Subgroups — #18072017

Order 4864.

574 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926600	C₂	1	1	yes	maximal, characteristic
2	C₂×#11926596	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926595	C₂	1	1	yes	maximal, characteristic
4	#14561470	C₂	1	1	yes	maximal, characteristic, complemented
5	#14561469	C₂	1	1	yes	maximal, characteristic, complemented
6	#14561469	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1277	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄²×D₁₉	C₂²	1	1	yes	characteristic
9	#11926600	—	2	2	no	—
10	#11926600	C₄	1	4	yes	—
11	#11926596	—	2	2	no	—
12	#11926596	C₄	1	4	yes	—
13	#11926595	—	2	2	no	—
14	#11926595	—	2	2	no	—
15	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₉⋊C₈	C₄	1	2	yes	—
18	C₂×(C₂×D₁₉)⋊C₈	—	2	1	no	—
19	C₂×(C₂×D₁₉)⋊C₈	—	2	1	no	—
20	C₂×(C₂×D₁₉)⋊C₈	C₄	1	2	yes	—
21	C₂×#10597272	—	2	1	no	—
22	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₈×Dic₁₉	—	2	1	no	—
24	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
27	C₂²×C₄×D₁₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×D₁₉	D₄	1	1	yes	characteristic
29	C₂²×C₄×D₁₉	Q₈	1	1	yes	characteristic
30	C₂²×C₈×C₁₉	—	2	1	no	—
31	C₂²×C₈×C₁₉	—	2	1	no	—
32	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
34	C₂×C₄×Dic₁₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×Dic₁₉	D₄	1	1	yes	characteristic
36	C₂×C₄×Dic₁₉	Q₈	1	1	yes	characteristic
37	C₂²×C₁₉⋊C₈	—	2	1	no	—
38	C₂²×C₁₉⋊C₈	—	2	1	no	—
39	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
40	C₄²×D₁₉	C₂×C₄	1	2	yes	—
41	C₄²×D₁₉	C₂×C₄	1	2	yes	—
42	C₄²×D₁₉	D₄	1	2	yes	—
43	C₄²×D₁₉	D₄	1	2	yes	—
44	C₁₉×C₄⋊C₈	—	2	1	no	—
45	C₁₉×C₄⋊C₈	—	2	1	no	—
46	C₁₉×C₄⋊C₈	C₂×C₄	1	2	yes	—
47	C₁₉×C₄⋊C₈	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₁₉	—	2	1	no	—
49	C₄×C₈×C₁₉	—	2	1	no	—
50	(C₂×D₁₉)⋊C₈	—	2	4	no	—
51	(C₂×D₁₉)⋊C₈	—	2	4	no	—
52	(C₂×D₁₉)⋊C₈	—	2	4	no	—
53	#10597272	—	2	4	no	—
54	Dic₁₉⋊C₈	—	2	4	no	—
55	C₈×Dic₁₉	—	2	4	no	—
56	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
57	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
58	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	2	yes	—
59	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	2	yes	—
60	C₄×C₁₉⋊C₈	—	2	1	no	—
61	C₄×C₁₉⋊C₈	—	2	1	no	—
62	C₂³×D₁₉	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
66	C₂²×Dic₁₉	C₄⋊C₄	1	1	yes	characteristic
67	C₂²×Dic₁₉	C₄⋊C₄	1	1	yes	characteristic
68	C₂²×Dic₁₉	C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×D₁₉	—	2	2	no	—
70	C₂×C₄×D₁₉	—	2	2	no	—
71	C₂×C₄×D₁₉	—	2	1	no	—
72	C₂×C₄×D₁₉	—	2	1	no	—
73	C₂×C₄×D₁₉	—	2	1	no	—
74	C₂×C₄×D₁₉	—	2	1	no	—
75	C₂×C₄×D₁₉	—	4	2	no	—
76	C₂×C₄×D₁₉	C₄²	1	2	yes	—
77	C₂×C₄×D₁₉	C₂²⋊C₄	1	2	yes	—
78	C₂×C₄×D₁₉	C₂²⋊C₄	1	2	yes	—
79	C₂×C₄×D₁₉	C₂²⋊C₄	1	2	yes	—
80	C₂×C₄×D₁₉	C₄⋊C₄	1	2	yes	—
81	C₂×C₄×D₁₉	C₄⋊C₄	1	2	yes	—
82	C₂×C₈×C₁₉	—	2	2	no	—
83	C₂×C₈×C₁₉	—	2	2	no	—
84	C₂×C₈×C₁₉	—	2	2	no	—
85	C₂×C₈×C₁₉	—	2	2	no	—
86	C₂×C₈×C₁₉	—	2	2	no	—
87	C₂×C₈×C₁₉	—	4	1	no	—
88	C₂×C₈×C₁₉	—	4	1	no	—
89	C₂×C₈×C₁₉	—	4	1	no	—
90	C₂×C₈×C₁₉	—	4	1	no	—
91	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
95	C₄×Dic₁₉	C₄²	1	2	yes	—
96	C₄×Dic₁₉	C₂²⋊C₄	1	2	yes	—
97	C₄×Dic₁₉	C₂²⋊C₄	1	2	yes	—
98	C₄×Dic₁₉	C₂²⋊C₄	1	2	yes	—
99	C₄×Dic₁₉	C₄⋊C₄	1	2	yes	—
100	C₄×Dic₁₉	C₄⋊C₄	1	2	yes	—
101	C₂×C₁₉⋊C₈	—	2	2	no	—
102	C₂×C₁₉⋊C₈	—	2	2	no	—
103	C₂×C₁₉⋊C₈	—	2	2	no	—
104	C₂×C₁₉⋊C₈	—	2	2	no	—
105	C₂×C₁₉⋊C₈	—	2	2	no	—
106	C₂×C₁₉⋊C₈	—	4	1	no	—
107	C₂×C₁₉⋊C₈	—	4	1	no	—
108	C₂×C₁₉⋊C₈	—	4	1	no	—
109	C₂×C₁₉⋊C₈	—	4	1	no	—
110	#7878	—	19	1	no	maximal, Sylow
111	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂²×D₁₉	—	2	2	no	—
113	C₂²×D₁₉	—	2	1	no	—
114	C₂²×D₁₉	—	2	1	no	—
115	C₂²×D₁₉	—	2	2	no	—
116	C₂²×D₁₉	#95	1	2	yes	—
117	C₂×C₄×C₁₉	—	2	2	no	—
118	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
119	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₁₉	#131	1	2	yes	—
125	C₂×C₄×C₁₉	#131	1	2	yes	—
126	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
127	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
128	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
129	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
130	C₂×Dic₁₉	—	2	2	no	—
131	C₂×Dic₁₉	—	2	1	no	—
132	C₂×Dic₁₉	—	2	1	no	—
133	C₂×Dic₁₉	—	2	2	no	—
134	C₂×Dic₁₉	#95	1	2	yes	—
135	C₂×Dic₁₉	#95	1	2	yes	—
136	C₂×Dic₁₉	#95	1	2	yes	—
137	C₄×D₁₉	—	2	2	no	—
138	C₄×D₁₉	—	2	2	no	—
139	C₄×D₁₉	—	2	4	no	—
140	C₄×D₁₉	—	2	2	no	—
141	C₄×D₁₉	—	2	2	no	—
142	C₄×D₁₉	—	2	2	no	—
143	C₄×D₁₉	—	2	2	no	—
144	C₄×D₁₉	—	4	2	no	—
145	C₄×D₁₉	—	4	2	no	—
146	C₈×C₁₉	—	4	2	no	—
147	C₈×C₁₉	—	4	2	no	—
148	C₈×C₁₉	—	4	2	no	—
149	C₁₉⋊C₈	—	4	2	no	—
150	C₁₉⋊C₈	—	4	2	no	—
151	C₁₉⋊C₈	—	4	2	no	—
152	C₂×#432	—	19	1	no	—
153	C₂×#431	—	19	1	no	—
154	C₂×#431	—	19	1	no	—
155	#1277	—	19	1	no	—
156	#1277	—	19	1	no	—
157	#1277	—	19	1	no	—
158	#1277	D₁₉	1	1	yes	characteristic, complemented
159	C₂²×C₁₉	C₂×#95	1	1	yes	characteristic
160	C₂²×C₁₉	#424	1	2	yes	—
161	C₂²×C₁₉	#427	1	1	yes	characteristic
162	C₂²×C₁₉	#427	1	1	yes	characteristic
163	C₂²×C₁₉	#430	1	2	yes	—
164	C₂×D₁₉	—	2	2	no	—
165	C₂×D₁₉	—	2	2	no	—
166	C₂×D₁₉	—	2	2	no	—
167	C₂×D₁₉	—	4	2	no	—
168	C₂×D₁₉	—	4	2	no	—
169	C₄×C₁₉	—	2	1	no	—
170	C₄×C₁₉	—	2	1	no	—
171	C₄×C₁₉	#405	1	1	yes	characteristic
172	C₄×C₁₉	#405	1	1	yes	characteristic
173	C₄×C₁₉	#408	1	2	yes	—
174	C₄×C₁₉	#416	1	1	yes	characteristic
175	C₄×C₁₉	#416	1	1	yes	characteristic
176	C₄×C₁₉	#416	1	1	yes	characteristic
177	C₄×C₁₉	#416	1	1	yes	characteristic
178	Dic₁₉	—	2	2	no	—
179	Dic₁₉	—	2	2	no	—
180	Dic₁₉	—	2	2	no	—
181	C₂²×C₄²	—	19	1	no	—
182	#432	—	19	4	no	—
183	#432	—	38	2	no	—
184	#431	—	19	4	no	—
185	#431	—	38	2	no	—
186	#431	—	38	2	no	—
187	#431	—	38	2	no	—
188	C₂×C₄⋊C₈	—	19	1	no	—
189	C₂×C₄⋊C₈	—	19	1	no	—
190	C₂×C₄⋊C₈	—	19	2	no	—
191	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
192	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
193	C₂×C₂²⋊C₈	—	19	2	no	—
194	C₂×C₂²⋊C₈	—	38	1	no	—
195	C₂×C₂²⋊C₈	—	38	1	no	—
196	C₂×#97	—	38	1	no	—
197	C₂×C₄×C₈	—	19	1	no	—
198	C₂×C₄×C₈	—	38	1	no	—
199	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
200	C₂×C₁₉	#1572	1	2	yes	—
201	C₂×C₁₉	#1572	1	2	yes	—
202	C₂×C₁₉	#1729	1	1	yes	characteristic
203	C₂×C₁₉	#1735	1	1	yes	characteristic
204	C₂×C₁₉	#1738	1	1	yes	characteristic
205	D₁₉	—	4	2	no	—
206	C₂³×C₄	—	19	1	no	—
207	C₂³×C₄	—	19	1	no	—
208	C₂³×C₄	—	19	1	no	—
209	C₂²×C₈	—	2	1	no	—
210	C₂²×C₈	—	2	1	no	—
211	C₂²×C₈	—	19	2	no	—
212	C₂²×C₈	—	38	1	no	—
213	C₂²×C₈	—	38	1	no	—
214	C₂²×C₈	C₄×D₁₉	1	2	yes	—
215	C₂×C₄²	—	19	1	no	—
216	C₂×C₄²	—	19	1	no	—
217	C₂×C₄²	—	19	1	no	—
218	C₂×C₄²	—	19	2	no	—
219	C₂×C₄²	—	19	2	no	—
220	C₂×C₄²	—	19	2	no	—
221	C₂×C₄²	—	19	2	no	—
222	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
223	C₄⋊C₈	—	2	1	no	—
224	C₄⋊C₈	—	2	1	no	—
225	C₄⋊C₈	—	19	2	no	—
226	C₄⋊C₈	—	19	2	no	—
227	C₄⋊C₈	—	38	4	no	—
228	C₄⋊C₈	—	38	1	no	—
229	C₄⋊C₈	—	38	1	no	—
230	C₄⋊C₈	C₄×D₁₉	1	2	yes	—
231	C₄⋊C₈	C₄×D₁₉	1	2	yes	—
232	C₂²⋊C₈	—	38	4	no	—
233	C₂²⋊C₈	—	38	4	no	—
234	C₂²⋊C₈	—	38	4	no	—
235	#97	—	38	4	no	—
236	C₄×C₈	—	2	1	no	—
237	C₄×C₈	—	2	1	no	—
238	C₄×C₈	—	38	4	no	—
239	C₄×C₈	—	38	1	no	—
240	C₄×C₈	—	38	1	no	—
241	C₁₉	#7878	1	1	yes	characteristic, complemented, Sylow
242	C₂⁴	—	19	1	no	—
243	C₂²×C₄	—	19	1	no	—
244	C₂²×C₄	—	19	1	no	—
245	C₂²×C₄	—	19	1	no	—
246	C₂²×C₄	—	19	2	no	—
247	C₂²×C₄	—	19	2	no	—
248	C₂²×C₄	—	19	2	no	—
249	C₂²×C₄	—	19	2	no	—
250	C₂²×C₄	—	19	2	no	—
251	C₂²×C₄	—	19	2	no	—
252	C₂²×C₄	—	38	2	no	—
253	C₂²×C₄	—	38	2	no	—
254	C₂²×C₄	—	38	1	no	—
255	C₂²×C₄	—	38	1	no	—
256	C₂²×C₄	—	38	1	no	—
257	C₂²×C₄	—	38	1	no	—
258	C₂²×C₄	—	76	2	no	—
259	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
260	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
261	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	4	1	no	—
268	C₂×C₈	—	4	1	no	—
269	C₂×C₈	—	4	1	no	—
270	C₂×C₈	—	4	1	no	—
271	C₂×C₈	—	38	2	no	—
272	C₂×C₈	—	38	2	no	—
273	C₂×C₈	—	38	2	no	—
274	C₂×C₈	—	38	2	no	—
275	C₂×C₈	—	38	2	no	—
276	C₂×C₈	—	76	1	no	—
277	C₂×C₈	—	76	1	no	—
278	C₂×C₈	—	76	1	no	—
279	C₂×C₈	—	76	1	no	—
280	C₄²	—	19	2	no	—
281	C₄²	—	19	2	no	—
282	C₄²	—	19	2	no	—
283	C₄²	—	19	2	no	—
284	C₄²	—	19	2	no	—
285	C₄²	—	19	2	no	—
286	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
287	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
288	C₄²	D₄×D₁₉	1	1	yes	characteristic
289	C₄²	D₄×D₁₉	1	1	yes	characteristic
290	C₂³	—	19	2	no	—
291	C₂³	—	38	2	no	—
292	C₂³	—	38	1	no	—
293	C₂³	—	38	1	no	—
294	C₂³	—	38	2	no	—
295	C₂³	C₄⋊C₄×D₁₉	1	1	yes	characteristic
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	19	2	no	—
298	C₂×C₄	—	19	2	no	—
299	C₂×C₄	—	19	2	no	—
300	C₂×C₄	—	38	2	no	—
301	C₂×C₄	—	38	1	no	—
302	C₂×C₄	—	38	1	no	—
303	C₂×C₄	—	38	2	no	—
304	C₂×C₄	—	38	2	no	—
305	C₂×C₄	—	38	2	no	—
306	C₂×C₄	—	38	2	no	—
307	C₂×C₄	—	38	2	no	—
308	C₂×C₄	—	38	2	no	—
309	C₂×C₄	—	38	2	no	—
310	C₂×C₄	—	38	4	no	—
311	C₂×C₄	—	76	2	no	—
312	C₂×C₄	—	76	2	no	—
313	C₂×C₄	C₄²×D₁₉	1	1	yes	characteristic
314	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
315	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
316	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
317	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
318	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
319	C₂×C₄	#10597347	1	2	yes	—
320	C₂×C₄	#10597353	1	2	yes	—
321	C₂×C₄	#10597358	1	1	yes	characteristic
322	C₂×C₄	#10597362	1	1	yes	characteristic
323	C₂×C₄	#10597362	1	1	yes	characteristic
324	C₂×C₄	#10597365	1	1	yes	characteristic
325	C₈	—	4	2	no	—
326	C₈	—	4	2	no	—
327	C₈	—	4	2	no	—
328	C₈	—	76	2	no	—
329	C₈	—	76	2	no	—
330	C₈	—	76	2	no	—
331	C₂²	—	38	2	no	—
332	C₂²	—	38	2	no	—
333	C₂²	—	38	2	no	—
334	C₂²	—	76	2	no	—
335	C₂²	—	76	2	no	—
336	C₂²	#95×D₁₉	1	1	yes	characteristic
337	C₂²	#11926569	1	2	yes	—
338	C₂²	#11926579	1	1	yes	characteristic
339	C₂²	#11926581	1	1	yes	characteristic
340	C₂²	#11926591	1	2	yes	—
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	38	2	no	—
344	C₄	—	38	2	no	—
345	C₄	—	38	2	no	—
346	C₄	#11926498	1	1	yes	characteristic
347	C₄	#11926501	1	1	yes	characteristic
348	C₄	#11926509	1	2	yes	—
349	C₄	#11926537	1	1	yes	characteristic
350	C₄	#11926537	1	1	yes	characteristic
351	C₄	#11926539	1	1	yes	characteristic
352	C₄	#11926539	1	1	yes	characteristic
353	C₂	—	76	2	no	—
354	C₂	#14562412	1	2	yes	—
355	C₂	#14562414	1	2	yes	—
356	C₂	#14563330	1	1	yes	characteristic
357	C₂	#14563350	1	1	yes	characteristic
358	C₂	#14563360	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₉
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1277	C₂
socle	C₂³×C₁₉	C₂×C₄⋊C₄
prime radical	C₂³×D₁₉	C₄⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1277	D₁₉
hypocenter	C₁₉	#7878
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1277

Complements

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

N	H ≅ G/N	type
4. #14561470	C₂	semidirect
5. #14561469	C₂	semidirect
6. #14561469	C₂	semidirect
7. C₁₉×#1277	C₂	semidirect
158. #1277	D₁₉	semidirect
241. C₁₉	#7878	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926600	2
C₂×#11926596	2
C₂×#11926595	2
#14561470	2
#14561469	2
C₁₉×#1277	2
#7878	19