Subgroups — #18090444

Order 4864.

670 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14572199	C₂	1	1	yes	maximal, characteristic
2	C₂×#11926314	C₂	1	2	yes	maximal
3	#14563374	C₂	1	1	yes	maximal, characteristic
4	C₂×#11926313	C₂	1	2	yes	maximal
5	C₁₉×#2317	C₂	1	1	yes	maximal, characteristic
6	C₂²×#64520	C₂²	1	2	yes	—
7	C₂²×#64520	C₂²	1	2	yes	—
8	#11926571	—	2	4	no	—
9	#11926568	—	2	2	no	—
10	#11926314	—	2	4	no	—
11	#11926313	—	2	4	no	—
12	#11926313	—	2	1	no	—
13	#11926313	—	2	1	no	—
14	C₂×C₁₉×#117	C₂²	1	1	yes	characteristic
15	C₂×C₁₉×#117	C₂²	1	1	yes	characteristic
16	C₂²×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
17	C₁₉×#389	—	2	2	no	—
18	C₁₉×#385	C₄	1	4	yes	—
19	C₁₉×#384	—	2	1	no	—
20	C₁₉×#382	—	2	1	no	—
21	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
22	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
23	C₁₉×#117	—	2	2	no	—
24	C₁₉×#117	—	2	1	no	—
25	C₁₉×#117	—	2	1	no	—
26	C₁₉×#117	—	2	2	no	—
27	C₁₉×#117	D₄	1	2	yes	—
28	C₁₉×#117	Q₈	1	2	yes	—
29	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
30	C₂×C₄⋊C₄×C₁₉	—	2	2	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	—	2	4	no	—
35	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
36	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
37	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
38	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₁₉	—	2	1	no	—
40	C₂×C₄²×C₁₉	D₄	1	2	yes	—
41	C₂×#64520	—	2	2	no	—
42	C₂×#64520	—	2	2	no	—
43	C₂×#64520	—	2	2	no	—
44	C₂×#64520	—	2	2	no	—
45	C₂×#64520	—	2	2	no	—
46	C₂×#64520	—	2	2	no	—
47	C₂×#64520	—	2	2	no	—
48	C₂×#64520	—	2	2	no	—
49	C₂×#64520	—	2	2	no	—
50	C₂×#64520	—	2	2	no	—
51	C₂×#64520	D₄	1	4	yes	—
52	C₂×#64520	Q₈	1	4	yes	—
53	C₂²×C₁₉⋊C₈	—	2	2	no	—
54	C₂²×C₁₉⋊C₈	—	2	2	no	—
55	C₁₉×#95	—	2	4	no	—
56	C₄⋊(C₁₉⋊C₈)	—	2	8	no	—
57	C₂⁴×C₁₉	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	—	2	2	no	—
59	C₂²×C₄×C₁₉	—	2	2	no	—
60	C₂²×C₄×C₁₉	—	2	4	no	—
61	C₂²×C₄×C₁₉	—	8	2	no	—
62	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
71	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
72	C₂²×C₄×C₁₉	#41	1	2	yes	—
73	C₂²×C₄×C₁₉	#41	1	2	yes	—
74	C₄⋊C₄×C₁₉	—	2	2	no	—
75	C₄⋊C₄×C₁₉	—	2	2	no	—
76	C₄⋊C₄×C₁₉	—	2	2	no	—
77	C₄⋊C₄×C₁₉	—	2	2	no	—
78	C₄⋊C₄×C₁₉	—	4	1	no	—
79	C₄⋊C₄×C₁₉	—	4	2	no	—
80	C₄⋊C₄×C₁₉	—	4	1	no	—
81	C₄⋊C₄×C₁₉	—	8	4	no	—
82	C₂²⋊C₄×C₁₉	—	2	2	no	—
83	C₂²⋊C₄×C₁₉	—	2	4	no	—
84	C₂²⋊C₄×C₁₉	C₄⋊C₄	1	4	yes	—
85	C₄²×C₁₉	—	2	2	no	—
86	C₄²×C₁₉	—	2	2	no	—
87	C₄²×C₁₉	—	2	2	no	—
88	C₄²×C₁₉	—	2	2	no	—
89	#64520	—	2	4	no	—
90	#64520	—	2	4	no	—
91	#64520	—	2	4	no	—
92	#64520	—	2	4	no	—
93	#64520	—	4	2	no	—
94	#64520	—	4	2	no	—
95	#64520	—	4	2	no	—
96	#64520	—	4	2	no	—
97	C₂×C₁₉⋊C₈	—	2	4	no	—
98	C₂×C₁₉⋊C₈	—	2	4	no	—
99	C₂×C₁₉⋊C₈	—	4	2	no	—
100	C₂×C₁₉⋊C₈	—	4	2	no	—
101	C₂×C₁₉⋊C₈	—	4	2	no	—
102	C₂×C₁₉⋊C₈	—	4	2	no	—
103	#11206	—	19	1	no	maximal, Sylow
104	C₂³×C₁₉	—	4	1	no	—
105	C₂³×C₁₉	—	4	1	no	—
106	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂³×C₁₉	#117	1	1	yes	characteristic
109	C₂³×C₁₉	C₄×D₄	1	2	yes	—
110	C₂³×C₁₉	C₄×Q₈	1	2	yes	—
111	C₂×C₄×C₁₉	—	2	4	no	—
112	C₂×C₄×C₁₉	—	2	2	no	—
113	C₂×C₄×C₁₉	—	2	2	no	—
114	C₂×C₄×C₁₉	—	4	2	no	—
115	C₂×C₄×C₁₉	—	4	2	no	—
116	C₂×C₄×C₁₉	—	4	2	no	—
117	C₂×C₄×C₁₉	—	4	2	no	—
118	C₂×C₄×C₁₉	—	4	4	no	—
119	C₂×C₄×C₁₉	—	4	2	no	—
120	C₂×C₄×C₁₉	—	8	4	no	—
121	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
122	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
123	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
124	C₂×C₄×C₁₉	#121	1	2	yes	—
125	C₂×C₄×C₁₉	#121	1	2	yes	—
126	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
127	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
128	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
129	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
130	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
131	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
132	C₂×C₄×C₁₉	#123	1	2	yes	—
133	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
134	C₂×C₄×C₁₉	#125	1	2	yes	—
135	C₂×C₄×C₁₉	#125	1	2	yes	—
136	C₂×C₄×C₁₉	#126	1	2	yes	—
137	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
138	C₂×C₄×C₁₉	C₄⋊Q₈	1	2	yes	—
139	C₁₉⋊C₈	—	4	4	no	—
140	C₁₉⋊C₈	—	4	4	no	—
141	#2905	—	19	1	no	—
142	#2317	D₁₉	1	1	yes	characteristic
143	C₂×#344	—	19	2	no	—
144	#1742	—	19	1	no	—
145	C₂×#343	—	19	2	no	—
146	C₂²×C₁₉	—	2	1	no	—
147	C₂²×C₁₉	—	2	2	no	—
148	C₂²×C₁₉	—	2	1	no	—
149	C₂²×C₁₉	—	2	2	no	—
150	C₂²×C₁₉	—	4	1	no	—
151	C₂²×C₁₉	—	4	1	no	—
152	C₂²×C₁₉	—	4	2	no	—
153	C₂²×C₁₉	#380	1	1	yes	characteristic
154	C₂²×C₁₉	#386	1	1	yes	characteristic
155	C₂²×C₁₉	#387	1	2	yes	—
156	C₂²×C₁₉	#389	1	2	yes	—
157	C₂²×C₁₉	#391	1	1	yes	characteristic
158	C₄×C₁₉	—	4	4	no	—
159	C₄×C₁₉	—	4	2	no	—
160	C₄×C₁₉	—	4	2	no	—
161	C₄×C₁₉	—	8	2	no	—
162	C₄×C₁₉	#394	1	1	yes	characteristic
163	C₄×C₁₉	#395	1	1	yes	characteristic
164	C₄×C₁₉	#396	1	2	yes	—
165	C₄×C₁₉	#398	1	1	yes	characteristic
166	C₄×C₁₉	#398	1	2	yes	—
167	C₄×C₁₉	#400	1	1	yes	characteristic
168	C₂²×M₁₆	—	19	2	no	—
169	C₂²×M₁₆	—	19	2	no	—
170	C₂×#117	C₂×D₁₉	1	1	yes	characteristic
171	C₂×#117	C₂×D₁₉	1	1	yes	characteristic
172	C₂²×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
173	#424	—	38	4	no	—
174	#423	—	38	2	no	—
175	#389	—	2	2	no	—
176	#385	Dic₁₉	1	4	yes	—
177	#384	—	2	1	no	—
178	#382	—	2	1	no	—
179	#344	—	38	4	no	—
180	#343	—	38	4	no	—
181	#343	—	38	1	no	—
182	#343	—	38	1	no	—
183	C₂×C₁₉	—	2	1	no	—
184	C₂×C₁₉	—	2	2	no	—
185	C₂×C₁₉	—	2	2	no	—
186	C₂×C₁₉	—	2	1	no	—
187	C₂×C₁₉	#1441	1	1	yes	characteristic
188	C₂×C₁₉	#1914	1	1	yes	characteristic
189	C₂×C₁₉	#1915	1	1	yes	characteristic
190	C₂³×C₄	C₂×Dic₁₉	1	2	yes	—
191	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
192	C₂×M₁₆	—	19	4	no	—
193	C₂×M₁₆	—	19	4	no	—
194	C₂×M₁₆	—	38	2	no	—
195	C₂×M₁₆	—	38	2	no	—
196	C₂×M₁₆	—	38	2	no	—
197	C₂×M₁₆	—	38	2	no	—
198	C₂×M₁₆	—	38	2	no	—
199	C₂×M₁₆	—	38	2	no	—
200	C₂×M₁₆	—	38	2	no	—
201	C₂×M₁₆	—	38	2	no	—
202	C₂×M₁₆	—	38	2	no	—
203	C₂×M₁₆	—	38	2	no	—
204	C₂²×C₈	—	38	2	no	—
205	C₂²×C₈	—	38	2	no	—
206	#117	—	2	1	no	—
207	#117	—	2	1	no	—
208	#117	—	2	2	no	—
209	#117	—	2	2	no	—
210	#117	Dic₃₈	1	2	yes	—
211	#117	D₇₆	1	2	yes	—
212	C₂×C₄⋊C₄	—	2	1	no	—
213	C₂×C₄⋊C₄	—	2	2	no	—
214	C₂×C₄⋊C₄	—	2	2	no	—
215	C₂×C₄⋊C₄	—	2	1	no	—
216	C₂×C₄⋊C₄	—	2	1	no	—
217	C₂×C₄⋊C₄	—	2	4	no	—
218	C₂×C₄⋊C₄	C₂²⋊D₁₉	1	1	yes	characteristic
219	C₂×C₄⋊C₄	C₂²⋊D₁₉	1	1	yes	characteristic
220	C₂×C₂²⋊C₄	C₂×Dic₁₉	1	2	yes	—
221	C₂×C₂²⋊C₄	C₂×Dic₁₉	1	2	yes	—
222	C₂×C₄²	—	2	1	no	—
223	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
224	C₄⋊C₈	—	38	8	no	—
225	#95	—	2	4	no	—
226	C₁₉	#11206	1	1	yes	characteristic, complemented, Sylow
227	C₂⁴	C₂²×Dic₁₉	1	1	yes	characteristic
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	—	8	2	no	—
232	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
233	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
234	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
235	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
236	C₂²×C₄	#64541	1	1	yes	characteristic
237	C₂²×C₄	#64541	1	1	yes	characteristic
238	C₂²×C₄	D₄×D₁₉	1	2	yes	—
239	C₂²×C₄	#64543	1	2	yes	—
240	C₂²×C₄	Q₈×D₁₉	1	2	yes	—
241	C₂²×C₄	#64545	1	2	yes	—
242	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
243	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
244	M₁₆	—	38	4	no	—
245	M₁₆	—	38	4	no	—
246	M₁₆	—	38	4	no	—
247	M₁₆	—	38	4	no	—
248	M₁₆	—	76	2	no	—
249	M₁₆	—	76	2	no	—
250	M₁₆	—	76	2	no	—
251	M₁₆	—	76	2	no	—
252	C₂×C₈	—	38	4	no	—
253	C₂×C₈	—	38	4	no	—
254	C₂×C₈	—	76	2	no	—
255	C₂×C₈	—	76	2	no	—
256	C₂×C₈	—	76	2	no	—
257	C₂×C₈	—	76	2	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	4	1	no	—
263	C₄⋊C₄	—	4	1	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	—	8	4	no	—
266	C₂²⋊C₄	—	2	4	no	—
267	C₂²⋊C₄	—	2	2	no	—
268	C₂²⋊C₄	C₄⋊Dic₁₉	1	4	yes	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	2	2	no	—
273	C₂³	—	4	1	no	—
274	C₂³	—	4	1	no	—
275	C₂³	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
276	C₂³	#10597372	1	1	yes	characteristic
277	C₂³	D₄×Dic₁₉	1	2	yes	—
278	C₂³	Q₈×Dic₁₉	1	2	yes	—
279	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	8	4	no	—
290	C₂×C₄	#10597334	1	2	yes	—
291	C₂×C₄	#10597335	1	2	yes	—
292	C₂×C₄	#10597336	1	2	yes	—
293	C₂×C₄	#10597340	1	2	yes	—
294	C₂×C₄	#10597341	1	2	yes	—
295	C₂×C₄	#10597342	1	2	yes	—
296	C₂×C₄	#10597343	1	2	yes	—
297	C₂×C₄	#10597344	1	2	yes	—
298	C₂×C₄	#10597370	1	1	yes	characteristic
299	C₂×C₄	#10597370	1	1	yes	characteristic
300	C₂×C₄	#10597376	1	1	yes	characteristic
301	C₂×C₄	#10597376	1	1	yes	characteristic
302	C₂×C₄	#10597381	1	1	yes	characteristic
303	C₂×C₄	#10597383	1	1	yes	characteristic
304	C₂×C₄	#10597385	1	2	yes	—
305	C₂×C₄	#10597390	1	1	yes	characteristic
306	C₂×C₄	#10597392	1	2	yes	—
307	C₂×C₄	#10597393	1	1	yes	characteristic
308	C₈	—	76	4	no	—
309	C₈	—	76	4	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	4	2	no	—
315	C₂²	—	4	1	no	—
316	C₂²	—	4	1	no	—
317	C₂²	#11926404	1	1	yes	characteristic
318	C₂²	#11926424	1	1	yes	characteristic
319	C₂²	#11926430	1	2	yes	—
320	C₂²	#11926443	1	2	yes	—
321	C₂²	#11926451	1	1	yes	characteristic
322	C₄	—	4	4	no	—
323	C₄	—	4	2	no	—
324	C₄	—	4	2	no	—
325	C₄	—	8	2	no	—
326	C₄	#11926461	1	1	yes	characteristic
327	C₄	#11926464	1	1	yes	characteristic
328	C₄	#11926466	1	2	yes	—
329	C₄	#11926474	1	1	yes	characteristic
330	C₄	#11926477	1	2	yes	—
331	C₄	#11926483	1	1	yes	characteristic
332	C₂	—	2	1	no	—
333	C₂	—	2	2	no	—
334	C₂	—	2	2	no	—
335	C₂	—	2	1	no	—
336	C₂	#14561888	1	1	yes	characteristic
337	C₂	#14564183	1	1	yes	characteristic
338	C₂	#14564188	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11926424
commutator, prime radical	C₂⁴×C₁₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₁₉×#2317	C₂
socle	C₂²×C₁₉	#386
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	#2317	D₁₉
hypocenter	C₁₉	#11206
LCS	G ▷ C₂⁴×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2317

Complements

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

N	H ≅ G/N	type
226. C₁₉	#11206	semidirect

Maximal subgroups

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

subgroup	index
#14572199	2
C₂×#11926314	2
#14563374	2
C₂×#11926313	2
C₁₉×#2317	2
#11206	19