Subgroups — #18080773

Order 4864.

378 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#10597280	C₂	1	1	yes	maximal, characteristic
2	#14562385	C₂	1	1	yes	maximal, characteristic, complemented
3	#14562027	C₂	1	1	yes	maximal, characteristic
4	C₂×#11926271	C₂	1	1	yes	maximal, characteristic
5	M₁₆×C₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#14561471	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1283	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₇₆	C₂²	1	1	yes	characteristic
9	#11926597	C₂²	1	1	yes	characteristic
10	C₂×#10597280	C₄	1	1	yes	characteristic
11	C₂×#10597280	C₄	1	1	yes	characteristic
12	C₄×#64520	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
14	#11926312	C₄	1	1	yes	characteristic
15	#11926312	C₄	1	1	yes	characteristic
16	#11926296	—	2	1	no	—
17	#11926294	—	2	1	no	—
18	#11926273	—	2	1	no	—
19	#11926273	—	2	1	no	—
20	#11926271	—	2	1	no	—
21	#11926271	—	2	1	no	—
22	#11926265	—	2	1	no	—
23	C₈×C₁₉⋊C₈	—	2	1	no	—
24	C₁₉×#431	C₂²	1	1	yes	characteristic
25	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
26	C₄×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
27	C₂²×D₇₆	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×D₁₉	—	2	1	no	—
29	C₂×M₁₆×C₁₉	C₂×C₄	1	1	yes	characteristic
30	C₂×M₁₆×C₁₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₉	—	2	1	no	—
32	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
33	C₂×(C₂×D₁₉)⋊C₄	—	2	1	no	—
34	C₂×C₄⋊Dic₁₉	C₂×C₄	1	1	yes	characteristic
35	C₂×#64520	C₂×C₄	1	1	yes	characteristic
36	C₂×#64520	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₁₉⋊C₈	—	2	1	no	—
38	C₄×D₇₆	—	2	1	no	—
39	C₄×D₇₆	—	2	1	no	—
40	C₄×D₇₆	—	2	1	no	—
41	C₄×D₇₆	—	2	1	no	—
42	C₁₉×C₄⋊C₈	—	2	1	no	—
43	C₁₉×C₄⋊C₈	—	2	1	no	—
44	C₁₉×C₄⋊C₈	—	2	1	no	—
45	C₁₉×C₂²⋊C₈	—	2	1	no	—
46	C₁₉×#97	—	2	1	no	—
47	C₄×C₈×C₁₉	—	2	1	no	—
48	C₄×C₈×C₁₉	—	2	1	no	—
49	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
50	#10597280	—	2	2	no	—
51	#10597280	C₈	1	4	yes	complemented
52	C₄⋊(C₁₉⋊C₈)	D₄	1	1	yes	characteristic
53	C₄⋊(C₁₉⋊C₈)	D₄	1	1	yes	characteristic
54	#10597260	—	2	1	no	—
55	C₄×C₁₉⋊C₈	—	2	1	no	—
56	C₄×C₁₉⋊C₈	—	2	1	no	—
57	C₄×C₁₉⋊C₈	—	2	1	no	—
58	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
59	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
60	C₂³×D₁₉	—	2	1	no	—
61	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
64	C₂²×Dic₁₉	—	2	1	no	—
65	C₂×D₇₆	—	2	1	no	—
66	C₂×D₇₆	—	2	1	no	—
67	C₂×D₇₆	—	2	1	no	—
68	C₂×D₇₆	—	2	1	no	—
69	C₂×D₇₆	—	2	1	no	—
70	C₂×D₇₆	C₂×C₈	1	2	yes	—
71	C₂×C₄×D₁₉	—	4	1	no	—
72	C₂×C₄×D₁₉	—	4	1	no	—
73	C₂×C₄×D₁₉	—	8	1	no	—
74	C₂×C₄×D₁₉	—	8	1	no	—
75	M₁₆×C₁₉	—	2	1	no	—
76	M₁₆×C₁₉	—	2	1	no	—
77	M₁₆×C₁₉	C₂×C₈	1	2	yes	—
78	M₁₆×C₁₉	M₁₆	1	2	yes	—
79	C₂×C₈×C₁₉	—	2	1	no	—
80	C₂×C₈×C₁₉	—	2	1	no	—
81	C₂×C₈×C₁₉	—	4	1	no	—
82	C₂×C₈×C₁₉	—	4	1	no	—
83	C₂×C₈×C₁₉	—	8	1	no	—
84	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₁₉	#41	1	1	yes	characteristic
87	C₄²×C₁₉	#41	1	1	yes	characteristic
88	(C₂×D₁₉)⋊C₄	—	4	1	no	—
89	(C₂×D₁₉)⋊C₄	—	4	1	no	—
90	C₄⋊Dic₁₉	—	2	1	no	—
91	C₄⋊Dic₁₉	—	2	1	no	—
92	#64520	—	2	1	no	—
93	#64520	—	2	1	no	—
94	#64520	—	2	1	no	—
95	#64520	C₂×C₈	1	2	yes	—
96	C₂×C₁₉⋊C₈	—	2	1	no	—
97	C₂×C₁₉⋊C₈	—	2	1	no	—
98	C₂×C₁₉⋊C₈	—	2	1	no	—
99	C₂×C₁₉⋊C₈	—	2	1	no	—
100	C₂×C₁₉⋊C₈	—	2	1	no	—
101	C₂×C₁₉⋊C₈	—	2	1	no	—
102	C₂×C₁₉⋊C₈	—	4	2	no	—
103	#9550	—	19	1	no	maximal, Sylow
104	C₂³×C₁₉	#117	1	1	yes	characteristic
105	C₂²×D₁₉	—	2	2	no	—
106	C₂²×D₁₉	—	4	1	no	—
107	C₂²×D₁₉	—	4	1	no	—
108	C₂²×D₁₉	—	8	1	no	—
109	C₂²×D₁₉	—	8	1	no	—
110	C₂×C₄×C₁₉	—	2	1	no	—
111	C₂×C₄×C₁₉	—	2	1	no	—
112	C₂×C₄×C₁₉	—	4	1	no	—
113	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
116	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
117	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
118	C₂×C₄×C₁₉	#126	1	1	yes	characteristic
119	C₂×C₄×C₁₉	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
121	C₂×C₄×C₁₉	#131	1	1	yes	characteristic
122	C₂×C₄×C₁₉	#131	1	1	yes	characteristic
123	C₂×Dic₁₉	—	4	1	no	—
124	C₂×Dic₁₉	—	8	1	no	—
125	D₇₆	—	2	2	no	—
126	D₇₆	—	2	2	no	—
127	D₇₆	—	4	1	no	—
128	D₇₆	—	4	1	no	—
129	C₄×D₁₉	—	8	2	no	—
130	C₄×D₁₉	—	8	1	no	—
131	C₄×D₁₉	—	8	1	no	—
132	C₈×C₁₉	—	2	2	no	—
133	C₈×C₁₉	—	2	2	no	—
134	C₈×C₁₉	—	8	1	no	—
135	C₈×C₁₉	—	8	1	no	—
136	C₁₉⋊C₈	—	2	2	no	—
137	C₁₉⋊C₈	—	2	2	no	—
138	C₁₉⋊C₈	—	4	1	no	—
139	C₁₉⋊C₈	—	4	1	no	—
140	C₁₉⋊C₈	—	4	1	no	—
141	C₁₉⋊C₈	—	4	1	no	—
142	C₄×#100	—	19	1	no	—
143	#1565	—	19	1	no	—
144	#1478	—	19	1	no	—
145	C₂×D₄⋊C₈	—	19	1	no	—
146	C₈×M₁₆	—	19	1	no	—
147	#1283	D₁₉	1	1	yes	characteristic, complemented
148	#1278	—	19	1	no	—
149	C₂²×C₁₉	—	2	1	no	—
150	C₂²×C₁₉	—	2	1	no	—
151	C₂²×C₁₉	#388	1	1	yes	characteristic
152	C₂²×C₁₉	#431	1	1	yes	characteristic
153	C₂²×C₁₉	#433	1	1	yes	characteristic
154	C₂×D₁₉	—	4	2	no	—
155	C₂×D₁₉	—	8	2	no	—
156	C₂×D₁₉	—	8	1	no	—
157	C₂×D₁₉	—	8	1	no	—
158	C₂×D₁₉	—	8	2	no	—
159	C₄×C₁₉	—	2	1	no	—
160	C₄×C₁₉	—	2	1	no	—
161	C₄×C₁₉	—	2	1	no	—
162	C₄×C₁₉	C₈×D₄	1	1	yes	characteristic
163	C₄×C₁₉	C₈×D₄	1	1	yes	characteristic
164	C₄×C₁₉	#435	1	1	yes	characteristic
165	C₄×C₁₉	#436	1	1	yes	characteristic
166	C₄×C₁₉	#471	1	1	yes	characteristic
167	C₄×C₁₉	#472	1	1	yes	characteristic
168	Dic₁₉	—	8	1	no	—
169	C₂×C₄×D₄	—	19	1	no	—
170	#431	—	19	1	no	—
171	#431	C₂×D₁₉	1	1	yes	characteristic
172	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
173	C₂×#100	—	19	1	no	—
174	C₂×#100	—	19	1	no	—
175	C₄×M₁₆	—	19	1	no	—
176	C₄×M₁₆	C₂×D₁₉	1	1	yes	characteristic
177	C₂×C₄×C₈	—	19	1	no	—
178	#343	—	19	1	no	—
179	#343	—	19	1	no	—
180	#335	—	38	1	no	—
181	#334	—	38	1	no	—
182	D₄⋊C₈	—	38	1	no	—
183	D₄⋊C₈	—	38	1	no	—
184	D₄⋊C₈	—	38	1	no	—
185	D₄⋊C₈	—	38	1	no	—
186	#322	—	38	1	no	—
187	C₈²	—	38	1	no	—
188	C₂×C₁₉	—	2	1	no	—
189	C₂×C₁₉	—	2	1	no	—
190	C₂×C₁₉	#1923	1	1	yes	characteristic
191	C₂×C₁₉	#1931	1	1	yes	characteristic
192	C₂×C₁₉	#1932	1	1	yes	characteristic
193	D₁₉	—	8	2	no	—
194	C₂²×D₄	—	19	1	no	—
195	C₂³×C₄	—	38	1	no	—
196	C₂×M₁₆	—	19	1	no	—
197	C₂×M₁₆	—	19	1	no	—
198	C₂×M₁₆	C₄×D₁₉	1	1	yes	characteristic
199	C₂×M₁₆	C₄×D₁₉	1	1	yes	characteristic
200	C₂²×C₈	—	2	1	no	—
201	C₂²×C₈	—	38	1	no	—
202	C₄×D₄	—	38	1	no	—
203	C₄×D₄	—	38	1	no	—
204	C₄×D₄	—	38	1	no	—
205	C₄×D₄	—	38	1	no	—
206	C₂×C₄⋊C₄	—	19	1	no	—
207	C₂×C₂²⋊C₄	—	38	1	no	—
208	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
209	C₄⋊C₈	—	2	1	no	—
210	C₄⋊C₈	—	2	1	no	—
211	C₄⋊C₈	—	2	1	no	—
212	C₄⋊C₈	—	19	1	no	—
213	C₄⋊C₈	—	19	1	no	—
214	#100	—	19	4	no	—
215	#100	—	38	2	no	—
216	C₂²⋊C₈	—	2	1	no	—
217	C₂²⋊C₈	—	38	1	no	—
218	#97	—	2	1	no	—
219	#97	—	38	1	no	—
220	C₄×C₈	—	2	1	no	—
221	C₄×C₈	—	2	1	no	—
222	C₄×C₈	—	19	1	no	—
223	C₄×C₈	—	19	1	no	—
224	C₄×C₈	—	38	1	no	—
225	C₄×C₈	—	38	1	no	—
226	C₄×C₈	—	38	1	no	—
227	C₁₉	#9550	1	1	yes	characteristic, complemented, Sylow
228	C₂⁴	—	38	1	no	—
229	C₂×D₄	—	19	2	no	—
230	C₂×D₄	—	38	1	no	—
231	C₂×D₄	—	38	1	no	—
232	C₂×D₄	—	38	1	no	—
233	C₂×D₄	—	38	1	no	—
234	C₂×D₄	—	38	1	no	—
235	C₂²×C₄	—	38	1	no	—
236	C₂²×C₄	—	76	1	no	—
237	C₂²×C₄	—	76	1	no	—
238	C₂²×C₄	—	152	1	no	—
239	C₂²×C₄	—	152	1	no	—
240	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
241	C₂²×C₄	#64541	1	1	yes	characteristic
242	C₂²×C₄	#64541	1	1	yes	characteristic
243	M₁₆	—	2	1	no	—
244	M₁₆	—	2	1	no	—
245	M₁₆	—	19	2	no	—
246	M₁₆	—	38	1	no	—
247	M₁₆	—	38	1	no	—
248	M₁₆	—	38	1	no	—
249	M₁₆	C₈×D₁₉	1	2	yes	—
250	M₁₆	#64515	1	2	yes	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	4	1	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	—	8	1	no	—
256	C₂×C₈	—	38	1	no	—
257	C₂×C₈	—	38	1	no	—
258	C₂×C₈	—	38	1	no	—
259	C₂×C₈	—	38	1	no	—
260	C₂×C₈	—	38	1	no	—
261	C₂×C₈	—	38	1	no	—
262	C₂×C₈	—	76	2	no	—
263	C₄⋊C₄	—	38	1	no	—
264	C₄⋊C₄	—	38	1	no	—
265	C₂²⋊C₄	—	76	1	no	—
266	C₂²⋊C₄	—	76	1	no	—
267	C₄²	D₄×D₁₉	1	1	yes	characteristic
268	C₄²	D₄×D₁₉	1	1	yes	characteristic
269	C₄²	#64543	1	1	yes	characteristic
270	C₄²	#64545	1	1	yes	characteristic
271	C₂³	—	38	2	no	—
272	C₂³	—	76	1	no	—
273	C₂³	—	76	1	no	—
274	C₂³	—	152	1	no	—
275	C₂³	—	152	1	no	—
276	C₂³	#10597318	1	1	yes	characteristic
277	D₄	—	38	2	no	—
278	D₄	—	38	2	no	—
279	D₄	—	76	1	no	—
280	D₄	—	76	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	4	1	no	—
284	C₂×C₄	—	76	1	no	—
285	C₂×C₄	—	152	2	no	—
286	C₂×C₄	—	152	1	no	—
287	C₂×C₄	—	152	1	no	—
288	C₂×C₄	—	152	1	no	—
289	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
290	C₂×C₄	#10597330	1	1	yes	characteristic
291	C₂×C₄	#10597331	1	1	yes	characteristic
292	C₂×C₄	#10597339	1	1	yes	characteristic
293	C₂×C₄	#10597340	1	1	yes	characteristic
294	C₂×C₄	#10597344	1	1	yes	characteristic
295	C₂×C₄	C₂×C₈×D₁₉	1	1	yes	characteristic
296	C₂×C₄	C₂×#64515	1	1	yes	characteristic
297	C₂×C₄	#10597353	1	1	yes	characteristic
298	C₂×C₄	#10597353	1	1	yes	characteristic
299	C₈	—	2	2	no	—
300	C₈	—	2	2	no	—
301	C₈	—	8	1	no	—
302	C₈	—	8	1	no	—
303	C₈	—	38	2	no	—
304	C₈	—	38	2	no	—
305	C₈	—	76	1	no	—
306	C₈	—	76	1	no	—
307	C₈	—	76	1	no	—
308	C₈	—	76	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	76	2	no	—
312	C₂²	—	152	2	no	—
313	C₂²	—	152	2	no	—
314	C₂²	—	152	1	no	—
315	C₂²	—	152	1	no	—
316	C₂²	#11926433	1	1	yes	characteristic
317	C₂²	#11926595	1	1	yes	characteristic
318	C₂²	#11926606	1	1	yes	characteristic
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	152	1	no	—
323	C₄	(C₂²⋊D₁₉)⋊C₈	1	1	yes	characteristic
324	C₄	#11926609	1	1	yes	characteristic
325	C₄	#11926612	1	1	yes	characteristic
326	C₄	#11926621	1	1	yes	characteristic
327	C₄	#11926810	1	1	yes	characteristic
328	C₄	#11926813	1	1	yes	characteristic
329	C₂	—	2	1	no	—
330	C₂	—	2	1	no	—
331	C₂	—	152	2	no	—
332	C₂	#14564239	1	1	yes	characteristic
333	C₂	#14564295	1	1	yes	characteristic
334	C₂	#14564302	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	#388
prime radical	C₂²×D₇₆	C₂×C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1283	D₁₉
hypocenter	C₁₉	#9550
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. #14562385	C₂	semidirect
5. M₁₆×C₁₉⋊C₈	C₂	semidirect
6. #14561471	C₂	semidirect
7. C₁₉×#1283	C₂	semidirect
51. #10597280	C₈	semidirect
147. #1283	D₁₉	semidirect
227. C₁₉	#9550	semidirect

Maximal subgroups

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

subgroup	index
C₄×#10597280	2
#14562385	2
#14562027	2
C₂×#11926271	2
M₁₆×C₁₉⋊C₈	2
#14561471	2
C₁₉×#1283	2
#9550	19