Subgroups — C₂×C₁₉×#2185 #18039471

Order 4864.

1178 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₁₉×SD₃₂	C₂	1	2	yes	maximal, complemented
3	C₂×C₁₉×#466	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#2185	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₁₉×#358	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₉×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×Q₈×C₁₉	—	2	1	no	—
8	C₂²×Q₁₆×C₁₉	—	2	8	no	—
9	C₂²×Q₁₆×C₁₉	—	2	1	no	—
10	C₂²×Q₁₆×C₁₉	—	2	2	no	—
11	C₂²×Q₁₆×C₁₉	C₂²	1	2	yes	—
12	C₂²×Q₁₆×C₁₉	C₂²	1	4	yes	complemented
13	C₂²×D₈×C₁₉	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
15	C₂×C₁₉×#121	—	2	1	no	—
16	C₂×C₁₉×SD₃₂	—	2	8	no	—
17	C₂×C₁₉×SD₃₂	C₂²	1	8	yes	complemented
18	C₂²×C₁₆×C₁₉	C₂²	1	2	yes	—
19	C₁₉×#466	—	2	2	no	—
20	C₁₉×#466	C₂²	1	4	yes	—
21	C₂×C₁₉×#107	C₂²	1	1	yes	characteristic
22	C₂×C₁₉×#102	—	2	1	no	—
23	C₁₉×#358	C₂²	1	8	yes	complemented
24	C₁₉×C₂²⋊C₁₆	C₂²	1	4	yes	complemented
25	C₂²×Q₈×C₁₉	—	2	8	no	—
26	C₂²×Q₈×C₁₉	—	2	2	no	—
27	C₂²×Q₈×C₁₉	—	2	4	no	—
28	C₂²×Q₈×C₁₉	—	4	4	no	—
29	C₂²×Q₈×C₁₉	—	4	2	no	—
30	C₂²×Q₈×C₁₉	—	4	1	no	—
31	C₂²×D₄×C₁₉	—	2	1	no	—
32	C₂²×D₄×C₁₉	—	4	1	no	—
33	C₂³×C₄×C₁₉	—	4	1	no	—
34	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
35	C₂×Q₁₆×C₁₉	—	2	8	no	—
36	C₂×Q₁₆×C₁₉	—	2	16	no	—
37	C₂×Q₁₆×C₁₉	—	2	16	no	—
38	C₂×Q₁₆×C₁₉	—	2	2	no	—
39	C₂×Q₁₆×C₁₉	—	2	4	no	—
40	C₂×Q₁₆×C₁₉	—	2	4	no	—
41	C₂×Q₁₆×C₁₉	D₄	1	8	yes	complemented
42	C₂×Q₁₆×C₁₉	C₂³	1	4	yes	—
43	C₂×D₈×C₁₉	—	2	4	no	—
44	C₂×D₈×C₁₉	—	2	1	no	—
45	C₂×D₈×C₁₉	C₂³	1	2	yes	—
46	C₂²×C₈×C₁₉	—	2	4	no	—
47	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₁₉	D₄	1	2	yes	—
49	C₂²×C₈×C₁₉	C₂³	1	2	yes	—
50	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
51	C₁₉×#121	—	2	4	no	—
52	C₁₉×#121	—	4	2	no	—
53	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
54	C₂×C₂²⋊C₄×C₁₉	—	4	1	no	—
55	C₁₉×SD₃₂	—	2	16	no	—
56	C₂×C₁₆×C₁₉	—	2	4	no	—
57	C₂×C₁₆×C₁₉	C₂³	1	4	yes	—
58	C₁₉×#107	—	2	1	no	—
59	C₁₉×#107	C₂³	1	2	yes	—
60	C₁₉×#102	—	2	4	no	—
61	C₂⁴×C₁₉	—	4	1	no	—
62	C₂⁴×C₁₉	D₈	1	1	yes	characteristic
63	C₂×Q₈×C₁₉	—	2	16	no	—
64	C₂×Q₈×C₁₉	—	2	8	no	—
65	C₂×Q₈×C₁₉	—	2	4	no	—
66	C₂×Q₈×C₁₉	—	4	8	no	—
67	C₂×Q₈×C₁₉	—	4	8	no	—
68	C₂×Q₈×C₁₉	—	4	16	no	—
69	C₂×Q₈×C₁₉	—	4	4	no	—
70	C₂×Q₈×C₁₉	—	4	2	no	—
71	C₂×Q₈×C₁₉	—	4	4	no	—
72	C₂×D₄×C₁₉	—	2	2	no	—
73	C₂×D₄×C₁₉	—	4	4	no	—
74	C₂×D₄×C₁₉	—	4	2	no	—
75	C₂×D₄×C₁₉	—	4	1	no	—
76	C₂×D₄×C₁₉	—	4	2	no	—
77	C₂×D₄×C₁₉	—	8	4	no	—
78	C₂²×C₄×C₁₉	—	2	2	no	—
79	C₂²×C₄×C₁₉	—	2	2	no	—
80	C₂²×C₄×C₁₉	—	4	8	no	—
81	C₂²×C₄×C₁₉	—	4	4	no	—
82	C₂²×C₄×C₁₉	—	4	2	no	—
83	C₂²×C₄×C₁₉	—	4	1	no	—
84	C₂²×C₄×C₁₉	D₈	1	2	yes	—
85	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
86	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
88	Q₁₆×C₁₉	—	2	16	no	—
89	Q₁₆×C₁₉	—	2	8	no	—
90	Q₁₆×C₁₉	—	2	4	no	—
91	Q₁₆×C₁₉	C₂×D₄	1	8	yes	complemented
92	D₈×C₁₉	—	2	4	no	—
93	D₈×C₁₉	—	4	2	no	—
94	C₂×C₈×C₁₉	—	2	2	no	—
95	C₂×C₈×C₁₉	—	2	4	no	—
96	C₂×C₈×C₁₉	—	2	4	no	—
97	C₂×C₈×C₁₉	C₂×D₄	1	2	yes	—
98	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₁₉	C₂×D₄	1	4	yes	—
100	C₂×C₈×C₁₉	C₂⁴	1	1	yes	characteristic
101	C₄⋊C₄×C₁₉	—	2	2	no	—
102	C₄⋊C₄×C₁₉	—	4	1	no	—
103	C₂²⋊C₄×C₁₉	—	4	4	no	—
104	C₁₆×C₁₉	—	2	4	no	—
105	C₂×#2185	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
106	C₂³×C₁₉	—	2	2	no	—
107	C₂³×C₁₉	—	2	2	no	—
108	C₂³×C₁₉	—	4	2	no	—
109	C₂³×C₁₉	—	8	2	no	—
110	C₂³×C₁₉	—	8	2	no	—
111	C₂³×C₁₉	—	8	2	no	—
112	C₂³×C₁₉	SD₃₂	1	4	yes	complemented
113	C₂³×C₁₉	C₂×D₈	1	2	yes	—
114	C₂³×C₁₉	C₂×D₈	1	1	yes	characteristic
115	Q₈×C₁₉	—	2	8	no	—
116	Q₈×C₁₉	—	4	16	no	—
117	Q₈×C₁₉	—	4	4	no	—
118	Q₈×C₁₉	—	4	8	no	—
119	D₄×C₁₉	—	4	2	no	—
120	D₄×C₁₉	—	4	2	no	—
121	D₄×C₁₉	—	8	8	no	—
122	D₄×C₁₉	—	8	2	no	—
123	C₂×C₄×C₁₉	—	2	2	no	—
124	C₂×C₄×C₁₉	—	2	2	no	—
125	C₂×C₄×C₁₉	—	2	2	no	—
126	C₂×C₄×C₁₉	—	2	4	no	—
127	C₂×C₄×C₁₉	—	4	16	no	—
128	C₂×C₄×C₁₉	—	4	4	no	—
129	C₂×C₄×C₁₉	—	4	8	no	—
130	C₂×C₄×C₁₉	—	4	2	no	—
131	C₂×C₄×C₁₉	—	8	2	no	—
132	C₂×C₄×C₁₉	#120	1	2	yes	—
133	C₂×C₄×C₁₉	C₂×D₈	1	2	yes	—
134	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
136	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
137	C₈×C₁₉	—	2	2	no	—
138	C₈×C₁₉	#120	1	2	yes	—
139	C₈×C₁₉	C₂²×D₄	1	2	yes	—
140	C₂³×Q₁₆	C₂×C₁₉	1	1	yes	characteristic, complemented
141	C₂²×SD₃₂	C₂×C₁₉	1	2	yes	complemented
142	C₂×#466	C₂×C₁₉	1	1	yes	characteristic
143	#2185	C₂×C₁₉	1	8	yes	direct product factor
144	C₂×#358	C₂×C₁₉	1	2	yes	complemented
145	C₂×C₂²⋊C₁₆	C₂×C₁₉	1	1	yes	characteristic, complemented
146	C₂²×C₁₉	—	2	2	no	—
147	C₂²×C₁₉	—	2	8	no	—
148	C₂²×C₁₉	—	2	2	no	—
149	C₂²×C₁₉	—	8	2	no	—
150	C₂²×C₁₉	—	8	2	no	—
151	C₂²×C₁₉	—	8	2	no	—
152	C₂²×C₁₉	—	16	4	no	—
153	C₂²×C₁₉	#447	1	2	yes	—
154	C₂²×C₁₉	C₂×SD₃₂	1	2	yes	—
155	C₂²×C₁₉	C₂×SD₃₂	1	4	yes	complemented
156	C₂²×C₁₉	#510	1	2	yes	—
157	C₂²×C₁₉	C₂²×D₈	1	1	yes	characteristic
158	C₄×C₁₉	—	2	2	no	—
159	C₄×C₁₉	—	4	8	no	—
160	C₄×C₁₉	—	8	2	no	—
161	C₄×C₁₉	#447	1	2	yes	—
162	C₄×C₁₉	C₂×#120	1	1	yes	characteristic
163	C₄×C₁₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
164	C₂³×Q₈	—	2	1	no	—
165	C₂²×Q₁₆	—	2	8	no	—
166	C₂²×Q₁₆	—	2	1	no	—
167	C₂²×Q₁₆	—	2	2	no	—
168	C₂²×Q₁₆	C₂²×C₁₉	1	2	yes	—
169	C₂²×Q₁₆	C₂²×C₁₉	1	4	yes	complemented
170	C₂²×D₈	C₂²×C₁₉	1	1	yes	characteristic
171	C₂³×C₈	C₂²×C₁₉	1	1	yes	characteristic
172	C₂×#121	—	2	1	no	—
173	C₂×SD₃₂	—	2	8	no	—
174	C₂×SD₃₂	C₂²×C₁₉	1	8	yes	complemented
175	C₂²×C₁₆	C₂²×C₁₉	1	2	yes	—
176	#466	—	2	2	no	—
177	#466	C₂²×C₁₉	1	4	yes	—
178	C₂×#107	C₂²×C₁₉	1	1	yes	characteristic
179	C₂×#102	—	2	1	no	—
180	#358	C₂²×C₁₉	1	8	yes	complemented
181	C₂²⋊C₁₆	C₂²×C₁₉	1	4	yes	complemented
182	C₂×C₁₉	—	2	4	no	—
183	C₂×C₁₉	—	16	2	no	—
184	C₂×C₁₉	#2185	1	4	yes	direct product factor
185	C₂×C₁₉	C₂×#447	1	1	yes	characteristic
186	C₂×C₁₉	C₂²×SD₃₂	1	1	yes	characteristic
187	C₂×C₁₉	C₂×#510	1	1	yes	characteristic
188	C₂²×Q₈	—	2	8	no	—
189	C₂²×Q₈	—	2	2	no	—
190	C₂²×Q₈	—	2	4	no	—
191	C₂²×Q₈	—	4	4	no	—
192	C₂²×Q₈	—	4	1	no	—
193	C₂²×Q₈	—	4	2	no	—
194	C₂²×D₄	—	2	1	no	—
195	C₂²×D₄	—	4	1	no	—
196	C₂³×C₄	—	4	1	no	—
197	C₂³×C₄	D₄×C₁₉	1	1	yes	characteristic
198	C₂×Q₁₆	—	2	8	no	—
199	C₂×Q₁₆	—	2	16	no	—
200	C₂×Q₁₆	—	2	16	no	—
201	C₂×Q₁₆	—	2	2	no	—
202	C₂×Q₁₆	—	2	4	no	—
203	C₂×Q₁₆	—	2	4	no	—
204	C₂×Q₁₆	D₄×C₁₉	1	8	yes	complemented
205	C₂×Q₁₆	C₂³×C₁₉	1	4	yes	—
206	C₂×D₈	—	2	4	no	—
207	C₂×D₈	—	2	1	no	—
208	C₂×D₈	C₂³×C₁₉	1	2	yes	—
209	C₂²×C₈	—	2	4	no	—
210	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
211	C₂²×C₈	D₄×C₁₉	1	2	yes	—
212	C₂²×C₈	C₂³×C₁₉	1	2	yes	—
213	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
214	#121	—	2	4	no	—
215	#121	—	4	2	no	—
216	C₂×C₄⋊C₄	—	2	1	no	—
217	C₂×C₂²⋊C₄	—	4	1	no	—
218	SD₃₂	—	2	16	no	—
219	C₂×C₁₆	—	2	4	no	—
220	C₂×C₁₆	C₂³×C₁₉	1	4	yes	—
221	#107	—	2	1	no	—
222	#107	C₂³×C₁₉	1	2	yes	—
223	#102	—	2	4	no	—
224	C₁₉	C₂×#2185	1	1	yes	characteristic, direct product factor, Sylow
225	C₂⁴	—	4	1	no	—
226	C₂⁴	D₈×C₁₉	1	1	yes	characteristic
227	C₂×Q₈	—	2	16	no	—
228	C₂×Q₈	—	2	8	no	—
229	C₂×Q₈	—	2	4	no	—
230	C₂×Q₈	—	4	8	no	—
231	C₂×Q₈	—	4	16	no	—
232	C₂×Q₈	—	4	8	no	—
233	C₂×Q₈	—	4	4	no	—
234	C₂×Q₈	—	4	2	no	—
235	C₂×Q₈	—	4	4	no	—
236	C₂×D₄	—	2	2	no	—
237	C₂×D₄	—	4	4	no	—
238	C₂×D₄	—	4	2	no	—
239	C₂×D₄	—	4	1	no	—
240	C₂×D₄	—	4	2	no	—
241	C₂×D₄	—	8	4	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	4	8	no	—
245	C₂²×C₄	—	4	4	no	—
246	C₂²×C₄	—	4	2	no	—
247	C₂²×C₄	—	4	1	no	—
248	C₂²×C₄	D₈×C₁₉	1	1	yes	characteristic
249	C₂²×C₄	D₈×C₁₉	1	2	yes	—
250	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
251	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
252	Q₁₆	—	2	16	no	—
253	Q₁₆	—	2	8	no	—
254	Q₁₆	—	2	4	no	—
255	Q₁₆	C₂×D₄×C₁₉	1	8	yes	complemented
256	D₈	—	2	4	no	—
257	D₈	—	4	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	4	no	—
260	C₂×C₈	—	2	4	no	—
261	C₂×C₈	C₂×D₄×C₁₉	1	2	yes	—
262	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
263	C₂×C₈	C₂×D₄×C₁₉	1	4	yes	—
264	C₂×C₈	C₂⁴×C₁₉	1	1	yes	characteristic
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₂²⋊C₄	—	4	4	no	—
268	C₁₆	—	2	4	no	—
269	C₂³	—	2	2	no	—
270	C₂³	—	2	2	no	—
271	C₂³	—	4	2	no	—
272	C₂³	—	8	2	no	—
273	C₂³	—	8	2	no	—
274	C₂³	—	8	2	no	—
275	C₂³	C₁₉×SD₃₂	1	4	yes	complemented
276	C₂³	C₂×D₈×C₁₉	1	2	yes	—
277	C₂³	C₂×D₈×C₁₉	1	1	yes	characteristic
278	Q₈	—	2	8	no	—
279	Q₈	—	4	16	no	—
280	Q₈	—	4	8	no	—
281	Q₈	—	4	4	no	—
282	D₄	—	4	2	no	—
283	D₄	—	4	2	no	—
284	D₄	—	8	8	no	—
285	D₄	—	8	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	4	16	no	—
291	C₂×C₄	—	4	8	no	—
292	C₂×C₄	—	4	4	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	8	2	no	—
295	C₂×C₄	C₁₉×#120	1	2	yes	—
296	C₂×C₄	C₂×D₈×C₁₉	1	2	yes	—
297	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
298	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	2	yes	—
299	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
300	C₈	—	2	2	no	—
301	C₈	C₁₉×#120	1	2	yes	—
302	C₈	C₂²×D₄×C₁₉	1	2	yes	—
303	C₂²	—	2	8	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	8	2	no	—
307	C₂²	—	8	2	no	—
308	C₂²	—	8	2	no	—
309	C₂²	—	16	4	no	—
310	C₂²	C₁₉×#447	1	2	yes	—
311	C₂²	C₂×C₁₉×SD₃₂	1	2	yes	—
312	C₂²	C₂×C₁₉×SD₃₂	1	4	yes	complemented
313	C₂²	C₁₉×#510	1	2	yes	—
314	C₂²	C₂²×D₈×C₁₉	1	1	yes	characteristic
315	C₄	—	2	2	no	—
316	C₄	—	4	8	no	—
317	C₄	—	8	2	no	—
318	C₄	C₁₉×#447	1	2	yes	—
319	C₄	C₂×C₁₉×#120	1	1	yes	characteristic
320	C₄	C₂×C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
321	C₂	—	2	4	no	—
322	C₂	—	16	2	no	—
323	C₂	C₁₉×#2185	1	4	yes	direct product factor
324	C₂	C₂×C₁₉×#447	1	1	yes	characteristic
325	C₂	C₂²×C₁₉×SD₃₂	1	1	yes	characteristic
326	C₂	C₂×C₁₉×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₉	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₉
prime radical	C₂×C₁₉×#466	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ C₂³×C₄×C₁₉ ◁ C₂³×C₈×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×Q₁₆×C₁₉	C₂	semidirect
2. C₂²×C₁₉×SD₃₂	C₂	semidirect
4. C₁₉×#2185	C₂	direct (N × H)
5. C₂×C₁₉×#358	C₂	semidirect
6. C₂×C₁₉×C₂²⋊C₁₆	C₂	semidirect
12. C₂²×Q₁₆×C₁₉	C₂²	semidirect
17. C₂×C₁₉×SD₃₂	C₂²	semidirect
23. C₁₉×#358	C₂²	semidirect
24. C₁₉×C₂²⋊C₁₆	C₂²	semidirect
41. C₂×Q₁₆×C₁₉	D₄	semidirect
91. Q₁₆×C₁₉	C₂×D₄	semidirect
105. C₂×#2185	C₁₉	direct (N × H)
112. C₂³×C₁₉	SD₃₂	semidirect
140. C₂³×Q₁₆	C₂×C₁₉	semidirect
141. C₂²×SD₃₂	C₂×C₁₉	semidirect
143. #2185	C₂×C₁₉	direct (N × H)
144. C₂×#358	C₂×C₁₉	semidirect
145. C₂×C₂²⋊C₁₆	C₂×C₁₉	semidirect
155. C₂²×C₁₉	C₂×SD₃₂	semidirect
169. C₂²×Q₁₆	C₂²×C₁₉	semidirect
174. C₂×SD₃₂	C₂²×C₁₉	semidirect
180. #358	C₂²×C₁₉	semidirect
181. C₂²⋊C₁₆	C₂²×C₁₉	semidirect
184. C₂×C₁₉	#2185	direct (N × H)
204. C₂×Q₁₆	D₄×C₁₉	semidirect
224. C₁₉	C₂×#2185	direct (N × H)
255. Q₁₆	C₂×D₄×C₁₉	semidirect
275. C₂³	C₁₉×SD₃₂	semidirect
312. C₂²	C₂×C₁₉×SD₃₂	semidirect
323. C₂	C₁₉×#2185	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×Q₁₆×C₁₉	2
C₂²×C₁₉×SD₃₂	2
C₂×C₁₉×#466	2
C₁₉×#2185	2
C₂×C₁₉×#358	2
C₂×C₁₉×C₂²⋊C₁₆	2
C₂×#2185	19

Subgroups — C2×C19×#2185 #18039471

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₉×#2185 #18039471