Subgroups — #18082370

Order 4864.

510 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926947	C₂	1	1	yes	maximal, characteristic
2	#14564597	C₂	1	1	yes	maximal, characteristic
3	#14564595	C₂	1	1	yes	maximal, characteristic
4	#14564437	C₂	1	1	yes	maximal, characteristic
5	#14563791	C₂	1	1	yes	maximal, characteristic
6	C₂×#11926266	C₂	1	1	yes	maximal, characteristic
7	C₁₉×#1956	C₂	1	1	yes	maximal, characteristic
8	C₂×#10597315	C₂²	1	1	yes	characteristic
9	#11926947	—	2	2	no	—
10	#11926947	C₄	1	4	yes	complemented
11	C₂×#10597264	—	2	2	no	—
12	C₂×#10597265	—	2	2	no	—
13	C₂×#10597273	—	2	2	no	—
14	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
16	#11926414	C₂²	1	1	yes	characteristic
17	#11926306	—	2	2	no	—
18	#11926306	—	2	2	no	—
19	#11926266	—	2	2	no	—
20	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
21	C₂×C₁₉×#103	—	2	2	no	—
22	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
23	C₁₉×#384	C₂²	1	1	yes	characteristic
24	C₂²×Q₈×C₁₉	—	2	1	no	—
25	C₂²×Dic₃₈	—	2	1	no	—
26	C₂²×C₈×C₁₉	D₄	1	2	yes	—
27	C₁₉×C₄⋊Q₈	—	2	1	no	—
28	C₁₉×C₄⋊Q₈	—	2	1	no	—
29	C₁₉×C₄⋊Q₈	—	2	1	no	—
30	C₁₉×C₄⋊Q₈	C₂×C₄	1	2	yes	—
31	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
32	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
33	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
34	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
35	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
36	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
37	C₂×C₄⋊Dic₁₉	—	2	1	no	—
38	C₂×C₄⋊Dic₁₉	—	2	1	no	—
39	C₂×C₄⋊Dic₁₉	—	2	1	no	—
40	C₂×C₄⋊Dic₁₉	—	2	1	no	—
41	C₂×C₄⋊Dic₁₉	—	2	1	no	—
42	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
43	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
44	#10597315	—	2	1	no	—
45	#10597315	—	2	1	no	—
46	#10597315	—	2	1	no	—
47	#10597315	C₂×C₄	1	2	yes	—
48	C₁₉×#103	—	4	4	no	—
49	C₄×C₈×C₁₉	—	2	1	no	—
50	C₄×C₈×C₁₉	—	2	1	no	—
51	#10597273	—	4	4	no	—
52	#10597265	—	4	4	no	—
53	#10597264	—	2	8	no	—
54	C₄×C₁₉⋊C₈	—	2	1	no	—
55	C₄×C₁₉⋊C₈	—	2	1	no	—
56	C₄×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
57	C₄×C₁₉⋊C₈	D₄	1	2	yes	—
58	C₂×Q₈×C₁₉	—	2	2	no	—
59	C₂×Q₈×C₁₉	—	4	1	no	—
60	C₂×Q₈×C₁₉	—	4	2	no	—
61	C₂×Q₈×C₁₉	—	4	2	no	—
62	C₂²×C₄×C₁₉	—	4	1	no	—
63	C₂²×C₄×C₁₉	—	4	1	no	—
64	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
67	C₂²×Dic₁₉	—	4	1	no	—
68	C₂²×Dic₁₉	—	4	1	no	—
69	C₂×Dic₃₈	—	2	2	no	—
70	C₂×Dic₃₈	—	4	2	no	—
71	C₂×Dic₃₈	—	4	2	no	—
72	C₂×Dic₃₈	—	4	1	no	—
73	C₂×C₈×C₁₉	—	2	2	no	—
74	C₂×C₈×C₁₉	—	2	2	no	—
75	C₂×C₈×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	1	no	—
80	C₄⋊C₄×C₁₉	—	4	2	no	—
81	C₄⋊C₄×C₁₉	—	4	2	no	—
82	C₄⋊C₄×C₁₉	—	4	2	no	—
83	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₁₉	#41	1	1	yes	characteristic
86	C₄²×C₁₉	#41	1	1	yes	characteristic
87	C₄⋊Dic₁₉	—	2	2	no	—
88	C₄⋊Dic₁₉	—	2	2	no	—
89	C₄⋊Dic₁₉	—	4	1	no	—
90	C₄⋊Dic₁₉	—	4	1	no	—
91	C₄⋊Dic₁₉	—	4	2	no	—
92	C₄⋊Dic₁₉	—	4	2	no	—
93	C₄⋊Dic₁₉	—	4	2	no	—
94	C₂×C₁₉⋊C₈	—	2	2	no	—
95	C₂×C₁₉⋊C₈	—	2	2	no	—
96	C₂×C₁₉⋊C₈	—	2	2	no	—
97	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	4	yes	—
98	C₂×C₁₉⋊C₈	SD₁₆	1	4	yes	—
99	C₂×C₁₉⋊C₈	Q₁₆	1	4	yes	—
100	#9836	—	19	1	no	maximal, Sylow
101	C₂³×C₁₉	#127	1	1	yes	characteristic
102	Q₈×C₁₉	—	4	4	no	—
103	Q₈×C₁₉	—	8	2	no	—
104	C₂×C₄×C₁₉	—	4	2	no	—
105	C₂×C₄×C₁₉	—	8	1	no	—
106	C₂×C₄×C₁₉	—	8	1	no	—
107	C₂×C₄×C₁₉	—	8	1	no	—
108	C₂×C₄×C₁₉	—	8	1	no	—
109	C₂×C₄×C₁₉	—	8	1	no	—
110	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
114	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
115	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
116	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
117	C₂×C₄×C₁₉	C₂×Q₁₆	1	2	yes	—
118	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
119	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
120	C₂×C₄×C₁₉	#137	1	2	yes	—
121	C₂×C₄×C₁₉	#137	1	2	yes	—
122	C₂×Dic₁₉	—	4	2	no	—
123	C₂×Dic₁₉	—	8	1	no	—
124	C₂×Dic₁₉	—	8	1	no	—
125	C₂×Dic₁₉	—	8	1	no	—
126	C₂×Dic₁₉	—	8	1	no	—
127	C₂×Dic₁₉	—	8	1	no	—
128	Dic₃₈	—	4	4	no	—
129	Dic₃₈	—	8	2	no	—
130	C₈×C₁₉	—	4	2	no	—
131	C₁₉⋊C₈	—	4	2	no	—
132	C₁₉⋊C₈	#103	1	8	yes	complemented
133	C₂×#500	—	19	1	no	—
134	#1990	—	19	1	no	—
135	#1990	—	19	1	no	—
136	#1956	—	19	1	no	—
137	#1956	D₁₉	1	1	yes	characteristic
138	#1844	—	19	1	no	—
139	C₂×#322	—	19	1	no	—
140	C₂²×C₁₉	#390	1	1	yes	characteristic
141	C₂²×C₁₉	#492	1	1	yes	characteristic
142	C₂²×C₁₉	#494	1	1	yes	characteristic
143	C₂²×C₁₉	#496	1	1	yes	characteristic
144	C₂²×C₁₉	#496	1	1	yes	characteristic
145	C₂²×C₁₉	#497	1	1	yes	characteristic
146	C₂²×C₁₉	#497	1	1	yes	characteristic
147	C₄×C₁₉	—	8	2	no	—
148	C₄×C₁₉	—	16	1	no	—
149	C₄×C₁₉	C₂×#103	1	2	yes	—
150	C₄×C₁₉	#441	1	2	yes	—
151	C₄×C₁₉	#442	1	2	yes	—
152	C₄×C₁₉	#469	1	2	yes	—
153	C₄×C₁₉	#470	1	2	yes	—
154	C₄×C₁₉	#487	1	2	yes	—
155	Dic₁₉	—	8	2	no	—
156	Dic₁₉	—	16	1	no	—
157	C₂×C₄⋊Q₈	—	19	1	no	—
158	C₂×C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
159	#500	—	19	4	no	—
160	#500	—	38	2	no	—
161	C₂×#107	—	38	2	no	—
162	C₂×#106	—	38	2	no	—
163	C₂×#103	—	2	2	no	—
164	C₂×#103	—	38	2	no	—
165	C₂×C₄×C₈	—	19	1	no	—
166	C₂×C₄×C₈	—	19	1	no	—
167	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
168	#384	—	19	1	no	—
169	#384	C₂×D₁₉	1	1	yes	characteristic
170	#340	—	38	2	no	—
171	#340	—	38	2	no	—
172	#322	—	38	2	no	—
173	C₂×C₁₉	#1714	1	1	yes	characteristic
174	C₂×C₁₉	#1714	1	1	yes	characteristic
175	C₂×C₁₉	#1723	1	1	yes	characteristic
176	C₂×C₁₉	#1723	1	1	yes	characteristic
177	C₂×C₁₉	#1956	1	1	yes	characteristic
178	C₂×C₁₉	#1971	1	1	yes	characteristic
179	C₂×C₁₉	#1972	1	1	yes	characteristic
180	C₂²×Q₈	—	2	1	no	—
181	C₂²×Q₈	—	38	1	no	—
182	C₂²×C₈	—	19	2	no	—
183	C₂²×C₈	—	19	2	no	—
184	C₂²×C₈	C₂²⋊D₁₉	1	2	yes	—
185	C₄⋊Q₈	—	2	1	no	—
186	C₄⋊Q₈	—	2	1	no	—
187	C₄⋊Q₈	—	2	1	no	—
188	C₄⋊Q₈	—	19	2	no	—
189	C₄⋊Q₈	—	38	1	no	—
190	C₄⋊Q₈	—	38	1	no	—
191	C₄⋊Q₈	—	38	1	no	—
192	C₄⋊Q₈	C₄×D₁₉	1	2	yes	—
193	C₂×C₄⋊C₄	—	2	1	no	—
194	C₂×C₄⋊C₄	—	2	1	no	—
195	C₂×C₄⋊C₄	—	2	1	no	—
196	C₂×C₄⋊C₄	—	2	1	no	—
197	C₂×C₄⋊C₄	—	2	1	no	—
198	C₂×C₄⋊C₄	—	38	1	no	—
199	C₂×C₄⋊C₄	—	38	1	no	—
200	C₂×C₄⋊C₄	—	38	1	no	—
201	C₂×C₄⋊C₄	—	38	1	no	—
202	C₂×C₄⋊C₄	—	38	1	no	—
203	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
204	#107	—	38	8	no	—
205	#106	—	76	4	no	—
206	#103	—	4	4	no	—
207	#103	—	76	4	no	—
208	C₄×C₈	—	2	1	no	—
209	C₄×C₈	—	2	1	no	—
210	C₄×C₈	—	19	2	no	—
211	C₄×C₈	—	19	2	no	—
212	C₄×C₈	—	38	1	no	—
213	C₄×C₈	—	38	1	no	—
214	C₁₉	#9836	1	1	yes	characteristic, complemented, Sylow
215	C₂×Q₈	—	2	2	no	—
216	C₂×Q₈	—	4	1	no	—
217	C₂×Q₈	—	4	2	no	—
218	C₂×Q₈	—	4	2	no	—
219	C₂×Q₈	—	38	2	no	—
220	C₂×Q₈	—	76	1	no	—
221	C₂×Q₈	—	76	2	no	—
222	C₂×Q₈	—	76	2	no	—
223	C₂²×C₄	—	4	1	no	—
224	C₂²×C₄	—	4	1	no	—
225	C₂²×C₄	—	76	1	no	—
226	C₂²×C₄	—	76	1	no	—
227	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
228	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
229	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	19	4	no	—
234	C₂×C₈	—	19	4	no	—
235	C₂×C₈	—	19	4	no	—
236	C₂×C₈	—	38	2	no	—
237	C₂×C₈	—	38	2	no	—
238	C₂×C₈	—	38	2	no	—
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	4	1	no	—
242	C₄⋊C₄	—	4	1	no	—
243	C₄⋊C₄	—	4	2	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₄⋊C₄	—	4	2	no	—
246	C₄⋊C₄	—	38	2	no	—
247	C₄⋊C₄	—	38	2	no	—
248	C₄⋊C₄	—	76	1	no	—
249	C₄⋊C₄	—	76	1	no	—
250	C₄⋊C₄	—	76	2	no	—
251	C₄⋊C₄	—	76	2	no	—
252	C₄⋊C₄	—	76	2	no	—
253	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
254	C₄²	#64541	1	1	yes	characteristic
255	C₄²	D₄×D₁₉	1	1	yes	characteristic
256	C₄²	#64543	1	1	yes	characteristic
257	C₂³	#10597386	1	1	yes	characteristic
258	Q₈	—	4	4	no	—
259	Q₈	—	8	2	no	—
260	Q₈	—	76	4	no	—
261	Q₈	—	152	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	8	1	no	—
264	C₂×C₄	—	8	1	no	—
265	C₂×C₄	—	8	1	no	—
266	C₂×C₄	—	8	1	no	—
267	C₂×C₄	—	8	1	no	—
268	C₂×C₄	—	76	2	no	—
269	C₂×C₄	—	152	1	no	—
270	C₂×C₄	—	152	1	no	—
271	C₂×C₄	—	152	1	no	—
272	C₂×C₄	—	152	1	no	—
273	C₂×C₄	—	152	1	no	—
274	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
275	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
276	C₂×C₄	#10597330	1	1	yes	characteristic
277	C₂×C₄	#10597331	1	1	yes	characteristic
278	C₂×C₄	SD₁₆×D₁₉	1	2	yes	—
279	C₂×C₄	#10597361	1	2	yes	—
280	C₂×C₄	Q₁₆×D₁₉	1	2	yes	—
281	C₂×C₄	#10597364	1	2	yes	—
282	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
283	C₂×C₄	#10597378	1	2	yes	—
284	C₂×C₄	#10597385	1	1	yes	characteristic
285	C₂×C₄	#10597395	1	2	yes	—
286	C₈	—	4	2	no	—
287	C₈	—	19	8	no	—
288	C₈	—	76	2	no	—
289	C₂²	#11926444	1	1	yes	characteristic
290	C₂²	#11926915	1	1	yes	characteristic
291	C₂²	#11926922	1	1	yes	characteristic
292	C₂²	#11926931	1	1	yes	characteristic
293	C₂²	#11926934	1	1	yes	characteristic
294	C₂²	#11926935	1	1	yes	characteristic
295	C₂²	#11926937	1	1	yes	characteristic
296	C₄	—	8	2	no	—
297	C₄	—	16	1	no	—
298	C₄	—	304	1	no	—
299	C₄	—	152	2	no	—
300	C₄	#103×D₁₉	1	2	yes	—
301	C₄	#11926648	1	2	yes	—
302	C₄	#11926654	1	2	yes	—
303	C₄	#11926795	1	2	yes	—
304	C₄	#11926801	1	2	yes	—
305	C₄	#11926893	1	2	yes	—
306	C₂	#14563255	1	1	yes	characteristic
307	C₂	#14563259	1	1	yes	characteristic
308	C₂	#14563310	1	1	yes	characteristic
309	C₂	#14563314	1	1	yes	characteristic
310	C₂	#14564435	1	1	yes	characteristic
311	C₂	#14564503	1	1	yes	characteristic
312	C₂	#14564510	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10597386
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1956	C₂
socle, prime radical	C₂³×C₁₉	#127
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1956	D₁₉
hypocenter	C₁₉	#9836
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1956

Complements

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

N	H ≅ G/N	type
10. #11926947	C₄	semidirect
132. C₁₉⋊C₈	#103	semidirect
214. C₁₉	#9836	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926947	2
#14564597	2
#14564595	2
#14564437	2
#14563791	2
C₂×#11926266	2
C₁₉×#1956	2
#9836	19