Subgroups — #18075727

Order 4864.

408 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	#14572591	C₂	1	1	yes	maximal, characteristic
2	C₄×#10597279	C₂	1	1	yes	maximal, characteristic, complemented
3	#14562393	C₂	1	1	yes	maximal, characteristic, complemented
4	#14562075	C₂	1	1	yes	maximal, characteristic
5	#14562072	C₂	1	1	yes	maximal, characteristic
6	#14561526	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1299	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#64541	C₂²	1	1	yes	characteristic
9	#11926668	C₄	1	1	yes	characteristic
10	#11926668	C₄	1	1	yes	characteristic
11	#11926609	C₄	1	1	yes	characteristic
12	#11926609	C₄	1	1	yes	characteristic
13	(C₂²⋊D₁₉)⋊C₈	—	2	1	no	—
14	(C₂²⋊D₁₉)⋊C₈	—	2	1	no	—
15	#11926595	—	2	1	no	—
16	C₂×#10597279	—	2	1	no	—
17	#11926568	C₂²	1	1	yes	characteristic
18	C₄×#64520	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
20	C₄×C₈×D₁₉	—	2	1	no	—
21	#11926308	—	2	1	no	—
22	#11926296	—	2	1	no	—
23	#11926296	—	2	1	no	—
24	#11926276	—	2	1	no	—
25	#11926274	—	2	1	no	—
26	#11926273	—	2	1	no	—
27	#11926271	—	2	1	no	—
28	C₁₉×#431	C₂²	1	1	yes	characteristic
29	C₁₉×#423	C₂²	1	1	yes	characteristic
30	C₄×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
31	C₂×#64541	Q₈	1	1	yes	characteristic
32	C₂×M₁₆×C₁₉	—	2	1	no	—
33	C₂×M₁₆×C₁₉	—	2	1	no	—
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	Q₈	1	1	yes	characteristic
38	C₂×#64520	—	2	1	no	—
39	C₂×#64520	—	2	1	no	—
40	C₂²×C₁₉⋊C₈	D₄	1	1	yes	characteristic
41	C₂²×C₁₉⋊C₈	D₄	1	1	yes	characteristic
42	C₂×C₈×D₁₉	—	2	1	no	—
43	C₂×C₈×D₁₉	—	2	1	no	—
44	C₄×D₇₆	C₂×C₄	1	1	yes	characteristic
45	C₄×D₇₆	D₄	1	1	yes	characteristic
46	#10597318	—	2	1	no	—
47	C₄²×D₁₉	—	2	1	no	—
48	C₄×Dic₃₈	C₂×C₄	1	1	yes	characteristic
49	C₄×Dic₃₈	D₄	1	1	yes	characteristic
50	C₁₉×C₄⋊C₈	—	2	1	no	—
51	C₁₉×C₄⋊C₈	—	2	1	no	—
52	C₁₉×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
53	C₁₉×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
54	C₁₉×C₂²⋊C₈	—	2	1	no	—
55	C₁₉×#97	—	2	1	no	—
56	C₄×C₈×C₁₉	—	2	1	no	—
57	C₄×C₈×C₁₉	—	2	1	no	—
58	#10597279	—	2	4	no	—
59	(C₂×D₁₉)⋊C₈	—	2	1	no	—
60	(C₂×D₁₉)⋊C₈	—	2	1	no	—
61	Dic₁₉⋊C₈	—	2	1	no	—
62	Dic₁₉⋊C₈	—	2	1	no	—
63	C₈×Dic₁₉	—	2	1	no	—
64	C₈×Dic₁₉	—	2	1	no	—
65	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
66	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
67	#10597260	—	2	1	no	—
68	C₄×C₁₉⋊C₈	—	2	1	no	—
69	C₄×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
70	C₄×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
71	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
72	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
75	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
76	C₂×C₂²⋊D₁₉	—	2	1	no	—
77	#64541	—	2	2	no	—
78	#64541	—	2	2	no	—
79	C₂×D₇₆	C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×D₁₉	—	2	1	no	—
81	C₂×C₄×D₁₉	—	2	1	no	—
82	C₂×C₄×D₁₉	—	2	1	no	—
83	C₂×Dic₃₈	C₄⋊C₄	1	1	yes	characteristic
84	M₁₆×C₁₉	—	2	2	no	—
85	M₁₆×C₁₉	—	4	1	no	—
86	M₁₆×C₁₉	—	8	1	no	—
87	C₂×C₈×C₁₉	—	2	1	no	—
88	C₂×C₈×C₁₉	—	2	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	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₁₉	#41	1	1	yes	characteristic
95	C₂²⋊Dic₁₉	—	2	1	no	—
96	(C₂×D₁₉)⋊C₄	—	2	1	no	—
97	(C₂×D₁₉)⋊C₄	—	2	1	no	—
98	C₄⋊Dic₁₉	C₄⋊C₄	1	1	yes	characteristic
99	C₄⋊Dic₁₉	C₄⋊C₄	1	1	yes	characteristic
100	Dic₁₉⋊C₄	—	2	1	no	—
101	Dic₁₉⋊C₄	—	2	1	no	—
102	C₄×Dic₁₉	—	2	1	no	—
103	C₄×Dic₁₉	—	2	1	no	—
104	C₄×Dic₁₉	—	2	1	no	—
105	#64520	—	2	2	no	—
106	#64520	—	4	1	no	—
107	#64520	—	8	1	no	—
108	C₂×C₁₉⋊C₈	—	2	2	no	—
109	C₂×C₁₉⋊C₈	—	4	1	no	—
110	C₂×C₁₉⋊C₈	—	4	1	no	—
111	C₂×C₁₉⋊C₈	—	4	1	no	—
112	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
116	C₈×D₁₉	—	4	2	no	—
117	C₈×D₁₉	—	4	2	no	—
118	#8671	—	19	1	no	maximal, Sylow
119	C₂³×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂²×D₁₉	—	2	1	no	—
121	C₂×C₄×C₁₉	—	2	1	no	—
122	C₂×C₄×C₁₉	—	2	1	no	—
123	C₂×C₄×C₁₉	—	2	1	no	—
124	C₂×C₄×C₁₉	—	2	1	no	—
125	C₂×C₄×C₁₉	—	2	1	no	—
126	C₂×C₄×C₁₉	—	2	1	no	—
127	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
130	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
131	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
132	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
133	C₂²⋊D₁₉	—	2	4	no	—
134	C₂×Dic₁₉	—	2	1	no	—
135	C₂×Dic₁₉	—	2	1	no	—
136	C₂×Dic₁₉	—	2	1	no	—
137	D₇₆	—	2	1	no	—
138	D₇₆	#108	1	2	yes	complemented
139	C₄×D₁₉	—	2	4	no	—
140	C₄×D₁₉	—	4	1	no	—
141	C₄×D₁₉	—	4	1	no	—
142	C₄×D₁₉	—	4	1	no	—
143	C₄×D₁₉	—	4	1	no	—
144	Dic₃₈	—	2	1	no	—
145	Dic₃₈	#108	1	2	yes	complemented
146	C₈×C₁₉	—	4	2	no	—
147	C₈×C₁₉	—	4	1	no	—
148	C₈×C₁₉	—	4	1	no	—
149	C₈×C₁₉	—	8	1	no	—
150	C₁₉⋊C₈	—	2	1	no	—
151	C₁₉⋊C₈	—	2	1	no	—
152	C₁₉⋊C₈	—	2	1	no	—
153	C₁₉⋊C₈	—	4	2	no	—
154	C₁₉⋊C₈	—	8	1	no	—
155	C₁₉⋊C₈	#104	1	2	yes	—
156	C₈×#41	—	19	1	no	—
157	C₄×#108	—	19	1	no	—
158	#1567	—	19	1	no	—
159	#1489	—	19	1	no	—
160	#1489	—	19	1	no	—
161	#1299	—	19	1	no	—
162	#1299	D₁₉	1	1	yes	characteristic, complemented
163	C₂²×C₁₉	—	2	1	no	—
164	C₂²×C₁₉	—	2	1	no	—
165	C₂²×C₁₉	—	2	1	no	—
166	C₂²×C₁₉	#380	1	1	yes	characteristic
167	C₂×D₁₉	—	2	2	no	—
168	C₂×D₁₉	—	4	1	no	—
169	C₂×D₁₉	—	4	1	no	—
170	C₄×C₁₉	—	2	1	no	—
171	C₄×C₁₉	—	2	1	no	—
172	C₄×C₁₉	—	2	1	no	—
173	C₄×C₁₉	C₂×#104	1	1	yes	characteristic
174	C₄×C₁₉	#421	1	1	yes	characteristic
175	C₄×C₁₉	C₂×#108	1	1	yes	characteristic
176	C₄×C₁₉	C₂×#108	1	1	yes	characteristic
177	C₄×C₁₉	#472	1	1	yes	characteristic
178	C₄×C₁₉	#473	1	1	yes	characteristic
179	Dic₁₉	—	2	2	no	—
180	Dic₁₉	—	4	1	no	—
181	Dic₁₉	—	4	1	no	—
182	C₄×#41	—	19	1	no	—
183	C₈×Q₈	—	19	1	no	—
184	C₈×Q₈	—	19	1	no	—
185	C₈×D₄	—	19	1	no	—
186	C₈×D₄	—	19	1	no	—
187	C₈×D₄	—	38	1	no	—
188	C₈×D₄	—	38	1	no	—
189	#431	—	38	1	no	—
190	#431	C₂×D₁₉	1	1	yes	characteristic
191	C₂×#108	—	38	1	no	—
192	#423	—	19	1	no	—
193	#423	C₂×D₁₉	1	1	yes	characteristic
194	C₄×M₁₆	—	19	1	no	—
195	C₄×M₁₆	C₂×D₁₉	1	1	yes	characteristic
196	C₂×C₄×C₈	—	19	1	no	—
197	C₂×C₄×C₈	—	38	1	no	—
198	#341	—	38	1	no	—
199	#335	—	38	1	no	—
200	#335	—	38	1	no	—
201	Q₈⋊C₈	—	38	1	no	—
202	Q₈⋊C₈	—	38	1	no	—
203	D₄⋊C₈	—	38	1	no	—
204	D₄⋊C₈	—	38	1	no	—
205	C₂×C₁₉	—	4	1	no	—
206	C₂×C₁₉	#1806	1	1	yes	characteristic
207	C₂×C₁₉	#1810	1	1	yes	characteristic
208	C₂×C₁₉	#1816	1	1	yes	characteristic
209	D₁₉	—	4	2	no	—
210	C₂×#41	—	19	1	no	—
211	C₂×M₁₆	—	2	1	no	—
212	C₂×M₁₆	—	2	1	no	—
213	C₂×M₁₆	—	38	1	no	—
214	C₂×M₁₆	—	38	1	no	—
215	C₂²×C₈	—	19	1	no	—
216	C₂²×C₈	—	19	1	no	—
217	C₂²×C₈	—	38	1	no	—
218	C₂²×C₈	—	38	1	no	—
219	C₄×Q₈	—	19	1	no	—
220	C₄×Q₈	—	19	1	no	—
221	C₄×D₄	—	19	1	no	—
222	C₄×D₄	—	19	1	no	—
223	C₄×D₄	—	38	1	no	—
224	C₄×D₄	—	38	1	no	—
225	#117	—	19	1	no	—
226	#117	—	38	1	no	—
227	C₂×C₄²	—	38	1	no	—
228	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
229	#108	—	38	4	no	—
230	C₄⋊C₈	—	2	1	no	—
231	C₄⋊C₈	—	2	1	no	—
232	C₄⋊C₈	—	38	1	no	—
233	C₄⋊C₈	—	38	1	no	—
234	C₄⋊C₈	—	38	1	no	—
235	C₄⋊C₈	—	38	1	no	—
236	C₄⋊C₈	C₄×D₁₉	1	1	yes	characteristic
237	C₄⋊C₈	C₄×D₁₉	1	1	yes	characteristic
238	C₂²⋊C₈	—	2	1	no	—
239	C₂²⋊C₈	—	38	1	no	—
240	C₂²⋊C₈	—	38	1	no	—
241	#97	—	2	1	no	—
242	#97	—	38	1	no	—
243	C₄×C₈	—	2	1	no	—
244	C₄×C₈	—	2	1	no	—
245	C₄×C₈	—	19	1	no	—
246	C₄×C₈	—	19	1	no	—
247	C₄×C₈	—	19	1	no	—
248	C₄×C₈	—	19	1	no	—
249	C₄×C₈	—	38	1	no	—
250	C₄×C₈	—	38	1	no	—
251	C₄×C₈	—	38	1	no	—
252	C₁₉	#8671	1	1	yes	characteristic, complemented, Sylow
253	#41	—	38	2	no	—
254	#41	—	38	2	no	—
255	C₂×Q₈	—	19	1	no	—
256	C₂×D₄	—	19	1	no	—
257	C₂×D₄	—	38	1	no	—
258	C₂²×C₄	—	38	1	no	—
259	C₂²×C₄	—	38	1	no	—
260	C₂²×C₄	—	38	1	no	—
261	C₂²×C₄	#64541	1	1	yes	characteristic
262	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
263	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
264	M₁₆	—	2	2	no	—
265	M₁₆	—	4	1	no	—
266	M₁₆	—	8	1	no	—
267	M₁₆	—	38	2	no	—
268	M₁₆	—	76	1	no	—
269	M₁₆	—	152	1	no	—
270	C₂×C₈	—	2	1	no	—
271	C₂×C₈	—	2	1	no	—
272	C₂×C₈	—	4	1	no	—
273	C₂×C₈	—	4	1	no	—
274	C₂×C₈	—	19	1	no	—
275	C₂×C₈	—	19	1	no	—
276	C₂×C₈	—	19	1	no	—
277	C₂×C₈	—	19	1	no	—
278	C₂×C₈	—	38	2	no	—
279	C₂×C₈	—	76	2	no	—
280	C₂×C₈	—	76	2	no	—
281	C₂×C₈	—	76	1	no	—
282	C₂×C₈	—	76	1	no	—
283	C₂×C₈	—	76	1	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	—	38	1	no	—
292	C₄²	—	38	1	no	—
293	C₄²	—	38	1	no	—
294	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
295	C₄²	#64541	1	1	yes	characteristic
296	C₄²	D₄×D₁₉	1	1	yes	characteristic
297	C₄²	D₄×D₁₉	1	1	yes	characteristic
298	C₂³	—	38	1	no	—
299	C₂³	#10597342	1	1	yes	characteristic
300	Q₈	—	19	2	no	—
301	Q₈	—	38	1	no	—
302	D₄	—	19	2	no	—
303	D₄	—	38	4	no	—
304	D₄	—	38	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	38	4	no	—
312	C₂×C₄	—	38	1	no	—
313	C₂×C₄	—	38	1	no	—
314	C₂×C₄	—	38	1	no	—
315	C₂×C₄	—	76	1	no	—
316	C₂×C₄	—	76	1	no	—
317	C₂×C₄	—	76	1	no	—
318	C₂×C₄	—	76	1	no	—
319	C₂×C₄	#10597318	1	1	yes	characteristic
320	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
321	C₂×C₄	#10597330	1	1	yes	characteristic
322	C₂×C₄	#10597330	1	1	yes	characteristic
323	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
324	C₂×C₄	#10597342	1	1	yes	characteristic
325	C₈	—	4	2	no	—
326	C₈	—	4	1	no	—
327	C₈	—	4	1	no	—
328	C₈	—	8	1	no	—
329	C₈	—	19	2	no	—
330	C₈	—	38	1	no	—
331	C₈	—	38	1	no	—
332	C₈	—	38	1	no	—
333	C₈	—	76	2	no	—
334	C₈	—	152	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	2	1	no	—
338	C₂²	—	38	2	no	—
339	C₂²	—	76	1	no	—
340	C₂²	—	76	1	no	—
341	C₂²	#11926403	1	1	yes	characteristic
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	2	1	no	—
345	C₄	—	38	2	no	—
346	C₄	—	76	1	no	—
347	C₄	—	76	1	no	—
348	C₄	#104×D₁₉	1	1	yes	characteristic
349	C₄	#11926559	1	1	yes	characteristic
350	C₄	#108×D₁₉	1	1	yes	characteristic
351	C₄	#108×D₁₉	1	1	yes	characteristic
352	C₄	#11926813	1	1	yes	characteristic
353	C₄	#11926818	1	1	yes	characteristic
354	C₂	—	4	1	no	—
355	C₂	—	76	2	no	—
356	C₂	#14563654	1	1	yes	characteristic
357	C₂	#14563672	1	1	yes	characteristic
358	C₂	#14563698	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10597330
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1299	C₂
socle	C₂²×C₁₉	#380
prime radical	C₂×#64541	Q₈
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1299	D₁₉
hypocenter	C₁₉	#8671
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1299

Complements

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

N	H ≅ G/N	type
2. C₄×#10597279	C₂	semidirect
3. #14562393	C₂	semidirect
6. #14561526	C₂	semidirect
7. C₁₉×#1299	C₂	semidirect
138. D₇₆	#108	semidirect
145. Dic₃₈	#108	semidirect
162. #1299	D₁₉	semidirect
252. C₁₉	#8671	semidirect

Maximal subgroups

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

subgroup	index
#14572591	2
C₄×#10597279	2
#14562393	2
#14562075	2
#14562072	2
#14561526	2
C₁₉×#1299	2
#8671	19