Subgroups — #18073592

Order 4864.

436 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14572610	C₂	1	1	yes	maximal, characteristic
2	#14562221	C₂	1	1	yes	maximal, characteristic
3	#14562119	C₂	1	1	yes	maximal, characteristic
4	#14561924	C₂	1	1	yes	maximal, characteristic, complemented
5	#14561534	C₂	1	1	yes	maximal, characteristic, complemented
6	#14561528	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1726	C₂	1	1	yes	maximal, characteristic, complemented
8	#11927078	C₂²	1	1	yes	characteristic
9	#11926674	C₄	1	1	yes	characteristic, complemented
10	C₈×Dic₃₈	C₄	1	1	yes	characteristic, complemented
11	#11926623	C₄	1	1	yes	characteristic, complemented
12	#11926615	—	2	1	no	—
13	C₈×D₇₆	C₄	1	1	yes	characteristic, complemented
14	C₈×C₂²⋊D₁₉	—	2	1	no	—
15	#11926603	—	2	1	no	—
16	#11926601	C₂²	1	1	yes	characteristic
17	#11926597	C₂²	1	1	yes	characteristic
18	#11926498	—	2	1	no	—
19	C₄×#64520	C₂²	1	1	yes	characteristic
20	#11926285	C₄	1	2	yes	complemented
21	#11926283	C₄	1	2	yes	complemented
22	#11926280	C₄	1	2	yes	—
23	#11926277	C₄	1	2	yes	—
24	#11926266	—	2	1	no	—
25	#11926265	—	2	1	no	—
26	C₂×C₁₉×#97	—	2	1	no	—
27	C₂×C₁₉×#97	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₁₉	—	2	1	no	—
29	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
30	C₁₉×#376	C₂²	1	1	yes	characteristic
31	C₂²×C₈×C₁₉	—	2	1	no	—
32	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₉	—	2	1	no	—
35	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
36	#10597376	C₂×C₄	1	1	yes	characteristic
37	#10597375	—	2	1	no	—
38	C₄×C₂²⋊D₁₉	—	2	1	no	—
39	#10597370	C₂×C₄	1	1	yes	characteristic
40	C₂×#64520	—	2	1	no	—
41	C₂×#64515	—	2	1	no	—
42	C₂×C₈×D₁₉	—	2	1	no	—
43	#10597322	—	2	1	no	—
44	#10597321	C₂×C₄	1	1	yes	characteristic
45	C₄×D₇₆	C₂×C₄	1	1	yes	characteristic
46	#10597318	—	2	1	no	—
47	#10597316	C₂×C₄	1	1	yes	characteristic
48	C₄×Dic₃₈	C₂×C₄	1	1	yes	characteristic
49	C₁₉×#97	—	2	4	no	—
50	C₁₉×#97	—	2	1	no	—
51	C₁₉×#97	C₂×C₄	1	2	yes	—
52	C₄×C₈×C₁₉	—	2	4	no	—
53	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
55	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
56	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
57	C₁₉×#95	—	2	1	no	—
58	C₁₉×#95	C₂×C₄	1	2	yes	—
59	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
60	C₂²⋊(C₁₉⋊C₈)	C₂×C₄	1	2	yes	—
61	(C₂×D₁₉)⋊C₈	—	2	1	no	—
62	(C₂×D₁₉)⋊C₈	—	2	1	no	—
63	#10597272	—	2	1	no	—
64	Dic₁₉⋊C₈	—	2	1	no	—
65	Dic₁₉⋊C₈	—	2	1	no	—
66	C₈×Dic₁₉	—	2	1	no	—
67	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
68	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	1	yes	characteristic
69	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	1	yes	characteristic
70	#10597260	—	2	1	no	—
71	#10597260	C₂×C₄	1	2	yes	—
72	C₄×C₁₉⋊C₈	—	2	1	no	—
73	C₄×C₁₉⋊C₈	—	2	1	no	—
74	C₂²×C₄×C₁₉	—	2	2	no	—
75	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
78	C₂×C₂²⋊D₁₉	—	2	1	no	—
79	C₂×D₇₆	C₄²	1	1	yes	characteristic
80	C₂×C₄×D₁₉	—	2	1	no	—
81	C₂×Dic₃₈	C₄²	1	1	yes	characteristic
82	C₂×C₈×C₁₉	—	2	2	no	—
83	C₂×C₈×C₁₉	—	2	2	no	—
84	C₂×C₈×C₁₉	—	2	2	no	—
85	C₂×C₈×C₁₉	—	2	2	no	—
86	C₂×C₈×C₁₉	—	2	2	no	—
87	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
91	C₄²×C₁₉	—	2	2	no	—
92	C₄²×C₁₉	—	2	2	no	—
93	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₁₉	#41	1	1	yes	characteristic
97	C₂²⋊Dic₁₉	—	2	1	no	—
98	(C₂×D₁₉)⋊C₄	—	2	1	no	—
99	(C₂×D₁₉)⋊C₄	—	2	1	no	—
100	(C₂×D₁₉)⋊C₄	—	2	1	no	—
101	(C₂×D₁₉)⋊C₄	—	2	1	no	—
102	C₄⋊Dic₁₉	C₄²	1	1	yes	characteristic
103	C₄⋊Dic₁₉	C₄²	1	1	yes	characteristic
104	Dic₁₉⋊C₄	—	2	1	no	—
105	Dic₁₉⋊C₄	—	2	1	no	—
106	Dic₁₉⋊C₄	—	2	1	no	—
107	Dic₁₉⋊C₄	—	2	1	no	—
108	C₄×Dic₁₉	—	2	1	no	—
109	#64520	—	4	2	no	—
110	C₂×C₁₉⋊C₈	—	2	1	no	—
111	C₂×C₁₉⋊C₈	—	2	1	no	—
112	C₂×C₁₉⋊C₈	—	2	2	no	—
113	C₂×C₁₉⋊C₈	—	2	2	no	—
114	#64515	—	4	2	no	—
115	C₈×D₁₉	—	4	2	no	—
116	#8249	—	19	1	no	maximal, Sylow
117	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
118	C₂²×D₁₉	—	2	1	no	—
119	C₂×C₄×C₁₉	—	2	2	no	—
120	C₂×C₄×C₁₉	—	2	1	no	—
121	C₂×C₄×C₁₉	—	2	1	no	—
122	C₂×C₄×C₁₉	—	2	4	no	—
123	C₂×C₄×C₁₉	—	4	1	no	—
124	C₂×C₄×C₁₉	—	4	2	no	—
125	C₂×C₄×C₁₉	—	4	2	no	—
126	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
127	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
129	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
132	C₂²⋊D₁₉	—	4	2	no	—
133	C₂×Dic₁₉	—	2	1	no	—
134	C₂×Dic₁₉	—	2	1	no	—
135	C₂×Dic₁₉	—	2	1	no	—
136	D₇₆	—	4	1	no	—
137	C₄×D₁₉	—	4	1	no	—
138	C₄×D₁₉	—	4	1	no	—
139	Dic₃₈	—	4	1	no	—
140	C₈×C₁₉	—	2	4	no	—
141	C₈×C₁₉	—	2	1	no	—
142	C₈×C₁₉	—	2	1	no	—
143	C₈×C₁₉	#104	1	2	yes	—
144	C₈×C₁₉	#104	1	2	yes	—
145	C₁₉⋊C₈	—	4	2	no	—
146	C₁₉⋊C₈	—	4	2	no	—
147	C₁₉⋊C₈	—	4	1	no	—
148	C₁₉⋊C₈	—	4	1	no	—
149	#2966	—	19	1	no	—
150	#1726	D₁₉	1	1	yes	characteristic, complemented
151	#1524	—	19	1	no	—
152	#1500	—	19	1	no	—
153	#1451	—	19	1	no	—
154	#1302	—	19	1	no	—
155	#1300	—	19	1	no	—
156	C₂²×C₁₉	—	2	1	no	—
157	C₂²×C₁₉	—	2	1	no	—
158	C₂²×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
159	C₂²×C₁₉	#444	1	2	yes	—
160	C₂×D₁₉	—	4	1	no	—
161	C₂×D₁₉	—	4	1	no	—
162	C₂×D₁₉	—	4	1	no	—
163	C₄×C₁₉	—	2	1	no	—
164	C₄×C₁₉	—	2	1	no	—
165	C₄×C₁₉	—	2	1	no	—
166	C₄×C₁₉	—	2	1	no	—
167	C₄×C₁₉	—	2	1	no	—
168	C₄×C₁₉	—	4	4	no	—
169	C₄×C₁₉	C₂×#104	1	1	yes	characteristic
170	C₄×C₁₉	C₂×#104	1	1	yes	characteristic
171	Dic₁₉	—	4	1	no	—
172	Dic₁₉	—	4	1	no	—
173	Dic₁₉	—	4	1	no	—
174	#529	—	19	1	no	—
175	#446	—	19	1	no	—
176	C₈×Q₈	—	19	1	no	—
177	#436	—	19	1	no	—
178	#435	—	38	1	no	—
179	C₈×D₄	—	19	1	no	—
180	C₈×D₄	—	38	1	no	—
181	#433	—	38	1	no	—
182	#432	—	19	1	no	—
183	#431	—	19	1	no	—
184	#405	—	38	1	no	—
185	C₄×M₁₆	—	19	1	no	—
186	C₂×#97	—	2	1	no	—
187	C₂×#97	C₂×D₁₉	1	1	yes	characteristic
188	C₂×C₄×C₈	—	2	1	no	—
189	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
190	#376	C₂×D₁₉	1	1	yes	characteristic
191	#330	—	19	2	no	—
192	#329	—	19	2	no	—
193	#328	—	19	2	no	—
194	#327	—	19	2	no	—
195	#322	—	38	1	no	—
196	#322	—	38	1	no	—
197	C₂×C₁₉	—	2	2	no	—
198	C₂×C₁₉	#1757	1	1	yes	characteristic
199	C₂×C₁₉	C₄×#104	1	1	yes	characteristic
200	C₂×C₁₉	#1765	1	1	yes	characteristic
201	D₁₉	—	8	1	no	—
202	C₂×M₁₆	—	38	1	no	—
203	C₂×M₁₆	—	38	1	no	—
204	C₂²×C₈	—	2	1	no	—
205	C₂²×C₈	—	38	1	no	—
206	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
207	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
208	#126	—	38	1	no	—
209	#125	—	19	1	no	—
210	#124	—	19	1	no	—
211	#123	—	38	1	no	—
212	C₂²⋊Q₈	—	19	1	no	—
213	#121	—	19	1	no	—
214	C₄×Q₈	—	19	1	no	—
215	C₄×D₄	—	19	1	no	—
216	C₄×D₄	—	38	1	no	—
217	#117	—	38	1	no	—
218	C₂×C₄²	—	2	1	no	—
219	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
220	C₄⋊C₈	—	19	1	no	—
221	C₄⋊C₈	—	19	1	no	—
222	C₄⋊C₈	—	38	1	no	—
223	C₄⋊C₈	—	38	1	no	—
224	C₄⋊C₈	—	38	1	no	—
225	C₂²⋊C₈	—	19	2	no	—
226	C₂²⋊C₈	—	38	1	no	—
227	C₂²⋊C₈	—	38	1	no	—
228	C₂²⋊C₈	—	38	1	no	—
229	#97	—	2	4	no	—
230	#97	—	2	1	no	—
231	#97	—	19	2	no	—
232	#97	—	38	1	no	—
233	#97	—	38	1	no	—
234	#97	C₄×D₁₉	1	2	yes	—
235	C₄×C₈	—	2	4	no	—
236	C₄×C₈	—	38	1	no	—
237	C₄×C₈	—	38	1	no	—
238	C₄×C₈	—	38	1	no	—
239	C₄×C₈	C₄×D₁₉	1	1	yes	characteristic
240	C₄×C₈	C₄×D₁₉	1	1	yes	characteristic
241	C₄×C₈	D₇₆	1	1	yes	characteristic
242	C₄×C₈	D₇₆	1	1	yes	characteristic
243	#95	—	2	1	no	—
244	#95	C₄×D₁₉	1	2	yes	—
245	C₁₉	#8249	1	1	yes	characteristic, complemented, Sylow
246	C₂×Q₈	—	19	1	no	—
247	C₂×D₄	—	19	1	no	—
248	C₂×D₄	—	38	1	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	38	1	no	—
251	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
252	C₂²×C₄	#64541	1	1	yes	characteristic
253	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
254	M₁₆	—	76	2	no	—
255	M₁₆	—	76	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	38	2	no	—
262	C₂×C₈	—	38	2	no	—
263	C₂×C₈	—	38	1	no	—
264	C₂×C₈	—	38	1	no	—
265	C₂×C₈	—	76	2	no	—
266	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
267	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
268	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
269	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
270	C₄⋊C₄	—	19	1	no	—
271	C₄⋊C₄	—	19	1	no	—
272	C₄⋊C₄	—	38	1	no	—
273	C₄⋊C₄	—	38	1	no	—
274	C₄⋊C₄	—	38	1	no	—
275	C₄⋊C₄	—	38	1	no	—
276	C₂²⋊C₄	—	38	1	no	—
277	C₂²⋊C₄	—	38	1	no	—
278	C₂²⋊C₄	—	38	1	no	—
279	C₂²⋊C₄	—	38	1	no	—
280	C₂²⋊C₄	—	38	1	no	—
281	C₄²	—	2	2	no	—
282	C₄²	—	2	2	no	—
283	C₄²	—	38	1	no	—
284	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
285	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
286	C₄²	C₂×D₇₆	1	1	yes	characteristic
287	C₄²	#64541	1	1	yes	characteristic
288	C₂³	—	38	1	no	—
289	C₂³	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
290	Q₈	—	76	1	no	—
291	D₄	—	76	1	no	—
292	D₄	—	76	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	38	1	no	—
301	C₂×C₄	—	38	1	no	—
302	C₂×C₄	—	38	1	no	—
303	C₂×C₄	—	76	1	no	—
304	C₂×C₄	—	76	1	no	—
305	C₂×C₄	C₄²×D₁₉	1	1	yes	characteristic
306	C₂×C₄	#10597318	1	1	yes	characteristic
307	C₂×C₄	C₄×D₇₆	1	1	yes	characteristic
308	C₂×C₄	C₄×D₇₆	1	1	yes	characteristic
309	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
310	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
311	C₈	—	2	4	no	—
312	C₈	—	2	1	no	—
313	C₈	—	2	1	no	—
314	C₈	—	76	2	no	—
315	C₈	—	76	2	no	—
316	C₈	—	76	1	no	—
317	C₈	—	76	1	no	—
318	C₈	#10597262	1	2	yes	—
319	C₈	#10597262	1	2	yes	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	76	1	no	—
323	C₂²	—	76	1	no	—
324	C₂²	—	76	1	no	—
325	C₂²	C₄×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
326	C₂²	#11926664	1	2	yes	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	4	4	no	—
333	C₄	—	76	1	no	—
334	C₄	—	76	1	no	—
335	C₄	—	76	1	no	—
336	C₄	C₂×#10597262	1	1	yes	characteristic
337	C₄	C₂×#10597262	1	1	yes	characteristic
338	C₂	—	2	2	no	—
339	C₂	—	152	1	no	—
340	C₂	#14563436	1	1	yes	characteristic
341	C₂	C₄×#10597262	1	1	yes	characteristic
342	C₂	#14563481	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1726	C₂
socle	C₂²×C₁₉	C₄×C₂²⋊C₄
prime radical	#10597376	C₂×C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1726	D₁₉
hypocenter	C₁₉	#8249
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #1726

Complements

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

N	H ≅ G/N	type
4. #14561924	C₂	semidirect
5. #14561534	C₂	semidirect
6. #14561528	C₂	semidirect
7. C₁₉×#1726	C₂	semidirect
9. #11926674	C₄	semidirect
10. C₈×Dic₃₈	C₄	semidirect
11. #11926623	C₄	semidirect
13. C₈×D₇₆	C₄	semidirect
20. #11926285	C₄	semidirect
21. #11926283	C₄	semidirect
150. #1726	D₁₉	semidirect
245. C₁₉	#8249	semidirect

Maximal subgroups

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

subgroup	index
#14572610	2
#14562221	2
#14562119	2
#14561924	2
#14561534	2
#14561528	2
C₁₉×#1726	2
#8249	19