Subgroups — #14574363

Order 2432.

370 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927079	C₂	1	1	yes	maximal, characteristic, complemented
2	#11927073	C₂	1	1	yes	maximal, characteristic, complemented
3	#11926943	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926926	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926899	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926801	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926796	C₂	1	1	yes	maximal, characteristic
8	#11926793	C₂	1	1	yes	maximal, characteristic, complemented
9	#11926789	C₂	1	1	yes	maximal, characteristic, complemented
10	#11926653	C₂	1	1	yes	maximal, characteristic, complemented
11	#11926648	C₂	1	1	yes	maximal, characteristic, complemented
12	#11926647	C₂	1	1	yes	maximal, characteristic, complemented
13	#11926641	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#64515	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₉×#488	C₂	1	1	yes	maximal, characteristic, complemented
16	C₁₉×#125	C₂²	1	1	yes	characteristic
17	C₁₉×#124	C₂²	1	1	yes	characteristic
18	#10597392	C₂²	1	1	yes	characteristic
19	Q₈×Dic₁₉	C₂²	1	1	yes	characteristic
20	C₂×#64528	C₂²	1	1	yes	characteristic
21	C₂×#64527	C₂²	1	1	yes	characteristic
22	#10597384	C₂²	1	1	yes	characteristic
23	D₄×Dic₁₉	C₂²	1	1	yes	characteristic
24	C₂×#64526	C₂²	1	1	yes	characteristic
25	C₂×#64525	C₂²	1	1	yes	characteristic
26	C₂×#64515	C₂²	1	1	yes	characteristic
27	C₂×#64515	C₂²	1	1	yes	characteristic
28	#10597344	—	2	1	no	—
29	#10597343	C₂²	1	1	yes	characteristic
30	#10597341	C₂²	1	1	yes	characteristic
31	#10597340	—	2	1	no	—
32	#10597339	—	2	1	no	—
33	#10597339	C₂²	1	1	yes	characteristic
34	#10597338	—	2	1	no	—
35	#10597333	C₂²	1	1	yes	characteristic
36	#10597329	—	2	1	no	—
37	(C₂²⋊D₁₉)⋊C₄	—	2	1	no	—
38	#10597325	—	2	1	no	—
39	#10597323	—	2	1	no	—
40	#10597321	C₂²	1	1	yes	characteristic
41	C₄²×D₁₉	C₂²	1	1	yes	characteristic
42	#10597316	C₂²	1	1	yes	characteristic
43	C₁₉×#103	C₂²	1	1	yes	characteristic
44	C₁₉×#103	C₂²	1	1	yes	characteristic
45	C₁₉×#102	C₂²	1	1	yes	characteristic
46	C₁₉×#102	C₂²	1	1	yes	characteristic
47	C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
48	#10597278	C₂²	1	1	yes	characteristic
49	#10597278	C₂²	1	1	yes	characteristic
50	#10597273	C₂²	1	1	yes	characteristic
51	#10597273	C₂²	1	1	yes	characteristic
52	#10597272	C₂²	1	1	yes	characteristic
53	#10597272	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×Q₈×C₁₉	C₂³	1	1	yes	characteristic
60	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
61	C₂×C₂²⋊D₁₉	—	2	1	no	—
62	C₂²×Dic₁₉	—	2	1	no	—
63	C₂×D₇₆	—	2	1	no	—
64	C₂×D₇₆	C₂³	1	1	yes	characteristic
65	C₂×C₄×D₁₉	—	2	1	no	—
66	C₂×C₄×D₁₉	D₄	1	1	yes	characteristic
67	C₂×C₄×D₁₉	D₄	1	1	yes	characteristic
68	C₂×C₄×D₁₉	C₂³	1	1	yes	characteristic
69	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
72	C₄⋊C₄×C₁₉	—	2	1	no	—
73	C₄⋊C₄×C₁₉	—	2	1	no	—
74	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
75	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
76	C₂²⋊C₄×C₁₉	—	2	1	no	—
77	C₂²⋊C₄×C₁₉	—	2	1	no	—
78	C₄²×C₁₉	C₂³	1	1	yes	characteristic
79	C₂²⋊Dic₁₉	—	2	1	no	—
80	C₂²⋊Dic₁₉	—	2	1	no	—
81	#64528	—	2	2	no	—
82	#64527	—	2	2	no	—
83	#64526	—	2	2	no	—
84	#64525	—	2	2	no	—
85	(C₂×D₁₉)⋊C₄	—	2	1	no	—
86	(C₂×D₁₉)⋊C₄	—	2	1	no	—
87	(C₂×D₁₉)⋊C₄	—	2	1	no	—
88	(C₂×D₁₉)⋊C₄	—	2	1	no	—
89	(C₂×D₁₉)⋊C₄	—	2	1	no	—
90	(C₂×D₁₉)⋊C₄	—	2	1	no	—
91	C₄⋊Dic₁₉	—	2	1	no	—
92	C₄⋊Dic₁₉	—	2	1	no	—
93	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
94	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
95	Dic₁₉⋊C₄	—	2	1	no	—
96	Dic₁₉⋊C₄	—	2	1	no	—
97	Dic₁₉⋊C₄	—	2	1	no	—
98	Dic₁₉⋊C₄	—	2	1	no	—
99	C₄×Dic₁₉	—	2	1	no	—
100	C₄×Dic₁₉	—	2	1	no	—
101	C₄×Dic₁₉	D₄	1	1	yes	characteristic
102	C₄×Dic₁₉	D₄	1	1	yes	characteristic
103	C₄×Dic₁₉	C₂³	1	1	yes	characteristic
104	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
105	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
106	#64515	—	2	2	no	—
107	#64515	—	2	2	no	—
108	C₂³×C₁₉	—	2	1	no	—
109	C₂²×D₁₉	—	2	1	no	—
110	C₂²×D₁₉	C₂×D₄	1	1	yes	characteristic
111	Q₈×C₁₉	—	2	1	no	—
112	Q₈×C₁₉	—	2	1	no	—
113	D₄×C₁₉	—	2	1	no	—
114	D₄×C₁₉	—	2	1	no	—
115	C₂×C₄×C₁₉	—	2	1	no	—
116	C₂×C₄×C₁₉	—	2	1	no	—
117	C₂×C₄×C₁₉	—	2	1	no	—
118	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
121	C₂²⋊D₁₉	—	4	2	no	—
122	C₂×Dic₁₉	—	2	1	no	—
123	C₂×Dic₁₉	—	2	1	no	—
124	C₂×Dic₁₉	—	2	1	no	—
125	C₂×Dic₁₉	—	4	2	no	—
126	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
127	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
128	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
129	D₇₆	—	2	1	no	—
130	D₇₆	—	2	1	no	—
131	D₇₆	—	4	2	no	—
132	C₄×D₁₉	—	2	1	no	—
133	C₄×D₁₉	—	2	1	no	—
134	C₄×D₁₉	—	4	1	no	—
135	C₄×D₁₉	—	4	1	no	—
136	C₄×D₁₉	—	4	2	no	—
137	Dic₃₈	—	2	1	no	—
138	Dic₃₈	—	2	1	no	—
139	C₈×C₁₉	—	2	1	no	—
140	C₈×C₁₉	—	2	1	no	—
141	C₁₉⋊C₈	#41	1	2	yes	—
142	C₁₉⋊C₈	#41	1	2	yes	—
143	#3183	—	19	1	no	maximal, Sylow
144	C₂²×C₁₉	—	4	1	no	—
145	C₂²×C₁₉	—	4	1	no	—
146	C₂²×C₁₉	—	4	1	no	—
147	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
148	C₂×D₁₉	—	2	1	no	—
149	C₂×D₁₉	—	2	1	no	—
150	C₂×D₁₉	—	2	1	no	—
151	C₂×D₁₉	—	4	1	no	—
152	C₂×D₁₉	—	4	1	no	—
153	C₂×D₁₉	—	4	1	no	—
154	C₄×C₁₉	—	2	1	no	—
155	C₄×C₁₉	—	2	1	no	—
156	C₄×C₁₉	—	4	1	no	—
157	C₄×C₁₉	—	4	1	no	—
158	C₄×C₁₉	—	4	1	no	—
159	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
160	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
161	Dic₁₉	—	2	1	no	—
162	Dic₁₉	—	2	1	no	—
163	Dic₁₉	—	2	1	no	—
164	Dic₁₉	—	4	1	no	—
165	Dic₁₉	—	4	1	no	—
166	Dic₁₉	—	4	1	no	—
167	#529	—	19	1	no	—
168	#529	—	19	1	no	—
169	#499	—	19	1	no	—
170	#495	—	19	1	no	—
171	#488	—	19	1	no	—
172	#488	D₁₉	1	1	yes	characteristic, complemented
173	#470	—	19	1	no	—
174	#469	—	19	1	no	—
175	#468	—	19	1	no	—
176	#468	—	19	1	no	—
177	#442	—	19	1	no	—
178	#441	—	19	1	no	—
179	#440	—	19	1	no	—
180	#440	—	19	1	no	—
181	C₄×M₁₆	—	19	1	no	—
182	C₂×C₁₉	—	8	1	no	—
183	C₂×C₁₉	#532	1	1	yes	characteristic
184	C₂×C₁₉	#575	1	1	yes	characteristic
185	C₂×C₁₉	#575	1	1	yes	characteristic
186	D₁₉	—	4	1	no	—
187	D₁₉	—	8	1	no	—
188	C₂×Q₁₆	—	19	1	no	—
189	C₂×SD₁₆	—	19	1	no	—
190	C₂×SD₁₆	—	19	1	no	—
191	C₂×D₈	—	19	1	no	—
192	C₂×M₁₆	—	19	1	no	—
193	C₂×M₁₆	—	19	1	no	—
194	#126	—	38	1	no	—
195	#126	—	38	1	no	—
196	#125	—	19	1	no	—
197	#125	C₂×D₁₉	1	1	yes	characteristic
198	#124	—	19	1	no	—
199	#124	C₂×D₁₉	1	1	yes	characteristic
200	#123	—	38	1	no	—
201	#123	—	38	1	no	—
202	C₂²⋊Q₈	—	19	1	no	—
203	C₂²⋊Q₈	—	19	1	no	—
204	#121	—	19	1	no	—
205	#121	—	19	1	no	—
206	C₄×Q₈	—	19	1	no	—
207	C₄×Q₈	—	19	1	no	—
208	C₄×D₄	—	19	1	no	—
209	C₄×D₄	—	19	1	no	—
210	C₄×D₄	—	38	1	no	—
211	C₄×D₄	—	38	1	no	—
212	#117	—	38	1	no	—
213	#117	—	38	1	no	—
214	C₂×C₄²	—	19	1	no	—
215	#107	—	19	1	no	—
216	#107	—	19	1	no	—
217	#106	—	19	1	no	—
218	#106	—	19	1	no	—
219	#103	—	19	1	no	—
220	#103	—	19	1	no	—
221	#103	C₂×D₁₉	1	1	yes	characteristic
222	#103	C₂×D₁₉	1	1	yes	characteristic
223	#102	—	19	1	no	—
224	#102	—	19	1	no	—
225	#102	C₂×D₁₉	1	1	yes	characteristic
226	#102	C₂×D₁₉	1	1	yes	characteristic
227	#97	—	19	1	no	—
228	#97	—	19	1	no	—
229	C₄×C₈	—	19	1	no	—
230	C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
231	C₁₉	#3183	1	1	yes	characteristic, complemented, Sylow
232	C₂×Q₈	—	19	1	no	—
233	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
234	C₂×D₄	—	19	1	no	—
235	C₂×D₄	—	38	1	no	—
236	C₂×D₄	—	38	1	no	—
237	C₂×D₄	C₂²×D₁₉	1	1	yes	characteristic
238	C₂²×C₄	—	19	1	no	—
239	C₂²×C₄	—	19	1	no	—
240	C₂²×C₄	—	19	1	no	—
241	C₂²×C₄	—	38	1	no	—
242	C₂²×C₄	—	38	1	no	—
243	Q₁₆	—	38	2	no	—
244	SD₁₆	—	38	2	no	—
245	SD₁₆	—	38	2	no	—
246	D₈	—	38	2	no	—
247	M₁₆	—	38	2	no	—
248	M₁₆	—	38	2	no	—
249	C₂×C₈	—	19	1	no	—
250	C₂×C₈	—	19	1	no	—
251	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
252	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	—	19	1	no	—
256	C₄⋊C₄	—	19	1	no	—
257	C₄⋊C₄	—	38	1	no	—
258	C₄⋊C₄	—	38	1	no	—
259	C₄⋊C₄	—	38	1	no	—
260	C₄⋊C₄	—	38	1	no	—
261	C₄⋊C₄	—	38	1	no	—
262	C₄⋊C₄	—	38	1	no	—
263	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
264	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	2	1	no	—
267	C₂²⋊C₄	—	38	1	no	—
268	C₂²⋊C₄	—	38	1	no	—
269	C₂²⋊C₄	—	38	1	no	—
270	C₂²⋊C₄	—	38	1	no	—
271	C₂²⋊C₄	—	38	1	no	—
272	C₂²⋊C₄	—	38	1	no	—
273	C₂²⋊C₄	—	38	1	no	—
274	C₂²⋊C₄	—	38	1	no	—
275	C₄²	—	19	1	no	—
276	C₄²	—	19	1	no	—
277	C₄²	—	19	1	no	—
278	C₄²	—	38	1	no	—
279	C₄²	—	38	1	no	—
280	C₄²	C₂²×D₁₉	1	1	yes	characteristic
281	C₂³	—	2	1	no	—
282	C₂³	—	19	1	no	—
283	C₂³	—	38	1	no	—
284	Q₈	—	2	1	no	—
285	Q₈	—	2	1	no	—
286	Q₈	—	38	1	no	—
287	Q₈	—	38	1	no	—
288	D₄	—	2	1	no	—
289	D₄	—	2	1	no	—
290	D₄	—	38	1	no	—
291	D₄	—	38	1	no	—
292	D₄	—	76	2	no	—
293	D₄	—	76	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	19	1	no	—
298	C₂×C₄	—	19	1	no	—
299	C₂×C₄	—	19	1	no	—
300	C₂×C₄	—	38	1	no	—
301	C₂×C₄	—	38	1	no	—
302	C₂×C₄	—	38	1	no	—
303	C₂×C₄	—	38	1	no	—
304	C₂×C₄	—	38	1	no	—
305	C₂×C₄	—	76	1	no	—
306	C₂×C₄	—	76	1	no	—
307	C₂×C₄	—	76	2	no	—
308	C₂×C₄	—	76	2	no	—
309	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
310	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
311	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
312	C₈	—	2	1	no	—
313	C₈	—	2	1	no	—
314	C₈	—	19	2	no	—
315	C₈	—	19	2	no	—
316	C₂²	—	4	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	38	1	no	—
320	C₂²	—	38	1	no	—
321	C₂²	—	38	1	no	—
322	C₂²	—	76	1	no	—
323	C₂²	—	76	1	no	—
324	C₂²	—	76	1	no	—
325	C₂²	C₂×D₄×D₁₉	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	4	1	no	—
329	C₄	—	4	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	38	1	no	—
332	C₄	—	38	1	no	—
333	C₄	—	38	1	no	—
334	C₄	—	76	1	no	—
335	C₄	—	76	1	no	—
336	C₄	—	76	1	no	—
337	C₄	#41×D₁₉	1	1	yes	characteristic
338	C₄	#41×D₁₉	1	1	yes	characteristic
339	C₂	—	8	1	no	—
340	C₂	—	76	1	no	—
341	C₂	—	152	1	no	—
342	C₂	#11927093	1	1	yes	characteristic
343	C₂	#11927341	1	1	yes	characteristic
344	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×#488	C₂
socle	C₂²×C₁₉	C₂²×D₄
prime radical	#11926796	C₂
hypercenter	#488	D₁₉
hypocenter	C₁₉	#3183
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #488

Complements

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

N	H ≅ G/N	type
1. #11927079	C₂	semidirect
2. #11927073	C₂	semidirect
3. #11926943	C₂	semidirect
4. #11926926	C₂	semidirect
5. #11926899	C₂	semidirect
6. #11926801	C₂	semidirect
8. #11926793	C₂	semidirect
9. #11926789	C₂	semidirect
10. #11926653	C₂	semidirect
11. #11926648	C₂	semidirect
12. #11926647	C₂	semidirect
13. #11926641	C₂	semidirect
14. C₄×#64515	C₂	semidirect
15. C₁₉×#488	C₂	semidirect
172. #488	D₁₉	semidirect
231. C₁₉	#3183	semidirect

Maximal subgroups

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

subgroup	index
#11927079	2
#11927073	2
#11926943	2
#11926926	2
#11926899	2
#11926801	2
#11926796	2
#11926793	2
#11926789	2
#11926653	2
#11926648	2
#11926647	2
#11926641	2
C₄×#64515	2
C₁₉×#488	2
#3183	19