Subgroups — #18082625

Order 4864.

598 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14571987	C₂	1	1	yes	maximal, characteristic
2	#14563397	C₂	1	1	yes	maximal, characteristic
3	#14561514	C₂	1	4	yes	maximal
4	C₁₉×#1434	C₂	1	1	yes	maximal, characteristic
5	C₂×#10597315	C₄	1	1	yes	characteristic, complemented
6	C₂×#10597370	—	2	1	no	—
7	C₂×C₂²⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
8	C₂×#10597260	—	2	1	no	—
9	#11926450	—	2	1	no	—
10	#11926442	—	2	1	no	—
11	#11926442	C₄	1	2	yes	complemented
12	#11926414	C₄	1	1	yes	characteristic, complemented
13	#11926404	—	2	1	no	—
14	#11926404	C₂²	1	2	yes	—
15	#11926299	—	2	1	no	—
16	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
17	C₂×C₁₉×#95	—	2	1	no	—
18	C₂×C₁₉×#95	—	2	1	no	—
19	C₂×C₁₉×#95	C₂²	1	2	yes	—
20	C₂³×C₄×C₁₉	—	2	2	no	—
21	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
24	C₂²×Dic₃₈	—	2	1	no	—
25	C₂×C₄²×C₁₉	—	4	2	no	—
26	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
30	C₂×C₂²⋊Dic₁₉	—	2	2	no	—
31	C₂×C₄⋊Dic₁₉	—	2	1	no	—
32	C₂×C₄⋊Dic₁₉	—	2	1	no	—
33	C₂×C₄⋊Dic₁₉	—	2	1	no	—
34	C₂×C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
35	C₂×C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
36	#10597370	—	4	4	no	—
37	C₂×Dic₁₉⋊C₄	—	2	2	no	—
38	C₂²×C₁₉⋊C₈	—	2	2	no	—
39	#10597315	—	4	2	no	—
40	C₁₉×#95	—	2	4	no	—
41	C₁₉×#95	—	2	4	no	—
42	C₁₉×#95	—	4	2	no	—
43	C₁₉⋊#95	—	2	2	no	—
44	C₁₉⋊#95	—	2	2	no	—
45	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
46	C₂²⋊(C₁₉⋊C₈)	Q₈	1	4	yes	complemented
47	#10597260	—	4	2	no	—
48	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₉	—	2	2	no	—
50	C₂²×C₄×C₁₉	—	2	2	no	—
51	C₂²×C₄×C₁₉	—	2	2	no	—
52	C₂²×C₄×C₁₉	—	2	2	no	—
53	C₂²×C₄×C₁₉	—	2	4	no	—
54	C₂²×C₄×C₁₉	—	4	1	no	—
55	C₂²×C₄×C₁₉	—	4	1	no	—
56	C₂²×C₄×C₁₉	—	4	1	no	—
57	C₂²×C₄×C₁₉	—	4	1	no	—
58	C₂²×C₄×C₁₉	—	4	2	no	—
59	C₂²×C₄×C₁₉	—	4	2	no	—
60	C₂²×C₄×C₁₉	—	4	2	no	—
61	C₂²×C₄×C₁₉	—	4	2	no	—
62	C₂²×C₄×C₁₉	—	4	2	no	—
63	C₂²×C₄×C₁₉	—	4	2	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	1	yes	characteristic
70	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
71	C₂²×C₄×C₁₉	#41	1	2	yes	—
72	C₂²×Dic₁₉	—	2	2	no	—
73	C₂²×Dic₁₉	—	2	2	no	—
74	C₂×Dic₃₈	—	4	2	no	—
75	C₂×Dic₃₈	—	4	2	no	—
76	C₂×Dic₃₈	—	4	2	no	—
77	C₄²×C₁₉	—	2	2	no	—
78	C₄²×C₁₉	—	2	2	no	—
79	C₄²×C₁₉	—	4	1	no	—
80	C₄²×C₁₉	—	4	1	no	—
81	C₂²⋊Dic₁₉	—	4	4	no	—
82	C₄⋊Dic₁₉	—	4	2	no	—
83	C₄⋊Dic₁₉	—	4	2	no	—
84	C₄⋊Dic₁₉	—	4	2	no	—
85	C₄⋊Dic₁₉	—	4	2	no	—
86	C₄⋊Dic₁₉	—	4	2	no	—
87	Dic₁₉⋊C₄	—	4	4	no	—
88	C₂×C₁₉⋊C₈	—	2	4	no	—
89	C₂×C₁₉⋊C₈	—	4	2	no	—
90	C₂×C₁₉⋊C₈	—	4	2	no	—
91	#9889	—	19	1	no	maximal, Sylow
92	C₂³×C₁₉	—	2	1	no	—
93	C₂³×C₁₉	—	2	1	no	—
94	C₂³×C₁₉	—	2	1	no	—
95	C₂³×C₁₉	—	2	1	no	—
96	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂³×C₁₉	#124	1	1	yes	characteristic
98	C₂³×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
99	C₂³×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
100	C₂³×C₁₉	#137	1	2	yes	—
101	C₂×C₄×C₁₉	—	2	1	no	—
102	C₂×C₄×C₁₉	—	2	1	no	—
103	C₂×C₄×C₁₉	—	2	1	no	—
104	C₂×C₄×C₁₉	—	2	1	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	—	4	2	no	—
116	C₂×C₄×C₁₉	—	4	2	no	—
117	C₂×C₄×C₁₉	—	4	4	no	—
118	C₂×C₄×C₁₉	—	4	4	no	—
119	C₂×C₄×C₁₉	—	4	4	no	—
120	C₂×C₄×C₁₉	—	4	4	no	—
121	C₂×C₄×C₁₉	—	4	4	no	—
122	C₂×C₄×C₁₉	—	8	2	no	—
123	C₂×C₄×C₁₉	—	8	2	no	—
124	C₂×C₄×C₁₉	—	8	2	no	—
125	C₂×C₄×C₁₉	—	8	2	no	—
126	C₂×C₄×C₁₉	#99	1	2	yes	—
127	C₂×C₄×C₁₉	#104	1	2	yes	—
128	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
129	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
130	C₂×Dic₁₉	—	4	2	no	—
131	C₂×Dic₁₉	—	4	2	no	—
132	C₂×Dic₁₉	—	4	2	no	—
133	C₂×Dic₁₉	—	4	2	no	—
134	C₂×Dic₁₉	—	4	2	no	—
135	C₂×Dic₁₉	—	4	2	no	—
136	Dic₃₈	—	8	4	no	—
137	C₁₉⋊C₈	—	4	4	no	—
138	#2857	—	19	1	no	—
139	#1749	—	19	1	no	—
140	#1434	D₁₉	1	1	yes	characteristic
141	#1294	—	19	4	no	—
142	C₂²×C₁₉	—	2	1	no	—
143	C₂²×C₁₉	—	2	1	no	—
144	C₂²×C₁₉	—	2	1	no	—
145	C₂²×C₁₉	—	2	1	no	—
146	C₂²×C₁₉	—	2	2	no	—
147	C₂²×C₁₉	—	2	2	no	—
148	C₂²×C₁₉	—	2	2	no	—
149	C₂²×C₁₉	—	2	2	no	—
150	C₂²×C₁₉	#391	1	1	yes	characteristic
151	C₂²×C₁₉	C₂×#99	1	1	yes	characteristic
152	C₂²×C₁₉	#410	1	1	yes	characteristic
153	C₂²×C₁₉	#418	1	1	yes	characteristic
154	C₂²×C₁₉	#419	1	1	yes	characteristic
155	C₂²×C₁₉	C₂×#104	1	1	yes	characteristic
156	C₂²×C₁₉	#421	1	1	yes	characteristic
157	C₂²×C₁₉	#489	1	2	yes	—
158	C₂²×C₁₉	#501	1	2	yes	—
159	C₄×C₁₉	—	2	2	no	—
160	C₄×C₁₉	—	2	2	no	—
161	C₄×C₁₉	—	2	2	no	—
162	C₄×C₁₉	—	2	2	no	—
163	C₄×C₁₉	—	4	1	no	—
164	C₄×C₁₉	—	4	1	no	—
165	C₄×C₁₉	—	8	4	no	—
166	Dic₁₉	—	8	2	no	—
167	Dic₁₉	—	8	2	no	—
168	C₂×C₄⋊Q₈	—	19	1	no	—
169	C₂×C₂²⋊Q₈	—	38	1	no	—
170	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
171	C₂×C₂²⋊C₈	—	19	2	no	—
172	C₂×#97	—	38	1	no	—
173	#391	—	38	1	no	—
174	#389	—	19	2	no	—
175	#389	—	38	1	no	—
176	#384	—	19	1	no	—
177	#380	—	19	2	no	—
178	#380	—	38	1	no	—
179	C₂×#95	—	2	1	no	—
180	C₂×#95	—	2	1	no	—
181	C₂×#95	C₂×D₁₉	1	2	yes	—
182	#336	—	38	1	no	—
183	C₂×C₁₉	—	2	2	no	—
184	C₂×C₁₉	—	2	2	no	—
185	C₂×C₁₉	#1505	1	1	yes	characteristic
186	C₂×C₁₉	#1508	1	1	yes	characteristic
187	C₂×C₁₉	#1510	1	1	yes	characteristic
188	C₂×C₁₉	#1515	1	1	yes	characteristic
189	C₂×C₁₉	#1986	1	1	yes	characteristic
190	C₂×C₁₉	#1993	1	1	yes	characteristic
191	C₂×C₁₉	#1996	1	1	yes	characteristic
192	C₂²×Q₈	—	38	1	no	—
193	C₂³×C₄	—	2	2	no	—
194	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
195	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
196	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
197	C₂²×C₈	—	38	2	no	—
198	C₄⋊Q₈	—	76	2	no	—
199	C₂²⋊Q₈	—	76	4	no	—
200	C₂×C₄⋊C₄	—	19	2	no	—
201	C₂×C₄⋊C₄	—	19	2	no	—
202	C₂×C₄⋊C₄	—	38	1	no	—
203	C₂×C₄⋊C₄	—	38	2	no	—
204	C₂×C₄⋊C₄	—	38	1	no	—
205	C₂×C₄⋊C₄	—	38	1	no	—
206	C₂×C₂²⋊C₄	—	38	2	no	—
207	C₂×C₄²	—	4	2	no	—
208	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
209	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
210	C₂×C₄²	D₇₆	1	1	yes	characteristic
211	C₂×C₄²	D₇₆	1	1	yes	characteristic
212	C₂²⋊C₈	—	19	4	no	—
213	C₂²⋊C₈	—	38	2	no	—
214	#97	—	76	2	no	—
215	#95	—	2	4	no	—
216	#95	—	2	4	no	—
217	#95	—	4	2	no	—
218	#95	—	38	2	no	—
219	#95	—	38	2	no	—
220	C₁₉	#9889	1	1	yes	characteristic, complemented, Sylow
221	C₂⁴	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
222	C₂×Q₈	—	76	2	no	—
223	C₂×Q₈	—	76	2	no	—
224	C₂×Q₈	—	76	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	—	4	1	no	—
231	C₂²×C₄	—	4	1	no	—
232	C₂²×C₄	—	4	1	no	—
233	C₂²×C₄	—	4	1	no	—
234	C₂²×C₄	—	4	2	no	—
235	C₂²×C₄	—	4	2	no	—
236	C₂²×C₄	—	4	2	no	—
237	C₂²×C₄	—	4	2	no	—
238	C₂²×C₄	—	4	2	no	—
239	C₂²×C₄	—	4	2	no	—
240	C₂²×C₄	—	38	2	no	—
241	C₂²×C₄	—	38	2	no	—
242	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
243	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
244	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
245	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
246	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
247	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
248	C₂²×C₄	#64543	1	2	yes	—
249	C₂²×C₄	Q₈×D₁₉	1	2	yes	—
250	C₂×C₈	—	38	4	no	—
251	C₂×C₈	—	76	2	no	—
252	C₂×C₈	—	76	2	no	—
253	C₄⋊C₄	—	76	2	no	—
254	C₄⋊C₄	—	76	2	no	—
255	C₄⋊C₄	—	76	4	no	—
256	C₄⋊C₄	—	76	2	no	—
257	C₄⋊C₄	—	76	2	no	—
258	C₄⋊C₄	—	76	2	no	—
259	C₂²⋊C₄	—	76	4	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	4	1	no	—
263	C₄²	—	4	1	no	—
264	C₂³	—	2	1	no	—
265	C₂³	—	2	1	no	—
266	C₂³	—	2	1	no	—
267	C₂³	—	2	1	no	—
268	C₂³	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
269	C₂³	#10597382	1	1	yes	characteristic
270	C₂³	#10597390	1	1	yes	characteristic
271	C₂³	#10597395	1	2	yes	—
272	C₂³	#10597397	1	2	yes	—
273	Q₈	—	152	4	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	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	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	—	4	4	no	—
292	C₂×C₄	—	4	4	no	—
293	C₂×C₄	—	4	4	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	—	8	2	no	—
296	C₂×C₄	—	8	2	no	—
297	C₂×C₄	—	8	2	no	—
298	C₂×C₄	—	8	2	no	—
299	C₂×C₄	—	76	2	no	—
300	C₂×C₄	—	76	2	no	—
301	C₂×C₄	—	76	2	no	—
302	C₂×C₄	—	76	2	no	—
303	C₂×C₄	—	76	2	no	—
304	C₂×C₄	—	76	2	no	—
305	C₂×C₄	#10597262	1	2	yes	—
306	C₂×C₄	#10597263	1	2	yes	—
307	C₂×C₄	#10597333	1	2	yes	—
308	C₂×C₄	#10597343	1	2	yes	—
309	C₈	—	76	4	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	#11926449	1	1	yes	characteristic
319	C₂²	C₂×#10597263	1	1	yes	characteristic
320	C₂²	#11926517	1	1	yes	characteristic
321	C₂²	#11926548	1	1	yes	characteristic
322	C₂²	#11926552	1	1	yes	characteristic
323	C₂²	C₂×#10597262	1	1	yes	characteristic
324	C₂²	#11926562	1	1	yes	characteristic
325	C₂²	#11926903	1	2	yes	—
326	C₂²	#11926954	1	2	yes	—
327	C₄	—	2	2	no	—
328	C₄	—	2	2	no	—
329	C₄	—	2	2	no	—
330	C₄	—	2	2	no	—
331	C₄	—	4	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	8	4	no	—
334	C₄	—	152	2	no	—
335	C₄	—	152	2	no	—
336	C₂	—	2	2	no	—
337	C₂	—	2	2	no	—
338	C₂	#14562142	1	1	yes	characteristic
339	C₂	#14562154	1	1	yes	characteristic
340	C₂	#14562163	1	1	yes	characteristic
341	C₂	#14562185	1	1	yes	characteristic
342	C₂	#14564578	1	1	yes	characteristic
343	C₂	#14564607	1	1	yes	characteristic
344	C₂	#14564619	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₁₉)⋊C₄
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₁₉×#1434	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₉	C₂×D₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	#1434	D₁₉
hypocenter	C₁₉	#9889
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #1434

Complements

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

N	H ≅ G/N	type
5. C₂×#10597315	C₄	semidirect
11. #11926442	C₄	semidirect
12. #11926414	C₄	semidirect
46. C₂²⋊(C₁₉⋊C₈)	Q₈	semidirect
220. C₁₉	#9889	semidirect

Maximal subgroups

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

subgroup	index
#14571987	2
#14563397	2
#14561514	2
C₁₉×#1434	2
#9889	19