Subgroups — #14574529

Order 2432.

378 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927120	C₂	1	1	yes	maximal, characteristic
2	#11927117	C₂	1	1	yes	maximal, characteristic
3	#11926888	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926887	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926885	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926803	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926801	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926786	C₂	1	1	yes	maximal, characteristic, complemented
9	#11926784	C₂	1	1	yes	maximal, characteristic, complemented
10	#11926702	C₂	1	1	yes	maximal, characteristic, complemented
11	#11926701	C₂	1	1	yes	maximal, characteristic, complemented
12	#11926701	C₂	1	1	yes	maximal, characteristic, complemented
13	#11926699	C₂	1	1	yes	maximal, characteristic, complemented
14	#11926509	C₂	1	1	yes	maximal, characteristic
15	C₁₉×#485	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₁₉	C₂²	1	1	yes	characteristic
17	C₂×#64541	C₂²	1	1	yes	characteristic
18	C₂²×Dic₃₈	C₂²	1	1	yes	characteristic
19	C₁₉×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
20	C₁₉×#117	C₂²	1	1	yes	characteristic
21	#10597392	C₂²	1	1	yes	characteristic
22	C₂×#64528	C₂²	1	1	yes	characteristic
23	C₂×#64528	C₂²	1	1	yes	characteristic
24	#10597370	C₂²	1	1	yes	characteristic
25	C₂×Dic₇₆	C₂²	1	1	yes	characteristic
26	C₂×Dic₇₆	C₂²	1	1	yes	characteristic
27	C₂×#64515	C₂²	1	1	yes	characteristic
28	C₂×C₈×D₁₉	C₂²	1	1	yes	characteristic
29	#10597343	C₂²	1	1	yes	characteristic
30	#10597343	C₂²	1	1	yes	characteristic
31	#10597343	C₂²	1	1	yes	characteristic
32	#10597343	C₂²	1	1	yes	characteristic
33	#10597343	C₂²	1	1	yes	characteristic
34	#10597340	—	2	1	no	—
35	#10597338	C₂²	1	1	yes	characteristic
36	#10597334	—	2	1	no	—
37	#10597332	—	2	1	no	—
38	#10597332	—	2	1	no	—
39	#10597331	—	2	1	no	—
40	#10597329	—	2	1	no	—
41	#10597321	—	2	1	no	—
42	#10597315	—	2	1	no	—
43	C₁₉×#107	C₂²	1	1	yes	characteristic
44	C₁₉×#106	C₂²	1	1	yes	characteristic
45	C₁₉×#103	C₂²	1	1	yes	characteristic
46	C₁₉×#103	C₂²	1	1	yes	characteristic
47	C₁₉×C₂²⋊C₈	C₂²	1	1	yes	characteristic
48	C₂²⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
49	(C₂×D₁₉)⋊C₈	C₂²	1	1	yes	characteristic
50	(C₂×D₁₉)⋊C₈	C₂²	1	1	yes	characteristic
51	#10597273	C₂²	1	1	yes	characteristic
52	#10597273	C₂²	1	1	yes	characteristic
53	#10597267	C₂²	1	1	yes	characteristic
54	#10597267	C₂²	1	1	yes	characteristic
55	#10597267	C₂²	1	1	yes	characteristic
56	#10597267	C₂²	1	1	yes	characteristic
57	#10597265	C₂²	1	1	yes	characteristic
58	#10597264	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
61	C₂×C₂²⋊D₁₉	D₄	1	1	yes	characteristic
62	C₂×C₂²⋊D₁₉	D₄	1	1	yes	characteristic
63	C₂²×Dic₁₉	—	2	1	no	—
64	Q₈×D₁₉	—	4	1	no	—
65	Q₈×D₁₉	—	4	1	no	—
66	#64541	—	4	1	no	—
67	#64541	—	4	1	no	—
68	C₂×D₇₆	D₄	1	1	yes	characteristic
69	C₂×C₄×D₁₉	—	2	1	no	—
70	C₂×C₄×D₁₉	C₂³	1	1	yes	characteristic
71	C₂×C₄×D₁₉	C₂³	1	1	yes	characteristic
72	C₂×Dic₃₈	—	2	1	no	—
73	C₂×Dic₃₈	—	2	1	no	—
74	C₂×Dic₃₈	—	4	1	no	—
75	C₂×Dic₃₈	—	4	1	no	—
76	C₂×Dic₃₈	D₄	1	1	yes	characteristic
77	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
78	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
79	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
80	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
81	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₁₉	—	2	1	no	—
83	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
84	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
85	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
86	C₂²⋊C₄×C₁₉	—	2	1	no	—
87	C₂²⋊C₄×C₁₉	—	2	1	no	—
88	C₄²×C₁₉	—	2	1	no	—
89	C₂²⋊Dic₁₉	—	2	1	no	—
90	#64528	—	4	1	no	—
91	#64528	—	4	1	no	—
92	(C₂×D₁₉)⋊C₄	—	2	1	no	—
93	(C₂×D₁₉)⋊C₄	—	2	1	no	—
94	(C₂×D₁₉)⋊C₄	—	2	1	no	—
95	(C₂×D₁₉)⋊C₄	—	2	1	no	—
96	(C₂×D₁₉)⋊C₄	—	2	1	no	—
97	(C₂×D₁₉)⋊C₄	—	2	1	no	—
98	(C₂×D₁₉)⋊C₄	—	2	1	no	—
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	—	2	1	no	—
104	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
105	Dic₁₉⋊C₄	—	2	1	no	—
106	Dic₁₉⋊C₄	—	2	1	no	—
107	C₄×Dic₁₉	—	2	1	no	—
108	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
109	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
110	Dic₇₆	—	4	1	no	—
111	Dic₇₆	—	4	1	no	—
112	#64515	—	4	1	no	—
113	C₈×D₁₉	—	4	1	no	—
114	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
115	C₂²×D₁₉	C₂×D₄	1	1	yes	characteristic
116	C₂²×D₁₉	C₂×D₄	1	1	yes	characteristic
117	Q₈×C₁₉	—	2	1	no	—
118	Q₈×C₁₉	—	2	1	no	—
119	C₂×C₄×C₁₉	—	2	1	no	—
120	C₂×C₄×C₁₉	—	2	1	no	—
121	C₂×C₄×C₁₉	—	2	1	no	—
122	C₂×C₄×C₁₉	—	2	1	no	—
123	C₂×C₄×C₁₉	—	2	1	no	—
124	C₂×C₄×C₁₉	—	2	1	no	—
125	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
127	C₂²⋊D₁₉	—	4	1	no	—
128	C₂²⋊D₁₉	—	4	1	no	—
129	C₂×Dic₁₉	—	2	1	no	—
130	C₂×Dic₁₉	—	2	1	no	—
131	C₂×Dic₁₉	—	2	1	no	—
132	C₂×Dic₁₉	—	2	1	no	—
133	C₂×Dic₁₉	—	4	2	no	—
134	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
135	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
136	D₇₆	—	4	1	no	—
137	C₄×D₁₉	—	2	1	no	—
138	C₄×D₁₉	—	2	1	no	—
139	C₄×D₁₉	—	2	1	no	—
140	C₄×D₁₉	—	2	1	no	—
141	C₄×D₁₉	—	4	2	no	—
142	Dic₃₈	—	2	1	no	—
143	Dic₃₈	—	2	1	no	—
144	Dic₃₈	—	2	1	no	—
145	Dic₃₈	—	2	1	no	—
146	Dic₃₈	—	2	1	no	—
147	Dic₃₈	—	2	1	no	—
148	Dic₃₈	—	4	1	no	—
149	Dic₃₈	—	4	2	no	—
150	Dic₃₈	—	4	2	no	—
151	C₈×C₁₉	—	2	1	no	—
152	C₈×C₁₉	—	2	1	no	—
153	C₁₉⋊C₈	—	2	1	no	—
154	C₁₉⋊C₈	—	2	1	no	—
155	#3202	—	19	1	no	maximal, Sylow
156	C₂²×C₁₉	—	2	1	no	—
157	C₂²×C₁₉	—	2	1	no	—
158	C₂²×C₁₉	—	2	1	no	—
159	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
160	C₂×D₁₉	—	2	1	no	—
161	C₂×D₁₉	—	2	1	no	—
162	C₂×D₁₉	—	2	1	no	—
163	C₂×D₁₉	—	2	1	no	—
164	C₂×D₁₉	—	2	1	no	—
165	C₂×D₁₉	—	2	1	no	—
166	C₄×C₁₉	—	2	1	no	—
167	C₄×C₁₉	—	4	1	no	—
168	C₄×C₁₉	—	4	1	no	—
169	C₄×C₁₉	—	4	1	no	—
170	C₄×C₁₉	—	4	1	no	—
171	C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
172	C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
173	Dic₁₉	—	2	1	no	—
174	Dic₁₉	—	2	1	no	—
175	Dic₁₉	—	4	1	no	—
176	Dic₁₉	—	4	1	no	—
177	Dic₁₉	—	4	1	no	—
178	Dic₁₉	—	4	1	no	—
179	#536	—	19	1	no	—
180	#536	—	19	1	no	—
181	#485	—	19	1	no	—
182	#485	—	19	1	no	—
183	#485	—	19	1	no	—
184	#485	D₁₉	1	1	yes	characteristic, complemented
185	#470	—	19	1	no	—
186	#470	—	19	1	no	—
187	#467	—	19	1	no	—
188	#467	—	19	1	no	—
189	#451	—	19	1	no	—
190	#451	—	19	1	no	—
191	#451	—	19	1	no	—
192	#451	—	19	1	no	—
193	#408	—	19	1	no	—
194	C₂×C₁₉	—	4	1	no	—
195	C₂×C₁₉	#534	1	1	yes	characteristic
196	C₂×C₁₉	#578	1	1	yes	characteristic
197	C₂×C₁₉	#578	1	1	yes	characteristic
198	D₁₉	—	4	1	no	—
199	D₁₉	—	4	1	no	—
200	C₂×#41	—	19	1	no	—
201	C₂²×Q₈	—	19	1	no	—
202	C₂²×Q₈	—	19	1	no	—
203	C₂×Q₁₆	—	19	1	no	—
204	C₂×Q₁₆	—	19	1	no	—
205	C₂×Q₁₆	—	19	1	no	—
206	C₂×Q₁₆	—	19	1	no	—
207	C₂×M₁₆	—	19	1	no	—
208	C₂²×C₈	—	19	1	no	—
209	C₄⋊Q₈	—	38	1	no	—
210	C₄⋊Q₈	—	38	1	no	—
211	#124	—	38	1	no	—
212	#124	—	38	1	no	—
213	#123	—	38	1	no	—
214	#123	—	38	1	no	—
215	#123	—	38	1	no	—
216	#123	—	38	1	no	—
217	C₂²⋊Q₈	—	19	1	no	—
218	C₂²⋊Q₈	—	19	1	no	—
219	C₂²⋊Q₈	—	19	1	no	—
220	C₂²⋊Q₈	—	19	1	no	—
221	C₂²⋊Q₈	—	19	1	no	—
222	C₂²⋊Q₈	—	19	1	no	—
223	C₂²⋊Q₈	—	19	1	no	—
224	C₂²⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
225	#117	—	19	1	no	—
226	#117	C₂×D₁₉	1	1	yes	characteristic
227	#107	—	19	1	no	—
228	#107	C₂×D₁₉	1	1	yes	characteristic
229	#106	—	19	1	no	—
230	#106	C₂×D₁₉	1	1	yes	characteristic
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	#103	—	19	1	no	—
237	#103	C₂×D₁₉	1	1	yes	characteristic
238	#103	C₂×D₁₉	1	1	yes	characteristic
239	C₂²⋊C₈	—	19	1	no	—
240	C₂²⋊C₈	—	19	1	no	—
241	C₂²⋊C₈	—	19	1	no	—
242	C₂²⋊C₈	C₂×D₁₉	1	1	yes	characteristic
243	C₁₉	#3202	1	1	yes	characteristic, complemented, Sylow
244	#41	—	76	1	no	—
245	#41	—	76	1	no	—
246	C₂×Q₈	—	19	1	no	—
247	C₂×Q₈	—	19	1	no	—
248	C₂×Q₈	—	19	1	no	—
249	C₂×Q₈	—	19	1	no	—
250	C₂×Q₈	—	38	1	no	—
251	C₂×Q₈	—	38	1	no	—
252	C₂×Q₈	—	76	1	no	—
253	C₂×Q₈	—	76	1	no	—
254	C₂×Q₈	—	76	1	no	—
255	C₂×Q₈	—	76	1	no	—
256	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
257	C₂×D₄	—	19	1	no	—
258	C₂×D₄	—	19	1	no	—
259	C₂×D₄	—	19	1	no	—
260	C₂²×C₄	—	19	1	no	—
261	C₂²×C₄	—	19	1	no	—
262	C₂²×C₄	—	38	1	no	—
263	C₂²×C₄	—	38	1	no	—
264	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
265	Q₁₆	—	76	1	no	—
266	Q₁₆	—	76	1	no	—
267	Q₁₆	—	76	1	no	—
268	Q₁₆	—	76	1	no	—
269	M₁₆	—	76	1	no	—
270	C₂×C₈	—	19	1	no	—
271	C₂×C₈	—	19	1	no	—
272	C₂×C₈	—	76	1	no	—
273	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
274	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
275	C₄⋊C₄	—	2	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	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
285	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
286	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
287	C₂²⋊C₄	—	2	1	no	—
288	C₂²⋊C₄	—	2	1	no	—
289	C₂²⋊C₄	—	38	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	C₂²⋊C₄	—	38	1	no	—
296	C₂²⋊C₄	—	38	1	no	—
297	C₄²	—	2	1	no	—
298	C₄²	—	38	1	no	—
299	C₂³	—	19	1	no	—
300	C₂³	—	19	1	no	—
301	C₂³	D₄×D₁₉	1	1	yes	characteristic
302	Q₈	—	2	1	no	—
303	Q₈	—	2	1	no	—
304	Q₈	—	38	1	no	—
305	Q₈	—	38	1	no	—
306	Q₈	—	38	1	no	—
307	Q₈	—	38	1	no	—
308	Q₈	—	38	1	no	—
309	Q₈	—	38	1	no	—
310	Q₈	—	76	1	no	—
311	Q₈	—	76	2	no	—
312	Q₈	—	76	2	no	—
313	D₄	—	76	1	no	—
314	D₄	—	76	1	no	—
315	D₄	—	76	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	19	1	no	—
323	C₂×C₄	—	19	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	—	38	1	no	—
329	C₂×C₄	—	38	1	no	—
330	C₂×C₄	—	38	1	no	—
331	C₂×C₄	—	38	1	no	—
332	C₂×C₄	—	76	2	no	—
333	C₂×C₄	—	76	2	no	—
334	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
335	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
336	C₈	—	2	1	no	—
337	C₈	—	2	1	no	—
338	C₈	—	38	1	no	—
339	C₈	—	38	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	38	1	no	—
344	C₂²	—	38	1	no	—
345	C₂²	—	38	1	no	—
346	C₂²	—	38	1	no	—
347	C₂²	—	38	1	no	—
348	C₂²	—	38	1	no	—
349	C₂²	C₂×D₄×D₁₉	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	4	1	no	—
352	C₄	—	4	1	no	—
353	C₄	—	4	1	no	—
354	C₄	—	4	1	no	—
355	C₄	—	38	1	no	—
356	C₄	—	38	1	no	—
357	C₄	—	76	1	no	—
358	C₄	—	76	1	no	—
359	C₄	—	76	1	no	—
360	C₄	—	76	1	no	—
361	C₄	#10597435	1	1	yes	characteristic
362	C₄	#10597435	1	1	yes	characteristic
363	C₂	—	4	1	no	—
364	C₂	—	76	1	no	—
365	C₂	—	76	1	no	—
366	C₂	#11927105	1	1	yes	characteristic
367	C₂	#11927358	1	1	yes	characteristic
368	C₂	#11927360	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×#485	C₂
socle	C₂²×C₁₉	C₂²×D₄
prime radical	C₂×#64541	C₂²
hypercenter	#485	D₁₉
hypocenter	C₁₉	#3202
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #485

Complements

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

N	H ≅ G/N	type
3. #11926888	C₂	semidirect
4. #11926887	C₂	semidirect
5. #11926885	C₂	semidirect
6. #11926803	C₂	semidirect
7. #11926801	C₂	semidirect
8. #11926786	C₂	semidirect
9. #11926784	C₂	semidirect
10. #11926702	C₂	semidirect
11. #11926701	C₂	semidirect
12. #11926701	C₂	semidirect
13. #11926699	C₂	semidirect
15. C₁₉×#485	C₂	semidirect
184. #485	D₁₉	semidirect
243. C₁₉	#3202	semidirect

Maximal subgroups

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

subgroup	index
#11927120	2
#11927117	2
#11926888	2
#11926887	2
#11926885	2
#11926803	2
#11926801	2
#11926786	2
#11926784	2
#11926702	2
#11926701	2
#11926699	2
#11926509	2
C₁₉×#485	2
#3202	19