	semidirect
4. C₁₉×#2346	C₂	semidirect
6. C₁₉×#2319	C₂	semidirect
7. C₁₉×#2283	C₂	semidirect
8. C₁₉×#2283	C₂	semidirect
9. C₁₉×#2276	C₂	semidirect
11. C₄×#41×C₁₉	C₄	semidirect
105. #34197	C₁₉	direct (N × H)
135. #2357	C₂×C₁₉	semidirect
136. #2351	C₂×C₁₉	semidirect
137. #2346	C₂×C₁₉	semidirect
139. #2319	C₂×C₁₉	semidirect
140. #2283	C₂×C₁₉	semidirect
141. #2283	C₂×C₁₉	semidirect
142. #2276	C₂×C₁₉	semidirect
160. C₄×#41	C₄×C₁₉	semidirect
226. C₁₉	#34197	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×C₄×#41×C₁₉	2
C₁₉×#2357	2
C₁₉×#2351	2
C₁₉×#2346	2
C₁₉×#2341	2
C₁₉×#2319	2
C₁₉×#2283	2
C₁₉×#2276	2
#34197	19

Subgroups — C19×#34197 #18040156

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#34197 #18040156