Subgroups — #18075417

Order 4864.

462 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#10597353	C₂	1	1	yes	maximal, characteristic
2	#14562381	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11926296	C₂	1	1	yes	maximal, characteristic, complemented
4	#14562023	C₂	1	1	yes	maximal, characteristic
5	#14562020	C₂	1	1	yes	maximal, characteristic
6	#14561486	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1283	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10597353	C₄	1	1	yes	characteristic
9	C₂×#10597353	C₄	1	1	yes	characteristic
10	C₄×#64541	C₂²	1	1	yes	characteristic
11	#11926597	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
13	#11926502	C₄	1	1	yes	characteristic
14	#11926502	C₄	1	1	yes	characteristic
15	#11926501	—	2	1	no	—
16	#11926501	—	2	1	no	—
17	C₄×#64515	—	2	1	no	—
18	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
19	C₄×C₈×D₁₉	—	2	1	no	—
20	#11926296	—	2	1	no	—
21	#11926296	—	2	1	no	—
22	#11926296	—	2	1	no	—
23	#11926294	—	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	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
30	C₄×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
31	C₂×#64541	C₂×C₄	1	1	yes	characteristic
32	C₂×M₁₆×C₁₉	C₂×C₄	1	1	yes	characteristic
33	C₂×M₁₆×C₁₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₉	—	2	1	no	—
35	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
36	C₄×C₂²⋊D₁₉	—	2	1	no	—
37	C₄×C₂²⋊D₁₉	—	2	1	no	—
38	#10597372	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₁₉⋊C₈	—	2	1	no	—
40	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
42	#10597353	—	2	2	no	—
43	#10597353	—	2	2	no	—
44	#10597353	—	2	2	no	—
45	#10597353	C₈	1	4	yes	complemented
46	C₂×#64515	—	2	1	no	—
47	C₂×#64515	—	2	1	no	—
48	C₂×C₈×D₁₉	—	2	1	no	—
49	C₂×C₈×D₁₉	—	2	1	no	—
50	C₄×D₇₆	D₄	1	1	yes	characteristic
51	C₄×D₇₆	Q₈	1	1	yes	characteristic
52	#10597318	—	2	1	no	—
53	C₄²×D₁₉	—	2	1	no	—
54	C₄×Dic₃₈	D₄	1	1	yes	characteristic
55	C₄×Dic₃₈	Q₈	1	1	yes	characteristic
56	C₁₉×C₄⋊C₈	—	2	1	no	—
57	C₁₉×C₄⋊C₈	—	2	1	no	—
58	C₁₉×C₄⋊C₈	—	2	1	no	—
59	C₁₉×C₂²⋊C₈	—	2	1	no	—
60	C₁₉×#97	—	2	1	no	—
61	C₄×C₈×C₁₉	—	2	1	no	—
62	C₄×C₈×C₁₉	—	2	1	no	—
63	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
64	#10597272	—	2	1	no	—
65	#10597272	—	2	1	no	—
66	C₈×Dic₁₉	—	2	1	no	—
67	C₈×Dic₁₉	—	2	1	no	—
68	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
69	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
70	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
71	C₄×C₁₉⋊C₈	—	2	1	no	—
72	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
73	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
74	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
75	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
78	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
79	C₂×C₂²⋊D₁₉	—	2	1	no	—
80	#64541	—	2	1	no	—
81	#64541	—	2	1	no	—
82	#64541	C₂×C₈	1	2	yes	—
83	#64541	M₁₆	1	2	yes	—
84	C₂×D₇₆	C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×D₁₉	—	2	1	no	—
86	C₂×C₄×D₁₉	—	2	1	no	—
87	C₂×C₄×D₁₉	—	2	1	no	—
88	C₂×Dic₃₈	C₄⋊C₄	1	1	yes	characteristic
89	M₁₆×C₁₉	—	2	1	no	—
90	M₁₆×C₁₉	—	2	1	no	—
91	M₁₆×C₁₉	C₂×C₈	1	2	yes	—
92	M₁₆×C₁₉	M₁₆	1	2	yes	—
93	C₂×C₈×C₁₉	—	2	1	no	—
94	C₂×C₈×C₁₉	—	2	1	no	—
95	C₂×C₈×C₁₉	—	4	1	no	—
96	C₂×C₈×C₁₉	—	4	1	no	—
97	C₂×C₈×C₁₉	—	8	1	no	—
98	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
102	C₂²⋊Dic₁₉	—	2	1	no	—
103	(C₂×D₁₉)⋊C₄	—	2	1	no	—
104	(C₂×D₁₉)⋊C₄	—	2	1	no	—
105	C₄⋊Dic₁₉	C₄⋊C₄	1	1	yes	characteristic
106	C₄⋊Dic₁₉	C₄⋊C₄	1	1	yes	characteristic
107	Dic₁₉⋊C₄	—	2	1	no	—
108	Dic₁₉⋊C₄	—	2	1	no	—
109	C₄×Dic₁₉	—	2	1	no	—
110	C₄×Dic₁₉	—	2	1	no	—
111	C₄×Dic₁₉	—	2	1	no	—
112	C₂×C₁₉⋊C₈	—	2	1	no	—
113	C₂×C₁₉⋊C₈	—	2	1	no	—
114	C₂×C₁₉⋊C₈	—	4	1	no	—
115	C₂×C₁₉⋊C₈	—	4	1	no	—
116	C₂×C₁₉⋊C₈	—	8	1	no	—
117	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₁₉⋊C₈	C₂×C₈	1	2	yes	—
122	C₂×C₁₉⋊C₈	M₁₆	1	2	yes	—
123	#64515	—	2	4	no	—
124	#64515	—	2	4	no	—
125	C₈×D₁₉	—	2	4	no	—
126	C₈×D₁₉	—	2	4	no	—
127	#8605	—	19	1	no	maximal, Sylow
128	C₂³×C₁₉	#117	1	1	yes	characteristic
129	C₂²×D₁₉	—	2	1	no	—
130	C₂×C₄×C₁₉	—	2	1	no	—
131	C₂×C₄×C₁₉	—	2	1	no	—
132	C₂×C₄×C₁₉	—	4	1	no	—
133	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
136	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
137	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
138	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
139	C₂×C₄×C₁₉	C₂²×C₈	1	1	yes	characteristic
140	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
141	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
142	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
143	C₂²⋊D₁₉	—	2	2	no	—
144	C₂²⋊D₁₉	—	2	2	no	—
145	C₂×Dic₁₉	—	2	1	no	—
146	C₂×Dic₁₉	—	2	1	no	—
147	C₂×Dic₁₉	—	2	1	no	—
148	D₇₆	—	2	1	no	—
149	D₇₆	C₄⋊C₈	1	2	yes	—
150	C₄×D₁₉	—	2	2	no	—
151	C₄×D₁₉	—	2	2	no	—
152	C₄×D₁₉	—	4	2	no	—
153	C₄×D₁₉	—	4	1	no	—
154	C₄×D₁₉	—	4	1	no	—
155	Dic₃₈	—	2	1	no	—
156	Dic₃₈	C₄⋊C₈	1	2	yes	—
157	C₈×C₁₉	—	2	2	no	—
158	C₈×C₁₉	—	2	2	no	—
159	C₈×C₁₉	—	8	1	no	—
160	C₈×C₁₉	—	8	1	no	—
161	C₁₉⋊C₈	—	2	1	no	—
162	C₁₉⋊C₈	—	2	1	no	—
163	C₁₉⋊C₈	—	8	1	no	—
164	C₁₉⋊C₈	—	8	1	no	—
165	C₁₉⋊C₈	C₂²⋊C₈	1	2	yes	—
166	C₁₉⋊C₈	C₂²⋊C₈	1	2	yes	—
167	C₄×#131	—	19	1	no	—
168	#1564	—	19	1	no	—
169	C₂×#335	—	19	1	no	—
170	#1477	—	19	1	no	—
171	#1477	—	19	1	no	—
172	#1283	—	19	1	no	—
173	#1283	D₁₉	1	1	yes	characteristic, complemented
174	C₂²×C₁₉	—	2	1	no	—
175	C₂²×C₁₉	—	2	1	no	—
176	C₂²×C₁₉	#380	1	1	yes	characteristic
177	C₂²×C₁₉	#431	1	1	yes	characteristic
178	C₂²×C₁₉	#432	1	1	yes	characteristic
179	C₂×D₁₉	—	2	2	no	—
180	C₂×D₁₉	—	4	1	no	—
181	C₂×D₁₉	—	4	1	no	—
182	C₄×C₁₉	—	2	1	no	—
183	C₄×C₁₉	—	2	1	no	—
184	C₄×C₁₉	—	2	1	no	—
185	C₄×C₁₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
186	C₄×C₁₉	#407	1	1	yes	characteristic
187	C₄×C₁₉	C₂×C₄⋊C₈	1	1	yes	characteristic
188	C₄×C₁₉	#423	1	1	yes	characteristic
189	C₄×C₁₉	#472	1	1	yes	characteristic
190	C₄×C₁₉	#473	1	1	yes	characteristic
191	Dic₁₉	—	2	2	no	—
192	Dic₁₉	—	4	1	no	—
193	Dic₁₉	—	4	1	no	—
194	C₂×#131	—	19	1	no	—
195	C₂×#131	—	19	1	no	—
196	C₄×#41	—	19	1	no	—
197	#431	—	19	1	no	—
198	#431	C₂×D₁₉	1	1	yes	characteristic
199	C₂×C₄⋊C₈	—	19	1	no	—
200	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
201	#405	—	19	1	no	—
202	#405	—	19	1	no	—
203	#405	—	38	1	no	—
204	#405	—	38	1	no	—
205	C₄×M₁₆	—	38	1	no	—
206	C₄×M₁₆	C₂×D₁₉	1	1	yes	characteristic
207	C₂×C₄×C₈	—	19	1	no	—
208	C₂×C₄×C₈	—	38	1	no	—
209	#335	—	38	1	no	—
210	#335	—	38	1	no	—
211	#335	—	38	1	no	—
212	#334	—	38	1	no	—
213	Q₈⋊C₈	—	38	1	no	—
214	Q₈⋊C₈	—	38	1	no	—
215	D₄⋊C₈	—	38	1	no	—
216	D₄⋊C₈	—	38	1	no	—
217	C₂×C₁₉	—	2	1	no	—
218	C₂×C₁₉	—	2	1	no	—
219	C₂×C₁₉	#1797	1	1	yes	characteristic
220	C₂×C₁₉	#1814	1	1	yes	characteristic
221	C₂×C₁₉	#1816	1	1	yes	characteristic
222	D₁₉	—	4	2	no	—
223	C₂×#41	—	19	1	no	—
224	#131	—	19	4	no	—
225	#131	—	38	2	no	—
226	#131	—	38	2	no	—
227	#131	—	38	2	no	—
228	C₂×M₁₆	—	38	1	no	—
229	C₂×M₁₆	—	38	1	no	—
230	C₂×M₁₆	C₄×D₁₉	1	1	yes	characteristic
231	C₂×M₁₆	C₄×D₁₉	1	1	yes	characteristic
232	C₂²×C₈	—	2	1	no	—
233	C₂²×C₈	—	19	1	no	—
234	C₂²×C₈	—	19	1	no	—
235	C₂²×C₈	—	38	1	no	—
236	C₂²×C₈	—	38	1	no	—
237	C₂²×C₈	—	38	1	no	—
238	C₄×Q₈	—	19	1	no	—
239	C₄×Q₈	—	19	1	no	—
240	C₄×D₄	—	19	1	no	—
241	C₄×D₄	—	19	1	no	—
242	C₄×D₄	—	38	1	no	—
243	C₄×D₄	—	38	1	no	—
244	#117	—	19	1	no	—
245	#117	—	38	1	no	—
246	C₂×C₄²	—	38	1	no	—
247	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
248	C₄⋊C₈	—	2	1	no	—
249	C₄⋊C₈	—	2	1	no	—
250	C₄⋊C₈	—	2	1	no	—
251	C₄⋊C₈	—	38	1	no	—
252	C₄⋊C₈	—	38	1	no	—
253	C₄⋊C₈	—	38	1	no	—
254	C₂²⋊C₈	—	2	1	no	—
255	C₂²⋊C₈	—	38	1	no	—
256	#97	—	2	1	no	—
257	#97	—	38	1	no	—
258	#97	—	38	1	no	—
259	C₄×C₈	—	2	1	no	—
260	C₄×C₈	—	2	1	no	—
261	C₄×C₈	—	19	1	no	—
262	C₄×C₈	—	19	1	no	—
263	C₄×C₈	—	19	1	no	—
264	C₄×C₈	—	19	1	no	—
265	C₄×C₈	—	38	1	no	—
266	C₄×C₈	—	38	1	no	—
267	C₄×C₈	—	38	1	no	—
268	C₁₉	#8605	1	1	yes	characteristic, complemented, Sylow
269	#41	—	19	2	no	—
270	#41	—	19	2	no	—
271	#41	—	38	1	no	—
272	#41	—	38	1	no	—
273	C₂×Q₈	—	19	1	no	—
274	C₂×D₄	—	19	1	no	—
275	C₂×D₄	—	38	1	no	—
276	C₂²×C₄	—	38	1	no	—
277	C₂²×C₄	—	38	1	no	—
278	C₂²×C₄	—	38	1	no	—
279	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
280	C₂²×C₄	#64541	1	1	yes	characteristic
281	C₂²×C₄	#64541	1	1	yes	characteristic
282	M₁₆	—	2	1	no	—
283	M₁₆	—	2	1	no	—
284	M₁₆	—	38	4	no	—
285	M₁₆	—	38	4	no	—
286	M₁₆	C₈×D₁₉	1	2	yes	—
287	M₁₆	#64515	1	2	yes	—
288	C₂×C₈	—	2	1	no	—
289	C₂×C₈	—	2	1	no	—
290	C₂×C₈	—	4	1	no	—
291	C₂×C₈	—	4	1	no	—
292	C₂×C₈	—	8	1	no	—
293	C₂×C₈	—	19	1	no	—
294	C₂×C₈	—	19	1	no	—
295	C₂×C₈	—	19	2	no	—
296	C₂×C₈	—	19	2	no	—
297	C₂×C₈	—	19	1	no	—
298	C₂×C₈	—	19	1	no	—
299	C₂×C₈	—	38	4	no	—
300	C₂×C₈	—	38	4	no	—
301	C₂×C₈	—	38	1	no	—
302	C₂×C₈	—	38	1	no	—
303	C₂×C₈	—	76	1	no	—
304	C₂×C₈	—	76	1	no	—
305	C₂×C₈	—	152	1	no	—
306	C₄⋊C₄	—	19	1	no	—
307	C₄⋊C₄	—	19	1	no	—
308	C₄⋊C₄	—	38	1	no	—
309	C₄⋊C₄	—	38	1	no	—
310	C₂²⋊C₄	—	38	1	no	—
311	C₂²⋊C₄	—	38	1	no	—
312	C₂²⋊C₄	—	38	1	no	—
313	C₄²	—	38	1	no	—
314	C₄²	—	38	1	no	—
315	C₄²	—	38	1	no	—
316	C₄²	D₄×D₁₉	1	1	yes	characteristic
317	C₄²	D₄×D₁₉	1	1	yes	characteristic
318	C₄²	D₄×D₁₉	1	1	yes	characteristic
319	C₄²	Q₈×D₁₉	1	1	yes	characteristic
320	C₂³	—	38	1	no	—
321	C₂³	#10597318	1	1	yes	characteristic
322	Q₈	—	19	2	no	—
323	Q₈	—	38	1	no	—
324	D₄	—	19	2	no	—
325	D₄	—	38	2	no	—
326	D₄	—	38	2	no	—
327	D₄	—	38	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	38	1	no	—
332	C₂×C₄	—	38	2	no	—
333	C₂×C₄	—	38	2	no	—
334	C₂×C₄	—	38	1	no	—
335	C₂×C₄	—	38	1	no	—
336	C₂×C₄	—	76	2	no	—
337	C₂×C₄	—	76	1	no	—
338	C₂×C₄	—	76	1	no	—
339	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
340	C₂×C₄	#10597330	1	1	yes	characteristic
341	C₂×C₄	#10597330	1	1	yes	characteristic
342	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
343	C₂×C₄	#10597342	1	1	yes	characteristic
344	C₂×C₄	#10597342	1	1	yes	characteristic
345	C₂×C₄	C₂×C₈×D₁₉	1	1	yes	characteristic
346	C₂×C₄	C₂×#64515	1	1	yes	characteristic
347	C₂×C₄	M₁₆×D₁₉	1	1	yes	characteristic
348	C₂×C₄	M₁₆×D₁₉	1	1	yes	characteristic
349	C₈	—	2	2	no	—
350	C₈	—	2	2	no	—
351	C₈	—	8	1	no	—
352	C₈	—	8	1	no	—
353	C₈	—	19	2	no	—
354	C₈	—	19	2	no	—
355	C₈	—	38	1	no	—
356	C₈	—	38	1	no	—
357	C₈	—	152	1	no	—
358	C₈	—	152	1	no	—
359	C₂²	—	2	1	no	—
360	C₂²	—	2	1	no	—
361	C₂²	—	38	2	no	—
362	C₂²	—	76	1	no	—
363	C₂²	—	76	1	no	—
364	C₂²	#11926403	1	1	yes	characteristic
365	C₂²	#11926595	1	1	yes	characteristic
366	C₂²	#11926599	1	1	yes	characteristic
367	C₄	—	2	1	no	—
368	C₄	—	2	1	no	—
369	C₄	—	2	1	no	—
370	C₄	—	38	2	no	—
371	C₄	—	76	1	no	—
372	C₄	—	76	1	no	—
373	C₄	C₂²⋊C₈×D₁₉	1	1	yes	characteristic
374	C₄	#11926506	1	1	yes	characteristic
375	C₄	C₄⋊C₈×D₁₉	1	1	yes	characteristic
376	C₄	#11926566	1	1	yes	characteristic
377	C₄	#11926813	1	1	yes	characteristic
378	C₄	#11926818	1	1	yes	characteristic
379	C₂	—	2	1	no	—
380	C₂	—	2	1	no	—
381	C₂	—	76	2	no	—
382	C₂	#14563614	1	1	yes	characteristic
383	C₂	#14563688	1	1	yes	characteristic
384	C₂	#14563698	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10597330
commutator	C₂×C₄×C₁₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1283	C₂
socle	C₂²×C₁₉	#380
prime radical	C₂×#64541	C₂×C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1283	D₁₉
hypocenter	C₁₉	#8605
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1283

Complements

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

N	H ≅ G/N	type
2. #14562381	C₂	semidirect
3. C₂×#11926296	C₂	semidirect
6. #14561486	C₂	semidirect
7. C₁₉×#1283	C₂	semidirect
45. #10597353	C₈	semidirect
173. #1283	D₁₉	semidirect
268. C₁₉	#8605	semidirect

Maximal subgroups

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

subgroup	index
C₄×#10597353	2
#14562381	2
C₂×#11926296	2
#14562023	2
#14562020	2
#14561486	2
C₁₉×#1283	2
#8605	19