Subgroups — C₂×C₁₉×#2848 #18042571

Order 4864.

7502 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₄×C₁₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₉×C₂²⋊Q₈	C₂	1	3	yes	maximal, complemented
3	C₁₉×#2848	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₁₉×#385	C₂	1	6	yes	maximal, complemented
5	C₂×C₁₉×#380	C₂	1	3	yes	maximal, complemented
6	C₂×C₁₉×#379	C₂	1	2	yes	maximal
7	C₂⁶×C₁₉	C₂²	1	1	yes	characteristic
8	C₂³×Q₈×C₁₉	C₂²	1	1	yes	characteristic, complemented
9	C₂⁴×C₄×C₁₉	—	2	24	no	—
10	C₂⁴×C₄×C₁₉	C₂²	1	3	yes	—
11	C₂⁴×C₄×C₁₉	C₂²	1	8	yes	—
12	C₂⁴×C₄×C₁₉	C₂²	1	3	yes	—
13	C₂×C₁₉×C₂²⋊Q₈	—	2	24	no	—
14	C₂×C₁₉×C₂²⋊Q₈	C₂²	1	24	yes	complemented
15	C₂²×C₄⋊C₄×C₁₉	C₂²	1	3	yes	complemented
16	C₂²×C₄⋊C₄×C₁₉	C₂²	1	6	yes	complemented
17	C₂²×C₂²⋊C₄×C₁₉	C₂²	1	6	yes	—
18	C₂²×C₂²⋊C₄×C₁₉	C₂²	1	6	yes	—
19	C₁₉×#385	C₂²	1	48	yes	complemented
20	C₁₉×#380	C₂²	1	24	yes	complemented
21	C₁₉×#379	C₂²	1	16	yes	—
22	C₂×C₁₉×#95	C₂²	1	6	yes	complemented
23	C₂⁵×C₁₉	—	2	12	no	—
24	C₂⁵×C₁₉	—	2	12	no	—
25	C₂⁵×C₁₉	Q₈	1	8	yes	complemented
26	C₂⁵×C₁₉	C₂³	1	4	yes	—
27	C₂⁵×C₁₉	C₂³	1	3	yes	—
28	C₂²×Q₈×C₁₉	—	2	12	no	—
29	C₂²×Q₈×C₁₉	C₂³	1	4	yes	complemented
30	C₂³×C₄×C₁₉	—	2	128	no	—
31	C₂³×C₄×C₁₉	—	2	12	no	—
32	C₂³×C₄×C₁₉	—	2	24	no	—
33	C₂³×C₄×C₁₉	—	2	96	no	—
34	C₂³×C₄×C₁₉	—	2	24	no	—
35	C₂³×C₄×C₁₉	D₄	1	24	yes	—
36	C₂³×C₄×C₁₉	C₂³	1	12	yes	—
37	C₂³×C₄×C₁₉	C₂³	1	6	yes	—
38	C₂³×C₄×C₁₉	C₂³	1	2	yes	—
39	C₂³×C₄×C₁₉	C₂³	1	12	yes	—
40	C₂³×C₄×C₁₉	C₂³	1	3	yes	—
41	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
42	C₁₉×C₂²⋊Q₈	—	2	96	no	—
43	C₂×C₄⋊C₄×C₁₉	—	2	12	no	—
44	C₂×C₄⋊C₄×C₁₉	—	2	24	no	—
45	C₂×C₄⋊C₄×C₁₉	C₂³	1	12	yes	complemented
46	C₂×C₄⋊C₄×C₁₉	C₂³	1	24	yes	complemented
47	C₂×C₂²⋊C₄×C₁₉	—	2	24	no	—
48	C₂×C₂²⋊C₄×C₁₉	—	2	24	no	—
49	C₂×C₂²⋊C₄×C₁₉	C₂³	1	24	yes	—
50	C₂×C₂²⋊C₄×C₁₉	C₂³	1	24	yes	—
51	C₁₉×#95	C₂³	1	24	yes	complemented
52	C₂⁴×C₁₉	—	2	32	no	—
53	C₂⁴×C₁₉	—	2	96	no	—
54	C₂⁴×C₁₉	—	2	96	no	—
55	C₂⁴×C₁₉	—	2	24	no	—
56	C₂⁴×C₁₉	—	2	12	no	—
57	C₂⁴×C₁₉	—	2	24	no	—
58	C₂⁴×C₁₉	—	2	12	no	—
59	C₂⁴×C₁₉	C₂×D₄	1	12	yes	—
60	C₂⁴×C₁₉	C₂×Q₈	1	12	yes	—
61	C₂⁴×C₁₉	C₂×Q₈	1	16	yes	complemented
62	C₂⁴×C₁₉	#41	1	12	yes	—
63	C₂⁴×C₁₉	C₂⁴	1	6	yes	—
64	C₂⁴×C₁₉	C₂⁴	1	1	yes	characteristic
65	C₂×Q₈×C₁₉	—	2	24	no	—
66	C₂×Q₈×C₁₉	—	4	16	no	—
67	C₂²×C₄×C₁₉	—	2	96	no	—
68	C₂²×C₄×C₁₉	—	2	256	no	—
69	C₂²×C₄×C₁₉	—	2	24	no	—
70	C₂²×C₄×C₁₉	—	2	12	no	—
71	C₂²×C₄×C₁₉	—	2	96	no	—
72	C₂²×C₄×C₁₉	—	2	48	no	—
73	C₂²×C₄×C₁₉	—	2	48	no	—
74	C₂²×C₄×C₁₉	—	2	12	no	—
75	C₂²×C₄×C₁₉	—	2	12	no	—
76	C₂²×C₄×C₁₉	—	2	24	no	—
77	C₂²×C₄×C₁₉	C₂×D₄	1	12	yes	—
78	C₂²×C₄×C₁₉	C₂×D₄	1	48	yes	—
79	C₂²×C₄×C₁₉	C₂×D₄	1	12	yes	—
80	C₂²×C₄×C₁₉	C₂⁴	1	4	yes	—
81	C₂²×C₄×C₁₉	C₂⁴	1	12	yes	—
82	C₂²×C₄×C₁₉	C₂⁴	1	6	yes	—
83	C₂²×C₄×C₁₉	C₂⁴	1	2	yes	—
84	C₄⋊C₄×C₁₉	—	2	24	no	—
85	C₄⋊C₄×C₁₉	—	2	48	no	—
86	C₂²⋊C₄×C₁₉	—	2	48	no	—
87	C₂²⋊C₄×C₁₉	—	2	48	no	—
88	C₂×#2848	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂³×C₁₉	—	2	32	no	—
90	C₂³×C₁₉	—	2	256	no	—
91	C₂³×C₁₉	—	2	96	no	—
92	C₂³×C₁₉	—	2	24	no	—
93	C₂³×C₁₉	—	2	48	no	—
94	C₂³×C₁₉	—	2	96	no	—
95	C₂³×C₁₉	—	2	24	no	—
96	C₂³×C₁₉	—	2	48	no	—
97	C₂³×C₁₉	—	2	12	no	—
98	C₂³×C₁₉	—	2	12	no	—
99	C₂³×C₁₉	C₂²⋊Q₈	1	48	yes	complemented
100	C₂³×C₁₉	C₂²×D₄	1	12	yes	—
101	C₂³×C₁₉	C₂²×D₄	1	6	yes	—
102	C₂³×C₁₉	C₂²×Q₈	1	12	yes	—
103	C₂³×C₁₉	C₂²×Q₈	1	2	yes	—
104	C₂³×C₁₉	C₂×#41	1	12	yes	—
105	C₂³×C₁₉	C₂×#41	1	6	yes	—
106	C₂³×C₁₉	C₂⁵	1	1	yes	characteristic
107	Q₈×C₁₉	—	4	16	no	—
108	C₂×C₄×C₁₉	—	2	96	no	—
109	C₂×C₄×C₁₉	—	2	24	no	—
110	C₂×C₄×C₁₉	—	2	12	no	—
111	C₂×C₄×C₁₉	—	2	96	no	—
112	C₂×C₄×C₁₉	—	2	12	no	—
113	C₂×C₄×C₁₉	—	2	12	no	—
114	C₂×C₄×C₁₉	—	2	24	no	—
115	C₂×C₄×C₁₉	—	4	8	no	—
116	C₂×C₄×C₁₉	—	4	24	no	—
117	C₂×C₄×C₁₉	#120	1	16	yes	—
118	C₂×C₄×C₁₉	C₂²×D₄	1	12	yes	—
119	C₂×C₄×C₁₉	C₂²×D₄	1	12	yes	—
120	C₂⁵×C₄	C₂×C₁₉	1	1	yes	characteristic
121	C₂²×C₂²⋊Q₈	C₂×C₁₉	1	3	yes	complemented
122	#2848	C₂×C₁₉	1	16	yes	direct product factor
123	C₂×#385	C₂×C₁₉	1	6	yes	complemented
124	C₂×#380	C₂×C₁₉	1	3	yes	complemented
125	C₂×#379	C₂×C₁₉	1	2	yes	—
126	C₂²×C₁₉	—	2	128	no	—
127	C₂²×C₁₉	—	2	96	no	—
128	C₂²×C₁₉	—	2	12	no	—
129	C₂²×C₁₉	—	2	24	no	—
130	C₂²×C₁₉	—	2	12	no	—
131	C₂²×C₁₉	—	2	24	no	—
132	C₂²×C₁₉	C₂×#120	1	4	yes	—
133	C₂²×C₁₉	C₂×C₂²⋊Q₈	1	12	yes	—
134	C₂²×C₁₉	C₂×C₂²⋊Q₈	1	24	yes	complemented
135	C₂²×C₁₉	#525	1	12	yes	—
136	C₂²×C₁₉	C₂³×D₄	1	3	yes	—
137	C₂²×C₁₉	C₂³×Q₈	1	1	yes	characteristic
138	C₂²×C₁₉	C₂²×#41	1	3	yes	—
139	C₄×C₁₉	—	2	12	no	—
140	C₄×C₁₉	—	4	4	no	—
141	C₄×C₁₉	—	4	12	no	—
142	C₄×C₁₉	C₂×#120	1	8	yes	—
143	C₂⁶	C₂²×C₁₉	1	1	yes	characteristic
144	C₂³×Q₈	C₂²×C₁₉	1	1	yes	characteristic, complemented
145	C₂⁴×C₄	—	2	24	no	—
146	C₂⁴×C₄	C₂²×C₁₉	1	8	yes	—
147	C₂⁴×C₄	C₂²×C₁₉	1	3	yes	—
148	C₂⁴×C₄	C₂²×C₁₉	1	3	yes	—
149	C₂×C₂²⋊Q₈	—	2	24	no	—
150	C₂×C₂²⋊Q₈	C₂²×C₁₉	1	24	yes	complemented
151	C₂²×C₄⋊C₄	C₂²×C₁₉	1	3	yes	complemented
152	C₂²×C₄⋊C₄	C₂²×C₁₉	1	6	yes	complemented
153	C₂²×C₂²⋊C₄	C₂²×C₁₉	1	6	yes	—
154	C₂²×C₂²⋊C₄	C₂²×C₁₉	1	6	yes	—
155	#385	C₂²×C₁₉	1	48	yes	complemented
156	#380	C₂²×C₁₉	1	24	yes	complemented
157	#379	C₂²×C₁₉	1	16	yes	—
158	C₂×#95	C₂²×C₁₉	1	6	yes	complemented
159	C₂×C₁₉	—	2	24	no	—
160	C₂×C₁₉	#2848	1	8	yes	direct product factor
161	C₂×C₁₉	C₂²×#120	1	1	yes	characteristic
162	C₂×C₁₉	C₂²×C₂²⋊Q₈	1	3	yes	—
163	C₂×C₁₉	C₂×#525	1	3	yes	—
164	C₂⁵	—	2	12	no	—
165	C₂⁵	—	2	12	no	—
166	C₂⁵	Q₈×C₁₉	1	8	yes	complemented
167	C₂⁵	C₂³×C₁₉	1	4	yes	—
168	C₂⁵	C₂³×C₁₉	1	3	yes	—
169	C₂²×Q₈	—	2	12	no	—
170	C₂²×Q₈	C₂³×C₁₉	1	4	yes	complemented
171	C₂³×C₄	—	2	128	no	—
172	C₂³×C₄	—	2	12	no	—
173	C₂³×C₄	—	2	24	no	—
174	C₂³×C₄	—	2	96	no	—
175	C₂³×C₄	—	2	24	no	—
176	C₂³×C₄	D₄×C₁₉	1	24	yes	—
177	C₂³×C₄	C₂³×C₁₉	1	2	yes	—
178	C₂³×C₄	C₂³×C₁₉	1	6	yes	—
179	C₂³×C₄	C₂³×C₁₉	1	12	yes	—
180	C₂³×C₄	C₂³×C₁₉	1	3	yes	—
181	C₂³×C₄	C₂³×C₁₉	1	12	yes	—
182	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
183	C₂²⋊Q₈	—	2	96	no	—
184	C₂×C₄⋊C₄	—	2	12	no	—
185	C₂×C₄⋊C₄	—	2	24	no	—
186	C₂×C₄⋊C₄	C₂³×C₁₉	1	12	yes	complemented
187	C₂×C₄⋊C₄	C₂³×C₁₉	1	24	yes	complemented
188	C₂×C₂²⋊C₄	—	2	24	no	—
189	C₂×C₂²⋊C₄	—	2	24	no	—
190	C₂×C₂²⋊C₄	C₂³×C₁₉	1	24	yes	—
191	C₂×C₂²⋊C₄	C₂³×C₁₉	1	24	yes	—
192	#95	C₂³×C₁₉	1	24	yes	complemented
193	C₁₉	C₂×#2848	1	1	yes	characteristic, direct product factor, Sylow
194	C₂⁴	—	2	32	no	—
195	C₂⁴	—	2	96	no	—
196	C₂⁴	—	2	96	no	—
197	C₂⁴	—	2	24	no	—
198	C₂⁴	—	2	12	no	—
199	C₂⁴	—	2	24	no	—
200	C₂⁴	—	2	12	no	—
201	C₂⁴	C₂×D₄×C₁₉	1	12	yes	—
202	C₂⁴	C₂×Q₈×C₁₉	1	12	yes	—
203	C₂⁴	C₂×Q₈×C₁₉	1	16	yes	complemented
204	C₂⁴	#41×C₁₉	1	12	yes	—
205	C₂⁴	C₂⁴×C₁₉	1	6	yes	—
206	C₂⁴	C₂⁴×C₁₉	1	1	yes	characteristic
207	C₂×Q₈	—	2	24	no	—
208	C₂×Q₈	—	4	16	no	—
209	C₂²×C₄	—	2	96	no	—
210	C₂²×C₄	—	2	256	no	—
211	C₂²×C₄	—	2	24	no	—
212	C₂²×C₄	—	2	12	no	—
213	C₂²×C₄	—	2	96	no	—
214	C₂²×C₄	—	2	48	no	—
215	C₂²×C₄	—	2	48	no	—
216	C₂²×C₄	—	2	12	no	—
217	C₂²×C₄	—	2	12	no	—
218	C₂²×C₄	—	2	24	no	—
219	C₂²×C₄	C₂×D₄×C₁₉	1	12	yes	—
220	C₂²×C₄	C₂×D₄×C₁₉	1	48	yes	—
221	C₂²×C₄	C₂×D₄×C₁₉	1	12	yes	—
222	C₂²×C₄	C₂⁴×C₁₉	1	12	yes	—
223	C₂²×C₄	C₂⁴×C₁₉	1	4	yes	—
224	C₂²×C₄	C₂⁴×C₁₉	1	6	yes	—
225	C₂²×C₄	C₂⁴×C₁₉	1	2	yes	—
226	C₄⋊C₄	—	2	24	no	—
227	C₄⋊C₄	—	2	48	no	—
228	C₂²⋊C₄	—	2	48	no	—
229	C₂²⋊C₄	—	2	48	no	—
230	C₂³	—	2	32	no	—
231	C₂³	—	2	96	no	—
232	C₂³	—	2	256	no	—
233	C₂³	—	2	24	no	—
234	C₂³	—	2	48	no	—
235	C₂³	—	2	96	no	—
236	C₂³	—	2	24	no	—
237	C₂³	—	2	48	no	—
238	C₂³	—	2	12	no	—
239	C₂³	—	2	12	no	—
240	C₂³	C₁₉×C₂²⋊Q₈	1	48	yes	complemented
241	C₂³	C₂²×D₄×C₁₉	1	12	yes	—
242	C₂³	C₂²×D₄×C₁₉	1	6	yes	—
243	C₂³	C₂²×Q₈×C₁₉	1	12	yes	—
244	C₂³	C₂²×Q₈×C₁₉	1	2	yes	—
245	C₂³	C₂×#41×C₁₉	1	12	yes	—
246	C₂³	C₂×#41×C₁₉	1	6	yes	—
247	C₂³	C₂⁵×C₁₉	1	1	yes	characteristic
248	Q₈	—	4	16	no	—
249	C₂×C₄	—	2	96	no	—
250	C₂×C₄	—	2	24	no	—
251	C₂×C₄	—	2	12	no	—
252	C₂×C₄	—	2	96	no	—
253	C₂×C₄	—	2	12	no	—
254	C₂×C₄	—	2	12	no	—
255	C₂×C₄	—	2	24	no	—
256	C₂×C₄	—	4	8	no	—
257	C₂×C₄	—	4	24	no	—
258	C₂×C₄	C₁₉×#120	1	16	yes	—
259	C₂×C₄	C₂²×D₄×C₁₉	1	12	yes	—
260	C₂×C₄	C₂²×D₄×C₁₉	1	12	yes	—
261	C₂²	—	2	128	no	—
262	C₂²	—	2	12	no	—
263	C₂²	—	2	24	no	—
264	C₂²	—	2	12	no	—
265	C₂²	—	2	24	no	—
266	C₂²	—	2	96	no	—
267	C₂²	C₂×C₁₉×#120	1	4	yes	—
268	C₂²	C₂×C₁₉×C₂²⋊Q₈	1	12	yes	—
269	C₂²	C₂×C₁₉×C₂²⋊Q₈	1	24	yes	complemented
270	C₂²	C₁₉×#525	1	12	yes	—
271	C₂²	C₂³×D₄×C₁₉	1	3	yes	—
272	C₂²	C₂³×Q₈×C₁₉	1	1	yes	characteristic
273	C₂²	C₂²×#41×C₁₉	1	3	yes	—
274	C₄	—	2	12	no	—
275	C₄	—	4	4	no	—
276	C₄	—	4	12	no	—
277	C₄	C₂×C₁₉×#120	1	8	yes	—
278	C₂	—	2	24	no	—
279	C₂	C₁₉×#2848	1	8	yes	direct product factor
280	C₂	C₂²×C₁₉×#120	1	1	yes	characteristic
281	C₂	C₂²×C₁₉×C₂²⋊Q₈	1	3	yes	—
282	C₂	C₂×C₁₉×#525	1	3	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₁₉	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₁₉
prime radical	C₂⁶×C₁₉	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₁₉×C₂²⋊Q₈	C₂	semidirect
3. C₁₉×#2848	C₂	direct (N × H)
4. C₂×C₁₉×#385	C₂	semidirect
5. C₂×C₁₉×#380	C₂	semidirect
8. C₂³×Q₈×C₁₉	C₂²	semidirect
14. C₂×C₁₉×C₂²⋊Q₈	C₂²	semidirect
15. C₂²×C₄⋊C₄×C₁₉	C₂²	semidirect
16. C₂²×C₄⋊C₄×C₁₉	C₂²	semidirect
19. C₁₉×#385	C₂²	semidirect
20. C₁₉×#380	C₂²	semidirect
22. C₂×C₁₉×#95	C₂²	semidirect
25. C₂⁵×C₁₉	Q₈	semidirect
29. C₂²×Q₈×C₁₉	C₂³	semidirect
45. C₂×C₄⋊C₄×C₁₉	C₂³	semidirect
46. C₂×C₄⋊C₄×C₁₉	C₂³	semidirect
51. C₁₉×#95	C₂³	semidirect
61. C₂⁴×C₁₉	C₂×Q₈	semidirect
88. C₂×#2848	C₁₉	direct (N × H)
99. C₂³×C₁₉	C₂²⋊Q₈	semidirect
121. C₂²×C₂²⋊Q₈	C₂×C₁₉	semidirect
122. #2848	C₂×C₁₉	direct (N × H)
123. C₂×#385	C₂×C₁₉	semidirect
124. C₂×#380	C₂×C₁₉	semidirect
134. C₂²×C₁₉	C₂×C₂²⋊Q₈	semidirect
144. C₂³×Q₈	C₂²×C₁₉	semidirect
150. C₂×C₂²⋊Q₈	C₂²×C₁₉	semidirect
151. C₂²×C₄⋊C₄	C₂²×C₁₉	semidirect
152. C₂²×C₄⋊C₄	C₂²×C₁₉	semidirect
155. #385	C₂²×C₁₉	semidirect
156. #380	C₂²×C₁₉	semidirect
158. C₂×#95	C₂²×C₁₉	semidirect
160. C₂×C₁₉	#2848	direct (N × H)
166. C₂⁵	Q₈×C₁₉	semidirect
170. C₂²×Q₈	C₂³×C₁₉	semidirect
186. C₂×C₄⋊C₄	C₂³×C₁₉	semidirect
187. C₂×C₄⋊C₄	C₂³×C₁₉	semidirect
192. #95	C₂³×C₁₉	semidirect
193. C₁₉	C₂×#2848	direct (N × H)
203. C₂⁴	C₂×Q₈×C₁₉	semidirect
240. C₂³	C₁₉×C₂²⋊Q₈	semidirect
269. C₂²	C₂×C₁₉×C₂²⋊Q₈	semidirect
279. C₂	C₁₉×#2848	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×C₄×C₁₉	2
C₂²×C₁₉×C₂²⋊Q₈	2
C₁₉×#2848	2
C₂×C₁₉×#385	2
C₂×C₁₉×#380	2
C₂×C₁₉×#379	2
C₂×#2848	19

Subgroups — C2×C19×#2848 #18042571

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₉×#2848 #18042571