Subgroups — #18082283

Order 4864.

494 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926894	C₂	1	1	yes	maximal, characteristic
2	#14564597	C₂	1	1	yes	maximal, characteristic
3	#14564596	C₂	1	1	yes	maximal, characteristic
4	#14564436	C₂	1	1	yes	maximal, characteristic
5	#14563762	C₂	1	1	yes	maximal, characteristic
6	C₂×C₈×C₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic
7	C₁₉×#1990	C₂	1	1	yes	maximal, characteristic
8	C₂×#10597315	C₂²	1	1	yes	characteristic
9	#11926894	—	2	2	no	—
10	#11926894	C₄	1	4	yes	complemented
11	C₂×#10597292	—	2	2	no	—
12	C₂×#10597267	—	2	2	no	—
13	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
15	#11926414	C₂²	1	1	yes	characteristic
16	#11926307	—	2	2	no	—
17	#11926306	—	2	2	no	—
18	#11926306	—	2	2	no	—
19	C₈×C₁₉⋊C₈	—	2	2	no	—
20	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
22	C₁₉×#384	C₂²	1	1	yes	characteristic
23	C₁₉×#340	—	2	2	no	—
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	C₄×C₈×C₁₉	—	2	1	no	—
49	C₄×C₈×C₁₉	—	2	1	no	—
50	#10597292	—	2	8	no	—
51	#10597267	—	4	4	no	—
52	C₄×C₁₉⋊C₈	—	2	1	no	—
53	C₄×C₁₉⋊C₈	—	2	1	no	—
54	C₄×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
55	C₄×C₁₉⋊C₈	D₄	1	2	yes	—
56	C₂×Q₈×C₁₉	—	2	2	no	—
57	C₂×Q₈×C₁₉	—	4	1	no	—
58	C₂×Q₈×C₁₉	—	4	2	no	—
59	C₂×Q₈×C₁₉	—	4	2	no	—
60	C₂²×C₄×C₁₉	—	4	1	no	—
61	C₂²×C₄×C₁₉	—	4	1	no	—
62	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×Dic₁₉	—	4	1	no	—
66	C₂²×Dic₁₉	—	4	1	no	—
67	C₂×Dic₃₈	—	2	2	no	—
68	C₂×Dic₃₈	—	4	2	no	—
69	C₂×Dic₃₈	—	4	2	no	—
70	C₂×Dic₃₈	—	4	1	no	—
71	C₂×C₈×C₁₉	—	2	2	no	—
72	C₂×C₈×C₁₉	—	2	2	no	—
73	C₂×C₈×C₁₉	Q₁₆	1	4	yes	—
74	C₄⋊C₄×C₁₉	—	2	2	no	—
75	C₄⋊C₄×C₁₉	—	2	2	no	—
76	C₄⋊C₄×C₁₉	—	4	1	no	—
77	C₄⋊C₄×C₁₉	—	4	1	no	—
78	C₄⋊C₄×C₁₉	—	4	2	no	—
79	C₄⋊C₄×C₁₉	—	4	2	no	—
80	C₄⋊C₄×C₁₉	—	4	2	no	—
81	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₁₉	#41	1	1	yes	characteristic
84	C₄²×C₁₉	#41	1	1	yes	characteristic
85	C₄⋊Dic₁₉	—	2	2	no	—
86	C₄⋊Dic₁₉	—	2	2	no	—
87	C₄⋊Dic₁₉	—	4	1	no	—
88	C₄⋊Dic₁₉	—	4	1	no	—
89	C₄⋊Dic₁₉	—	4	2	no	—
90	C₄⋊Dic₁₉	—	4	2	no	—
91	C₄⋊Dic₁₉	—	4	2	no	—
92	C₂×C₁₉⋊C₈	—	2	2	no	—
93	C₂×C₁₉⋊C₈	—	2	2	no	—
94	C₂×C₁₉⋊C₈	—	2	2	no	—
95	C₂×C₁₉⋊C₈	—	2	2	no	—
96	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	4	yes	—
97	C₂×C₁₉⋊C₈	SD₁₆	1	4	yes	—
98	#9821	—	19	1	no	maximal, Sylow
99	C₂³×C₁₉	#127	1	1	yes	characteristic
100	Q₈×C₁₉	—	4	4	no	—
101	Q₈×C₁₉	—	8	2	no	—
102	C₂×C₄×C₁₉	—	4	2	no	—
103	C₂×C₄×C₁₉	—	8	1	no	—
104	C₂×C₄×C₁₉	—	8	1	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	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
112	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
113	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
114	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
115	C₂×C₄×C₁₉	C₂×Q₁₆	1	2	yes	—
116	C₂×C₄×C₁₉	#135	1	2	yes	—
117	C₂×C₄×C₁₉	#135	1	2	yes	—
118	C₂×C₄×C₁₉	#135	1	2	yes	—
119	C₂×C₄×C₁₉	#135	1	2	yes	—
120	C₂×Dic₁₉	—	4	2	no	—
121	C₂×Dic₁₉	—	8	1	no	—
122	C₂×Dic₁₉	—	8	1	no	—
123	C₂×Dic₁₉	—	8	1	no	—
124	C₂×Dic₁₉	—	8	1	no	—
125	C₂×Dic₁₉	—	8	1	no	—
126	Dic₃₈	—	4	4	no	—
127	Dic₃₈	—	8	2	no	—
128	C₈×C₁₉	—	2	4	no	—
129	C₁₉⋊C₈	—	2	4	no	—
130	C₁₉⋊C₈	—	2	4	no	—
131	C₂×#487	—	19	1	no	—
132	#1990	—	19	1	no	—
133	#1990	—	19	1	no	—
134	#1990	D₁₉	1	1	yes	characteristic
135	#1956	—	19	1	no	—
136	#1834	—	19	1	no	—
137	C₂×C₈²	—	19	1	no	—
138	C₂²×C₁₉	#390	1	1	yes	characteristic
139	C₂²×C₁₉	#492	1	1	yes	characteristic
140	C₂²×C₁₉	#494	1	1	yes	characteristic
141	C₂²×C₁₉	#495	1	1	yes	characteristic
142	C₂²×C₁₉	#495	1	1	yes	characteristic
143	C₂²×C₁₉	#495	1	1	yes	characteristic
144	C₂²×C₁₉	#495	1	1	yes	characteristic
145	C₄×C₁₉	—	8	2	no	—
146	C₄×C₁₉	—	16	1	no	—
147	C₄×C₁₉	#416	1	2	yes	—
148	C₄×C₁₉	C₄×SD₁₆	1	2	yes	—
149	C₄×C₁₉	C₄×Q₁₆	1	2	yes	—
150	C₄×C₁₉	#465	1	2	yes	—
151	C₄×C₁₉	#467	1	2	yes	—
152	C₄×C₁₉	#488	1	2	yes	—
153	Dic₁₉	—	8	2	no	—
154	Dic₁₉	—	16	1	no	—
155	C₂×C₄⋊Q₈	—	19	1	no	—
156	C₂×C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
157	#487	—	19	4	no	—
158	#487	—	38	2	no	—
159	C₂×#103	—	38	2	no	—
160	C₂×#103	—	38	2	no	—
161	C₂×C₄×C₈	—	19	1	no	—
162	C₂×C₄×C₈	—	19	1	no	—
163	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
164	#384	—	19	1	no	—
165	#384	C₂×D₁₉	1	1	yes	characteristic
166	#340	—	2	2	no	—
167	#340	—	38	2	no	—
168	#340	—	38	2	no	—
169	#340	—	38	2	no	—
170	C₈²	—	38	2	no	—
171	C₂×C₁₉	#1707	1	1	yes	characteristic
172	C₂×C₁₉	#1708	1	1	yes	characteristic
173	C₂×C₁₉	#1710	1	1	yes	characteristic
174	C₂×C₁₉	#1711	1	1	yes	characteristic
175	C₂×C₁₉	#1958	1	1	yes	characteristic
176	C₂×C₁₉	#1968	1	1	yes	characteristic
177	C₂×C₁₉	#1969	1	1	yes	characteristic
178	C₂²×Q₈	—	2	1	no	—
179	C₂²×Q₈	—	38	1	no	—
180	C₂²×C₈	—	19	2	no	—
181	C₂²×C₈	—	19	2	no	—
182	C₂²×C₈	C₂²⋊D₁₉	1	2	yes	—
183	C₄⋊Q₈	—	2	1	no	—
184	C₄⋊Q₈	—	2	1	no	—
185	C₄⋊Q₈	—	2	1	no	—
186	C₄⋊Q₈	—	19	2	no	—
187	C₄⋊Q₈	—	38	1	no	—
188	C₄⋊Q₈	—	38	1	no	—
189	C₄⋊Q₈	—	38	1	no	—
190	C₄⋊Q₈	C₄×D₁₉	1	2	yes	—
191	C₂×C₄⋊C₄	—	2	1	no	—
192	C₂×C₄⋊C₄	—	2	1	no	—
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	—	38	1	no	—
197	C₂×C₄⋊C₄	—	38	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×C₄²	C₂²×D₁₉	1	1	yes	characteristic
202	#103	—	38	8	no	—
203	#103	—	76	4	no	—
204	C₄×C₈	—	2	1	no	—
205	C₄×C₈	—	2	1	no	—
206	C₄×C₈	—	19	2	no	—
207	C₄×C₈	—	19	2	no	—
208	C₄×C₈	—	38	1	no	—
209	C₄×C₈	—	38	1	no	—
210	C₁₉	#9821	1	1	yes	characteristic, complemented, Sylow
211	C₂×Q₈	—	2	2	no	—
212	C₂×Q₈	—	4	1	no	—
213	C₂×Q₈	—	4	2	no	—
214	C₂×Q₈	—	4	2	no	—
215	C₂×Q₈	—	38	2	no	—
216	C₂×Q₈	—	76	1	no	—
217	C₂×Q₈	—	76	2	no	—
218	C₂×Q₈	—	76	2	no	—
219	C₂²×C₄	—	4	1	no	—
220	C₂²×C₄	—	4	1	no	—
221	C₂²×C₄	—	76	1	no	—
222	C₂²×C₄	—	76	1	no	—
223	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
224	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
225	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	19	4	no	—
229	C₂×C₈	—	19	4	no	—
230	C₂×C₈	—	38	2	no	—
231	C₂×C₈	—	38	2	no	—
232	C₂×C₈	—	38	2	no	—
233	C₂×C₈	—	38	2	no	—
234	C₂×C₈	#64528	1	4	yes	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	4	1	no	—
238	C₄⋊C₄	—	4	1	no	—
239	C₄⋊C₄	—	4	2	no	—
240	C₄⋊C₄	—	4	2	no	—
241	C₄⋊C₄	—	4	2	no	—
242	C₄⋊C₄	—	38	2	no	—
243	C₄⋊C₄	—	38	2	no	—
244	C₄⋊C₄	—	76	1	no	—
245	C₄⋊C₄	—	76	1	no	—
246	C₄⋊C₄	—	76	2	no	—
247	C₄⋊C₄	—	76	2	no	—
248	C₄⋊C₄	—	76	2	no	—
249	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
250	C₄²	#64541	1	1	yes	characteristic
251	C₄²	D₄×D₁₉	1	1	yes	characteristic
252	C₄²	#64543	1	1	yes	characteristic
253	C₂³	#10597386	1	1	yes	characteristic
254	Q₈	—	4	4	no	—
255	Q₈	—	8	2	no	—
256	Q₈	—	76	4	no	—
257	Q₈	—	152	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	8	1	no	—
260	C₂×C₄	—	8	1	no	—
261	C₂×C₄	—	8	1	no	—
262	C₂×C₄	—	8	1	no	—
263	C₂×C₄	—	8	1	no	—
264	C₂×C₄	—	76	2	no	—
265	C₂×C₄	—	152	1	no	—
266	C₂×C₄	—	152	1	no	—
267	C₂×C₄	—	152	1	no	—
268	C₂×C₄	—	152	1	no	—
269	C₂×C₄	—	152	1	no	—
270	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
271	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
272	C₂×C₄	#10597330	1	1	yes	characteristic
273	C₂×C₄	#10597331	1	1	yes	characteristic
274	C₂×C₄	#10597358	1	2	yes	—
275	C₂×C₄	SD₁₆×D₁₉	1	2	yes	—
276	C₂×C₄	#10597362	1	2	yes	—
277	C₂×C₄	#10597365	1	2	yes	—
278	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
279	C₂×C₄	#10597385	1	1	yes	characteristic
280	C₂×C₄	C₂×#64528	1	2	yes	—
281	C₂×C₄	#10597396	1	2	yes	—
282	C₈	—	2	4	no	—
283	C₈	—	38	4	no	—
284	C₈	—	38	4	no	—
285	C₂²	#11926444	1	1	yes	characteristic
286	C₂²	#11926915	1	1	yes	characteristic
287	C₂²	#11926923	1	1	yes	characteristic
288	C₂²	#11926925	1	1	yes	characteristic
289	C₂²	#11926926	1	1	yes	characteristic
290	C₂²	#11926927	1	1	yes	characteristic
291	C₂²	#11926929	1	1	yes	characteristic
292	C₄	—	8	2	no	—
293	C₄	—	16	1	no	—
294	C₄	—	304	1	no	—
295	C₄	—	152	2	no	—
296	C₄	#11926537	1	2	yes	—
297	C₄	#11926629	1	2	yes	—
298	C₄	C₄×#64528	1	2	yes	—
299	C₄	#11926776	1	2	yes	—
300	C₄	#11926783	1	2	yes	—
301	C₄	#11926897	1	2	yes	—
302	C₂	#14563230	1	1	yes	characteristic
303	C₂	#14563235	1	1	yes	characteristic
304	C₂	#14563244	1	1	yes	characteristic
305	C₂	#14563248	1	1	yes	characteristic
306	C₂	#14564445	1	1	yes	characteristic
307	C₂	#14564486	1	1	yes	characteristic
308	C₂	#14564495	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10597386
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1990	C₂
socle, prime radical	C₂³×C₁₉	#127
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1990	D₁₉
hypocenter	C₁₉	#9821
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1990

Complements

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

N	H ≅ G/N	type
10. #11926894	C₄	semidirect
210. C₁₉	#9821	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926894	2
#14564597	2
#14564596	2
#14564436	2
#14563762	2
C₂×C₈×C₁₉⋊C₈	2
C₁₉×#1990	2
#9821	19