Subgroups — #18081536

Order 4864.

630 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14563397	C₂	1	1	yes	maximal, characteristic
2	C₂×#11926281	C₂	1	4	yes	maximal
3	#14561862	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#2857	C₂	1	1	yes	maximal, characteristic
5	C₂×C₂²⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
6	C₂×#10597260	—	2	1	no	—
7	C₂×C₁₉⋊#95	—	2	1	no	—
8	C₂×C₁₉⋊#95	—	2	1	no	—
9	C₂×C₁₉⋊#95	C₂²	1	2	yes	—
10	#11926299	—	2	1	no	—
11	#11926281	—	2	8	no	—
12	C₂×C₁₉×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
13	C₂×C₁₉×C₂²⋊Q₈	C₂²	1	2	yes	—
14	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
15	C₁₉×#391	C₄	1	2	yes	complemented
16	C₁₉×#389	—	2	1	no	—
17	C₁₉×#389	—	2	1	no	—
18	C₁₉×#384	C₄	1	1	yes	characteristic, complemented
19	C₁₉×#380	—	2	1	no	—
20	C₁₉×#380	—	2	1	no	—
21	C₂²×Q₈×C₁₉	C₂×C₄	1	2	yes	—
22	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
25	C₂³×Dic₁₉	—	2	2	no	—
26	C₁₉×C₄⋊Q₈	—	4	2	no	—
27	C₁₉×C₂²⋊Q₈	—	2	4	no	—
28	C₁₉×C₂²⋊Q₈	—	2	4	no	—
29	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
30	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
31	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
32	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
33	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
34	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
35	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
36	C₂×C₄²×C₁₉	—	4	2	no	—
37	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
41	C₂²×C₁₉⋊C₈	—	2	2	no	—
42	C₁₉×#95	—	2	2	no	—
43	C₁₉×#95	—	2	2	no	—
44	C₁₉⋊#95	—	2	4	no	—
45	C₁₉⋊#95	—	2	4	no	—
46	C₁₉⋊#95	—	4	2	no	—
47	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
48	C₂²⋊(C₁₉⋊C₈)	D₄	1	4	yes	—
49	#10597260	—	4	2	no	—
50	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
51	C₂×Q₈×C₁₉	—	2	4	no	—
52	C₂×Q₈×C₁₉	—	4	2	no	—
53	C₂×Q₈×C₁₉	—	4	2	no	—
54	C₂²×C₄×C₁₉	—	2	2	no	—
55	C₂²×C₄×C₁₉	—	2	2	no	—
56	C₂²×C₄×C₁₉	—	2	2	no	—
57	C₂²×C₄×C₁₉	—	2	2	no	—
58	C₂²×C₄×C₁₉	—	2	2	no	—
59	C₂²×C₄×C₁₉	—	2	2	no	—
60	C₂²×C₄×C₁₉	—	4	1	no	—
61	C₂²×C₄×C₁₉	—	4	1	no	—
62	C₂²×C₄×C₁₉	—	4	1	no	—
63	C₂²×C₄×C₁₉	—	4	1	no	—
64	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₉	#41	1	2	yes	—
72	C₂²×Dic₁₉	—	2	4	no	—
73	C₂²×Dic₁₉	—	4	2	no	—
74	C₂²×Dic₁₉	—	4	2	no	—
75	C₂²×Dic₁₉	—	4	2	no	—
76	C₂²×Dic₁₉	—	4	2	no	—
77	C₂²×Dic₁₉	—	4	2	no	—
78	C₂²×Dic₁₉	—	4	2	no	—
79	C₄⋊C₄×C₁₉	—	2	4	no	—
80	C₄⋊C₄×C₁₉	—	4	2	no	—
81	C₄⋊C₄×C₁₉	—	4	2	no	—
82	C₄⋊C₄×C₁₉	—	4	2	no	—
83	C₄⋊C₄×C₁₉	—	4	2	no	—
84	C₄⋊C₄×C₁₉	—	4	4	no	—
85	C₂²⋊C₄×C₁₉	—	4	2	no	—
86	C₂²⋊C₄×C₁₉	—	4	2	no	—
87	C₄²×C₁₉	—	2	2	no	—
88	C₄²×C₁₉	—	2	2	no	—
89	C₄²×C₁₉	—	4	1	no	—
90	C₄²×C₁₉	—	4	1	no	—
91	C₂×C₁₉⋊C₈	—	2	4	no	—
92	C₂×C₁₉⋊C₈	—	4	2	no	—
93	C₂×C₁₉⋊C₈	—	4	2	no	—
94	#9683	—	19	1	no	maximal, Sylow
95	C₂³×C₁₉	—	2	1	no	—
96	C₂³×C₁₉	—	2	1	no	—
97	C₂³×C₁₉	—	2	1	no	—
98	C₂³×C₁₉	—	2	1	no	—
99	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂³×C₁₉	#124	1	1	yes	characteristic
101	C₂³×C₁₉	#127	1	1	yes	characteristic
102	C₂³×C₁₉	#137	1	2	yes	—
103	C₂³×C₁₉	#137	1	2	yes	—
104	Q₈×C₁₉	—	8	4	no	—
105	C₂×C₄×C₁₉	—	2	1	no	—
106	C₂×C₄×C₁₉	—	2	1	no	—
107	C₂×C₄×C₁₉	—	2	1	no	—
108	C₂×C₄×C₁₉	—	2	1	no	—
109	C₂×C₄×C₁₉	—	2	1	no	—
110	C₂×C₄×C₁₉	—	2	1	no	—
111	C₂×C₄×C₁₉	—	2	1	no	—
112	C₂×C₄×C₁₉	—	2	1	no	—
113	C₂×C₄×C₁₉	—	2	1	no	—
114	C₂×C₄×C₁₉	—	2	1	no	—
115	C₂×C₄×C₁₉	—	2	1	no	—
116	C₂×C₄×C₁₉	—	2	1	no	—
117	C₂×C₄×C₁₉	—	2	1	no	—
118	C₂×C₄×C₁₉	—	2	1	no	—
119	C₂×C₄×C₁₉	—	4	2	no	—
120	C₂×C₄×C₁₉	—	4	2	no	—
121	C₂×C₄×C₁₉	—	4	4	no	—
122	C₂×C₄×C₁₉	—	4	4	no	—
123	C₂×C₄×C₁₉	—	4	2	no	—
124	C₂×C₄×C₁₉	—	4	2	no	—
125	C₂×C₄×C₁₉	—	4	2	no	—
126	C₂×C₄×C₁₉	—	4	2	no	—
127	C₂×C₄×C₁₉	—	4	2	no	—
128	C₂×C₄×C₁₉	—	4	2	no	—
129	C₂×C₄×C₁₉	#99	1	2	yes	—
130	C₂×C₄×C₁₉	#104	1	2	yes	—
131	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
132	C₂×C₄×C₁₉	#121	1	2	yes	—
133	C₂×Dic₁₉	—	4	4	no	—
134	C₂×Dic₁₉	—	4	4	no	—
135	C₂×Dic₁₉	—	4	4	no	—
136	C₂×Dic₁₉	—	8	2	no	—
137	C₂×Dic₁₉	—	8	2	no	—
138	C₂×Dic₁₉	—	8	2	no	—
139	C₂×Dic₁₉	—	8	2	no	—
140	C₁₉⋊C₈	—	4	4	no	—
141	#2857	D₁₉	1	1	yes	characteristic
142	#1749	—	19	1	no	—
143	C₂×#328	—	19	4	no	—
144	#1434	—	19	1	no	—
145	C₂²×C₁₉	—	2	1	no	—
146	C₂²×C₁₉	—	2	1	no	—
147	C₂²×C₁₉	—	2	1	no	—
148	C₂²×C₁₉	—	2	1	no	—
149	C₂²×C₁₉	—	2	2	no	—
150	C₂²×C₁₉	—	2	2	no	—
151	C₂²×C₁₉	—	2	2	no	—
152	C₂²×C₁₉	—	2	2	no	—
153	C₂²×C₁₉	#390	1	1	yes	characteristic
154	C₂²×C₁₉	C₂×#99	1	1	yes	characteristic
155	C₂²×C₁₉	#410	1	1	yes	characteristic
156	C₂²×C₁₉	#419	1	1	yes	characteristic
157	C₂²×C₁₉	#419	1	1	yes	characteristic
158	C₂²×C₁₉	C₂×#104	1	1	yes	characteristic
159	C₂²×C₁₉	#421	1	1	yes	characteristic
160	C₂²×C₁₉	#491	1	2	yes	—
161	C₂²×C₁₉	#497	1	2	yes	—
162	C₄×C₁₉	—	2	2	no	—
163	C₄×C₁₉	—	2	2	no	—
164	C₄×C₁₉	—	2	2	no	—
165	C₄×C₁₉	—	2	2	no	—
166	C₄×C₁₉	—	4	1	no	—
167	C₄×C₁₉	—	4	1	no	—
168	C₄×C₁₉	—	8	2	no	—
169	C₄×C₁₉	—	8	2	no	—
170	Dic₁₉	—	8	4	no	—
171	C₂×C₄⋊Q₈	Dic₁₉	1	1	yes	characteristic, complemented
172	C₂×C₂²⋊Q₈	C₂×D₁₉	1	2	yes	—
173	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
174	C₂×C₂²⋊C₈	—	19	2	no	—
175	C₂×#97	—	38	1	no	—
176	#391	Dic₁₉	1	2	yes	complemented
177	#389	—	2	1	no	—
178	#389	—	2	1	no	—
179	#384	Dic₁₉	1	1	yes	characteristic, complemented
180	#380	—	2	1	no	—
181	#380	—	2	1	no	—
182	C₂×#95	—	19	2	no	—
183	C₂×#95	—	38	1	no	—
184	C₂×#95	—	38	1	no	—
185	#336	—	38	1	no	—
186	#328	—	38	8	no	—
187	C₂×C₁₉	—	2	2	no	—
188	C₂×C₁₉	—	2	2	no	—
189	C₂×C₁₉	#1506	1	1	yes	characteristic
190	C₂×C₁₉	#1508	1	1	yes	characteristic
191	C₂×C₁₉	#1511	1	1	yes	characteristic
192	C₂×C₁₉	#1515	1	1	yes	characteristic
193	C₂×C₁₉	#1960	1	1	yes	characteristic
194	C₂×C₁₉	#1964	1	1	yes	characteristic
195	C₂×C₁₉	#1980	1	1	yes	characteristic
196	C₂²×Q₈	C₂×Dic₁₉	1	2	yes	—
197	C₂³×C₄	—	38	2	no	—
198	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
199	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
200	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
201	C₂²×C₈	—	38	2	no	—
202	C₄⋊Q₈	—	4	2	no	—
203	C₂²⋊Q₈	—	2	4	no	—
204	C₂²⋊Q₈	—	2	4	no	—
205	C₂×C₄⋊C₄	—	2	1	no	—
206	C₂×C₄⋊C₄	—	2	1	no	—
207	C₂×C₄⋊C₄	—	2	1	no	—
208	C₂×C₄⋊C₄	—	2	1	no	—
209	C₂×C₄⋊C₄	—	2	2	no	—
210	C₂×C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
211	C₂×C₂²⋊C₄	—	2	2	no	—
212	C₂×C₄²	—	4	2	no	—
213	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
214	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
215	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
216	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
217	C₂²⋊C₈	—	19	4	no	—
218	C₂²⋊C₈	—	38	2	no	—
219	#97	—	76	2	no	—
220	#95	—	2	2	no	—
221	#95	—	2	2	no	—
222	#95	—	38	4	no	—
223	#95	—	38	4	no	—
224	#95	—	76	2	no	—
225	C₁₉	#9683	1	1	yes	characteristic, complemented, Sylow
226	C₂⁴	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
227	C₂×Q₈	—	2	4	no	—
228	C₂×Q₈	—	4	2	no	—
229	C₂×Q₈	—	4	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	4	1	no	—
237	C₂²×C₄	—	4	1	no	—
238	C₂²×C₄	—	4	1	no	—
239	C₂²×C₄	—	4	1	no	—
240	C₂²×C₄	—	38	4	no	—
241	C₂²×C₄	—	76	2	no	—
242	C₂²×C₄	—	76	2	no	—
243	C₂²×C₄	—	76	2	no	—
244	C₂²×C₄	—	76	2	no	—
245	C₂²×C₄	—	76	2	no	—
246	C₂²×C₄	—	76	2	no	—
247	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
248	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
249	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
250	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
251	C₂²×C₄	D₄×D₁₉	1	2	yes	—
252	C₂²×C₄	#64543	1	2	yes	—
253	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
254	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
255	C₂×C₈	—	38	4	no	—
256	C₂×C₈	—	76	2	no	—
257	C₂×C₈	—	76	2	no	—
258	C₄⋊C₄	—	2	4	no	—
259	C₄⋊C₄	—	4	2	no	—
260	C₄⋊C₄	—	4	2	no	—
261	C₄⋊C₄	—	4	4	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₂²⋊C₄	—	4	2	no	—
265	C₂²⋊C₄	—	4	2	no	—
266	C₄²	—	2	2	no	—
267	C₄²	—	2	2	no	—
268	C₄²	—	4	1	no	—
269	C₄²	—	4	1	no	—
270	C₂³	—	2	1	no	—
271	C₂³	—	2	1	no	—
272	C₂³	—	2	1	no	—
273	C₂³	—	2	1	no	—
274	C₂³	#10597382	1	1	yes	characteristic
275	C₂³	#10597386	1	1	yes	characteristic
276	C₂³	#10597388	1	2	yes	—
277	C₂³	#10597388	1	2	yes	—
278	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
279	Q₈	—	8	4	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	76	4	no	—
305	C₂×C₄	—	76	4	no	—
306	C₂×C₄	—	76	4	no	—
307	C₂×C₄	—	152	2	no	—
308	C₂×C₄	—	152	2	no	—
309	C₂×C₄	—	152	2	no	—
310	C₂×C₄	—	152	2	no	—
311	C₂×C₄	#10597291	1	2	yes	—
312	C₂×C₄	#10597294	1	2	yes	—
313	C₂×C₄	D₄×Dic₁₉	1	2	yes	—
314	C₂×C₄	#10597384	1	2	yes	—
315	C₈	—	76	4	no	—
316	C₂²	—	2	2	no	—
317	C₂²	—	2	2	no	—
318	C₂²	—	2	2	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	#11926447	1	1	yes	characteristic
325	C₂²	C₂×#10597291	1	1	yes	characteristic
326	C₂²	#11926518	1	1	yes	characteristic
327	C₂²	#11926553	1	1	yes	characteristic
328	C₂²	#11926553	1	1	yes	characteristic
329	C₂²	C₂×#10597294	1	1	yes	characteristic
330	C₂²	#11926561	1	1	yes	characteristic
331	C₂²	#11926912	1	2	yes	—
332	C₂²	#11926936	1	2	yes	—
333	C₄	—	2	2	no	—
334	C₄	—	2	2	no	—
335	C₄	—	2	2	no	—
336	C₄	—	2	2	no	—
337	C₄	—	4	1	no	—
338	C₄	—	4	1	no	—
339	C₄	—	8	2	no	—
340	C₄	—	8	2	no	—
341	C₄	—	152	4	no	—
342	C₂	—	2	2	no	—
343	C₂	—	2	2	no	—
344	C₂	#14562147	1	1	yes	characteristic
345	C₂	#14562155	1	1	yes	characteristic
346	C₂	#14562168	1	1	yes	characteristic
347	C₂	#14562186	1	1	yes	characteristic
348	C₂	#14564456	1	1	yes	characteristic
349	C₂	#14564472	1	1	yes	characteristic
350	C₂	#14564555	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₁₉
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₁₉×#2857	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₉	C₂×D₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	#2857	D₁₉
hypocenter	C₁₉	#9683
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #2857

Complements

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

N	H ≅ G/N	type
12. C₂×C₁₉×C₄⋊Q₈	C₄	semidirect
15. C₁₉×#391	C₄	semidirect
18. C₁₉×#384	C₄	semidirect
171. C₂×C₄⋊Q₈	Dic₁₉	semidirect
176. #391	Dic₁₉	semidirect
179. #384	Dic₁₉	semidirect
225. C₁₉	#9683	semidirect

Maximal subgroups

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

subgroup	index
#14563397	2
C₂×#11926281	2
#14561862	2
C₁₉×#2857	2
#9683	19