Subgroups — #14573452

Order 2432.

384 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#125×D₁₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#64543	C₂	1	1	yes	maximal, characteristic
3	#11926854	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926852	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926850	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926848	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926846	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926844	C₂	1	1	yes	maximal, characteristic, complemented
9	#11926842	C₂	1	1	yes	maximal, characteristic, complemented
10	#11926585	C₂	1	1	yes	maximal, characteristic, complemented
11	#11926583	C₂	1	1	yes	maximal, characteristic, complemented
12	#11926566	C₂	1	1	yes	maximal, characteristic, complemented
13	#11926542	C₂	1	1	yes	maximal, characteristic
14	#11926542	C₂	1	1	yes	maximal, characteristic
15	C₁₉×#478	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#64543	C₂²	1	1	yes	characteristic
17	C₁₉×#125	C₂²	1	1	yes	characteristic
18	C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
19	D₄×Dic₁₉	C₂²	1	1	yes	characteristic
20	C₄×C₂²⋊D₁₉	—	2	1	no	—
21	C₂×C₄×Dic₁₉	—	2	1	no	—
22	C₂×#64515	C₂²	1	1	yes	characteristic
23	C₂×#64515	C₂²	1	1	yes	characteristic
24	#10597338	C₂²	1	1	yes	characteristic
25	C₄⋊C₄×D₁₉	—	2	1	no	—
26	C₄⋊C₄×D₁₉	—	2	1	no	—
27	C₄⋊C₄×D₁₉	C₂²	1	1	yes	characteristic
28	C₄⋊C₄×D₁₉	C₂²	1	1	yes	characteristic
29	#10597336	C₂²	1	1	yes	characteristic
30	#10597336	C₂²	1	1	yes	characteristic
31	#10597335	—	2	1	no	—
32	#10597335	—	2	1	no	—
33	#10597333	C₂²	1	1	yes	characteristic
34	(C₂²⋊D₁₉)⋊C₄	—	2	1	no	—
35	#10597323	—	2	1	no	—
36	C₄²×D₁₉	C₂²	1	1	yes	characteristic
37	#10597316	C₂²	1	1	yes	characteristic
38	C₄×Dic₃₈	C₂²	1	1	yes	characteristic
39	C₁₉×#107	C₂²	1	1	yes	characteristic
40	C₁₉×#106	C₂²	1	1	yes	characteristic
41	C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
42	C₁₉×#102	C₂²	1	1	yes	characteristic
43	C₁₉×#102	C₂²	1	1	yes	characteristic
44	#10597289	C₂²	1	1	yes	characteristic
45	#10597289	C₂²	1	1	yes	characteristic
46	#10597275	C₂²	1	1	yes	characteristic
47	#10597274	C₂²	1	1	yes	characteristic
48	#10597273	C₂²	1	1	yes	characteristic
49	#10597273	C₂²	1	1	yes	characteristic
50	Dic₁₉⋊C₈	C₂²	1	1	yes	characteristic
51	Dic₁₉⋊C₈	C₂²	1	1	yes	characteristic
52	#10597267	C₂²	1	1	yes	characteristic
53	#10597267	C₂²	1	1	yes	characteristic
54	#10597265	C₂²	1	1	yes	characteristic
55	#10597265	C₂²	1	1	yes	characteristic
56	#10597264	C₂²	1	1	yes	characteristic
57	#10597264	C₂²	1	1	yes	characteristic
58	C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
59	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	—	2	1	no	—
61	C₂×C₂²⋊D₁₉	—	2	1	no	—
62	C₂²×Dic₁₉	—	2	1	no	—
63	C₂²×Dic₁₉	—	2	1	no	—
64	#64543	—	2	2	no	—
65	#64543	Q₈	1	4	yes	—
66	C₂×C₄×D₁₉	—	2	1	no	—
67	C₂×C₄×D₁₉	—	2	1	no	—
68	C₂×C₄×D₁₉	D₄	1	1	yes	characteristic
69	C₂×C₄×D₁₉	D₄	1	1	yes	characteristic
70	C₂×C₄×D₁₉	C₂³	1	1	yes	characteristic
71	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
73	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₁₉	—	2	1	no	—
75	C₄⋊C₄×C₁₉	—	2	1	no	—
76	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
79	C₂²⋊C₄×C₁₉	—	2	1	no	—
80	C₄²×C₁₉	C₂³	1	1	yes	characteristic
81	C₂²⋊Dic₁₉	—	2	1	no	—
82	(C₂×D₁₉)⋊C₄	—	2	1	no	—
83	C₄⋊Dic₁₉	—	2	1	no	—
84	C₄⋊Dic₁₉	—	2	1	no	—
85	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
86	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
87	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
88	Dic₁₉⋊C₄	—	2	1	no	—
89	Dic₁₉⋊C₄	—	2	1	no	—
90	Dic₁₉⋊C₄	—	2	1	no	—
91	Dic₁₉⋊C₄	—	2	1	no	—
92	Dic₁₉⋊C₄	—	2	1	no	—
93	Dic₁₉⋊C₄	—	2	1	no	—
94	Dic₁₉⋊C₄	—	2	1	no	—
95	Dic₁₉⋊C₄	—	2	1	no	—
96	C₄×Dic₁₉	—	2	1	no	—
97	C₄×Dic₁₉	—	2	1	no	—
98	C₄×Dic₁₉	—	2	1	no	—
99	C₄×Dic₁₉	D₄	1	1	yes	characteristic
100	C₄×Dic₁₉	D₄	1	1	yes	characteristic
101	C₄×Dic₁₉	C₂³	1	1	yes	characteristic
102	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
103	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
104	#64515	—	2	2	no	—
105	#64515	—	2	2	no	—
106	C₂³×C₁₉	—	2	1	no	—
107	C₂²×D₁₉	C₂×D₄	1	1	yes	characteristic
108	D₄×C₁₉	—	2	1	no	—
109	D₄×C₁₉	C₂×Q₈	1	2	yes	—
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	—	4	2	no	—
114	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
117	C₂²⋊D₁₉	—	2	4	no	—
118	C₂×Dic₁₉	—	2	1	no	—
119	C₂×Dic₁₉	—	2	4	no	—
120	C₂×Dic₁₉	—	2	1	no	—
121	C₂×Dic₁₉	—	2	1	no	—
122	C₂×Dic₁₉	—	2	1	no	—
123	C₂×Dic₁₉	—	4	2	no	—
124	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
127	C₄×D₁₉	—	4	1	no	—
128	C₄×D₁₉	—	4	1	no	—
129	C₄×D₁₉	—	4	2	no	—
130	C₄×D₁₉	—	4	2	no	—
131	C₄×D₁₉	C₂×Q₈	1	2	yes	—
132	C₄×D₁₉	#41	1	2	yes	—
133	Dic₃₈	—	2	1	no	—
134	Dic₃₈	C₂×Q₈	1	2	yes	—
135	C₈×C₁₉	—	2	1	no	—
136	C₈×C₁₉	—	2	1	no	—
137	C₁₉⋊C₈	—	2	1	no	—
138	C₁₉⋊C₈	—	2	1	no	—
139	#3080	—	19	1	no	maximal, Sylow
140	C₂²×C₁₉	—	2	2	no	—
141	C₂²×C₁₉	—	4	1	no	—
142	C₂²×C₁₉	—	4	1	no	—
143	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
144	C₂×D₁₉	—	2	1	no	—
145	C₂×D₁₉	—	2	1	no	—
146	C₂×D₁₉	C₂²⋊Q₈	1	2	yes	—
147	C₄×C₁₉	—	2	1	no	—
148	C₄×C₁₉	—	2	1	no	—
149	C₄×C₁₉	—	4	1	no	—
150	C₄×C₁₉	—	4	1	no	—
151	C₄×C₁₉	—	4	1	no	—
152	C₄×C₁₉	C₂²×Q₈	1	1	yes	characteristic
153	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
154	Dic₁₉	—	2	2	no	—
155	Dic₁₉	—	2	1	no	—
156	Dic₁₉	—	2	1	no	—
157	Dic₁₉	—	4	1	no	—
158	Dic₁₉	—	4	1	no	—
159	Dic₁₉	—	4	1	no	—
160	Dic₁₉	C₂²⋊Q₈	1	2	yes	—
161	C₂×#125	—	19	1	no	—
162	C₄×#41	—	19	1	no	—
163	#479	—	19	1	no	—
164	#479	—	19	1	no	—
165	#479	—	19	1	no	—
166	#479	—	19	1	no	—
167	#478	—	19	1	no	—
168	#478	—	19	1	no	—
169	#478	—	19	1	no	—
170	#478	D₁₉	1	1	yes	characteristic, complemented
171	#428	—	19	1	no	—
172	#428	—	19	1	no	—
173	#423	—	19	1	no	—
174	#417	—	19	1	no	—
175	#417	—	19	1	no	—
176	C₂×C₁₉	—	4	2	no	—
177	C₂×C₁₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
178	C₂×C₁₉	#575	1	1	yes	characteristic
179	C₂×C₁₉	#575	1	1	yes	characteristic
180	D₁₉	—	2	2	no	—
181	C₂×#41	—	19	1	no	—
182	C₂×M₁₆	—	19	1	no	—
183	C₂×M₁₆	—	19	1	no	—
184	#125	—	19	1	no	—
185	#125	—	19	1	no	—
186	#125	—	19	1	no	—
187	#125	—	38	1	no	—
188	#125	—	38	1	no	—
189	#125	C₂×D₁₉	1	1	yes	characteristic
190	C₄×Q₈	—	19	1	no	—
191	C₄×Q₈	—	19	1	no	—
192	C₄×D₄	—	19	1	no	—
193	C₄×D₄	—	38	1	no	—
194	C₄×D₄	—	38	1	no	—
195	C₄×D₄	C₂×D₁₉	1	1	yes	characteristic
196	#117	—	19	1	no	—
197	#117	—	38	1	no	—
198	C₂×C₄⋊C₄	—	19	1	no	—
199	C₂×C₄⋊C₄	—	19	1	no	—
200	C₂×C₄⋊C₄	—	38	1	no	—
201	C₂×C₄⋊C₄	—	38	1	no	—
202	C₂×C₄²	—	19	1	no	—
203	C₂×C₄²	—	38	1	no	—
204	#107	—	19	1	no	—
205	#107	—	19	1	no	—
206	#107	—	19	1	no	—
207	#107	C₂×D₁₉	1	1	yes	characteristic
208	#106	—	19	1	no	—
209	#106	—	19	1	no	—
210	#106	—	19	1	no	—
211	#106	C₂×D₁₉	1	1	yes	characteristic
212	C₄⋊C₈	—	19	1	no	—
213	C₄⋊C₈	—	19	1	no	—
214	C₄⋊C₈	—	19	1	no	—
215	C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
216	#103	—	19	1	no	—
217	#103	—	19	1	no	—
218	#103	—	19	1	no	—
219	#103	—	19	1	no	—
220	#102	—	19	1	no	—
221	#102	—	19	1	no	—
222	#102	C₂×D₁₉	1	1	yes	characteristic
223	#102	C₂×D₁₉	1	1	yes	characteristic
224	C₁₉	#3080	1	1	yes	characteristic, complemented, Sylow
225	#41	—	19	4	no	—
226	#41	—	38	2	no	—
227	C₂×Q₈	—	19	1	no	—
228	C₂×D₄	—	38	1	no	—
229	C₂×D₄	C₂²×D₁₉	1	1	yes	characteristic
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	—	19	1	no	—
232	C₂²×C₄	—	19	1	no	—
233	C₂²×C₄	—	19	1	no	—
234	C₂²×C₄	—	38	1	no	—
235	C₂²×C₄	—	38	1	no	—
236	C₂²×C₄	—	38	1	no	—
237	C₂²×C₄	—	38	1	no	—
238	M₁₆	—	38	2	no	—
239	M₁₆	—	38	2	no	—
240	C₂×C₈	—	19	1	no	—
241	C₂×C₈	—	19	1	no	—
242	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
243	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	19	1	no	—
247	C₄⋊C₄	—	19	1	no	—
248	C₄⋊C₄	—	19	1	no	—
249	C₄⋊C₄	—	38	1	no	—
250	C₄⋊C₄	—	38	1	no	—
251	C₄⋊C₄	—	38	1	no	—
252	C₄⋊C₄	—	38	1	no	—
253	C₄⋊C₄	—	38	1	no	—
254	C₄⋊C₄	—	38	1	no	—
255	C₄⋊C₄	—	38	1	no	—
256	C₄⋊C₄	—	38	1	no	—
257	C₄⋊C₄	—	38	1	no	—
258	C₄⋊C₄	—	38	1	no	—
259	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
260	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
261	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
262	C₂²⋊C₄	—	2	1	no	—
263	C₂²⋊C₄	—	38	1	no	—
264	C₂²⋊C₄	—	38	1	no	—
265	C₄²	—	19	1	no	—
266	C₄²	—	19	1	no	—
267	C₄²	—	19	1	no	—
268	C₄²	—	38	1	no	—
269	C₄²	—	38	1	no	—
270	C₄²	—	38	1	no	—
271	C₄²	C₂²×D₁₉	1	1	yes	characteristic
272	C₂³	—	2	1	no	—
273	C₂³	—	19	1	no	—
274	Q₈	—	19	2	no	—
275	Q₈	—	38	1	no	—
276	D₄	—	2	1	no	—
277	D₄	—	38	4	no	—
278	D₄	Q₈×D₁₉	1	2	yes	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	19	1	no	—
284	C₂×C₄	—	19	2	no	—
285	C₂×C₄	—	19	2	no	—
286	C₂×C₄	—	19	1	no	—
287	C₂×C₄	—	19	1	no	—
288	C₂×C₄	—	38	4	no	—
289	C₂×C₄	—	38	1	no	—
290	C₂×C₄	—	38	1	no	—
291	C₂×C₄	—	38	1	no	—
292	C₂×C₄	—	38	1	no	—
293	C₂×C₄	—	76	1	no	—
294	C₂×C₄	—	76	1	no	—
295	C₂×C₄	—	76	2	no	—
296	C₂×C₄	—	76	2	no	—
297	C₂×C₄	—	76	2	no	—
298	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
299	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
300	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
301	C₈	—	2	1	no	—
302	C₈	—	2	1	no	—
303	C₈	—	38	1	no	—
304	C₈	—	38	1	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	4	1	no	—
307	C₂²	—	4	1	no	—
308	C₂²	—	19	2	no	—
309	C₂²	—	38	1	no	—
310	C₂²	—	38	1	no	—
311	C₂²	C₂×D₄×D₁₉	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	4	1	no	—
315	C₄	—	4	1	no	—
316	C₄	—	4	1	no	—
317	C₄	—	19	2	no	—
318	C₄	—	38	1	no	—
319	C₄	—	38	1	no	—
320	C₄	—	38	2	no	—
321	C₄	—	76	1	no	—
322	C₄	—	76	1	no	—
323	C₄	—	76	1	no	—
324	C₄	C₂×Q₈×D₁₉	1	1	yes	characteristic
325	C₄	#41×D₁₉	1	1	yes	characteristic
326	C₂	—	4	2	no	—
327	C₂	—	38	2	no	—
328	C₂	C₂²⋊Q₈×D₁₉	1	1	yes	characteristic
329	C₂	#11927340	1	1	yes	characteristic
330	C₂	#11927341	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₉
commutator, cosocle	C₂×C₄×C₁₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉
Fitting	C₁₉×#478	C₂
socle	C₂²×C₁₉	C₂²×D₄
prime radical	C₂×#64543	C₂²
hypercenter	#478	D₁₉
hypocenter	C₁₉	#3080
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #478

Complements

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

N	H ≅ G/N	type
1. #125×D₁₉	C₂	semidirect
3. #11926854	C₂	semidirect
4. #11926852	C₂	semidirect
5. #11926850	C₂	semidirect
6. #11926848	C₂	semidirect
7. #11926846	C₂	semidirect
8. #11926844	C₂	semidirect
9. #11926842	C₂	semidirect
10. #11926585	C₂	semidirect
11. #11926583	C₂	semidirect
12. #11926566	C₂	semidirect
15. C₁₉×#478	C₂	semidirect
170. #478	D₁₉	semidirect
224. C₁₉	#3080	semidirect

Maximal subgroups

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

subgroup	index
#125×D₁₉	2
C₄×#64543	2
#11926854	2
#11926852	2
#11926850	2
#11926848	2
#11926846	2
#11926844	2
#11926842	2
#11926585	2
#11926583	2
#11926566	2
#11926542	2
C₁₉×#478	2
#3080	19