Subgroups — C₂×C₁₉×#3139 #18066573

Order 4864.

2118 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×C₁₉×#533	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#3139	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₁₉×#491	C₂	1	4	yes	maximal, complemented
5	C₂×C₁₉×#487	C₂	1	4	yes	maximal, complemented
6	C₂×C₁₉×#419	C₂	1	4	yes	maximal
7	C₂×C₄×M₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic
8	C₂³×Q₈×C₁₉	C₂²	1	2	yes	—
9	C₂²×M₁₆×C₁₉	C₂²	1	2	yes	—
10	C₁₉×#533	—	2	4	no	—
11	C₁₉×#533	C₂²	1	8	yes	—
12	C₂×C₁₉×C₄⋊Q₈	—	2	2	no	—
13	C₂×C₁₉×C₄⋊Q₈	—	2	4	no	—
14	C₂×C₁₉×C₄⋊Q₈	C₂²	1	2	yes	—
15	C₂×C₁₉×C₄⋊Q₈	C₂²	1	2	yes	—
16	C₂×C₁₉×C₄⋊Q₈	C₂²	1	8	yes	—
17	C₂×C₁₉×C₄⋊Q₈	C₂²	1	2	yes	—
18	C₂×C₁₉×#125	C₂²	1	2	yes	—
19	C₂×C₁₉×C₂²⋊Q₈	—	2	2	no	—
20	C₂×C₄×Q₈×C₁₉	—	2	2	no	—
21	C₂×C₁₉×#117	C₂²	1	2	yes	—
22	C₂²×C₄⋊C₄×C₁₉	—	2	2	no	—
23	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
24	C₁₉×#491	C₂²	1	32	yes	complemented
25	C₁₉×#487	C₂²	1	32	yes	complemented
26	C₁₉×#419	C₂²	1	32	yes	—
27	C₂×C₁₉×#103	C₂²	1	16	yes	—
28	C₄×M₁₆×C₁₉	C₂²	1	8	yes	—
29	C₂×C₁₉×#97	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₁₉	C₂²	1	2	yes	—
31	C₂²×Q₈×C₁₉	—	2	2	no	—
32	C₂²×Q₈×C₁₉	—	2	4	no	—
33	C₂²×Q₈×C₁₉	—	2	2	no	—
34	C₂²×Q₈×C₁₉	—	2	16	no	—
35	C₂²×Q₈×C₁₉	C₂³	1	4	yes	—
36	C₂²×Q₈×C₁₉	C₂³	1	8	yes	—
37	C₂³×C₄×C₁₉	—	2	2	no	—
38	C₂³×C₄×C₁₉	D₄	1	2	yes	—
39	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
40	C₂×M₁₆×C₁₉	—	2	8	no	—
41	C₂×M₁₆×C₁₉	C₂³	1	8	yes	—
42	C₂²×C₈×C₁₉	C₂³	1	4	yes	—
43	C₁₉×C₄⋊Q₈	—	2	16	no	—
44	C₁₉×C₄⋊Q₈	—	2	4	no	—
45	C₁₉×C₄⋊Q₈	—	2	4	no	—
46	C₁₉×C₄⋊Q₈	—	2	4	no	—
47	C₁₉×C₄⋊Q₈	C₂³	1	8	yes	—
48	C₁₉×C₄⋊Q₈	C₂³	1	8	yes	—
49	C₁₉×C₄⋊Q₈	C₂³	1	8	yes	—
50	C₁₉×#125	—	2	4	no	—
51	C₁₉×#125	C₂³	1	8	yes	—
52	C₁₉×C₂²⋊Q₈	—	2	16	no	—
53	C₄×Q₈×C₁₉	—	2	16	no	—
54	C₁₉×#117	—	2	4	no	—
55	C₁₉×#117	C₂³	1	8	yes	—
56	C₂×C₄⋊C₄×C₁₉	—	2	16	no	—
57	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
58	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
59	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
60	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
61	C₂×C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
62	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
63	C₂×C₄²×C₁₉	—	2	2	no	—
64	C₂×C₄²×C₁₉	D₄	1	2	yes	—
65	C₂×C₄²×C₁₉	D₄	1	4	yes	—
66	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₁₉	C₂³	1	4	yes	—
69	C₁₉×#103	C₂³	1	64	yes	—
70	C₁₉×#97	C₂³	1	8	yes	—
71	C₄×C₈×C₁₉	C₂³	1	8	yes	—
72	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
73	C₂×Q₈×C₁₉	—	2	8	no	—
74	C₂×Q₈×C₁₉	—	2	8	no	—
75	C₂×Q₈×C₁₉	—	2	4	no	—
76	C₂×Q₈×C₁₉	—	2	16	no	—
77	C₂×Q₈×C₁₉	—	2	32	no	—
78	C₂×Q₈×C₁₉	—	4	16	no	—
79	C₂×Q₈×C₁₉	—	4	8	no	—
80	C₂×Q₈×C₁₉	—	4	8	no	—
81	C₂×Q₈×C₁₉	C₂⁴	1	8	yes	—
82	C₂²×C₄×C₁₉	—	2	8	no	—
83	C₂²×C₄×C₁₉	—	2	8	no	—
84	C₂²×C₄×C₁₉	—	2	4	no	—
85	C₂²×C₄×C₁₉	—	2	4	no	—
86	C₂²×C₄×C₁₉	—	4	8	no	—
87	C₂²×C₄×C₁₉	C₂×D₄	1	8	yes	—
88	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₁₉	#41	1	8	yes	—
93	C₂²×C₄×C₁₉	C₂⁴	1	2	yes	—
94	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
95	M₁₆×C₁₉	—	2	16	no	—
96	C₂×C₈×C₁₉	—	2	8	no	—
97	C₂×C₈×C₁₉	C₂⁴	1	8	yes	—
98	C₄⋊C₄×C₁₉	—	2	16	no	—
99	C₄⋊C₄×C₁₉	—	2	16	no	—
100	C₄⋊C₄×C₁₉	—	2	16	no	—
101	C₄⋊C₄×C₁₉	—	2	8	no	—
102	C₄⋊C₄×C₁₉	—	2	4	no	—
103	C₄⋊C₄×C₁₉	C₂⁴	1	8	yes	—
104	C₂²⋊C₄×C₁₉	—	2	8	no	—
105	C₄²×C₁₉	—	2	8	no	—
106	C₄²×C₁₉	C₂×D₄	1	8	yes	—
107	C₄²×C₁₉	C₂×D₄	1	2	yes	—
108	C₄²×C₁₉	C₂×D₄	1	2	yes	—
109	C₄²×C₁₉	C₂⁴	1	2	yes	—
110	C₄²×C₁₉	C₂⁴	1	2	yes	—
111	C₂×#3139	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
112	C₂³×C₁₉	—	2	4	no	—
113	C₂³×C₁₉	#124	1	4	yes	—
114	C₂³×C₁₉	C₂²×D₄	1	2	yes	—
115	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
116	Q₈×C₁₉	—	2	16	no	—
117	Q₈×C₁₉	—	4	16	no	—
118	Q₈×C₁₉	—	4	8	no	—
119	Q₈×C₁₉	—	4	16	no	—
120	C₂×C₄×C₁₉	—	2	4	no	—
121	C₂×C₄×C₁₉	—	2	16	no	—
122	C₂×C₄×C₁₉	—	2	8	no	—
123	C₂×C₄×C₁₉	—	2	8	no	—
124	C₂×C₄×C₁₉	—	4	16	no	—
125	C₂×C₄×C₁₉	—	4	8	no	—
126	C₂×C₄×C₁₉	—	4	8	no	—
127	C₂×C₄×C₁₉	#124	1	8	yes	—
128	C₂×C₄×C₁₉	#124	1	4	yes	—
129	C₂×C₄×C₁₉	#137	1	8	yes	—
130	C₂×C₄×C₁₉	C₂²×D₄	1	4	yes	—
131	C₂×C₄×C₁₉	C₂²×D₄	1	2	yes	—
132	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₉	C₂²×D₄	1	4	yes	—
134	C₂×C₄×C₁₉	C₂×#41	1	4	yes	—
135	C₂×C₄×C₁₉	C₂×#41	1	8	yes	—
136	C₂×C₄×C₁₉	C₂⁵	1	1	yes	characteristic
137	C₈×C₁₉	—	2	8	no	—
138	C₂×#533	C₂×C₁₉	1	1	yes	characteristic
139	C₂²×C₄⋊Q₈	C₂×C₁₉	1	1	yes	characteristic
140	#3139	C₂×C₁₉	1	16	yes	direct product factor
141	C₂×#491	C₂×C₁₉	1	4	yes	complemented
142	C₂×#487	C₂×C₁₉	1	4	yes	complemented
143	C₂×#419	C₂×C₁₉	1	4	yes	—
144	C₂×C₄×M₁₆	C₂×C₁₉	1	1	yes	characteristic
145	C₂²×C₁₉	—	2	8	no	—
146	C₂²×C₁₉	—	2	4	no	—
147	C₂²×C₁₉	C₂×#124	1	4	yes	—
148	C₂²×C₁₉	C₂×#124	1	2	yes	—
149	C₂²×C₁₉	C₂×#137	1	4	yes	—
150	C₂²×C₁₉	C₂³×D₄	1	1	yes	characteristic
151	C₄×C₁₉	—	2	4	no	—
152	C₄×C₁₉	—	4	8	no	—
153	C₄×C₁₉	—	4	8	no	—
154	C₄×C₁₉	C₂×#124	1	4	yes	—
155	C₄×C₁₉	C₂×#124	1	2	yes	—
156	C₄×C₁₉	C₂×#137	1	8	yes	—
157	C₄×C₁₉	C₂²×#41	1	2	yes	—
158	C₂³×Q₈	C₂²×C₁₉	1	2	yes	—
159	C₂²×M₁₆	C₂²×C₁₉	1	2	yes	—
160	#533	—	2	4	no	—
161	#533	C₂²×C₁₉	1	8	yes	—
162	C₂×C₄⋊Q₈	—	2	2	no	—
163	C₂×C₄⋊Q₈	—	2	4	no	—
164	C₂×C₄⋊Q₈	C₂²×C₁₉	1	2	yes	—
165	C₂×C₄⋊Q₈	C₂²×C₁₉	1	2	yes	—
166	C₂×C₄⋊Q₈	C₂²×C₁₉	1	8	yes	—
167	C₂×C₄⋊Q₈	C₂²×C₁₉	1	2	yes	—
168	C₂×#125	C₂²×C₁₉	1	2	yes	—
169	C₂×C₂²⋊Q₈	—	2	2	no	—
170	C₂×C₄×Q₈	—	2	2	no	—
171	C₂×#117	C₂²×C₁₉	1	2	yes	—
172	C₂²×C₄⋊C₄	—	2	2	no	—
173	C₂²×C₄²	C₂²×C₁₉	1	1	yes	characteristic
174	#491	C₂²×C₁₉	1	32	yes	complemented
175	#487	C₂²×C₁₉	1	32	yes	complemented
176	#419	C₂²×C₁₉	1	32	yes	—
177	C₂×#103	C₂²×C₁₉	1	16	yes	—
178	C₄×M₁₆	C₂²×C₁₉	1	8	yes	—
179	C₂×#97	C₂²×C₁₉	1	2	yes	—
180	C₂×C₄×C₈	C₂²×C₁₉	1	2	yes	—
181	C₂×C₁₉	—	2	4	no	—
182	C₂×C₁₉	#3139	1	4	yes	direct product factor
183	C₂×C₁₉	C₂²×#124	1	1	yes	characteristic
184	C₂×C₁₉	C₂²×#137	1	2	yes	—
185	C₂²×Q₈	—	2	2	no	—
186	C₂²×Q₈	—	2	2	no	—
187	C₂²×Q₈	—	2	4	no	—
188	C₂²×Q₈	—	2	16	no	—
189	C₂²×Q₈	C₂³×C₁₉	1	4	yes	—
190	C₂²×Q₈	C₂³×C₁₉	1	8	yes	—
191	C₂³×C₄	—	2	2	no	—
192	C₂³×C₄	D₄×C₁₉	1	2	yes	—
193	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
194	C₂×M₁₆	—	2	8	no	—
195	C₂×M₁₆	C₂³×C₁₉	1	8	yes	—
196	C₂²×C₈	C₂³×C₁₉	1	4	yes	—
197	C₄⋊Q₈	—	2	16	no	—
198	C₄⋊Q₈	—	2	4	no	—
199	C₄⋊Q₈	—	2	4	no	—
200	C₄⋊Q₈	—	2	4	no	—
201	C₄⋊Q₈	C₂³×C₁₉	1	8	yes	—
202	C₄⋊Q₈	C₂³×C₁₉	1	8	yes	—
203	C₄⋊Q₈	C₂³×C₁₉	1	8	yes	—
204	#125	—	2	4	no	—
205	#125	C₂³×C₁₉	1	8	yes	—
206	C₂²⋊Q₈	—	2	16	no	—
207	C₄×Q₈	—	2	16	no	—
208	#117	—	2	4	no	—
209	#117	C₂³×C₁₉	1	8	yes	—
210	C₂×C₄⋊C₄	—	2	4	no	—
211	C₂×C₄⋊C₄	—	2	16	no	—
212	C₂×C₄⋊C₄	—	2	4	no	—
213	C₂×C₄⋊C₄	—	2	4	no	—
214	C₂×C₄⋊C₄	—	2	2	no	—
215	C₂×C₄⋊C₄	C₂³×C₁₉	1	4	yes	—
216	C₂×C₂²⋊C₄	—	2	2	no	—
217	C₂×C₄²	—	2	2	no	—
218	C₂×C₄²	D₄×C₁₉	1	2	yes	—
219	C₂×C₄²	D₄×C₁₉	1	4	yes	—
220	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
221	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
222	C₂×C₄²	C₂³×C₁₉	1	4	yes	—
223	#103	C₂³×C₁₉	1	64	yes	—
224	#97	C₂³×C₁₉	1	8	yes	—
225	C₄×C₈	C₂³×C₁₉	1	8	yes	—
226	C₁₉	C₂×#3139	1	1	yes	characteristic, direct product factor, Sylow
227	C₂⁴	C₂×D₄×C₁₉	1	1	yes	characteristic
228	C₂×Q₈	—	2	8	no	—
229	C₂×Q₈	—	2	8	no	—
230	C₂×Q₈	—	2	4	no	—
231	C₂×Q₈	—	2	16	no	—
232	C₂×Q₈	—	2	32	no	—
233	C₂×Q₈	—	4	16	no	—
234	C₂×Q₈	—	4	8	no	—
235	C₂×Q₈	—	4	8	no	—
236	C₂×Q₈	C₂⁴×C₁₉	1	8	yes	—
237	C₂²×C₄	—	2	8	no	—
238	C₂²×C₄	—	2	4	no	—
239	C₂²×C₄	—	2	8	no	—
240	C₂²×C₄	—	2	4	no	—
241	C₂²×C₄	—	4	8	no	—
242	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
243	C₂²×C₄	C₂×D₄×C₁₉	1	8	yes	—
244	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
245	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
246	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
247	C₂²×C₄	#41×C₁₉	1	8	yes	—
248	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
249	C₂²×C₄	C₂⁴×C₁₉	1	2	yes	—
250	M₁₆	—	2	16	no	—
251	C₂×C₈	—	2	8	no	—
252	C₂×C₈	C₂⁴×C₁₉	1	8	yes	—
253	C₄⋊C₄	—	2	16	no	—
254	C₄⋊C₄	—	2	16	no	—
255	C₄⋊C₄	—	2	16	no	—
256	C₄⋊C₄	—	2	8	no	—
257	C₄⋊C₄	—	2	4	no	—
258	C₄⋊C₄	C₂⁴×C₁₉	1	8	yes	—
259	C₂²⋊C₄	—	2	8	no	—
260	C₄²	—	2	8	no	—
261	C₄²	C₂×D₄×C₁₉	1	8	yes	—
262	C₄²	C₂×D₄×C₁₉	1	2	yes	—
263	C₄²	C₂×D₄×C₁₉	1	2	yes	—
264	C₄²	C₂⁴×C₁₉	1	2	yes	—
265	C₄²	C₂⁴×C₁₉	1	2	yes	—
266	C₂³	—	2	4	no	—
267	C₂³	C₁₉×#124	1	4	yes	—
268	C₂³	C₂²×D₄×C₁₉	1	2	yes	—
269	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
270	Q₈	—	2	16	no	—
271	Q₈	—	4	16	no	—
272	Q₈	—	4	8	no	—
273	Q₈	—	4	16	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	16	no	—
276	C₂×C₄	—	2	8	no	—
277	C₂×C₄	—	2	8	no	—
278	C₂×C₄	—	4	16	no	—
279	C₂×C₄	—	4	8	no	—
280	C₂×C₄	—	4	8	no	—
281	C₂×C₄	C₁₉×#124	1	8	yes	—
282	C₂×C₄	C₁₉×#124	1	4	yes	—
283	C₂×C₄	C₁₉×#137	1	8	yes	—
284	C₂×C₄	C₂²×D₄×C₁₉	1	4	yes	—
285	C₂×C₄	C₂²×D₄×C₁₉	1	2	yes	—
286	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
287	C₂×C₄	C₂²×D₄×C₁₉	1	4	yes	—
288	C₂×C₄	C₂×#41×C₁₉	1	4	yes	—
289	C₂×C₄	C₂×#41×C₁₉	1	8	yes	—
290	C₂×C₄	C₂⁵×C₁₉	1	1	yes	characteristic
291	C₈	—	2	8	no	—
292	C₂²	—	2	4	no	—
293	C₂²	—	2	8	no	—
294	C₂²	C₂×C₁₉×#124	1	4	yes	—
295	C₂²	C₂×C₁₉×#124	1	2	yes	—
296	C₂²	C₂×C₁₉×#137	1	4	yes	—
297	C₂²	C₂³×D₄×C₁₉	1	1	yes	characteristic
298	C₄	—	2	4	no	—
299	C₄	—	4	8	no	—
300	C₄	—	4	8	no	—
301	C₄	C₂×C₁₉×#124	1	4	yes	—
302	C₄	C₂×C₁₉×#124	1	2	yes	—
303	C₄	C₂×C₁₉×#137	1	8	yes	—
304	C₄	C₂²×#41×C₁₉	1	2	yes	—
305	C₂	—	2	4	no	—
306	C₂	C₁₉×#3139	1	4	yes	direct product factor
307	C₂	C₂²×C₁₉×#124	1	1	yes	characteristic
308	C₂	C₂²×C₁₉×#137	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₁₉
prime radical	C₂⁴×C₁₉	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ C₂²×C₄²×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₉×#3139	C₂	direct (N × H)
4. C₂×C₁₉×#491	C₂	semidirect
5. C₂×C₁₉×#487	C₂	semidirect
24. C₁₉×#491	C₂²	semidirect
25. C₁₉×#487	C₂²	semidirect
111. C₂×#3139	C₁₉	direct (N × H)
140. #3139	C₂×C₁₉	direct (N × H)
141. C₂×#491	C₂×C₁₉	semidirect
142. C₂×#487	C₂×C₁₉	semidirect
174. #491	C₂²×C₁₉	semidirect
175. #487	C₂²×C₁₉	semidirect
182. C₂×C₁₉	#3139	direct (N × H)
226. C₁₉	C₂×#3139	direct (N × H)
306. C₂	C₁₉×#3139	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
C₂²×C₁₉×C₄⋊Q₈	2
C₁₉×#3139	2
C₂×C₁₉×#491	2
C₂×C₁₉×#487	2
C₂×C₁₉×#419	2
C₂×C₄×M₁₆×C₁₉	2
C₂×#3139	19

Subgroups — C2×C19×#3139 #18066573

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₉×#3139 #18066573