Subgroups — #18075802

Order 4864.

414 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	#14572346	C₂	1	1	yes	maximal, characteristic
2	#14563772	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11926294	C₂	1	1	yes	maximal, characteristic, complemented
4	#14562313	C₂	1	1	yes	maximal, characteristic
5	#14562310	C₂	1	1	yes	maximal, characteristic
6	#14561509	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1292	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10597353	—	2	1	no	—
9	#11927094	C₂²	1	1	yes	characteristic
10	#11926621	—	2	1	no	—
11	#11926621	C₄	1	2	yes	—
12	#11926609	—	2	1	no	—
13	#11926609	C₄	1	2	yes	—
14	#11926568	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
16	#11926512	—	2	1	no	—
17	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
18	#11926308	—	2	1	no	—
19	#11926308	—	2	1	no	—
20	#11926294	—	2	1	no	—
21	#11926294	—	2	1	no	—
22	#11926291	—	2	1	no	—
23	#11926289	—	2	1	no	—
24	#11926287	—	2	1	no	—
25	#11926287	—	2	1	no	—
26	C₁₉×#423	C₂²	1	1	yes	characteristic
27	C₁₉×#423	C₂²	1	1	yes	characteristic
28	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
29	C₂×#64541	—	2	1	no	—
30	C₂×M₁₆×C₁₉	—	2	1	no	—
31	C₂×M₁₆×C₁₉	—	2	1	no	—
32	C₂²×C₈×C₁₉	—	2	1	no	—
33	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
34	#10597376	—	2	1	no	—
35	#10597376	Q₈	1	2	yes	—
36	C₂×#64520	—	2	1	no	—
37	C₂²×C₁₉⋊C₈	—	2	1	no	—
38	C₂²×C₁₉⋊C₈	D₄	1	1	yes	characteristic
39	C₂²×C₁₉⋊C₈	D₄	1	1	yes	characteristic
40	#10597353	—	4	2	no	—
41	#10597353	—	4	2	no	—
42	C₂×#64515	—	2	2	no	—
43	C₂×C₈×D₁₉	—	2	2	no	—
44	#10597321	—	2	1	no	—
45	#10597320	D₄	1	1	yes	characteristic
46	C₄×D₇₆	—	2	1	no	—
47	C₄×D₇₆	C₂×C₄	1	2	yes	—
48	#10597315	D₄	1	1	yes	characteristic
49	C₁₉×C₄⋊C₈	—	2	1	no	—
50	C₁₉×C₄⋊C₈	—	2	1	no	—
51	C₁₉×C₄⋊C₈	—	2	1	no	—
52	C₁₉×C₄⋊C₈	—	2	1	no	—
53	C₁₉×C₄⋊C₈	—	2	1	no	—
54	C₁₉×C₄⋊C₈	C₂×C₄	1	2	yes	—
55	(C₂×D₁₉)⋊C₈	—	2	2	no	—
56	(C₂×D₁₉)⋊C₈	—	2	2	no	—
57	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
58	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
59	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
60	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
61	C₄×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
62	C₄×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
63	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
64	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
68	C₂×C₂²⋊D₁₉	—	2	2	no	—
69	#64541	—	4	1	no	—
70	#64541	—	4	1	no	—
71	#64541	—	4	1	no	—
72	#64541	—	4	1	no	—
73	C₂×D₇₆	—	2	1	no	—
74	C₂×D₇₆	—	2	1	no	—
75	C₂×D₇₆	C₄⋊C₄	1	2	yes	—
76	C₂×C₄×D₁₉	—	2	2	no	—
77	C₂×Dic₃₈	—	2	1	no	—
78	M₁₆×C₁₉	—	4	1	no	—
79	M₁₆×C₁₉	—	4	1	no	—
80	M₁₆×C₁₉	—	4	1	no	—
81	M₁₆×C₁₉	—	4	1	no	—
82	C₂×C₈×C₁₉	—	2	2	no	—
83	C₂×C₈×C₁₉	—	4	1	no	—
84	C₂×C₈×C₁₉	—	4	1	no	—
85	C₂×C₈×C₁₉	—	8	1	no	—
86	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₁₉	#41	1	1	yes	characteristic
90	(C₂×D₁₉)⋊C₄	—	2	2	no	—
91	(C₂×D₁₉)⋊C₄	—	2	2	no	—
92	C₄⋊Dic₁₉	—	2	1	no	—
93	C₄⋊Dic₁₉	C₄⋊C₄	1	2	yes	—
94	#64520	—	4	1	no	—
95	#64520	—	4	1	no	—
96	C₂×C₁₉⋊C₈	—	2	1	no	—
97	C₂×C₁₉⋊C₈	—	2	1	no	—
98	C₂×C₁₉⋊C₈	—	2	1	no	—
99	C₂×C₁₉⋊C₈	—	2	1	no	—
100	C₂×C₁₉⋊C₈	—	4	1	no	—
101	C₂×C₁₉⋊C₈	—	4	1	no	—
102	C₂×C₁₉⋊C₈	—	8	1	no	—
103	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
107	#64515	—	4	4	no	—
108	C₈×D₁₉	—	4	4	no	—
109	#8686	—	19	1	no	maximal, Sylow
110	C₂³×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
111	C₂²×D₁₉	—	2	2	no	—
112	C₂×C₄×C₁₉	—	2	1	no	—
113	C₂×C₄×C₁₉	—	2	1	no	—
114	C₂×C₄×C₁₉	—	4	1	no	—
115	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
118	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
119	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
120	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
121	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
122	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
123	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
124	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₂²⋊D₁₉	—	4	2	no	—
126	C₂²⋊D₁₉	—	4	2	no	—
127	C₂×Dic₁₉	—	2	2	no	—
128	D₇₆	—	4	2	no	—
129	D₇₆	—	4	1	no	—
130	D₇₆	—	4	1	no	—
131	D₇₆	—	4	1	no	—
132	D₇₆	—	4	1	no	—
133	C₄×D₁₉	—	4	2	no	—
134	C₄×D₁₉	—	4	2	no	—
135	Dic₃₈	—	4	1	no	—
136	Dic₃₈	—	4	1	no	—
137	C₈×C₁₉	—	4	2	no	—
138	C₈×C₁₉	—	8	1	no	—
139	C₈×C₁₉	—	8	1	no	—
140	C₁₉⋊C₈	—	2	1	no	—
141	C₁₉⋊C₈	—	2	1	no	—
142	C₁₉⋊C₈	—	2	1	no	—
143	C₁₉⋊C₈	—	2	1	no	—
144	C₁₉⋊C₈	—	8	1	no	—
145	C₁₉⋊C₈	—	8	1	no	—
146	#2931	—	19	1	no	—
147	#1837	—	19	1	no	—
148	C₂×#334	—	19	1	no	—
149	#1547	—	19	1	no	—
150	#1547	—	19	1	no	—
151	#1292	—	19	1	no	—
152	#1292	D₁₉	1	1	yes	characteristic, complemented
153	C₂²×C₁₉	—	2	1	no	—
154	C₂²×C₁₉	—	2	1	no	—
155	C₂²×C₁₉	#380	1	1	yes	characteristic
156	C₂²×C₁₉	#478	1	1	yes	characteristic
157	C₂²×C₁₉	#478	1	1	yes	characteristic
158	C₂×D₁₉	—	4	2	no	—
159	C₂×D₁₉	—	4	2	no	—
160	C₂×D₁₉	—	4	2	no	—
161	C₄×C₁₉	—	2	1	no	—
162	C₄×C₁₉	—	2	1	no	—
163	C₄×C₁₉	—	2	1	no	—
164	C₄×C₁₉	#416	1	1	yes	characteristic
165	C₄×C₁₉	#418	1	1	yes	characteristic
166	C₄×C₁₉	#463	1	1	yes	characteristic
167	C₄×C₁₉	#463	1	1	yes	characteristic
168	C₄×C₁₉	#471	1	1	yes	characteristic
169	C₄×C₁₉	#471	1	1	yes	characteristic
170	Dic₁₉	—	4	2	no	—
171	C₂×#131	—	38	1	no	—
172	#532	—	19	1	no	—
173	#436	—	19	2	no	—
174	#436	—	38	1	no	—
175	C₈×D₄	—	19	2	no	—
176	C₈×D₄	—	38	1	no	—
177	#423	—	19	1	no	—
178	#423	C₂×D₁₉	1	1	yes	characteristic
179	#423	C₂×D₁₉	1	1	yes	characteristic
180	C₂×C₄⋊C₈	—	19	1	no	—
181	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
182	#408	—	38	1	no	—
183	C₂×C₄×C₈	—	19	1	no	—
184	#341	—	38	1	no	—
185	#341	—	38	1	no	—
186	#334	—	38	1	no	—
187	#334	—	38	1	no	—
188	#332	—	38	1	no	—
189	#332	—	38	1	no	—
190	#331	—	38	1	no	—
191	#331	—	38	1	no	—
192	C₂×C₁₉	—	2	1	no	—
193	C₂×C₁₉	—	2	1	no	—
194	C₂×C₁₉	#1804	1	1	yes	characteristic
195	C₂×C₁₉	#1813	1	1	yes	characteristic
196	C₂×C₁₉	#1815	1	1	yes	characteristic
197	D₁₉	—	8	2	no	—
198	C₂×#41	—	38	1	no	—
199	#131	—	76	2	no	—
200	#131	—	76	2	no	—
201	C₂×M₁₆	—	2	1	no	—
202	C₂×M₁₆	—	2	1	no	—
203	C₂×M₁₆	—	38	2	no	—
204	C₂×M₁₆	—	38	1	no	—
205	C₂²×C₈	—	2	1	no	—
206	C₂²×C₈	—	19	1	no	—
207	C₂²×C₈	—	19	1	no	—
208	C₂²×C₈	—	38	2	no	—
209	C₂²×C₈	—	38	1	no	—
210	C₄⋊Q₈	—	19	1	no	—
211	#127	—	19	1	no	—
212	#124	—	38	1	no	—
213	#121	—	19	2	no	—
214	#121	—	38	1	no	—
215	C₄×D₄	—	19	2	no	—
216	C₄×D₄	—	38	1	no	—
217	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
218	C₄⋊C₈	—	2	1	no	—
219	C₄⋊C₈	—	2	1	no	—
220	C₄⋊C₈	—	2	1	no	—
221	C₄⋊C₈	—	2	1	no	—
222	C₄⋊C₈	—	2	1	no	—
223	C₄⋊C₈	—	38	1	no	—
224	C₄⋊C₈	—	38	1	no	—
225	C₄⋊C₈	—	38	1	no	—
226	C₄⋊C₈	—	38	1	no	—
227	C₄⋊C₈	C₄×D₁₉	1	2	yes	—
228	C₂²⋊C₈	—	38	2	no	—
229	C₂²⋊C₈	—	38	2	no	—
230	C₄×C₈	—	19	1	no	—
231	C₄×C₈	—	19	1	no	—
232	C₄×C₈	—	19	1	no	—
233	C₄×C₈	—	19	1	no	—
234	C₁₉	#8686	1	1	yes	characteristic, complemented, Sylow
235	#41	—	76	1	no	—
236	#41	—	76	1	no	—
237	#41	—	76	1	no	—
238	#41	—	76	1	no	—
239	C₂×Q₈	—	38	1	no	—
240	C₂×D₄	—	19	2	no	—
241	C₂×D₄	—	38	2	no	—
242	C₂×D₄	—	38	1	no	—
243	C₂×D₄	—	38	1	no	—
244	C₂²×C₄	—	38	2	no	—
245	C₂²×C₄	#64541	1	1	yes	characteristic
246	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
247	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
248	M₁₆	—	4	1	no	—
249	M₁₆	—	4	1	no	—
250	M₁₆	—	4	1	no	—
251	M₁₆	—	4	1	no	—
252	M₁₆	—	76	4	no	—
253	M₁₆	—	76	1	no	—
254	M₁₆	—	76	1	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	4	1	no	—
257	C₂×C₈	—	4	1	no	—
258	C₂×C₈	—	8	1	no	—
259	C₂×C₈	—	19	1	no	—
260	C₂×C₈	—	19	1	no	—
261	C₂×C₈	—	19	1	no	—
262	C₂×C₈	—	19	1	no	—
263	C₂×C₈	—	38	1	no	—
264	C₂×C₈	—	38	1	no	—
265	C₂×C₈	—	38	1	no	—
266	C₂×C₈	—	38	1	no	—
267	C₂×C₈	—	76	4	no	—
268	C₂×C₈	—	76	1	no	—
269	C₂×C₈	—	76	1	no	—
270	C₂×C₈	—	152	1	no	—
271	C₄⋊C₄	—	19	2	no	—
272	C₄⋊C₄	—	38	1	no	—
273	C₂²⋊C₄	—	38	2	no	—
274	C₂²⋊C₄	—	38	2	no	—
275	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
276	C₄²	#64541	1	1	yes	characteristic
277	C₄²	D₄×D₁₉	1	1	yes	characteristic
278	C₄²	D₄×D₁₉	1	1	yes	characteristic
279	C₂³	—	38	2	no	—
280	C₂³	#10597342	1	1	yes	characteristic
281	Q₈	—	76	1	no	—
282	Q₈	—	76	1	no	—
283	D₄	—	76	2	no	—
284	D₄	—	76	2	no	—
285	D₄	—	76	2	no	—
286	D₄	—	76	1	no	—
287	D₄	—	76	1	no	—
288	D₄	—	76	1	no	—
289	D₄	—	76	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	38	2	no	—
294	C₂×C₄	—	76	2	no	—
295	C₂×C₄	—	76	2	no	—
296	C₂×C₄	#10597318	1	1	yes	characteristic
297	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
298	C₂×C₄	#10597330	1	1	yes	characteristic
299	C₂×C₄	#10597330	1	1	yes	characteristic
300	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
301	C₂×C₄	#10597342	1	1	yes	characteristic
302	C₂×C₄	#10597357	1	1	yes	characteristic
303	C₂×C₄	#10597360	1	1	yes	characteristic
304	C₂×C₄	#10597362	1	1	yes	characteristic
305	C₂×C₄	#10597365	1	1	yes	characteristic
306	C₈	—	4	2	no	—
307	C₈	—	8	1	no	—
308	C₈	—	8	1	no	—
309	C₈	—	38	1	no	—
310	C₈	—	38	1	no	—
311	C₈	—	38	1	no	—
312	C₈	—	38	1	no	—
313	C₈	—	152	1	no	—
314	C₈	—	152	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	76	2	no	—
318	C₂²	—	76	2	no	—
319	C₂²	—	76	2	no	—
320	C₂²	#11926403	1	1	yes	characteristic
321	C₂²	#11926843	1	1	yes	characteristic
322	C₂²	#11926847	1	1	yes	characteristic
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	76	2	no	—
327	C₄	#11926539	1	1	yes	characteristic
328	C₄	#11926547	1	1	yes	characteristic
329	C₄	#11926758	1	1	yes	characteristic
330	C₄	#11926762	1	1	yes	characteristic
331	C₄	#11926806	1	1	yes	characteristic
332	C₄	#11926810	1	1	yes	characteristic
333	C₂	—	2	1	no	—
334	C₂	—	2	1	no	—
335	C₂	—	152	2	no	—
336	C₂	#14563644	1	1	yes	characteristic
337	C₂	#14563684	1	1	yes	characteristic
338	C₂	#14563693	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11926403
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1292	C₂
socle	C₂²×C₁₉	#380
prime radical	#11927094	C₂²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1292	D₁₉
hypocenter	C₁₉	#8686
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1292

Complements

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

N	H ≅ G/N	type
2. #14563772	C₂	semidirect
3. C₂×#11926294	C₂	semidirect
6. #14561509	C₂	semidirect
7. C₁₉×#1292	C₂	semidirect
152. #1292	D₁₉	semidirect
234. C₁₉	#8686	semidirect

Maximal subgroups

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

subgroup	index
#14572346	2
#14563772	2
C₂×#11926294	2
#14562313	2
#14562310	2
#14561509	2
C₁₉×#1292	2
#8686	19