Subgroups — #18078449

Order 4864.

566 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14573428	C₂	1	1	yes	maximal, characteristic
2	#14566546	C₂	1	1	yes	maximal, characteristic
3	#14562199	C₂	1	1	yes	maximal, characteristic, complemented
4	#14562025	C₂	1	1	yes	maximal, characteristic
5	#14561538	C₂	1	1	yes	maximal, characteristic, complemented
6	#14561494	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1503	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₇₆	—	2	1	no	—
9	C₂×C₄×D₇₆	C₂²	1	1	yes	characteristic
10	#11926841	—	2	2	no	—
11	#11926831	C₄	1	2	yes	complemented
12	#11926821	C₄	1	2	yes	complemented
13	#11926597	C₂²	1	1	yes	characteristic
14	#11926587	—	2	1	no	—
15	#11926568	C₂²	1	1	yes	characteristic
16	#11926548	—	2	1	no	—
17	#11926442	—	2	1	no	—
18	#11926415	—	2	2	no	—
19	#11926410	C₂²	1	1	yes	characteristic
20	#11926403	—	2	1	no	—
21	#95×D₁₉	—	2	1	no	—
22	#11926281	—	2	2	no	—
23	#11926271	—	2	2	no	—
24	C₁₉×#533	C₂²	1	1	yes	characteristic
25	C₁₉×#431	C₂²	1	1	yes	characteristic
26	C₄×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
27	C₁₉×#328	—	2	1	no	—
28	C₁₉×#328	—	2	1	no	—
29	C₂²×D₇₆	D₄	1	1	yes	characteristic
30	C₂²×C₄×D₁₉	—	2	1	no	—
31	C₂²×C₄×D₁₉	—	2	2	no	—
32	C₁₉×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic, complemented
33	C₁₉×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic, complemented
34	C₁₉×#125	—	2	1	no	—
35	C₁₉×C₂²⋊Q₈	—	2	1	no	—
36	C₁₉×C₂²⋊Q₈	—	2	1	no	—
37	C₄×Q₈×C₁₉	—	2	1	no	—
38	C₄×Q₈×C₁₉	—	2	1	no	—
39	C₁₉×#117	—	2	1	no	—
40	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
41	C₂×C₄²×C₁₉	—	2	1	no	—
42	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
43	C₂×(C₂×D₁₉)⋊C₄	—	2	1	no	—
44	C₂×(C₂×D₁₉)⋊C₄	—	2	2	no	—
45	C₂×C₄⋊Dic₁₉	—	2	1	no	—
46	C₂×C₄⋊Dic₁₉	D₄	1	1	yes	characteristic
47	C₂×Dic₁₉⋊C₄	—	2	2	no	—
48	C₂×#64520	—	2	1	no	—
49	C₄×D₇₆	—	2	1	no	—
50	C₄×D₇₆	—	2	1	no	—
51	C₄×D₇₆	—	4	2	no	—
52	C₄×D₇₆	—	4	2	no	—
53	C₄×D₇₆	C₂×C₄	1	2	yes	—
54	C₄×D₇₆	D₄	1	2	yes	—
55	C₁₉×C₄⋊C₈	—	2	1	no	—
56	C₁₉×C₂²⋊C₈	—	2	1	no	—
57	C₄×C₈×C₁₉	—	2	1	no	—
58	C₁₉×#95	D₄	1	1	yes	characteristic
59	C₁₉×#95	D₄	1	1	yes	characteristic
60	C₁₉⋊#95	—	2	2	no	—
61	C₁₉⋊#95	—	2	1	no	—
62	C₁₉⋊#95	D₄	1	2	yes	—
63	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
64	#10597266	—	2	2	no	—
65	#10597266	—	2	2	no	—
66	#10597265	—	2	2	no	—
67	#10597264	—	2	2	no	—
68	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
69	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
70	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	2	yes	complemented
71	C₄×C₁₉⋊C₈	—	2	1	no	—
72	C₂³×D₁₉	—	2	1	no	—
73	C₂×Q₈×C₁₉	—	2	1	no	—
74	C₂×Q₈×C₁₉	—	2	1	no	—
75	C₂²×C₄×C₁₉	—	2	1	no	—
76	C₂²×C₄×C₁₉	—	2	1	no	—
77	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
80	C₂²×Dic₁₉	—	2	1	no	—
81	C₂²×Dic₁₉	—	2	2	no	—
82	C₂×D₇₆	—	2	1	no	—
83	C₂×D₇₆	—	2	1	no	—
84	C₂×D₇₆	—	2	1	no	—
85	C₂×D₇₆	C₂²⋊C₄	1	2	yes	—
86	C₂×D₇₆	SD₁₆	1	2	yes	—
87	C₂×D₇₆	Q₁₆	1	2	yes	—
88	C₂×C₄×D₁₉	—	2	2	no	—
89	C₂×C₄×D₁₉	—	2	2	no	—
90	C₂×C₄×D₁₉	—	4	4	no	—
91	C₂×C₄×D₁₉	—	4	4	no	—
92	C₂×C₄×D₁₉	—	8	2	no	—
93	C₂×C₄×D₁₉	—	8	2	no	—
94	C₂×C₈×C₁₉	—	4	1	no	—
95	C₄⋊C₄×C₁₉	—	2	1	no	—
96	C₄⋊C₄×C₁₉	—	2	1	no	—
97	C₄⋊C₄×C₁₉	—	2	1	no	—
98	C₄⋊C₄×C₁₉	—	2	1	no	—
99	C₄⋊C₄×C₁₉	—	4	2	no	—
100	C₄⋊C₄×C₁₉	—	4	1	no	—
101	C₄⋊C₄×C₁₉	—	4	1	no	—
102	C₂²⋊C₄×C₁₉	—	4	1	no	—
103	C₄²×C₁₉	—	4	1	no	—
104	C₄²×C₁₉	—	4	1	no	—
105	C₄²×C₁₉	—	4	1	no	—
106	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₁₉	#41	1	1	yes	characteristic
109	C₄²×C₁₉	#41	1	1	yes	characteristic
110	(C₂×D₁₉)⋊C₄	—	2	2	no	—
111	(C₂×D₁₉)⋊C₄	—	2	2	no	—
112	(C₂×D₁₉)⋊C₄	—	4	4	no	—
113	C₄⋊Dic₁₉	—	2	1	no	—
114	C₄⋊Dic₁₉	—	4	1	no	—
115	C₄⋊Dic₁₉	—	4	1	no	—
116	C₄⋊Dic₁₉	C₂²⋊C₄	1	2	yes	—
117	Dic₁₉⋊C₄	—	8	2	no	—
118	#64520	—	4	1	no	—
119	#64520	—	4	1	no	—
120	C₂×C₁₉⋊C₈	—	2	2	no	—
121	C₂×C₁₉⋊C₈	—	4	1	no	—
122	#9175	—	19	1	no	maximal, Sylow
123	C₂³×C₁₉	#120	1	1	yes	characteristic
124	C₂²×D₁₉	—	2	2	no	—
125	C₂²×D₁₉	—	2	2	no	—
126	C₂²×D₁₉	—	2	2	no	—
127	C₂²×D₁₉	—	4	2	no	—
128	C₂²×D₁₉	—	4	2	no	—
129	Q₈×C₁₉	—	8	1	no	—
130	Q₈×C₁₉	—	8	1	no	—
131	C₂×C₄×C₁₉	—	2	2	no	—
132	C₂×C₄×C₁₉	—	4	1	no	—
133	C₂×C₄×C₁₉	—	4	1	no	—
134	C₂×C₄×C₁₉	—	4	1	no	—
135	C₂×C₄×C₁₉	—	4	1	no	—
136	C₂×C₄×C₁₉	—	4	1	no	—
137	C₂×C₄×C₁₉	—	4	1	no	—
138	C₂×C₄×C₁₉	—	4	1	no	—
139	C₂×C₄×C₁₉	—	4	1	no	—
140	C₂×C₄×C₁₉	—	4	1	no	—
141	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
144	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
145	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
146	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
147	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
148	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
149	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
150	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
151	C₂×Dic₁₉	—	2	2	no	—
152	C₂×Dic₁₉	—	4	2	no	—
153	C₂×Dic₁₉	—	4	2	no	—
154	C₂×Dic₁₉	—	4	2	no	—
155	C₂×Dic₁₉	—	4	2	no	—
156	D₇₆	—	2	2	no	—
157	D₇₆	—	2	2	no	—
158	D₇₆	—	2	2	no	—
159	D₇₆	#103	1	4	yes	—
160	C₄×D₁₉	—	8	2	no	—
161	C₄×D₁₉	—	8	2	no	—
162	C₄×D₁₉	—	8	8	no	—
163	C₈×C₁₉	—	8	1	no	—
164	C₁₉⋊C₈	—	4	2	no	—
165	C₁₉⋊C₈	—	8	1	no	—
166	#3077	—	19	1	no	—
167	#2349	—	19	1	no	—
168	#1518	—	19	1	no	—
169	#1503	D₁₉	1	1	yes	characteristic, complemented
170	#1478	—	19	1	no	—
171	#1303	—	19	1	no	—
172	#1286	—	19	1	no	—
173	C₂²×C₁₉	—	2	1	no	—
174	C₂²×C₁₉	—	2	1	no	—
175	C₂²×C₁₉	#386	1	1	yes	characteristic
176	C₂²×C₁₉	#450	1	1	yes	characteristic
177	C₂²×C₁₉	#451	1	1	yes	characteristic
178	C₂×D₁₉	—	2	4	no	—
179	C₂×D₁₉	—	4	4	no	—
180	C₂×D₁₉	—	4	4	no	—
181	C₂×D₁₉	—	4	2	no	—
182	C₂×D₁₉	—	4	2	no	—
183	C₄×C₁₉	—	2	1	no	—
184	C₄×C₁₉	—	2	1	no	—
185	C₄×C₁₉	—	2	1	no	—
186	C₄×C₁₉	—	2	1	no	—
187	C₄×C₁₉	—	4	2	no	—
188	C₄×C₁₉	—	8	1	no	—
189	C₄×C₁₉	—	8	1	no	—
190	C₄×C₁₉	—	8	1	no	—
191	C₄×C₁₉	C₂×#103	1	1	yes	characteristic
192	C₄×C₁₉	#417	1	1	yes	characteristic
193	C₄×C₁₉	#483	1	1	yes	characteristic
194	C₄×C₁₉	#484	1	1	yes	characteristic
195	Dic₁₉	—	4	2	no	—
196	Dic₁₉	—	8	2	no	—
197	#533	C₂×D₁₉	1	1	yes	characteristic
198	C₂×C₄×D₄	—	19	1	no	—
199	C₂×C₄×D₄	—	38	1	no	—
200	#478	—	38	2	no	—
201	#476	—	19	2	no	—
202	#474	—	19	2	no	—
203	#431	—	19	1	no	—
204	#431	C₂×D₁₉	1	1	yes	characteristic
205	#428	—	38	1	no	—
206	#423	—	19	1	no	—
207	#418	—	38	1	no	—
208	#389	—	38	1	no	—
209	#385	—	38	2	no	—
210	#382	—	19	1	no	—
211	#380	—	38	1	no	—
212	C₄×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
213	C₂×#95	—	38	1	no	—
214	#328	—	2	1	no	—
215	#328	—	2	1	no	—
216	#328	—	38	2	no	—
217	D₄⋊C₈	—	38	2	no	—
218	C₂×C₁₉	—	2	1	no	—
219	C₂×C₁₉	—	2	1	no	—
220	C₂×C₁₉	#1872	1	1	yes	characteristic
221	C₂×C₁₉	#1904	1	1	yes	characteristic
222	C₂×C₁₉	#1906	1	1	yes	characteristic
223	D₁₉	—	4	4	no	—
224	C₂²×D₄	—	19	1	no	—
225	C₂³×C₄	—	38	2	no	—
226	C₂³×C₄	—	38	1	no	—
227	C₂×M₁₆	—	38	1	no	—
228	C₄⋊Q₈	C₄×D₁₉	1	1	yes	characteristic, complemented
229	C₄⋊Q₈	C₄×D₁₉	1	1	yes	characteristic, complemented
230	#125	—	2	1	no	—
231	C₂²⋊Q₈	—	2	1	no	—
232	C₂²⋊Q₈	—	2	1	no	—
233	C₄×Q₈	—	2	1	no	—
234	C₄×Q₈	—	2	1	no	—
235	C₄×D₄	—	19	2	no	—
236	C₄×D₄	—	19	2	no	—
237	C₄×D₄	—	38	1	no	—
238	C₄×D₄	—	38	1	no	—
239	C₄×D₄	—	76	2	no	—
240	C₄×D₄	—	76	2	no	—
241	#117	—	2	1	no	—
242	C₂×C₄⋊C₄	—	2	1	no	—
243	C₂×C₄⋊C₄	—	19	1	no	—
244	C₂×C₄⋊C₄	—	38	1	no	—
245	C₂×C₄⋊C₄	—	38	2	no	—
246	C₂×C₂²⋊C₄	—	38	1	no	—
247	C₂×C₂²⋊C₄	—	38	2	no	—
248	C₂×C₄²	—	2	1	no	—
249	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
250	#107	—	38	2	no	—
251	#106	—	38	2	no	—
252	C₄⋊C₈	—	2	1	no	—
253	C₄⋊C₈	—	19	2	no	—
254	C₄⋊C₈	—	38	1	no	—
255	C₄⋊C₈	—	38	1	no	—
256	#102	—	38	2	no	—
257	#102	—	38	2	no	—
258	C₂²⋊C₈	—	2	1	no	—
259	C₂²⋊C₈	—	38	1	no	—
260	C₄×C₈	—	2	1	no	—
261	C₄×C₈	—	38	1	no	—
262	#95	—	19	2	no	—
263	#95	—	38	2	no	—
264	#95	—	38	1	no	—
265	#95	C₂²⋊D₁₉	1	1	yes	characteristic
266	#95	C₂²⋊D₁₉	1	1	yes	characteristic
267	C₁₉	#9175	1	1	yes	characteristic, complemented, Sylow
268	C₂⁴	—	38	1	no	—
269	C₂×Q₈	—	2	1	no	—
270	C₂×Q₈	—	2	1	no	—
271	C₂×D₄	—	19	2	no	—
272	C₂×D₄	—	19	2	no	—
273	C₂×D₄	—	19	2	no	—
274	C₂×D₄	—	38	1	no	—
275	C₂×D₄	—	38	1	no	—
276	C₂×D₄	—	38	1	no	—
277	C₂²×C₄	—	2	1	no	—
278	C₂²×C₄	—	2	1	no	—
279	C₂²×C₄	—	38	1	no	—
280	C₂²×C₄	—	38	2	no	—
281	C₂²×C₄	—	38	2	no	—
282	C₂²×C₄	—	38	2	no	—
283	C₂²×C₄	—	76	4	no	—
284	C₂²×C₄	—	76	4	no	—
285	C₂²×C₄	—	152	2	no	—
286	C₂²×C₄	—	152	2	no	—
287	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
288	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
289	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
290	M₁₆	—	76	1	no	—
291	M₁₆	—	76	1	no	—
292	C₂×C₈	—	4	1	no	—
293	C₂×C₈	—	38	2	no	—
294	C₂×C₈	—	76	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	—	4	2	no	—
300	C₄⋊C₄	—	4	1	no	—
301	C₄⋊C₄	—	4	1	no	—
302	C₄⋊C₄	—	19	2	no	—
303	C₄⋊C₄	—	38	1	no	—
304	C₄⋊C₄	—	76	1	no	—
305	C₄⋊C₄	—	76	1	no	—
306	C₄⋊C₄	—	152	2	no	—
307	C₂²⋊C₄	—	4	1	no	—
308	C₂²⋊C₄	—	38	2	no	—
309	C₂²⋊C₄	—	38	2	no	—
310	C₂²⋊C₄	—	76	4	no	—
311	C₄²	—	4	1	no	—
312	C₄²	—	4	1	no	—
313	C₄²	—	4	1	no	—
314	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
315	C₄²	#64541	1	1	yes	characteristic
316	C₄²	D₄×D₁₉	1	1	yes	characteristic
317	C₄²	#64543	1	1	yes	characteristic
318	C₂³	—	38	2	no	—
319	C₂³	—	38	2	no	—
320	C₂³	—	38	2	no	—
321	C₂³	—	76	2	no	—
322	C₂³	—	76	2	no	—
323	C₂³	#10597383	1	1	yes	characteristic
324	Q₈	—	8	1	no	—
325	Q₈	—	8	1	no	—
326	D₄	—	19	4	no	—
327	D₄	—	38	2	no	—
328	D₄	—	38	2	no	—
329	D₄	—	38	2	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	4	1	no	—
336	C₂×C₄	—	4	1	no	—
337	C₂×C₄	—	4	1	no	—
338	C₂×C₄	—	4	1	no	—
339	C₂×C₄	—	4	1	no	—
340	C₂×C₄	—	38	2	no	—
341	C₂×C₄	—	76	2	no	—
342	C₂×C₄	—	76	2	no	—
343	C₂×C₄	—	76	2	no	—
344	C₂×C₄	—	76	2	no	—
345	C₂×C₄	—	152	2	no	—
346	C₂×C₄	—	152	2	no	—
347	C₂×C₄	—	152	8	no	—
348	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
349	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
350	C₂×C₄	#10597329	1	1	yes	characteristic
351	C₂×C₄	#10597330	1	1	yes	characteristic
352	C₂×C₄	SD₁₆×D₁₉	1	1	yes	characteristic
353	C₂×C₄	#10597360	1	1	yes	characteristic
354	C₂×C₄	Q₁₆×D₁₉	1	1	yes	characteristic
355	C₂×C₄	#10597364	1	1	yes	characteristic
356	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
357	C₂×C₄	#10597384	1	1	yes	characteristic
358	C₈	—	8	1	no	—
359	C₈	—	76	2	no	—
360	C₈	—	152	1	no	—
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	38	4	no	—
364	C₂²	—	76	4	no	—
365	C₂²	—	76	4	no	—
366	C₂²	—	76	2	no	—
367	C₂²	—	76	2	no	—
368	C₂²	#11926420	1	1	yes	characteristic
369	C₂²	#11926693	1	1	yes	characteristic
370	C₂²	#11926698	1	1	yes	characteristic
371	C₄	—	2	1	no	—
372	C₄	—	2	1	no	—
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	4	2	no	—
376	C₄	—	8	1	no	—
377	C₄	—	8	1	no	—
378	C₄	—	8	1	no	—
379	C₄	—	76	2	no	—
380	C₄	—	152	2	no	—
381	C₄	#103×D₁₉	1	1	yes	characteristic
382	C₄	#11926544	1	1	yes	characteristic
383	C₄	#11926873	1	1	yes	characteristic
384	C₄	#11926878	1	1	yes	characteristic
385	C₂	—	2	1	no	—
386	C₂	—	2	1	no	—
387	C₂	—	76	4	no	—
388	C₂	#14563939	1	1	yes	characteristic
389	C₂	#14564125	1	1	yes	characteristic
390	C₂	#14564139	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11926420
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1503	C₂
socle	C₂²×C₁₉	#386
prime radical	C₂²×D₇₆	D₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1503	D₁₉
hypocenter	C₁₉	#9175
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1503

Complements

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

N	H ≅ G/N	type
3. #14562199	C₂	semidirect
5. #14561538	C₂	semidirect
6. #14561494	C₂	semidirect
7. C₁₉×#1503	C₂	semidirect
11. #11926831	C₄	semidirect
12. #11926821	C₄	semidirect
32. C₁₉×C₄⋊Q₈	C₂×C₄	semidirect
33. C₁₉×C₄⋊Q₈	C₂×C₄	semidirect
70. C₄⋊(C₁₉⋊C₈)	C₂×C₄	semidirect
169. #1503	D₁₉	semidirect
228. C₄⋊Q₈	C₄×D₁₉	semidirect
229. C₄⋊Q₈	C₄×D₁₉	semidirect
267. C₁₉	#9175	semidirect

Maximal subgroups

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

subgroup	index
#14573428	2
#14566546	2
#14562199	2
#14562025	2
#14561538	2
#14561494	2
C₁₉×#1503	2
#9175	19