Subgroups — #18075435

Order 4864.

550 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×C₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#14563662	C₂	1	2	yes	maximal, complemented
3	#14563623	C₂	1	1	yes	maximal, characteristic
4	#14561502	C₂	1	2	yes	maximal, complemented
5	C₁₉×#1799	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₁₉⋊C₈	C₄	1	1	yes	characteristic, complemented
7	C₂³×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
8	D₄×C₁₉⋊C₈	—	2	2	no	—
9	D₄×C₁₉⋊C₈	—	2	2	no	—
10	C₂×#10597264	—	2	2	no	—
11	C₂×C₄⋊(C₁₉⋊C₈)	C₄	1	1	yes	characteristic, complemented
12	C₂×#10597291	—	2	1	no	—
13	C₂×C₂²⋊(C₁₉⋊C₈)	C₄	1	1	yes	characteristic, complemented
14	C₂×C₂²⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₁₉⋊C₈	C₄	1	1	yes	characteristic, complemented
16	#11926404	C₂²	1	2	yes	—
17	#11926311	—	2	1	no	—
18	#11926306	—	2	2	no	—
19	C₂×C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
20	C₂×C₁₉×#99	—	2	1	no	—
21	C₁₉×#380	C₂²	1	2	yes	—
22	C₁₉×#342	—	2	1	no	—
23	C₂²×D₄×C₁₉	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
26	C₄×D₄×C₁₉	—	2	2	no	—
27	C₄×D₄×C₁₉	—	2	2	no	—
28	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
29	C₂×C₄⋊C₄×C₁₉	Q₈	1	1	yes	characteristic
30	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
31	C₂×C₂²⋊C₄×C₁₉	D₄	1	1	yes	characteristic
32	C₂×C₂²⋊C₄×C₁₉	Q₈	1	1	yes	characteristic
33	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₂²⋊Dic₁₉	—	2	2	no	—
35	C₂×C₄⋊Dic₁₉	—	2	2	no	—
36	C₂²×C₁₉⋊C₈	—	2	1	no	—
37	C₂²×C₁₉⋊C₈	—	2	1	no	—
38	C₂²×C₁₉⋊C₈	—	4	2	no	—
39	C₂²×C₁₉⋊C₈	—	4	2	no	—
40	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
45	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
46	C₁₉×#99	—	4	2	no	—
47	C₁₉×#95	—	2	2	no	—
48	#10597291	—	4	2	no	—
49	C₁₉⋊#95	—	2	2	no	—
50	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
51	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
52	#10597264	—	4	4	no	—
53	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
54	C₄×C₁₉⋊C₈	—	2	1	no	—
55	C₄×C₁₉⋊C₈	—	2	1	no	—
56	C₂⁴×C₁₉	C₄⋊C₄	1	1	yes	characteristic
57	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
58	C₂×D₄×C₁₉	—	2	2	no	—
59	C₂×D₄×C₁₉	—	2	2	no	—
60	C₂×D₄×C₁₉	D₈	1	2	yes	—
61	C₂×D₄×C₁₉	Q₁₆	1	2	yes	—
62	C₂²×C₄×C₁₉	—	2	1	no	—
63	C₂²×C₄×C₁₉	—	2	1	no	—
64	C₂²×C₄×C₁₉	—	4	1	no	—
65	C₂²×C₄×C₁₉	—	4	1	no	—
66	C₂²×C₄×C₁₉	—	4	1	no	—
67	C₂²×C₄×C₁₉	—	4	1	no	—
68	C₂²×C₄×C₁₉	—	4	2	no	—
69	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
75	C₂²×C₄×C₁₉	#41	1	2	yes	—
76	C₂²×Dic₁₉	—	4	2	no	—
77	C₄⋊C₄×C₁₉	—	2	2	no	—
78	C₄⋊C₄×C₁₉	—	4	2	no	—
79	C₄⋊C₄×C₁₉	—	4	2	no	—
80	C₂²⋊C₄×C₁₉	—	2	2	no	—
81	C₂²⋊C₄×C₁₉	—	2	2	no	—
82	C₂²⋊C₄×C₁₉	—	4	2	no	—
83	C₂²⋊C₄×C₁₉	—	4	2	no	—
84	C₄²×C₁₉	—	2	1	no	—
85	C₄²×C₁₉	—	2	1	no	—
86	C₂²⋊Dic₁₉	—	4	2	no	—
87	C₂²⋊Dic₁₉	—	4	2	no	—
88	C₄⋊Dic₁₉	—	4	2	no	—
89	C₄⋊Dic₁₉	—	4	2	no	—
90	C₂×C₁₉⋊C₈	—	2	4	no	—
91	C₂×C₁₉⋊C₈	—	2	2	no	—
92	C₂×C₁₉⋊C₈	—	2	1	no	—
93	C₂×C₁₉⋊C₈	—	2	1	no	—
94	C₂×C₁₉⋊C₈	—	4	1	no	—
95	C₂×C₁₉⋊C₈	—	4	2	no	—
96	C₂×C₁₉⋊C₈	—	4	2	no	—
97	C₂×C₁₉⋊C₈	—	4	2	no	—
98	C₂×C₁₉⋊C₈	—	4	1	no	—
99	C₂×C₁₉⋊C₈	—	4	1	no	—
100	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	2	yes	—
102	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
103	#8609	—	19	1	no	maximal, Sylow
104	C₂³×C₁₉	—	2	1	no	—
105	C₂³×C₁₉	—	2	1	no	—
106	C₂³×C₁₉	—	4	1	no	—
107	C₂³×C₁₉	—	4	1	no	—
108	C₂³×C₁₉	—	4	1	no	—
109	C₂³×C₁₉	—	4	1	no	—
110	C₂³×C₁₉	#107	1	2	yes	complemented
111	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂³×C₁₉	#121	1	2	yes	—
113	C₂³×C₁₉	C₂×D₈	1	1	yes	characteristic
114	C₂³×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
115	C₂³×C₁₉	#135	1	2	yes	—
116	D₄×C₁₉	—	2	4	no	—
117	D₄×C₁₉	—	2	4	no	—
118	C₂×C₄×C₁₉	—	2	1	no	—
119	C₂×C₄×C₁₉	—	2	1	no	—
120	C₂×C₄×C₁₉	—	2	1	no	—
121	C₂×C₄×C₁₉	—	2	2	no	—
122	C₂×C₄×C₁₉	—	2	1	no	—
123	C₂×C₄×C₁₉	—	2	2	no	—
124	C₂×C₄×C₁₉	—	2	1	no	—
125	C₂×C₄×C₁₉	—	2	1	no	—
126	C₂×C₄×C₁₉	—	4	1	no	—
127	C₂×C₄×C₁₉	—	4	2	no	—
128	C₂×C₄×C₁₉	—	4	2	no	—
129	C₂×C₄×C₁₉	—	4	1	no	—
130	C₂×C₄×C₁₉	—	4	1	no	—
131	C₂×C₄×C₁₉	—	4	1	no	—
132	C₂×C₄×C₁₉	—	4	1	no	—
133	C₂×C₄×C₁₉	—	8	2	no	—
134	C₂×C₄×C₁₉	—	8	2	no	—
135	C₂×C₄×C₁₉	—	8	2	no	—
136	C₂×C₄×C₁₉	#107	1	2	yes	complemented
137	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
139	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
140	C₂×Dic₁₉	—	8	2	no	—
141	C₂×Dic₁₉	—	8	2	no	—
142	C₂×Dic₁₉	—	8	2	no	—
143	C₁₉⋊C₈	—	2	1	no	—
144	C₁₉⋊C₈	—	2	2	no	—
145	C₁₉⋊C₈	—	4	1	no	—
146	C₁₉⋊C₈	—	4	1	no	—
147	C₁₉⋊C₈	#99	1	2	yes	complemented
148	C₂×C₈×D₄	—	19	1	no	—
149	#1808	—	19	2	no	—
150	#1799	—	19	1	no	—
151	#1799	D₁₉	1	1	yes	characteristic
152	#1289	—	19	2	no	—
153	C₂²×C₁₉	—	2	2	no	—
154	C₂²×C₁₉	—	2	1	no	—
155	C₂²×C₁₉	—	2	2	no	—
156	C₂²×C₁₉	—	2	1	no	—
157	C₂²×C₁₉	—	2	2	no	—
158	C₂²×C₁₉	—	4	1	no	—
159	C₂²×C₁₉	—	4	1	no	—
160	C₂²×C₁₉	—	4	2	no	—
161	C₂²×C₁₉	—	4	2	no	—
162	C₂²×C₁₉	—	4	2	no	—
163	C₂²×C₁₉	—	4	1	no	—
164	C₂²×C₁₉	—	4	1	no	—
165	C₂²×C₁₉	#380	1	1	yes	characteristic
166	C₂²×C₁₉	C₂×#107	1	1	yes	characteristic
167	C₂²×C₁₉	#427	1	1	yes	characteristic
168	C₂²×C₁₉	#466	1	2	yes	—
169	C₂²×C₁₉	#467	1	2	yes	—
170	C₄×C₁₉	—	2	1	no	—
171	C₄×C₁₉	—	2	2	no	—
172	C₄×C₁₉	—	2	2	no	—
173	C₄×C₁₉	—	4	1	no	—
174	C₄×C₁₉	—	4	1	no	—
175	C₄×C₁₉	—	16	2	no	—
176	C₄×C₁₉	C₂×#99	1	1	yes	characteristic
177	C₄×C₁₉	#410	1	1	yes	characteristic
178	Dic₁₉	—	16	2	no	—
179	C₂³×C₈	—	19	1	no	—
180	C₂³×C₈	—	19	1	no	—
181	C₂×C₄×D₄	C₂×D₁₉	1	1	yes	characteristic
182	C₈×D₄	—	38	2	no	—
183	C₈×D₄	—	38	2	no	—
184	C₂×#107	—	38	2	no	—
185	C₂×C₄⋊C₈	—	19	1	no	—
186	C₂×#99	—	2	1	no	—
187	C₂×#99	—	38	1	no	—
188	C₂×C₂²⋊C₈	—	19	1	no	—
189	C₂×C₂²⋊C₈	—	19	1	no	—
190	C₂×C₄×C₈	—	19	1	no	—
191	#380	—	19	2	no	—
192	#380	C₂×D₁₉	1	2	yes	—
193	#342	—	2	1	no	—
194	#342	—	38	1	no	—
195	#340	—	38	2	no	—
196	C₂×C₁₉	—	2	2	no	—
197	C₂×C₁₉	—	2	1	no	—
198	C₂×C₁₉	—	2	1	no	—
199	C₂×C₁₉	—	2	2	no	—
200	C₂×C₁₉	—	4	2	no	—
201	C₂×C₁₉	#1799	1	1	yes	characteristic
202	C₂×C₁₉	#1808	1	1	yes	characteristic
203	C₂×C₁₉	#1812	1	1	yes	characteristic
204	C₂²×D₄	C₂²⋊D₁₉	1	1	yes	characteristic
205	C₂³×C₄	C₄×D₁₉	1	1	yes	characteristic
206	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
207	C₂²×C₈	—	19	1	no	—
208	C₂²×C₈	—	19	2	no	—
209	C₂²×C₈	—	19	1	no	—
210	C₂²×C₈	—	19	2	no	—
211	C₂²×C₈	—	19	1	no	—
212	C₂²×C₈	—	19	1	no	—
213	C₂²×C₈	—	38	1	no	—
214	C₂²×C₈	—	38	1	no	—
215	C₂²×C₈	—	76	2	no	—
216	C₂²×C₈	—	76	2	no	—
217	C₄×D₄	—	2	2	no	—
218	C₄×D₄	—	2	2	no	—
219	C₂×C₄⋊C₄	—	2	2	no	—
220	C₂×C₄⋊C₄	—	38	2	no	—
221	C₂×C₄⋊C₄	Dic₃₈	1	1	yes	characteristic
222	C₂×C₂²⋊C₄	—	2	2	no	—
223	C₂×C₂²⋊C₄	—	38	2	no	—
224	C₂×C₂²⋊C₄	Dic₃₈	1	1	yes	characteristic
225	C₂×C₂²⋊C₄	C₂²⋊D₁₉	1	1	yes	characteristic
226	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
227	#107	—	76	4	no	—
228	C₄⋊C₈	—	38	2	no	—
229	#99	—	4	2	no	—
230	#99	—	76	2	no	—
231	C₂²⋊C₈	—	38	2	no	—
232	C₂²⋊C₈	—	38	2	no	—
233	C₄×C₈	—	38	1	no	—
234	C₄×C₈	—	38	1	no	—
235	#95	—	2	2	no	—
236	#95	—	38	2	no	—
237	C₁₉	#8609	1	1	yes	characteristic, complemented, Sylow
238	C₂⁴	Dic₁₉⋊C₄	1	1	yes	characteristic
239	C₂⁴	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
240	C₂×D₄	—	2	2	no	—
241	C₂×D₄	—	2	2	no	—
242	C₂×D₄	#64525	1	2	yes	—
243	C₂×D₄	#64528	1	2	yes	—
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	4	1	no	—
247	C₂²×C₄	—	4	1	no	—
248	C₂²×C₄	—	4	1	no	—
249	C₂²×C₄	—	4	1	no	—
250	C₂²×C₄	—	4	2	no	—
251	C₂²×C₄	—	76	2	no	—
252	C₂²×C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
253	C₂²×C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
254	C₂²×C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
255	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
256	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
257	C₂²×C₄	D₄×D₁₉	1	2	yes	—
258	C₂²×C₄	#64543	1	2	yes	—
259	C₂×C₈	—	19	1	no	—
260	C₂×C₈	—	19	1	no	—
261	C₂×C₈	—	19	2	no	—
262	C₂×C₈	—	38	4	no	—
263	C₂×C₈	—	38	2	no	—
264	C₂×C₈	—	38	1	no	—
265	C₂×C₈	—	38	1	no	—
266	C₂×C₈	—	76	2	no	—
267	C₂×C₈	—	76	2	no	—
268	C₂×C₈	—	76	1	no	—
269	C₂×C₈	—	76	2	no	—
270	C₂×C₈	—	76	1	no	—
271	C₂×C₈	—	76	1	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₄⋊C₄	—	4	2	no	—
275	C₄⋊C₄	—	76	2	no	—
276	C₄⋊C₄	—	76	2	no	—
277	C₂²⋊C₄	—	2	2	no	—
278	C₂²⋊C₄	—	2	2	no	—
279	C₂²⋊C₄	—	4	2	no	—
280	C₂²⋊C₄	—	4	2	no	—
281	C₂²⋊C₄	—	76	2	no	—
282	C₂²⋊C₄	—	76	2	no	—
283	C₄²	—	2	1	no	—
284	C₄²	—	2	1	no	—
285	C₂³	—	2	1	no	—
286	C₂³	—	2	1	no	—
287	C₂³	—	4	1	no	—
288	C₂³	—	4	1	no	—
289	C₂³	—	4	1	no	—
290	C₂³	—	4	1	no	—
291	C₂³	#10597264	1	2	yes	complemented
292	C₂³	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
293	C₂³	C₂×#64525	1	1	yes	characteristic
294	C₂³	#10597385	1	2	yes	—
295	C₂³	C₂×#64528	1	1	yes	characteristic
296	C₂³	#10597396	1	2	yes	—
297	D₄	—	2	4	no	—
298	D₄	—	2	4	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	8	2	no	—
315	C₂×C₄	—	8	2	no	—
316	C₂×C₄	—	8	2	no	—
317	C₂×C₄	—	152	2	no	—
318	C₂×C₄	—	152	2	no	—
319	C₂×C₄	—	152	2	no	—
320	C₂×C₄	#10597264	1	2	yes	complemented
321	C₂×C₄	#10597323	1	1	yes	characteristic
322	C₂×C₄	#10597324	1	2	yes	—
323	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
324	C₈	—	19	2	no	—
325	C₈	—	38	1	no	—
326	C₈	—	38	2	no	—
327	C₈	—	76	1	no	—
328	C₈	—	76	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	2	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	2	no	—
333	C₂²	—	2	2	no	—
334	C₂²	—	4	1	no	—
335	C₂²	—	4	1	no	—
336	C₂²	—	4	2	no	—
337	C₂²	—	4	2	no	—
338	C₂²	—	4	2	no	—
339	C₂²	—	4	1	no	—
340	C₂²	—	4	1	no	—
341	C₂²	#11926402	1	1	yes	characteristic
342	C₂²	C₂×#10597264	1	1	yes	characteristic
343	C₂²	#11926580	1	1	yes	characteristic
344	C₂²	#11926778	1	2	yes	—
345	C₂²	#11926783	1	2	yes	—
346	C₄	—	2	1	no	—
347	C₄	—	2	2	no	—
348	C₄	—	2	2	no	—
349	C₄	—	4	1	no	—
350	C₄	—	4	1	no	—
351	C₄	—	16	2	no	—
352	C₄	—	304	2	no	—
353	C₄	#99×D₁₉	1	1	yes	characteristic
354	C₄	#11926516	1	1	yes	characteristic
355	C₂	—	2	2	no	—
356	C₂	—	2	1	no	—
357	C₂	—	2	1	no	—
358	C₂	—	2	2	no	—
359	C₂	—	4	2	no	—
360	C₂	#14563621	1	1	yes	characteristic
361	C₂	#14563662	1	1	yes	characteristic
362	C₂	#14563679	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11926402
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₁₉×#1799	C₂
socle	C₂²×C₁₉	#380
prime radical	C₂²×D₄×C₁₉	D₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	#1799	D₁₉
hypocenter	C₁₉	#8609
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1799

Complements

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

N	H ≅ G/N	type
2. #14563662	C₂	semidirect
4. #14561502	C₂	semidirect
6. C₂³×C₁₉⋊C₈	C₄	semidirect
11. C₂×C₄⋊(C₁₉⋊C₈)	C₄	semidirect
13. C₂×C₂²⋊(C₁₉⋊C₈)	C₄	semidirect
15. C₂×C₄×C₁₉⋊C₈	C₄	semidirect
110. C₂³×C₁₉	#107	semidirect
136. C₂×C₄×C₁₉	#107	semidirect
147. C₁₉⋊C₈	#99	semidirect
237. C₁₉	#8609	semidirect
291. C₂³	#10597264	semidirect
320. C₂×C₄	#10597264	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×C₁₉⋊C₈	2
#14563662	2
#14563623	2
#14561502	2
C₁₉×#1799	2
#8609	19