Subgroups — #18076430

Order 4864.

446 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14572743	C₂	1	1	yes	maximal, characteristic
2	#14563752	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#10597262	C₂	1	1	yes	maximal, characteristic
4	#14562388	C₂	1	1	yes	maximal, characteristic, complemented
5	#14562023	C₂	1	1	yes	maximal, characteristic
6	#14561484	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1276	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#64541	C₂²	1	1	yes	characteristic
9	C₈×Dic₃₈	—	2	1	no	—
10	C₈×D₇₆	—	2	1	no	—
11	(C₂²⋊D₁₉)⋊C₈	—	2	2	no	—
12	#11926596	—	2	1	no	—
13	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×D₁₉	—	2	1	no	—
16	C₂×#10597262	C₄	1	1	yes	characteristic
17	C₂×#10597262	C₄	1	1	yes	characteristic
18	C₄×#64520	C₂²	1	1	yes	characteristic
19	#11926304	C₄	1	1	yes	characteristic
20	#11926304	C₄	1	1	yes	characteristic
21	#11926299	—	2	2	no	—
22	#11926296	—	2	1	no	—
23	#11926294	—	2	1	no	—
24	#11926276	—	2	1	no	—
25	#11926273	—	2	1	no	—
26	C₁₉×#431	C₂²	1	1	yes	characteristic
27	C₁₉×#431	C₂²	1	1	yes	characteristic
28	C₄³×C₁₉	C₂²	1	1	yes	characteristic
29	C₂×#64541	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₉	—	2	1	no	—
31	C₂×C₄²×C₁₉	—	2	1	no	—
32	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
35	C₄×C₂²⋊D₁₉	—	2	1	no	—
36	C₄×C₂²⋊D₁₉	—	2	1	no	—
37	#10597372	C₂×C₄	1	1	yes	characteristic
38	C₂×#64520	C₂×C₄	1	1	yes	characteristic
39	C₂×#64520	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₁₉⋊C₈	—	2	1	no	—
41	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
42	C₂×C₈×D₁₉	—	2	2	no	—
43	C₄×D₇₆	D₄	1	1	yes	characteristic
44	C₄×D₇₆	Q₈	1	1	yes	characteristic
45	#10597318	—	2	1	no	—
46	C₄²×D₁₉	—	2	1	no	—
47	C₄×Dic₃₈	D₄	1	1	yes	characteristic
48	C₄×Dic₃₈	Q₈	1	1	yes	characteristic
49	C₁₉×C₄⋊C₈	—	2	1	no	—
50	C₁₉×C₄⋊C₈	D₄	1	1	yes	characteristic
51	C₁₉×C₄⋊C₈	D₄	1	1	yes	characteristic
52	C₁₉×C₂²⋊C₈	—	2	1	no	—
53	C₁₉×C₂²⋊C₈	—	2	1	no	—
54	C₄×C₈×C₁₉	—	2	1	no	—
55	C₄×C₈×C₁₉	—	2	1	no	—
56	(C₂×D₁₉)⋊C₈	—	2	2	no	—
57	Dic₁₉⋊C₈	—	2	2	no	—
58	C₈×Dic₁₉	—	2	2	no	—
59	#10597262	—	2	2	no	—
60	#10597262	C₈	1	4	yes	complemented
61	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
62	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
63	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
64	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
65	#10597260	—	2	1	no	—
66	C₄×C₁₉⋊C₈	—	2	1	no	—
67	C₂²×C₄×C₁₉	—	2	1	no	—
68	C₂²×C₄×C₁₉	—	2	1	no	—
69	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
72	C₂×C₂²⋊D₁₉	—	2	1	no	—
73	#64541	—	2	1	no	—
74	#64541	—	2	1	no	—
75	#64541	C₂×C₈	1	2	yes	—
76	#64541	M₁₆	1	2	yes	—
77	C₂×D₇₆	C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×D₁₉	—	2	1	no	—
79	C₂×C₄×D₁₉	—	2	1	no	—
80	C₂×C₄×D₁₉	—	2	1	no	—
81	C₂×Dic₃₈	C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₁₉	—	2	1	no	—
83	C₂×C₈×C₁₉	—	2	1	no	—
84	C₂×C₈×C₁₉	—	4	1	no	—
85	C₄²×C₁₉	—	2	1	no	—
86	C₄²×C₁₉	—	2	1	no	—
87	C₄²×C₁₉	—	2	1	no	—
88	C₄²×C₁₉	—	4	1	no	—
89	C₄²×C₁₉	—	4	1	no	—
90	C₄²×C₁₉	—	4	2	no	—
91	C₄²×C₁₉	C₂×C₈	1	2	yes	complemented
92	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
95	C₄²×C₁₉	#41	1	1	yes	characteristic
96	C₂²⋊Dic₁₉	—	2	1	no	—
97	(C₂×D₁₉)⋊C₄	—	2	1	no	—
98	(C₂×D₁₉)⋊C₄	—	2	1	no	—
99	C₄⋊Dic₁₉	C₄⋊C₄	1	1	yes	characteristic
100	C₄⋊Dic₁₉	C₄⋊C₄	1	1	yes	characteristic
101	Dic₁₉⋊C₄	—	2	1	no	—
102	Dic₁₉⋊C₄	—	2	1	no	—
103	C₄×Dic₁₉	—	2	1	no	—
104	C₄×Dic₁₉	—	2	1	no	—
105	C₄×Dic₁₉	—	2	1	no	—
106	#64520	—	2	1	no	—
107	#64520	—	2	1	no	—
108	#64520	—	2	1	no	—
109	#64520	C₂×C₈	1	2	yes	complemented
110	C₂×C₁₉⋊C₈	—	2	1	no	—
111	C₂×C₁₉⋊C₈	—	2	1	no	—
112	C₂×C₁₉⋊C₈	—	2	2	no	—
113	C₂×C₁₉⋊C₈	—	4	2	no	—
114	C₂×C₁₉⋊C₈	—	4	2	no	—
115	C₂×C₁₉⋊C₈	—	4	1	no	—
116	C₈×D₁₉	—	8	2	no	—
117	#8840	—	19	1	no	maximal, Sylow
118	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
119	C₂²×D₁₉	—	2	1	no	—
120	C₂×C₄×C₁₉	—	2	1	no	—
121	C₂×C₄×C₁₉	—	2	1	no	—
122	C₂×C₄×C₁₉	—	2	2	no	—
123	C₂×C₄×C₁₉	—	2	2	no	—
124	C₂×C₄×C₁₉	—	4	1	no	—
125	C₂×C₄×C₁₉	—	4	1	no	—
126	C₂×C₄×C₁₉	—	4	1	no	—
127	C₂×C₄×C₁₉	—	4	1	no	—
128	C₂×C₄×C₁₉	—	4	1	no	—
129	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
132	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
133	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
134	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
135	C₂×C₄×C₁₉	C₂²×C₈	1	1	yes	characteristic
136	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
137	C₂×C₄×C₁₉	#131	1	1	yes	characteristic
138	C₂×C₄×C₁₉	#131	1	1	yes	characteristic
139	C₂²⋊D₁₉	—	2	2	no	—
140	C₂²⋊D₁₉	—	2	2	no	—
141	C₂×Dic₁₉	—	2	1	no	—
142	C₂×Dic₁₉	—	2	1	no	—
143	C₂×Dic₁₉	—	2	1	no	—
144	D₇₆	—	2	1	no	—
145	D₇₆	C₄⋊C₈	1	2	yes	—
146	C₄×D₁₉	—	2	2	no	—
147	C₄×D₁₉	—	2	2	no	—
148	C₄×D₁₉	—	4	1	no	—
149	C₄×D₁₉	—	4	1	no	—
150	C₄×D₁₉	—	8	1	no	—
151	Dic₃₈	—	2	1	no	—
152	Dic₃₈	C₄⋊C₈	1	2	yes	—
153	C₈×C₁₉	—	4	1	no	—
154	C₈×C₁₉	—	4	1	no	—
155	C₈×C₁₉	—	8	1	no	—
156	C₁₉⋊C₈	—	2	2	no	—
157	C₁₉⋊C₈	—	4	1	no	—
158	C₁₉⋊C₈	—	4	2	no	—
159	C₁₉⋊C₈	—	8	1	no	—
160	#2988	—	19	1	no	—
161	#1831	—	19	1	no	—
162	C₄×#104	—	19	1	no	—
163	#1565	—	19	1	no	—
164	#1477	—	19	1	no	—
165	#1283	—	19	1	no	—
166	#1276	D₁₉	1	1	yes	characteristic, complemented
167	C₂²×C₁₉	—	2	1	no	—
168	C₂²×C₁₉	—	2	1	no	—
169	C₂²×C₁₉	#385	1	1	yes	characteristic
170	C₂²×C₁₉	C₈×D₄	1	1	yes	characteristic
171	C₂²×C₁₉	#435	1	1	yes	characteristic
172	C₂×D₁₉	—	2	2	no	—
173	C₂×D₁₉	—	4	1	no	—
174	C₂×D₁₉	—	4	1	no	—
175	C₄×C₁₉	—	2	1	no	—
176	C₄×C₁₉	—	2	1	no	—
177	C₄×C₁₉	—	2	1	no	—
178	C₄×C₁₉	—	4	2	no	—
179	C₄×C₁₉	—	4	2	no	—
180	C₄×C₁₉	C₂×C₄⋊C₈	1	1	yes	characteristic
181	C₄×C₁₉	#424	1	1	yes	characteristic
182	C₄×C₁₉	C₈×D₄	1	1	yes	characteristic
183	C₄×C₁₉	#435	1	1	yes	characteristic
184	C₄×C₁₉	#454	1	1	yes	characteristic
185	C₄×C₁₉	#455	1	1	yes	characteristic
186	Dic₁₉	—	2	2	no	—
187	Dic₁₉	—	4	1	no	—
188	Dic₁₉	—	4	1	no	—
189	C₄×#41	—	19	1	no	—
190	C₈×Q₈	—	38	1	no	—
191	C₈×D₄	—	38	1	no	—
192	C₈×D₄	—	38	2	no	—
193	#431	—	38	1	no	—
194	#431	C₂×D₁₉	1	1	yes	characteristic
195	#431	C₂×D₁₉	1	1	yes	characteristic
196	C₂×C₄⋊C₈	—	19	1	no	—
197	C₂×C₄⋊C₈	—	19	1	no	—
198	C₂×C₄⋊C₈	—	38	1	no	—
199	C₂×#104	—	19	1	no	—
200	C₂×#104	—	19	1	no	—
201	C₄×M₁₆	—	19	1	no	—
202	C₄³	C₂×D₁₉	1	1	yes	characteristic
203	#339	—	19	1	no	—
204	#339	—	19	1	no	—
205	#336	—	38	2	no	—
206	#335	—	38	1	no	—
207	#334	—	38	1	no	—
208	Q₈⋊C₈	—	38	1	no	—
209	D₄⋊C₈	—	38	1	no	—
210	C₂×C₁₉	—	2	1	no	—
211	C₂×C₁₉	—	2	1	no	—
212	C₂×C₁₉	#1850	1	1	yes	characteristic
213	C₂×C₁₉	#1860	1	1	yes	characteristic
214	C₂×C₁₉	#1861	1	1	yes	characteristic
215	D₁₉	—	4	2	no	—
216	C₂×#41	—	19	1	no	—
217	C₂×M₁₆	—	19	1	no	—
218	C₂×M₁₆	—	19	1	no	—
219	C₂²×C₈	—	19	2	no	—
220	C₂²×C₈	—	38	2	no	—
221	C₂²×C₈	—	38	1	no	—
222	C₄×Q₈	—	19	1	no	—
223	C₄×Q₈	—	19	1	no	—
224	C₄×D₄	—	19	1	no	—
225	C₄×D₄	—	19	1	no	—
226	C₄×D₄	—	38	1	no	—
227	C₄×D₄	—	38	1	no	—
228	#117	—	19	1	no	—
229	#117	—	38	1	no	—
230	C₂×C₄²	—	2	1	no	—
231	C₂×C₄²	—	2	1	no	—
232	C₂×C₄²	—	38	1	no	—
233	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
234	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
235	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
236	C₄⋊C₈	—	2	1	no	—
237	C₄⋊C₈	—	38	2	no	—
238	C₄⋊C₈	—	38	1	no	—
239	C₄⋊C₈	—	38	1	no	—
240	C₄⋊C₈	—	38	1	no	—
241	C₄⋊C₈	—	38	1	no	—
242	C₄⋊C₈	D₇₆	1	1	yes	characteristic
243	C₄⋊C₈	D₇₆	1	1	yes	characteristic
244	#104	—	19	4	no	—
245	#104	—	38	2	no	—
246	C₂²⋊C₈	—	2	1	no	—
247	C₂²⋊C₈	—	2	1	no	—
248	C₂²⋊C₈	—	38	2	no	—
249	#97	—	38	1	no	—
250	C₄×C₈	—	2	1	no	—
251	C₄×C₈	—	2	1	no	—
252	C₄×C₈	—	38	2	no	—
253	C₄×C₈	—	38	1	no	—
254	C₁₉	#8840	1	1	yes	characteristic, complemented, Sylow
255	#41	—	19	2	no	—
256	#41	—	19	2	no	—
257	#41	—	38	1	no	—
258	#41	—	38	1	no	—
259	C₂×Q₈	—	19	1	no	—
260	C₂×D₄	—	19	1	no	—
261	C₂×D₄	—	38	1	no	—
262	C₂²×C₄	—	2	1	no	—
263	C₂²×C₄	—	2	1	no	—
264	C₂²×C₄	—	38	1	no	—
265	C₂²×C₄	—	38	1	no	—
266	C₂²×C₄	—	38	1	no	—
267	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
268	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
269	C₂²×C₄	#64543	1	1	yes	characteristic
270	M₁₆	—	19	2	no	—
271	M₁₆	—	38	1	no	—
272	M₁₆	—	38	1	no	—
273	M₁₆	—	38	1	no	—
274	C₂×C₈	—	2	1	no	—
275	C₂×C₈	—	2	1	no	—
276	C₂×C₈	—	4	1	no	—
277	C₂×C₈	—	38	1	no	—
278	C₂×C₈	—	38	1	no	—
279	C₂×C₈	—	38	2	no	—
280	C₂×C₈	—	76	2	no	—
281	C₂×C₈	—	76	2	no	—
282	C₂×C₈	—	76	1	no	—
283	C₂×C₈	—	152	2	no	—
284	C₄⋊C₄	—	19	1	no	—
285	C₄⋊C₄	—	19	1	no	—
286	C₄⋊C₄	—	38	1	no	—
287	C₄⋊C₄	—	38	1	no	—
288	C₂²⋊C₄	—	38	1	no	—
289	C₂²⋊C₄	—	38	1	no	—
290	C₂²⋊C₄	—	38	1	no	—
291	C₄²	—	2	1	no	—
292	C₄²	—	2	1	no	—
293	C₄²	—	2	1	no	—
294	C₄²	—	4	1	no	—
295	C₄²	—	4	1	no	—
296	C₄²	—	4	2	no	—
297	C₄²	—	38	1	no	—
298	C₄²	—	38	1	no	—
299	C₄²	—	38	1	no	—
300	C₄²	C₈×D₁₉	1	2	yes	complemented
301	C₄²	C₂×D₇₆	1	1	yes	characteristic
302	C₄²	#64541	1	1	yes	characteristic
303	C₄²	D₄×D₁₉	1	1	yes	characteristic
304	C₄²	Q₈×D₁₉	1	1	yes	characteristic
305	C₂³	—	38	1	no	—
306	C₂³	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
307	Q₈	—	19	2	no	—
308	Q₈	—	38	1	no	—
309	D₄	—	19	2	no	—
310	D₄	—	38	2	no	—
311	D₄	—	38	2	no	—
312	D₄	—	38	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	38	1	no	—
323	C₂×C₄	—	38	2	no	—
324	C₂×C₄	—	38	2	no	—
325	C₂×C₄	—	38	1	no	—
326	C₂×C₄	—	38	1	no	—
327	C₂×C₄	—	76	1	no	—
328	C₂×C₄	—	76	1	no	—
329	C₂×C₄	—	152	1	no	—
330	C₂×C₄	C₄×D₇₆	1	1	yes	characteristic
331	C₂×C₄	#10597328	1	1	yes	characteristic
332	C₂×C₄	#10597329	1	1	yes	characteristic
333	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
334	C₂×C₄	#10597342	1	1	yes	characteristic
335	C₂×C₄	#10597343	1	1	yes	characteristic
336	C₂×C₄	C₂×C₈×D₁₉	1	1	yes	characteristic
337	C₂×C₄	#10597347	1	1	yes	characteristic
338	C₂×C₄	M₁₆×D₁₉	1	1	yes	characteristic
339	C₂×C₄	#10597353	1	1	yes	characteristic
340	C₈	—	4	1	no	—
341	C₈	—	4	1	no	—
342	C₈	—	8	1	no	—
343	C₈	—	38	2	no	—
344	C₈	—	76	1	no	—
345	C₈	—	76	2	no	—
346	C₈	—	152	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	2	1	no	—
349	C₂²	—	38	2	no	—
350	C₂²	—	76	1	no	—
351	C₂²	—	76	1	no	—
352	C₂²	#11926418	1	1	yes	characteristic
353	C₂²	(C₂²⋊D₁₉)⋊C₈	1	1	yes	characteristic
354	C₂²	#11926618	1	1	yes	characteristic
355	C₄	—	2	1	no	—
356	C₄	—	2	1	no	—
357	C₄	—	2	1	no	—
358	C₄	—	4	2	no	—
359	C₄	—	4	2	no	—
360	C₄	—	38	2	no	—
361	C₄	—	76	1	no	—
362	C₄	—	76	1	no	—
363	C₄	C₄⋊C₈×D₁₉	1	1	yes	characteristic
364	C₄	#11926569	1	1	yes	characteristic
365	C₄	C₈×D₇₆	1	1	yes	characteristic
366	C₄	#11926617	1	1	yes	characteristic
367	C₄	#11926718	1	1	yes	characteristic
368	C₄	#11926723	1	1	yes	characteristic
369	C₂	—	2	1	no	—
370	C₂	—	2	1	no	—
371	C₂	—	76	2	no	—
372	C₂	#14563814	1	1	yes	characteristic
373	C₂	#14563869	1	1	yes	characteristic
374	C₂	#14563876	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10597328
commutator	C₂×C₄×C₁₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1276	C₂
socle	C₂²×C₁₉	#385
prime radical	C₂×#64541	C₂×C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1276	D₁₉
hypocenter	C₁₉	#8840
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1276

Complements

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

N	H ≅ G/N	type
2. #14563752	C₂	semidirect
4. #14562388	C₂	semidirect
6. #14561484	C₂	semidirect
7. C₁₉×#1276	C₂	semidirect
60. #10597262	C₈	semidirect
91. C₄²×C₁₉	C₂×C₈	semidirect
109. #64520	C₂×C₈	semidirect
166. #1276	D₁₉	semidirect
254. C₁₉	#8840	semidirect
300. C₄²	C₈×D₁₉	semidirect

Maximal subgroups

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

subgroup	index
#14572743	2
#14563752	2
C₄×#10597262	2
#14562388	2
#14562023	2
#14561484	2
C₁₉×#1276	2
#8840	19