Subgroups — #18079041

Order 4864.

492 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14574562	C₂	1	1	yes	maximal, characteristic
2	#14572381	C₂	1	1	yes	maximal, characteristic
3	#14564624	C₂	1	1	yes	maximal, characteristic, complemented
4	#14562310	C₂	1	1	yes	maximal, characteristic
5	C₂×#11926292	C₂	1	1	yes	maximal, characteristic, complemented
6	#14561531	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1283	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10597353	—	2	1	no	—
9	#11927094	C₂²	1	1	yes	characteristic
10	C₂×#10597315	C₂²	1	1	yes	characteristic
11	#11926917	—	2	1	no	—
12	#11926895	—	2	1	no	—
13	#11926750	—	2	2	no	—
14	#11926685	C₄	1	4	yes	—
15	#11926623	—	2	1	no	—
16	#11926617	—	2	2	no	—
17	C₈×D₇₆	—	2	1	no	—
18	#11926568	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
20	#11926512	—	2	1	no	—
21	#11926312	—	2	1	no	—
22	#11926312	—	2	1	no	—
23	#11926292	—	2	1	no	—
24	#11926292	—	2	1	no	—
25	#11926289	—	2	1	no	—
26	#11926287	—	2	1	no	—
27	C₁₉×#431	C₂²	1	1	yes	characteristic
28	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
29	C₄×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
30	C₂×#64541	—	2	1	no	—
31	C₂²×Dic₃₈	C₂×C₄	1	2	yes	—
32	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
33	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₉	—	2	1	no	—
35	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
36	#10597376	—	2	1	no	—
37	#10597376	C₂×C₄	1	2	yes	—
38	C₂×C₄⋊Dic₁₉	—	2	1	no	—
39	C₂×C₄⋊Dic₁₉	—	2	1	no	—
40	C₂×#64520	—	2	1	no	—
41	C₂²×C₁₉⋊C₈	—	2	1	no	—
42	#10597353	—	4	2	no	—
43	#10597353	—	4	2	no	—
44	C₂×#64516	—	2	4	no	—
45	C₂×#64515	—	2	2	no	—
46	C₂×C₈×D₁₉	—	2	2	no	—
47	#10597321	—	2	1	no	—
48	#10597320	D₄	1	1	yes	characteristic
49	C₄×D₇₆	—	2	1	no	—
50	C₄×D₇₆	D₄	1	2	yes	—
51	#10597315	—	2	1	no	—
52	#10597315	—	2	1	no	—
53	#10597315	—	2	1	no	—
54	#10597315	—	2	1	no	—
55	#10597315	D₄	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	C₂×C₄	1	2	yes	—
60	C₁₉×#97	—	2	1	no	—
61	C₄×C₈×C₁₉	—	2	1	no	—
62	C₄×C₈×C₁₉	—	2	1	no	—
63	(C₂×D₁₉)⋊C₈	—	2	2	no	—
64	(C₂×D₁₉)⋊C₈	—	2	2	no	—
65	#10597273	—	2	4	no	—
66	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
67	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
68	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
69	C₄⋊(C₁₉⋊C₈)	D₄	1	1	yes	characteristic
70	C₄⋊(C₁₉⋊C₈)	D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
74	C₂×C₂²⋊D₁₉	—	2	2	no	—
75	C₂²×Dic₁₉	—	2	2	no	—
76	#64541	—	4	1	no	—
77	#64541	—	4	1	no	—
78	#64541	—	4	1	no	—
79	#64541	—	4	1	no	—
80	C₂×D₇₆	—	2	1	no	—
81	C₂×D₇₆	—	2	1	no	—
82	C₂×D₇₆	C₂²⋊C₄	1	2	yes	—
83	C₂×C₄×D₁₉	—	2	2	no	—
84	C₂×Dic₃₈	—	2	1	no	—
85	C₂×Dic₃₈	—	2	2	no	—
86	C₂×Dic₃₈	—	2	2	no	—
87	C₂×Dic₃₈	—	8	2	no	—
88	M₁₆×C₁₉	—	2	1	no	—
89	M₁₆×C₁₉	—	2	1	no	—
90	M₁₆×C₁₉	—	2	1	no	—
91	M₁₆×C₁₉	SD₁₆	1	2	yes	complemented
92	C₂×C₈×C₁₉	—	2	1	no	—
93	C₂×C₈×C₁₉	—	2	2	no	—
94	C₂×C₈×C₁₉	—	2	1	no	—
95	C₂×C₈×C₁₉	—	4	1	no	—
96	C₂×C₈×C₁₉	—	8	1	no	—
97	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₁₉	#41	1	1	yes	characteristic
101	(C₂×D₁₉)⋊C₄	—	2	2	no	—
102	(C₂×D₁₉)⋊C₄	—	2	2	no	—
103	C₄⋊Dic₁₉	—	2	1	no	—
104	C₄⋊Dic₁₉	—	4	1	no	—
105	C₄⋊Dic₁₉	—	4	1	no	—
106	C₄⋊Dic₁₉	—	4	1	no	—
107	C₄⋊Dic₁₉	—	4	1	no	—
108	C₄⋊Dic₁₉	C₂²⋊C₄	1	2	yes	—
109	#64520	—	4	1	no	—
110	#64520	—	4	1	no	—
111	C₂×C₁₉⋊C₈	—	2	1	no	—
112	C₂×C₁₉⋊C₈	—	2	1	no	—
113	C₂×C₁₉⋊C₈	—	4	1	no	—
114	C₂×C₁₉⋊C₈	—	4	1	no	—
115	C₂×C₁₉⋊C₈	—	4	1	no	—
116	#64516	—	8	4	no	—
117	#64515	—	4	4	no	—
118	C₈×D₁₉	—	4	4	no	—
119	#9267	—	19	1	no	maximal, Sylow
120	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
121	C₂²×D₁₉	—	2	2	no	—
122	C₂×C₄×C₁₉	—	2	1	no	—
123	C₂×C₄×C₁₉	—	2	1	no	—
124	C₂×C₄×C₁₉	—	4	1	no	—
125	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
128	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
129	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
130	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
131	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
132	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
133	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
134	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
135	C₂²⋊D₁₉	—	4	2	no	—
136	C₂²⋊D₁₉	—	4	2	no	—
137	C₂×Dic₁₉	—	2	2	no	—
138	C₂×Dic₁₉	—	4	2	no	—
139	C₂×Dic₁₉	—	8	2	no	—
140	D₇₆	—	4	2	no	—
141	D₇₆	—	4	1	no	—
142	D₇₆	—	4	1	no	—
143	D₇₆	—	4	1	no	—
144	D₇₆	—	4	1	no	—
145	C₄×D₁₉	—	4	2	no	—
146	C₄×D₁₉	—	4	2	no	—
147	Dic₃₈	—	4	1	no	—
148	Dic₃₈	—	4	1	no	—
149	Dic₃₈	—	4	4	no	—
150	Dic₃₈	—	8	2	no	—
151	C₈×C₁₉	—	2	2	no	—
152	C₈×C₁₉	—	4	2	no	—
153	C₈×C₁₉	—	4	1	no	—
154	C₈×C₁₉	—	8	1	no	—
155	C₁₉⋊C₈	—	4	1	no	—
156	C₁₉⋊C₈	—	4	1	no	—
157	C₁₉⋊C₈	—	8	1	no	—
158	#3206	—	19	1	no	—
159	#2935	—	19	1	no	—
160	#1997	—	19	1	no	—
161	#1547	—	19	1	no	—
162	C₂×#333	—	19	1	no	—
163	#1301	—	19	1	no	—
164	#1283	D₁₉	1	1	yes	characteristic, complemented
165	C₂²×C₁₉	—	2	1	no	—
166	C₂²×C₁₉	—	2	1	no	—
167	C₂²×C₁₉	#386	1	1	yes	characteristic
168	C₂²×C₁₉	C₄×SD₁₆	1	1	yes	characteristic
169	C₂²×C₁₉	#440	1	1	yes	characteristic
170	C₂×D₁₉	—	4	2	no	—
171	C₂×D₁₉	—	4	2	no	—
172	C₂×D₁₉	—	4	2	no	—
173	C₄×C₁₉	—	2	1	no	—
174	C₄×C₁₉	—	2	1	no	—
175	C₄×C₁₉	—	2	1	no	—
176	C₄×C₁₉	#450	1	1	yes	characteristic
177	C₄×C₁₉	#452	1	1	yes	characteristic
178	C₄×C₁₉	#461	1	1	yes	characteristic
179	C₄×C₁₉	#463	1	1	yes	characteristic
180	C₄×C₁₉	#471	1	1	yes	characteristic
181	C₄×C₁₉	#473	1	1	yes	characteristic
182	Dic₁₉	—	4	2	no	—
183	Dic₁₉	—	8	2	no	—
184	C₂×#131	—	38	1	no	—
185	#532	—	19	1	no	—
186	C₂×C₄⋊Q₈	—	19	1	no	—
187	#492	—	38	1	no	—
188	#487	—	38	1	no	—
189	#461	—	38	2	no	—
190	#448	—	19	4	no	—
191	#436	—	38	1	no	—
192	#435	—	38	2	no	—
193	C₈×D₄	—	38	1	no	—
194	#431	C₂×D₁₉	1	1	yes	characteristic
195	#423	—	19	1	no	—
196	C₂×C₄⋊C₈	—	19	1	no	—
197	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
198	#408	—	38	1	no	—
199	C₄×M₁₆	C₂×D₁₉	1	1	yes	characteristic
200	#343	—	38	1	no	—
201	#343	—	38	1	no	—
202	#333	—	38	1	no	—
203	#333	—	38	1	no	—
204	#332	—	38	1	no	—
205	#331	—	38	1	no	—
206	C₂×C₁₉	—	2	1	no	—
207	C₂×C₁₉	—	2	1	no	—
208	C₂×C₁₉	#1877	1	1	yes	characteristic
209	C₂×C₁₉	#1908	1	1	yes	characteristic
210	C₂×C₁₉	#1914	1	1	yes	characteristic
211	D₁₉	—	8	2	no	—
212	C₂×#41	—	38	1	no	—
213	C₂²×Q₈	—	19	2	no	—
214	C₂×SD₁₆	—	38	4	no	—
215	#131	—	76	2	no	—
216	#131	—	76	2	no	—
217	C₂×M₁₆	—	38	2	no	—
218	C₂×M₁₆	—	38	1	no	—
219	C₂×M₁₆	D₇₆	1	1	yes	characteristic
220	C₂×M₁₆	D₇₆	1	1	yes	characteristic
221	C₂²×C₈	—	2	1	no	—
222	C₂²×C₈	—	38	2	no	—
223	C₂²×C₈	—	38	1	no	—
224	C₄⋊Q₈	—	19	1	no	—
225	C₄⋊Q₈	—	38	1	no	—
226	C₄⋊Q₈	—	38	1	no	—
227	C₄⋊Q₈	—	38	1	no	—
228	C₄⋊Q₈	—	38	1	no	—
229	#127	—	19	1	no	—
230	#124	—	38	1	no	—
231	#121	—	19	2	no	—
232	#121	—	38	1	no	—
233	C₄×D₄	—	19	2	no	—
234	C₄×D₄	—	38	1	no	—
235	C₂×C₄⋊C₄	—	38	1	no	—
236	C₂×C₄⋊C₄	—	38	1	no	—
237	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
238	C₄⋊C₈	—	2	1	no	—
239	C₄⋊C₈	—	2	1	no	—
240	C₄⋊C₈	—	2	1	no	—
241	C₄⋊C₈	—	19	1	no	—
242	C₄⋊C₈	—	19	1	no	—
243	C₄⋊C₈	—	38	1	no	—
244	C₄⋊C₈	—	38	1	no	—
245	C₄⋊C₈	—	38	1	no	—
246	#103	—	38	4	no	—
247	C₂²⋊C₈	—	38	2	no	—
248	C₂²⋊C₈	—	38	2	no	—
249	C₂²⋊C₈	C₄×D₁₉	1	2	yes	—
250	#97	—	2	1	no	—
251	C₄×C₈	—	2	1	no	—
252	C₄×C₈	—	2	1	no	—
253	C₁₉	#9267	1	1	yes	characteristic, complemented, Sylow
254	#41	—	76	1	no	—
255	#41	—	76	1	no	—
256	#41	—	76	1	no	—
257	#41	—	76	1	no	—
258	C₂×Q₈	—	38	1	no	—
259	C₂×Q₈	—	38	2	no	—
260	C₂×Q₈	—	38	2	no	—
261	C₂×Q₈	—	152	2	no	—
262	C₂×D₄	—	19	2	no	—
263	C₂×D₄	—	38	2	no	—
264	C₂×D₄	—	38	1	no	—
265	C₂×D₄	—	38	1	no	—
266	C₂²×C₄	—	38	2	no	—
267	C₂²×C₄	—	38	2	no	—
268	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
269	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
270	C₂²×C₄	#64541	1	1	yes	characteristic
271	SD₁₆	—	152	4	no	—
272	M₁₆	—	2	1	no	—
273	M₁₆	—	2	1	no	—
274	M₁₆	—	2	1	no	—
275	M₁₆	—	76	4	no	—
276	M₁₆	—	76	1	no	—
277	M₁₆	—	76	1	no	—
278	M₁₆	#64516	1	2	yes	complemented
279	C₂×C₈	—	2	1	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	2	1	no	—
282	C₂×C₈	—	4	1	no	—
283	C₂×C₈	—	8	1	no	—
284	C₂×C₈	—	38	1	no	—
285	C₂×C₈	—	38	1	no	—
286	C₂×C₈	—	76	4	no	—
287	C₂×C₈	—	76	1	no	—
288	C₂×C₈	—	76	1	no	—
289	C₂×C₈	—	76	1	no	—
290	C₄⋊C₄	—	19	2	no	—
291	C₄⋊C₄	—	38	1	no	—
292	C₄⋊C₄	—	76	1	no	—
293	C₄⋊C₄	—	76	1	no	—
294	C₄⋊C₄	—	76	1	no	—
295	C₄⋊C₄	—	76	1	no	—
296	C₂²⋊C₄	—	38	2	no	—
297	C₂²⋊C₄	—	38	2	no	—
298	C₄²	D₄×D₁₉	1	1	yes	characteristic
299	C₄²	D₄×D₁₉	1	1	yes	characteristic
300	C₄²	D₄×D₁₉	1	1	yes	characteristic
301	C₄²	#64545	1	1	yes	characteristic
302	C₂³	—	38	2	no	—
303	C₂³	C₄×D₇₆	1	1	yes	characteristic
304	Q₈	—	76	4	no	—
305	Q₈	—	76	1	no	—
306	Q₈	—	76	1	no	—
307	Q₈	—	152	2	no	—
308	D₄	—	76	2	no	—
309	D₄	—	76	2	no	—
310	D₄	—	76	2	no	—
311	D₄	—	76	1	no	—
312	D₄	—	76	1	no	—
313	D₄	—	76	1	no	—
314	D₄	—	76	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	38	2	no	—
319	C₂×C₄	—	76	2	no	—
320	C₂×C₄	—	76	2	no	—
321	C₂×C₄	—	76	2	no	—
322	C₂×C₄	—	152	2	no	—
323	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
324	C₂×C₄	#10597328	1	1	yes	characteristic
325	C₂×C₄	#10597330	1	1	yes	characteristic
326	C₂×C₄	#10597339	1	1	yes	characteristic
327	C₂×C₄	#10597340	1	1	yes	characteristic
328	C₂×C₄	#10597341	1	1	yes	characteristic
329	C₂×C₄	C₂×#64516	1	1	yes	characteristic
330	C₂×C₄	#10597350	1	1	yes	characteristic
331	C₂×C₄	#10597354	1	1	yes	characteristic
332	C₂×C₄	#10597355	1	1	yes	characteristic
333	C₈	—	2	2	no	—
334	C₈	—	4	2	no	—
335	C₈	—	4	1	no	—
336	C₈	—	8	1	no	—
337	C₈	—	76	1	no	—
338	C₈	—	76	1	no	—
339	C₈	—	152	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	76	2	no	—
343	C₂²	—	76	2	no	—
344	C₂²	—	76	2	no	—
345	C₂²	#11926422	1	1	yes	characteristic
346	C₂²	C₄×#64516	1	1	yes	characteristic
347	C₂²	#11926646	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	76	2	no	—
352	C₄	—	152	2	no	—
353	C₄	#11926697	1	1	yes	characteristic
354	C₄	#11926708	1	1	yes	characteristic
355	C₄	#11926750	1	1	yes	characteristic
356	C₄	#11926760	1	1	yes	characteristic
357	C₄	#11926806	1	1	yes	characteristic
358	C₄	#11926818	1	1	yes	characteristic
359	C₂	—	2	1	no	—
360	C₂	—	2	1	no	—
361	C₂	—	152	2	no	—
362	C₂	#14563968	1	1	yes	characteristic
363	C₂	#14564151	1	1	yes	characteristic
364	C₂	#14564181	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11926422
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1283	C₂
socle	C₂²×C₁₉	#386
prime radical	#11927094	C₂²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1283	D₁₉
hypocenter	C₁₉	#9267
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1283

Complements

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

N	H ≅ G/N	type
3. #14564624	C₂	semidirect
5. C₂×#11926292	C₂	semidirect
6. #14561531	C₂	semidirect
7. C₁₉×#1283	C₂	semidirect
91. M₁₆×C₁₉	SD₁₆	semidirect
164. #1283	D₁₉	semidirect
253. C₁₉	#9267	semidirect
278. M₁₆	#64516	semidirect

Maximal subgroups

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

subgroup	index
#14574562	2
#14572381	2
#14564624	2
#14562310	2
C₂×#11926292	2
#14561531	2
C₁₉×#1283	2
#9267	19