Subgroups — #18081690

Order 4864.

468 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14564580	C₂	1	1	yes	maximal, characteristic, complemented
2	#14564555	C₂	1	1	yes	maximal, characteristic
3	C₄×#10597274	C₂	1	1	yes	maximal, characteristic, complemented
4	#14562134	C₂	1	1	yes	maximal, characteristic, complemented
5	#14562130	C₂	1	1	yes	maximal, characteristic
6	C₈×#64520	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#3167	C₂	1	1	yes	maximal, characteristic
8	#11926597	C₂²	1	1	yes	characteristic
9	C₂×#10597274	—	2	1	no	—
10	C₂×#10597294	—	2	2	no	—
11	C₄×#64520	C₂²	1	1	yes	characteristic
12	C₄×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
14	#11926307	—	2	1	no	—
15	#11926303	—	2	2	no	—
16	#11926281	—	2	2	no	—
17	#11926278	—	2	2	no	—
18	#11926266	—	2	1	no	—
19	C₈×C₁₉⋊C₈	—	2	1	no	—
20	C₁₉×#533	C₂²	1	1	yes	characteristic
21	C₁₉×#532	C₂²	1	1	yes	characteristic
22	C₁₉×#498	C₄	1	1	yes	characteristic, complemented
23	C₁₉×#492	C₄	1	1	yes	characteristic, complemented
24	C₁₉×#487	C₄	1	1	yes	characteristic, complemented
25	C₁₉×#486	C₄	1	1	yes	characteristic, complemented
26	C₁₉×#465	—	2	1	no	—
27	C₁₉×#465	—	2	1	no	—
28	C₄×SD₁₆×C₁₉	—	2	1	no	—
29	C₄×SD₁₆×C₁₉	—	2	1	no	—
30	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
31	C₂×#41×C₁₉	—	2	1	no	—
32	C₂×SD₁₆×C₁₉	—	2	1	no	—
33	C₂×SD₁₆×C₁₉	—	2	1	no	—
34	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
36	C₁₉×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
37	C₁₉×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
38	C₁₉×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
39	C₁₉×#127	C₂×C₄	1	1	yes	characteristic
40	C₁₉×#125	—	2	1	no	—
41	C₁₉×#124	—	2	1	no	—
42	C₁₉×C₂²⋊Q₈	—	2	1	no	—
43	C₁₉×C₂²⋊Q₈	—	2	1	no	—
44	C₁₉×#121	—	2	1	no	—
45	C₁₉×#121	—	2	1	no	—
46	C₄×Q₈×C₁₉	—	2	1	no	—
47	C₄×Q₈×C₁₉	—	2	1	no	—
48	C₄×D₄×C₁₉	—	2	1	no	—
49	C₄×D₄×C₁₉	—	2	1	no	—
50	C₁₉×#117	—	2	1	no	—
51	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
52	C₂×C₄⋊Dic₁₉	—	2	1	no	—
53	C₂×C₄⋊Dic₁₉	—	2	1	no	—
54	C₂×C₄×Dic₁₉	—	2	1	no	—
55	C₂×C₄×Dic₁₉	—	2	1	no	—
56	C₂×#64520	D₄	1	2	yes	—
57	C₁₉×#106	—	2	1	no	—
58	C₁₉×#106	—	2	1	no	—
59	C₁₉×#103	—	2	1	no	—
60	C₁₉×#103	—	2	1	no	—
61	C₁₉×#102	—	2	1	no	—
62	C₁₉×#102	—	2	1	no	—
63	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
66	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
67	#10597294	—	4	4	no	—
68	C₁₉⋊#95	—	2	1	no	—
69	C₁₉⋊#95	—	2	1	no	—
70	C₂²⋊(C₁₉⋊C₈)	D₄	1	2	yes	—
71	#10597274	—	2	4	no	—
72	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
73	#10597260	—	2	1	no	—
74	C₄×C₁₉⋊C₈	—	2	1	no	—
75	C₄×C₁₉⋊C₈	—	2	1	no	—
76	#41×C₁₉	—	4	2	no	—
77	#41×C₁₉	—	4	2	no	—
78	C₂×Q₈×C₁₉	—	2	1	no	—
79	C₂×Q₈×C₁₉	—	2	1	no	—
80	C₂×Q₈×C₁₉	—	2	1	no	—
81	C₂×D₄×C₁₉	—	2	1	no	—
82	C₂×D₄×C₁₉	—	2	1	no	—
83	C₂×D₄×C₁₉	—	2	1	no	—
84	C₂×D₄×C₁₉	—	4	1	no	—
85	C₂²×C₄×C₁₉	—	4	1	no	—
86	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
89	C₂²×Dic₁₉	—	4	1	no	—
90	C₂²×Dic₁₉	—	4	1	no	—
91	SD₁₆×C₁₉	—	4	2	no	—
92	SD₁₆×C₁₉	—	4	2	no	—
93	C₂×C₈×C₁₉	—	2	2	no	—
94	C₂×C₈×C₁₉	—	4	1	no	—
95	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
99	C₄⋊C₄×C₁₉	—	2	1	no	—
100	C₄⋊C₄×C₁₉	—	2	1	no	—
101	C₄⋊C₄×C₁₉	—	2	1	no	—
102	C₄⋊C₄×C₁₉	—	2	1	no	—
103	C₄⋊C₄×C₁₉	—	2	1	no	—
104	C₄⋊C₄×C₁₉	—	2	1	no	—
105	C₄⋊C₄×C₁₉	—	4	1	no	—
106	C₄⋊C₄×C₁₉	—	4	1	no	—
107	C₂²⋊C₄×C₁₉	—	4	1	no	—
108	C₂²⋊C₄×C₁₉	—	4	1	no	—
109	C₂²⋊C₄×C₁₉	—	4	1	no	—
110	C₄²×C₁₉	—	4	1	no	—
111	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₁₉	#41	1	1	yes	characteristic
114	C₄²×C₁₉	#41	1	1	yes	characteristic
115	C₄⋊Dic₁₉	—	2	2	no	—
116	C₄⋊Dic₁₉	—	2	2	no	—
117	C₄⋊Dic₁₉	—	4	1	no	—
118	C₄⋊Dic₁₉	—	4	1	no	—
119	C₄×Dic₁₉	—	4	2	no	—
120	C₄×Dic₁₉	—	4	2	no	—
121	#64520	—	2	2	no	—
122	#64520	SD₁₆	1	4	yes	complemented
123	C₂×C₁₉⋊C₈	—	2	2	no	—
124	C₂×C₁₉⋊C₈	—	2	2	no	—
125	#9717	—	19	1	no	maximal, Sylow
126	C₂³×C₁₉	—	4	1	no	—
127	C₂³×C₁₉	#127	1	1	yes	characteristic
128	Q₈×C₁₉	—	4	1	no	—
129	Q₈×C₁₉	—	4	1	no	—
130	Q₈×C₁₉	—	4	2	no	—
131	Q₈×C₁₉	—	8	1	no	—
132	D₄×C₁₉	—	4	1	no	—
133	D₄×C₁₉	—	4	1	no	—
134	D₄×C₁₉	—	4	2	no	—
135	D₄×C₁₉	—	8	1	no	—
136	D₄×C₁₉	—	8	1	no	—
137	D₄×C₁₉	—	8	1	no	—
138	C₂×C₄×C₁₉	—	2	1	no	—
139	C₂×C₄×C₁₉	—	2	1	no	—
140	C₂×C₄×C₁₉	—	4	1	no	—
141	C₂×C₄×C₁₉	—	4	1	no	—
142	C₂×C₄×C₁₉	—	4	1	no	—
143	C₂×C₄×C₁₉	—	4	1	no	—
144	C₂×C₄×C₁₉	—	8	2	no	—
145	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
148	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
149	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
150	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
151	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
152	C₂×C₄×C₁₉	#135	1	2	yes	—
153	C₂×Dic₁₉	—	4	2	no	—
154	C₂×Dic₁₉	—	4	2	no	—
155	C₂×Dic₁₉	—	8	1	no	—
156	C₂×Dic₁₉	—	8	1	no	—
157	C₂×Dic₁₉	—	8	1	no	—
158	C₂×Dic₁₉	—	8	1	no	—
159	C₈×C₁₉	—	2	1	no	—
160	C₈×C₁₉	—	2	1	no	—
161	C₈×C₁₉	—	2	1	no	—
162	C₈×C₁₉	#104	1	2	yes	complemented
163	C₁₉⋊C₈	—	2	4	no	—
164	C₁₉⋊C₈	—	4	2	no	—
165	#3167	D₁₉	1	1	yes	characteristic
166	#1986	—	19	1	no	—
167	#1980	—	19	1	no	—
168	C₄×#106	—	19	1	no	—
169	#1503	—	19	1	no	—
170	#1502	—	19	1	no	—
171	C₈×M₁₆	—	19	1	no	—
172	C₂²×C₁₉	—	2	1	no	—
173	C₂²×C₁₉	—	2	1	no	—
174	C₂²×C₁₉	—	8	1	no	—
175	C₂²×C₁₉	—	8	1	no	—
176	C₂²×C₁₉	—	8	1	no	—
177	C₂²×C₁₉	#390	1	1	yes	characteristic
178	C₂²×C₁₉	#492	1	1	yes	characteristic
179	C₂²×C₁₉	#495	1	1	yes	characteristic
180	C₄×C₁₉	—	2	1	no	—
181	C₄×C₁₉	—	2	1	no	—
182	C₄×C₁₉	—	2	1	no	—
183	C₄×C₁₉	—	8	1	no	—
184	C₄×C₁₉	—	8	1	no	—
185	C₄×C₁₉	—	8	1	no	—
186	C₄×C₁₉	C₂×#104	1	1	yes	characteristic
187	C₄×C₁₉	#421	1	1	yes	characteristic
188	C₄×C₁₉	C₄×SD₁₆	1	2	yes	—
189	C₄×C₁₉	#465	1	2	yes	—
190	Dic₁₉	—	8	2	no	—
191	Dic₁₉	—	8	2	no	—
192	#533	C₂×D₁₉	1	1	yes	characteristic
193	#532	C₂×D₁₉	1	1	yes	characteristic
194	#498	Dic₁₉	1	1	yes	characteristic, complemented
195	#492	Dic₁₉	1	1	yes	characteristic, complemented
196	#487	Dic₁₉	1	1	yes	characteristic, complemented
197	#486	Dic₁₉	1	1	yes	characteristic, complemented
198	#465	—	2	1	no	—
199	#465	—	2	1	no	—
200	C₄×SD₁₆	—	2	1	no	—
201	C₄×SD₁₆	—	2	1	no	—
202	#431	—	19	1	no	—
203	C₂×#106	—	38	1	no	—
204	C₂×#104	—	38	2	no	—
205	C₄×M₁₆	—	19	1	no	—
206	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
207	C₄×C₄⋊C₄	—	19	1	no	—
208	C₄×C₄⋊C₄	—	19	1	no	—
209	#340	—	38	1	no	—
210	#339	—	38	2	no	—
211	#328	—	38	2	no	—
212	#327	—	38	2	no	—
213	#322	—	38	1	no	—
214	C₈²	—	38	1	no	—
215	C₂×C₁₉	—	2	1	no	—
216	C₂×C₁₉	—	2	1	no	—
217	C₂×C₁₉	—	16	1	no	—
218	C₂×C₁₉	#1968	1	1	yes	characteristic
219	C₂×C₁₉	#1976	1	1	yes	characteristic
220	C₂×C₁₉	#1980	1	1	yes	characteristic
221	C₂×#41	—	2	1	no	—
222	C₂×SD₁₆	—	2	1	no	—
223	C₂×SD₁₆	—	2	1	no	—
224	C₂×M₁₆	—	19	2	no	—
225	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
226	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
227	C₄⋊Q₈	C₂×Dic₁₉	1	1	yes	characteristic
228	C₄⋊Q₈	C₂×Dic₁₉	1	1	yes	characteristic
229	C₄⋊Q₈	C₂×Dic₁₉	1	1	yes	characteristic
230	#127	C₂×Dic₁₉	1	1	yes	characteristic
231	#125	—	2	1	no	—
232	#124	—	2	1	no	—
233	C₂²⋊Q₈	—	2	1	no	—
234	C₂²⋊Q₈	—	2	1	no	—
235	#121	—	2	1	no	—
236	#121	—	2	1	no	—
237	C₄×Q₈	—	2	1	no	—
238	C₄×Q₈	—	2	1	no	—
239	C₄×D₄	—	2	1	no	—
240	C₄×D₄	—	2	1	no	—
241	#117	—	2	1	no	—
242	C₂×C₄⋊C₄	—	38	1	no	—
243	C₂×C₄⋊C₄	—	38	1	no	—
244	C₂×C₄²	—	38	1	no	—
245	C₂×C₄²	—	38	1	no	—
246	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
247	#106	—	2	1	no	—
248	#106	—	2	1	no	—
249	#106	—	38	4	no	—
250	C₄⋊C₈	—	38	1	no	—
251	#104	—	76	4	no	—
252	#103	—	2	1	no	—
253	#103	—	2	1	no	—
254	#102	—	2	1	no	—
255	#102	—	2	1	no	—
256	C₂²⋊C₈	—	19	2	no	—
257	#97	—	38	1	no	—
258	C₄×C₈	—	38	1	no	—
259	C₄×C₈	—	38	1	no	—
260	C₄×C₈	C₂×Dic₁₉	1	1	yes	characteristic
261	C₄×C₈	C₂×Dic₁₉	1	1	yes	characteristic
262	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
263	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
264	#95	—	38	1	no	—
265	#95	—	38	1	no	—
266	C₁₉	#9717	1	1	yes	characteristic, complemented, Sylow
267	#41	—	4	2	no	—
268	#41	—	4	2	no	—
269	C₂×Q₈	—	2	1	no	—
270	C₂×Q₈	—	2	1	no	—
271	C₂×Q₈	—	2	1	no	—
272	C₂×D₄	—	2	1	no	—
273	C₂×D₄	—	2	1	no	—
274	C₂×D₄	—	2	1	no	—
275	C₂×D₄	—	4	1	no	—
276	C₂²×C₄	—	4	1	no	—
277	C₂²×C₄	—	76	1	no	—
278	C₂²×C₄	—	76	1	no	—
279	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
280	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
281	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
282	SD₁₆	—	4	2	no	—
283	SD₁₆	—	4	2	no	—
284	M₁₆	—	19	4	no	—
285	M₁₆	—	38	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	4	1	no	—
288	C₂×C₈	—	38	2	no	—
289	C₂×C₈	—	38	2	no	—
290	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
291	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
292	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
293	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
294	C₄⋊C₄	—	2	1	no	—
295	C₄⋊C₄	—	2	1	no	—
296	C₄⋊C₄	—	2	1	no	—
297	C₄⋊C₄	—	2	1	no	—
298	C₄⋊C₄	—	2	1	no	—
299	C₄⋊C₄	—	2	1	no	—
300	C₄⋊C₄	—	4	1	no	—
301	C₄⋊C₄	—	4	1	no	—
302	C₄⋊C₄	—	38	2	no	—
303	C₄⋊C₄	—	38	2	no	—
304	C₄⋊C₄	—	76	1	no	—
305	C₄⋊C₄	—	76	1	no	—
306	C₂²⋊C₄	—	4	1	no	—
307	C₂²⋊C₄	—	4	1	no	—
308	C₂²⋊C₄	—	4	1	no	—
309	C₄²	—	4	1	no	—
310	C₄²	—	76	2	no	—
311	C₄²	—	76	2	no	—
312	C₄²	#64543	1	1	yes	characteristic
313	C₄²	#64543	1	1	yes	characteristic
314	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
315	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
316	C₂³	—	4	1	no	—
317	C₂³	#10597386	1	1	yes	characteristic
318	Q₈	—	4	1	no	—
319	Q₈	—	4	1	no	—
320	Q₈	—	4	2	no	—
321	Q₈	—	8	1	no	—
322	D₄	—	4	1	no	—
323	D₄	—	4	1	no	—
324	D₄	—	4	2	no	—
325	D₄	—	8	1	no	—
326	D₄	—	8	1	no	—
327	D₄	—	8	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	—	8	2	no	—
335	C₂×C₄	—	76	2	no	—
336	C₂×C₄	—	76	2	no	—
337	C₂×C₄	—	152	1	no	—
338	C₂×C₄	—	152	1	no	—
339	C₂×C₄	—	152	1	no	—
340	C₂×C₄	—	152	1	no	—
341	C₂×C₄	SD₁₆×D₁₉	1	2	yes	—
342	C₂×C₄	#10597362	1	2	yes	—
343	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
344	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
345	C₂×C₄	#10597382	1	1	yes	characteristic
346	C₂×C₄	#10597384	1	1	yes	characteristic
347	C₂×C₄	#10597384	1	1	yes	characteristic
348	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
349	C₈	—	2	1	no	—
350	C₈	—	2	1	no	—
351	C₈	—	2	1	no	—
352	C₈	—	38	4	no	—
353	C₈	—	76	2	no	—
354	C₈	#10597294	1	2	yes	complemented
355	C₂²	—	2	1	no	—
356	C₂²	—	2	1	no	—
357	C₂²	—	8	1	no	—
358	C₂²	—	8	1	no	—
359	C₂²	—	8	1	no	—
360	C₂²	#11926447	1	1	yes	characteristic
361	C₂²	#11926915	1	1	yes	characteristic
362	C₂²	#11926925	1	1	yes	characteristic
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	8	1	no	—
367	C₄	—	8	1	no	—
368	C₄	—	8	1	no	—
369	C₄	—	152	2	no	—
370	C₄	—	152	2	no	—
371	C₄	C₂×#10597294	1	1	yes	characteristic
372	C₄	#11926561	1	1	yes	characteristic
373	C₄	SD₁₆×Dic₁₉	1	2	yes	—
374	C₄	#11926770	1	2	yes	—
375	C₂	—	2	1	no	—
376	C₂	—	2	1	no	—
377	C₂	—	16	1	no	—
378	C₂	#14564491	1	1	yes	characteristic
379	C₂	#14564534	1	1	yes	characteristic
380	C₂	#14564555	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10597384
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#3167	C₂
socle	C₂²×C₁₉	#390
prime radical	C₁₉×#532	C₂²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#3167	D₁₉
hypocenter	C₁₉	#9717
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3167

Complements

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

N	H ≅ G/N	type
1. #14564580	C₂	semidirect
3. C₄×#10597274	C₂	semidirect
4. #14562134	C₂	semidirect
6. C₈×#64520	C₂	semidirect
22. C₁₉×#498	C₄	semidirect
23. C₁₉×#492	C₄	semidirect
24. C₁₉×#487	C₄	semidirect
25. C₁₉×#486	C₄	semidirect
122. #64520	SD₁₆	semidirect
162. C₈×C₁₉	#104	semidirect
194. #498	Dic₁₉	semidirect
195. #492	Dic₁₉	semidirect
196. #487	Dic₁₉	semidirect
197. #486	Dic₁₉	semidirect
266. C₁₉	#9717	semidirect
354. C₈	#10597294	semidirect

Maximal subgroups

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

subgroup	index
#14564580	2
#14564555	2
C₄×#10597274	2
#14562134	2
#14562130	2
C₈×#64520	2
C₁₉×#3167	2
#9717	19