Subgroups — #14574737

Order 2432.

376 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927163	C₂	1	1	yes	maximal, characteristic, complemented
2	#11927114	C₂	1	1	yes	maximal, characteristic, complemented
3	#11926939	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926903	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926895	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926764	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926757	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926752	C₂	1	1	yes	maximal, characteristic, complemented
9	#11926747	C₂	1	1	yes	maximal, characteristic, complemented
10	#11926703	C₂	1	1	yes	maximal, characteristic, complemented
11	#11926698	C₂	1	1	yes	maximal, characteristic, complemented
12	#11926686	C₂	1	1	yes	maximal, characteristic
13	#11926681	C₂	1	1	yes	maximal, characteristic, complemented
14	#11926603	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₉×#495	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₁₉	C₂²	1	1	yes	characteristic
17	C₂×#64543	C₂²	1	1	yes	characteristic
18	C₂×Q₁₆×C₁₉	C₂²	1	1	yes	characteristic
19	C₂×SD₁₆×C₁₉	C₂²	1	1	yes	characteristic
20	C₂×SD₁₆×C₁₉	C₂²	1	1	yes	characteristic
21	C₂×D₈×C₁₉	C₂²	1	1	yes	characteristic
22	C₁₉×#124	C₂²	1	1	yes	characteristic
23	C₁₉×#124	C₂²	1	1	yes	characteristic
24	#10597392	C₂²	1	1	yes	characteristic
25	Q₈×Dic₁₉	C₂²	1	1	yes	characteristic
26	#10597390	C₂²	1	1	yes	characteristic
27	C₂×#64528	C₂²	1	1	yes	characteristic
28	C₂×#64528	C₂²	1	1	yes	characteristic
29	#10597384	C₂²	1	1	yes	characteristic
30	#10597382	C₂²	1	1	yes	characteristic
31	D₄×Dic₁₉	C₂²	1	1	yes	characteristic
32	C₂×#64526	C₂²	1	1	yes	characteristic
33	C₂×#64526	C₂²	1	1	yes	characteristic
34	#10597332	—	2	1	no	—
35	#10597331	—	2	1	no	—
36	#10597329	—	2	1	no	—
37	#10597329	—	2	1	no	—
38	#10597325	—	2	1	no	—
39	#10597324	—	2	1	no	—
40	#10597324	—	2	1	no	—
41	#10597324	—	2	1	no	—
42	#10597318	C₂²	1	1	yes	characteristic
43	#10597315	C₂²	1	1	yes	characteristic
44	#10597315	C₂²	1	1	yes	characteristic
45	C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
46	#10597292	C₂²	1	1	yes	characteristic
47	#10597292	C₂²	1	1	yes	characteristic
48	#10597289	C₂²	1	1	yes	characteristic
49	#10597289	C₂²	1	1	yes	characteristic
50	(C₂×D₁₉)⋊C₈	C₂²	1	1	yes	characteristic
51	(C₂×D₁₉)⋊C₈	C₂²	1	1	yes	characteristic
52	#10597273	C₂²	1	1	yes	characteristic
53	#10597273	C₂²	1	1	yes	characteristic
54	#10597273	C₂²	1	1	yes	characteristic
55	#10597273	C₂²	1	1	yes	characteristic
56	Dic₁₉⋊C₈	C₂²	1	1	yes	characteristic
57	Dic₁₉⋊C₈	C₂²	1	1	yes	characteristic
58	#10597260	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
60	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
61	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
62	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₁₉	—	2	1	no	—
64	C₂×C₂²⋊D₁₉	—	2	1	no	—
65	C₂²×Dic₁₉	—	2	1	no	—
66	C₂²×Dic₁₉	—	2	1	no	—
67	Q₈×D₁₉	—	4	1	no	—
68	Q₈×D₁₉	—	4	1	no	—
69	#64543	—	4	1	no	—
70	#64543	—	4	1	no	—
71	C₂×C₄×D₁₉	—	2	1	no	—
72	C₂×C₄×D₁₉	C₂³	1	1	yes	characteristic
73	C₂×Dic₃₈	—	2	1	no	—
74	C₂×Dic₃₈	—	2	1	no	—
75	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
76	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
77	Q₁₆×C₁₉	—	4	1	no	—
78	SD₁₆×C₁₉	—	4	1	no	—
79	SD₁₆×C₁₉	—	4	1	no	—
80	D₈×C₁₉	—	4	1	no	—
81	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
82	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
83	C₂²⋊C₄×C₁₉	—	2	1	no	—
84	C₂²⋊C₄×C₁₉	—	2	1	no	—
85	C₂²⋊C₄×C₁₉	—	2	1	no	—
86	C₂²⋊C₄×C₁₉	—	2	1	no	—
87	C₄²×C₁₉	C₂³	1	1	yes	characteristic
88	C₂²⋊Dic₁₉	—	2	1	no	—
89	C₂²⋊Dic₁₉	—	2	1	no	—
90	C₂²⋊Dic₁₉	—	2	1	no	—
91	C₂²⋊Dic₁₉	—	2	1	no	—
92	#64528	—	4	1	no	—
93	#64528	—	4	1	no	—
94	#64526	—	4	1	no	—
95	#64526	—	4	1	no	—
96	(C₂×D₁₉)⋊C₄	—	2	1	no	—
97	(C₂×D₁₉)⋊C₄	D₄	1	1	yes	characteristic
98	(C₂×D₁₉)⋊C₄	D₄	1	1	yes	characteristic
99	C₄⋊Dic₁₉	—	2	1	no	—
100	C₄⋊Dic₁₉	—	2	1	no	—
101	C₄⋊Dic₁₉	—	2	1	no	—
102	C₄⋊Dic₁₉	—	2	1	no	—
103	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
104	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
105	Dic₁₉⋊C₄	—	2	1	no	—
106	Dic₁₉⋊C₄	—	2	1	no	—
107	Dic₁₉⋊C₄	—	2	1	no	—
108	Dic₁₉⋊C₄	D₄	1	1	yes	characteristic
109	Dic₁₉⋊C₄	D₄	1	1	yes	characteristic
110	C₄×Dic₁₉	—	2	1	no	—
111	C₄×Dic₁₉	C₂³	1	1	yes	characteristic
112	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
113	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
114	C₂³×C₁₉	—	2	1	no	—
115	C₂³×C₁₉	—	2	1	no	—
116	C₂²×D₁₉	C₂×D₄	1	1	yes	characteristic
117	Q₈×C₁₉	—	2	1	no	—
118	Q₈×C₁₉	—	2	1	no	—
119	Q₈×C₁₉	—	2	1	no	—
120	Q₈×C₁₉	—	2	1	no	—
121	D₄×C₁₉	—	2	1	no	—
122	D₄×C₁₉	—	2	1	no	—
123	D₄×C₁₉	—	2	1	no	—
124	D₄×C₁₉	—	2	1	no	—
125	C₂×C₄×C₁₉	—	2	1	no	—
126	C₂×C₄×C₁₉	—	2	1	no	—
127	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
130	C₂²⋊D₁₉	—	4	2	no	—
131	C₂²⋊D₁₉	—	8	1	no	—
132	C₂×Dic₁₉	—	2	1	no	—
133	C₂×Dic₁₉	—	2	1	no	—
134	C₂×Dic₁₉	—	2	1	no	—
135	C₂×Dic₁₉	—	2	1	no	—
136	C₂×Dic₁₉	—	4	2	no	—
137	C₂×Dic₁₉	—	8	1	no	—
138	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
139	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
140	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
141	C₄×D₁₉	—	2	1	no	—
142	C₄×D₁₉	—	2	1	no	—
143	C₄×D₁₉	—	4	2	no	—
144	Dic₃₈	—	2	1	no	—
145	Dic₃₈	—	2	1	no	—
146	Dic₃₈	—	2	1	no	—
147	Dic₃₈	—	2	1	no	—
148	Dic₃₈	—	4	2	no	—
149	Dic₃₈	—	8	1	no	—
150	C₈×C₁₉	—	2	1	no	—
151	C₈×C₁₉	—	2	1	no	—
152	C₁₉⋊C₈	—	2	1	no	—
153	C₁₉⋊C₈	—	2	1	no	—
154	#3225	—	19	1	no	maximal, Sylow
155	C₂²×C₁₉	—	4	1	no	—
156	C₂²×C₁₉	—	4	1	no	—
157	C₂²×C₁₉	—	4	1	no	—
158	C₂²×C₁₉	—	4	1	no	—
159	C₂²×C₁₉	—	4	1	no	—
160	C₂²×C₁₉	—	4	1	no	—
161	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
162	C₂×D₁₉	—	2	1	no	—
163	C₂×D₁₉	—	2	1	no	—
164	C₂×D₁₉	—	2	1	no	—
165	C₄×C₁₉	—	2	1	no	—
166	C₄×C₁₉	—	2	1	no	—
167	C₄×C₁₉	—	4	1	no	—
168	C₄×C₁₉	—	4	1	no	—
169	C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
170	C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
171	Dic₁₉	—	2	1	no	—
172	Dic₁₉	—	2	1	no	—
173	Dic₁₉	—	2	1	no	—
174	Dic₁₉	—	4	1	no	—
175	Dic₁₉	—	4	1	no	—
176	Dic₁₉	—	4	1	no	—
177	Dic₁₉	—	4	1	no	—
178	#542	—	19	1	no	—
179	#536	—	19	1	no	—
180	#497	—	19	1	no	—
181	#495	D₁₉	1	1	yes	characteristic, complemented
182	#489	—	19	1	no	—
183	#487	—	19	1	no	—
184	#464	—	19	1	no	—
185	#463	—	19	1	no	—
186	#462	—	19	1	no	—
187	#461	—	19	1	no	—
188	#452	—	19	1	no	—
189	#451	—	19	1	no	—
190	#449	—	19	1	no	—
191	#448	—	19	1	no	—
192	#433	—	19	1	no	—
193	C₂×C₁₉	—	8	1	no	—
194	C₂×C₁₉	—	8	1	no	—
195	C₂×C₁₉	#535	1	1	yes	characteristic
196	C₂×C₁₉	#577	1	1	yes	characteristic
197	C₂×C₁₉	#578	1	1	yes	characteristic
198	D₁₉	—	4	1	no	—
199	C₂×#41	—	19	1	no	—
200	C₂²×Q₈	—	19	1	no	—
201	C₂×Q₁₆	—	19	1	no	—
202	C₂×Q₁₆	—	19	1	no	—
203	C₂×Q₁₆	C₂×D₁₉	1	1	yes	characteristic
204	C₂×SD₁₆	—	19	1	no	—
205	C₂×SD₁₆	—	19	1	no	—
206	C₂×SD₁₆	C₂×D₁₉	1	1	yes	characteristic
207	C₂×SD₁₆	C₂×D₁₉	1	1	yes	characteristic
208	C₂×D₈	C₂×D₁₉	1	1	yes	characteristic
209	C₄⋊Q₈	—	19	1	no	—
210	C₄⋊Q₈	—	19	1	no	—
211	C₄⋊Q₈	—	19	1	no	—
212	#126	—	38	1	no	—
213	#124	—	19	1	no	—
214	#124	—	38	1	no	—
215	#124	C₂×D₁₉	1	1	yes	characteristic
216	#124	C₂×D₁₉	1	1	yes	characteristic
217	#123	—	38	1	no	—
218	#123	—	38	1	no	—
219	#123	—	38	1	no	—
220	C₂²⋊Q₈	—	19	1	no	—
221	C₂²⋊Q₈	—	38	1	no	—
222	C₂²⋊Q₈	—	38	1	no	—
223	C₂²⋊Q₈	—	38	1	no	—
224	#121	—	19	1	no	—
225	C₄×Q₈	—	19	1	no	—
226	C₄×D₄	—	19	1	no	—
227	#117	—	19	1	no	—
228	C₄⋊C₈	—	19	1	no	—
229	C₄⋊C₈	—	19	1	no	—
230	#103	—	19	1	no	—
231	#103	—	19	1	no	—
232	#103	—	19	1	no	—
233	#103	—	19	1	no	—
234	#103	—	19	1	no	—
235	#103	—	19	1	no	—
236	#102	—	19	1	no	—
237	#102	—	19	1	no	—
238	C₂²⋊C₈	—	19	1	no	—
239	C₂²⋊C₈	—	19	1	no	—
240	#97	—	19	1	no	—
241	C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
242	C₁₉	#3225	1	1	yes	characteristic, complemented, Sylow
243	#41	—	76	1	no	—
244	#41	—	76	1	no	—
245	C₂×Q₈	—	19	1	no	—
246	C₂×Q₈	—	19	1	no	—
247	C₂×Q₈	—	38	1	no	—
248	C₂×Q₈	—	38	1	no	—
249	C₂×Q₈	—	76	1	no	—
250	C₂×Q₈	—	76	1	no	—
251	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
252	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
253	C₂×D₄	—	38	1	no	—
254	C₂×D₄	—	38	1	no	—
255	C₂×D₄	C₂²×D₁₉	1	1	yes	characteristic
256	C₂×D₄	C₂²×D₁₉	1	1	yes	characteristic
257	C₂²×C₄	—	19	1	no	—
258	C₂²×C₄	—	38	1	no	—
259	C₂²×C₄	—	38	1	no	—
260	C₂²×C₄	—	38	1	no	—
261	Q₁₆	—	4	1	no	—
262	Q₁₆	—	76	1	no	—
263	Q₁₆	—	76	1	no	—
264	SD₁₆	—	4	1	no	—
265	SD₁₆	—	4	1	no	—
266	SD₁₆	—	76	1	no	—
267	SD₁₆	—	76	1	no	—
268	D₈	—	4	1	no	—
269	C₂×C₈	—	19	1	no	—
270	C₂×C₈	—	19	1	no	—
271	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
272	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
273	C₄⋊C₄	—	19	1	no	—
274	C₄⋊C₄	—	19	1	no	—
275	C₄⋊C₄	—	19	1	no	—
276	C₄⋊C₄	—	19	1	no	—
277	C₄⋊C₄	—	38	1	no	—
278	C₄⋊C₄	—	38	1	no	—
279	C₄⋊C₄	—	38	1	no	—
280	C₄⋊C₄	—	38	1	no	—
281	C₄⋊C₄	—	38	1	no	—
282	C₄⋊C₄	—	38	1	no	—
283	C₄⋊C₄	—	38	1	no	—
284	C₂²⋊C₄	—	2	1	no	—
285	C₂²⋊C₄	—	2	1	no	—
286	C₂²⋊C₄	—	2	1	no	—
287	C₂²⋊C₄	—	2	1	no	—
288	C₂²⋊C₄	—	19	1	no	—
289	C₂²⋊C₄	—	19	1	no	—
290	C₂²⋊C₄	—	38	1	no	—
291	C₂²⋊C₄	—	38	1	no	—
292	C₂²⋊C₄	—	38	1	no	—
293	C₂²⋊C₄	—	38	1	no	—
294	C₂²⋊C₄	—	38	1	no	—
295	C₄²	—	19	1	no	—
296	C₄²	—	38	1	no	—
297	C₄²	C₂²×D₁₉	1	1	yes	characteristic
298	C₂³	—	2	1	no	—
299	C₂³	—	2	1	no	—
300	C₂³	—	19	1	no	—
301	Q₈	—	2	1	no	—
302	Q₈	—	2	1	no	—
303	Q₈	—	2	1	no	—
304	Q₈	—	2	1	no	—
305	Q₈	—	38	1	no	—
306	Q₈	—	38	1	no	—
307	Q₈	—	38	1	no	—
308	Q₈	—	38	1	no	—
309	Q₈	—	76	2	no	—
310	Q₈	—	152	1	no	—
311	D₄	—	2	1	no	—
312	D₄	—	2	1	no	—
313	D₄	—	2	1	no	—
314	D₄	—	2	1	no	—
315	D₄	—	76	2	no	—
316	D₄	—	152	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	19	1	no	—
320	C₂×C₄	—	19	1	no	—
321	C₂×C₄	—	19	1	no	—
322	C₂×C₄	—	38	1	no	—
323	C₂×C₄	—	38	1	no	—
324	C₂×C₄	—	38	1	no	—
325	C₂×C₄	—	38	1	no	—
326	C₂×C₄	—	38	1	no	—
327	C₂×C₄	—	38	1	no	—
328	C₂×C₄	—	76	2	no	—
329	C₂×C₄	—	76	2	no	—
330	C₂×C₄	—	152	1	no	—
331	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
332	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
333	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	—	38	1	no	—
337	C₈	—	38	1	no	—
338	C₂²	—	4	1	no	—
339	C₂²	—	4	1	no	—
340	C₂²	—	4	1	no	—
341	C₂²	—	4	1	no	—
342	C₂²	—	4	1	no	—
343	C₂²	—	4	1	no	—
344	C₂²	—	38	1	no	—
345	C₂²	—	38	1	no	—
346	C₂²	—	38	1	no	—
347	C₂²	C₂×D₄×D₁₉	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	4	1	no	—
351	C₄	—	4	1	no	—
352	C₄	—	38	1	no	—
353	C₄	—	38	1	no	—
354	C₄	—	38	1	no	—
355	C₄	—	76	1	no	—
356	C₄	—	76	1	no	—
357	C₄	—	76	1	no	—
358	C₄	—	76	1	no	—
359	C₄	#10597430	1	1	yes	characteristic
360	C₄	#10597430	1	1	yes	characteristic
361	C₂	—	8	1	no	—
362	C₂	—	8	1	no	—
363	C₂	—	76	1	no	—
364	C₂	#11927107	1	1	yes	characteristic
365	C₂	#11927351	1	1	yes	characteristic
366	C₂	#11927358	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×#495	C₂
socle	C₂²×C₁₉	C₂²×D₄
prime radical	#11926686	C₂
hypercenter	#495	D₁₉
hypocenter	C₁₉	#3225
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #495

Complements

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

N	H ≅ G/N	type
1. #11927163	C₂	semidirect
2. #11927114	C₂	semidirect
3. #11926939	C₂	semidirect
4. #11926903	C₂	semidirect
5. #11926895	C₂	semidirect
6. #11926764	C₂	semidirect
7. #11926757	C₂	semidirect
8. #11926752	C₂	semidirect
9. #11926747	C₂	semidirect
10. #11926703	C₂	semidirect
11. #11926698	C₂	semidirect
13. #11926681	C₂	semidirect
14. #11926603	C₂	semidirect
15. C₁₉×#495	C₂	semidirect
181. #495	D₁₉	semidirect
242. C₁₉	#3225	semidirect

Maximal subgroups

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

subgroup	index
#11927163	2
#11927114	2
#11926939	2
#11926903	2
#11926895	2
#11926764	2
#11926757	2
#11926752	2
#11926747	2
#11926703	2
#11926698	2
#11926686	2
#11926681	2
#11926603	2
C₁₉×#495	2
#3225	19