Subgroups — #14574511

Order 2432.

390 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927115	C₂	1	1	yes	maximal, characteristic, complemented
2	#11926882	C₂	1	1	yes	maximal, characteristic, complemented
3	#11926879	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926874	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926865	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926800	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926788	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926783	C₂	1	1	yes	maximal, characteristic, complemented
9	#11926771	C₂	1	1	yes	maximal, characteristic, complemented
10	#11926706	C₂	1	1	yes	maximal, characteristic, complemented
11	#11926706	C₂	1	1	yes	maximal, characteristic, complemented
12	#11926699	C₂	1	1	yes	maximal, characteristic, complemented
13	#11926682	C₂	1	1	yes	maximal, characteristic, complemented
14	#11926511	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₉×#537	C₂	1	1	yes	maximal, characteristic
16	C₂×#41×C₁₉	C₂²	1	1	yes	characteristic
17	C₂×#41×C₁₉	C₂²	1	1	yes	characteristic
18	C₂²×Dic₃₈	C₂²	1	1	yes	characteristic
19	C₁₉×C₂²⋊Q₈	—	2	1	no	—
20	C₁₉×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
21	C₁₉×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
22	C₁₉×#121	—	2	1	no	—
23	C₁₉×#121	—	2	1	no	—
24	C₁₉×#121	—	2	1	no	—
25	C₁₉×#121	C₂²	1	1	yes	characteristic
26	C₁₉×#121	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
28	#10597390	—	2	1	no	—
29	C₂×#64528	C₂²	1	1	yes	characteristic
30	C₂×#64528	C₂²	1	1	yes	characteristic
31	#10597382	—	2	1	no	—
32	#10597381	—	2	1	no	—
33	#10597381	—	2	1	no	—
34	C₂×#64526	C₂²	1	1	yes	characteristic
35	C₂×#64526	C₂²	1	1	yes	characteristic
36	#10597372	C₂²	1	1	yes	characteristic
37	#10597370	C₂²	1	1	yes	characteristic
38	#10597370	C₂²	1	1	yes	characteristic
39	#10597370	C₂²	1	1	yes	characteristic
40	#10597370	C₂²	1	1	yes	characteristic
41	C₂×#64520	C₂²	1	1	yes	characteristic
42	C₂²×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
43	#10597292	C₂²	1	1	yes	characteristic
44	#10597292	C₂²	1	1	yes	characteristic
45	#10597289	C₂²	1	1	yes	characteristic
46	#10597289	C₂²	1	1	yes	characteristic
47	C₂²⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
48	C₂²⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
49	C₂²⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
50	C₂²⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
51	#10597267	C₂²	1	1	yes	characteristic
52	#10597267	C₂²	1	1	yes	characteristic
53	#10597267	C₂²	1	1	yes	characteristic
54	#10597267	C₂²	1	1	yes	characteristic
55	#10597265	C₂²	1	1	yes	characteristic
56	#10597265	C₂²	1	1	yes	characteristic
57	#10597264	C₂²	1	1	yes	characteristic
58	#10597264	C₂²	1	1	yes	characteristic
59	#41×C₁₉	—	4	1	no	—
60	#41×C₁₉	—	4	1	no	—
61	#41×C₁₉	—	4	1	no	—
62	#41×C₁₉	—	4	1	no	—
63	C₂×Q₈×C₁₉	D₄	1	1	yes	characteristic
64	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
65	C₂×D₄×C₁₉	—	2	1	no	—
66	C₂×D₄×C₁₉	—	2	1	no	—
67	C₂×D₄×C₁₉	—	2	1	no	—
68	C₂×D₄×C₁₉	D₄	1	1	yes	characteristic
69	C₂×D₄×C₁₉	D₄	1	1	yes	characteristic
70	C₂×D₄×C₁₉	D₄	1	1	yes	characteristic
71	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
72	C₂²×C₄×C₁₉	—	2	1	no	—
73	C₂²×C₄×C₁₉	—	2	1	no	—
74	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
75	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
76	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
77	C₂²×Dic₁₉	—	2	1	no	—
78	C₂×Dic₃₈	—	2	1	no	—
79	C₂×Dic₃₈	—	4	1	no	—
80	C₂×Dic₃₈	—	4	1	no	—
81	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
82	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
83	C₄⋊C₄×C₁₉	—	2	1	no	—
84	C₄⋊C₄×C₁₉	—	2	1	no	—
85	C₄⋊C₄×C₁₉	—	2	1	no	—
86	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
87	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
88	C₂²⋊C₄×C₁₉	—	2	1	no	—
89	C₂²⋊C₄×C₁₉	—	2	1	no	—
90	C₂²⋊C₄×C₁₉	—	2	1	no	—
91	C₂²⋊C₄×C₁₉	—	2	1	no	—
92	C₂²⋊Dic₁₉	—	2	1	no	—
93	C₂²⋊Dic₁₉	—	2	1	no	—
94	C₂²⋊Dic₁₉	—	2	1	no	—
95	C₂²⋊Dic₁₉	—	2	1	no	—
96	C₂²⋊Dic₁₉	—	2	1	no	—
97	#64528	—	4	1	no	—
98	#64528	—	4	1	no	—
99	#64526	—	4	1	no	—
100	#64526	—	4	1	no	—
101	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
102	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
103	Dic₁₉⋊C₄	—	2	1	no	—
104	Dic₁₉⋊C₄	—	2	1	no	—
105	Dic₁₉⋊C₄	—	2	1	no	—
106	Dic₁₉⋊C₄	—	2	1	no	—
107	C₄×Dic₁₉	—	2	1	no	—
108	#64520	—	4	1	no	—
109	C₂×C₁₉⋊C₈	—	4	1	no	—
110	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
111	C₂×C₁₉⋊C₈	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	C₂×D₄	1	1	yes	characteristic
116	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
117	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
118	Q₈×C₁₉	—	2	1	no	—
119	Q₈×C₁₉	—	2	1	no	—
120	Q₈×C₁₉	—	4	1	no	—
121	D₄×C₁₉	—	2	1	no	—
122	D₄×C₁₉	—	2	1	no	—
123	D₄×C₁₉	—	4	1	no	—
124	D₄×C₁₉	—	4	1	no	—
125	D₄×C₁₉	—	4	1	no	—
126	D₄×C₁₉	—	4	2	no	—
127	D₄×C₁₉	—	8	1	no	—
128	D₄×C₁₉	—	8	1	no	—
129	C₂×C₄×C₁₉	—	2	1	no	—
130	C₂×C₄×C₁₉	—	2	1	no	—
131	C₂×C₄×C₁₉	—	2	1	no	—
132	C₂×C₄×C₁₉	—	2	1	no	—
133	C₂×C₄×C₁₉	—	2	1	no	—
134	C₂×C₄×C₁₉	—	2	1	no	—
135	C₂×C₄×C₁₉	—	2	1	no	—
136	C₂×C₄×C₁₉	—	2	1	no	—
137	C₂×C₄×C₁₉	—	2	1	no	—
138	C₂×C₄×C₁₉	—	4	2	no	—
139	C₂×C₄×C₁₉	—	8	1	no	—
140	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
141	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
142	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
143	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
144	C₂×Dic₁₉	—	2	1	no	—
145	C₂×Dic₁₉	—	2	1	no	—
146	C₂×Dic₁₉	—	2	1	no	—
147	C₂×Dic₁₉	—	2	1	no	—
148	C₂×Dic₁₉	—	4	2	no	—
149	Dic₃₈	—	2	1	no	—
150	Dic₃₈	—	2	1	no	—
151	Dic₃₈	—	2	1	no	—
152	Dic₃₈	—	2	1	no	—
153	Dic₃₈	—	4	2	no	—
154	C₁₉⋊C₈	—	2	1	no	—
155	C₁₉⋊C₈	—	2	1	no	—
156	C₁₉⋊C₈	—	2	1	no	—
157	C₁₉⋊C₈	—	2	1	no	—
158	#3201	—	19	1	no	maximal, Sylow
159	C₂²×C₁₉	—	2	1	no	—
160	C₂²×C₁₉	—	2	1	no	—
161	C₂²×C₁₉	—	2	1	no	—
162	C₂²×C₁₉	—	2	1	no	—
163	C₂²×C₁₉	—	2	1	no	—
164	C₂²×C₁₉	—	2	1	no	—
165	C₂²×C₁₉	—	2	1	no	—
166	C₂²×C₁₉	—	2	1	no	—
167	C₂²×C₁₉	—	2	1	no	—
168	C₂²×C₁₉	—	4	1	no	—
169	C₂²×C₁₉	—	4	1	no	—
170	C₂²×C₁₉	—	4	1	no	—
171	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
172	C₄×C₁₉	—	2	1	no	—
173	C₄×C₁₉	—	2	1	no	—
174	C₄×C₁₉	—	2	1	no	—
175	C₄×C₁₉	—	4	1	no	—
176	C₄×C₁₉	—	4	1	no	—
177	C₄×C₁₉	—	4	1	no	—
178	C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
179	C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
180	Dic₁₉	—	4	1	no	—
181	Dic₁₉	—	4	1	no	—
182	Dic₁₉	—	4	1	no	—
183	Dic₁₉	—	4	1	no	—
184	#537	D₁₉	1	1	yes	characteristic
185	#536	—	19	1	no	—
186	#485	—	19	1	no	—
187	#484	—	19	1	no	—
188	#483	—	19	1	no	—
189	#482	—	19	1	no	—
190	#470	—	19	1	no	—
191	#468	—	19	1	no	—
192	#467	—	19	1	no	—
193	#465	—	19	1	no	—
194	#452	—	19	1	no	—
195	#452	—	19	1	no	—
196	#451	—	19	1	no	—
197	#448	—	19	1	no	—
198	#408	—	19	1	no	—
199	C₂×C₁₉	—	4	1	no	—
200	C₂×C₁₉	—	4	1	no	—
201	C₂×C₁₉	—	4	1	no	—
202	C₂×C₁₉	—	8	1	no	—
203	C₂×C₁₉	#534	1	1	yes	characteristic
204	C₂×C₁₉	#577	1	1	yes	characteristic
205	C₂×C₁₉	#578	1	1	yes	characteristic
206	C₂×#41	C₂×D₁₉	1	1	yes	characteristic
207	C₂×#41	C₂×D₁₉	1	1	yes	characteristic
208	C₂²×Q₈	—	19	1	no	—
209	C₂×Q₁₆	—	19	1	no	—
210	C₂×Q₁₆	—	19	1	no	—
211	C₂×SD₁₆	—	19	1	no	—
212	C₂×SD₁₆	—	19	1	no	—
213	C₂×M₁₆	—	19	1	no	—
214	C₂²×C₈	—	19	1	no	—
215	C₄⋊Q₈	—	38	1	no	—
216	#124	—	38	1	no	—
217	#123	—	38	1	no	—
218	#123	—	38	1	no	—
219	C₂²⋊Q₈	—	2	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	C₂²⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
226	#121	—	2	1	no	—
227	#121	—	2	1	no	—
228	#121	—	2	1	no	—
229	#121	C₂×D₁₉	1	1	yes	characteristic
230	#121	C₂×D₁₉	1	1	yes	characteristic
231	#117	—	19	1	no	—
232	C₂×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
233	#107	—	19	1	no	—
234	#107	—	19	1	no	—
235	#106	—	19	1	no	—
236	#106	—	19	1	no	—
237	#103	—	19	1	no	—
238	#103	—	19	1	no	—
239	#103	—	19	1	no	—
240	#103	—	19	1	no	—
241	#103	—	19	1	no	—
242	#103	—	19	1	no	—
243	#102	—	19	1	no	—
244	#102	—	19	1	no	—
245	C₂²⋊C₈	—	19	1	no	—
246	C₂²⋊C₈	—	19	1	no	—
247	C₂²⋊C₈	—	19	1	no	—
248	C₂²⋊C₈	—	19	1	no	—
249	C₁₉	#3201	1	1	yes	characteristic, complemented, Sylow
250	#41	—	4	1	no	—
251	#41	—	4	1	no	—
252	#41	—	4	1	no	—
253	#41	—	4	1	no	—
254	C₂×Q₈	—	19	1	no	—
255	C₂×Q₈	—	19	1	no	—
256	C₂×Q₈	—	38	1	no	—
257	C₂×Q₈	—	76	1	no	—
258	C₂×Q₈	—	76	1	no	—
259	C₂×Q₈	C₂²⋊D₁₉	1	1	yes	characteristic
260	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
261	C₂×D₄	—	2	1	no	—
262	C₂×D₄	—	2	1	no	—
263	C₂×D₄	—	2	1	no	—
264	C₂×D₄	C₂²⋊D₁₉	1	1	yes	characteristic
265	C₂×D₄	C₂²⋊D₁₉	1	1	yes	characteristic
266	C₂×D₄	C₂²⋊D₁₉	1	1	yes	characteristic
267	C₂×D₄	C₂²×D₁₉	1	1	yes	characteristic
268	C₂²×C₄	—	2	1	no	—
269	C₂²×C₄	—	2	1	no	—
270	C₂²×C₄	—	38	1	no	—
271	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
272	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
273	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
274	Q₁₆	—	76	1	no	—
275	Q₁₆	—	76	1	no	—
276	SD₁₆	—	76	1	no	—
277	SD₁₆	—	76	1	no	—
278	M₁₆	—	76	1	no	—
279	C₂×C₈	—	19	1	no	—
280	C₂×C₈	—	19	1	no	—
281	C₂×C₈	—	19	1	no	—
282	C₂×C₈	—	19	1	no	—
283	C₂×C₈	—	76	1	no	—
284	C₄⋊C₄	—	2	1	no	—
285	C₄⋊C₄	—	2	1	no	—
286	C₄⋊C₄	—	2	1	no	—
287	C₄⋊C₄	—	19	1	no	—
288	C₄⋊C₄	—	19	1	no	—
289	C₄⋊C₄	—	38	1	no	—
290	C₄⋊C₄	—	38	1	no	—
291	C₄⋊C₄	—	38	1	no	—
292	C₄⋊C₄	—	38	1	no	—
293	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
294	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
295	C₂²⋊C₄	—	2	1	no	—
296	C₂²⋊C₄	—	2	1	no	—
297	C₂²⋊C₄	—	2	1	no	—
298	C₂²⋊C₄	—	2	1	no	—
299	C₂²⋊C₄	—	38	1	no	—
300	C₂²⋊C₄	—	38	1	no	—
301	C₂²⋊C₄	—	38	1	no	—
302	C₂²⋊C₄	—	38	1	no	—
303	C₂²⋊C₄	—	38	1	no	—
304	C₄²	—	38	1	no	—
305	C₂³	—	2	1	no	—
306	C₂³	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
307	C₂³	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
308	C₂³	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
309	Q₈	—	2	1	no	—
310	Q₈	—	2	1	no	—
311	Q₈	—	4	1	no	—
312	Q₈	—	38	1	no	—
313	Q₈	—	38	1	no	—
314	Q₈	—	38	1	no	—
315	Q₈	—	38	1	no	—
316	Q₈	—	76	2	no	—
317	D₄	—	2	1	no	—
318	D₄	—	2	1	no	—
319	D₄	—	4	1	no	—
320	D₄	—	4	1	no	—
321	D₄	—	4	1	no	—
322	D₄	—	4	2	no	—
323	D₄	—	8	1	no	—
324	D₄	—	8	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	2	1	no	—
334	C₂×C₄	—	4	2	no	—
335	C₂×C₄	—	8	1	no	—
336	C₂×C₄	—	38	1	no	—
337	C₂×C₄	—	38	1	no	—
338	C₂×C₄	—	38	1	no	—
339	C₂×C₄	—	38	1	no	—
340	C₂×C₄	—	76	2	no	—
341	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
342	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
343	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
344	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
345	C₈	—	38	1	no	—
346	C₈	—	38	1	no	—
347	C₈	—	38	1	no	—
348	C₈	—	38	1	no	—
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	2	1	no	—
353	C₂²	—	2	1	no	—
354	C₂²	—	2	1	no	—
355	C₂²	—	2	1	no	—
356	C₂²	—	2	1	no	—
357	C₂²	—	2	1	no	—
358	C₂²	—	4	1	no	—
359	C₂²	—	4	1	no	—
360	C₂²	—	4	1	no	—
361	C₂²	C₂²×C₂²⋊D₁₉	1	1	yes	characteristic
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	4	1	no	—
366	C₄	—	4	1	no	—
367	C₄	—	4	1	no	—
368	C₄	—	76	1	no	—
369	C₄	—	76	1	no	—
370	C₄	—	76	1	no	—
371	C₄	—	76	1	no	—
372	C₄	#10597430	1	1	yes	characteristic
373	C₄	#10597430	1	1	yes	characteristic
374	C₂	—	4	1	no	—
375	C₂	—	4	1	no	—
376	C₂	—	4	1	no	—
377	C₂	—	8	1	no	—
378	C₂	#11927103	1	1	yes	characteristic
379	C₂	#11927355	1	1	yes	characteristic
380	C₂	#11927359	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₉
commutator, cosocle	C₂×C₄×C₁₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉
Fitting, prime radical	C₁₉×#537	C₂
socle	C₂²×C₁₉	C₂²×D₄
hypercenter	#537	D₁₉
hypocenter	C₁₉	#3201
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ #537

Complements

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

N	H ≅ G/N	type
1. #11927115	C₂	semidirect
2. #11926882	C₂	semidirect
3. #11926879	C₂	semidirect
4. #11926874	C₂	semidirect
5. #11926865	C₂	semidirect
6. #11926800	C₂	semidirect
7. #11926788	C₂	semidirect
8. #11926783	C₂	semidirect
9. #11926771	C₂	semidirect
10. #11926706	C₂	semidirect
11. #11926706	C₂	semidirect
12. #11926699	C₂	semidirect
13. #11926682	C₂	semidirect
14. #11926511	C₂	semidirect
249. C₁₉	#3201	semidirect

Maximal subgroups

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

subgroup	index
#11927115	2
#11926882	2
#11926879	2
#11926874	2
#11926865	2
#11926800	2
#11926788	2
#11926783	2
#11926771	2
#11926706	2
#11926699	2
#11926682	2
#11926511	2
C₁₉×#537	2
#3201	19