Subgroups — #18079045

Order 4864.

492 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14574560	C₂	1	1	yes	maximal, characteristic
2	#14572381	C₂	1	1	yes	maximal, characteristic
3	#14562313	C₂	1	1	yes	maximal, characteristic
4	C₂×#11926293	C₂	1	1	yes	maximal, characteristic, complemented
5	#14561530	C₂	1	1	yes	maximal, characteristic, complemented
6	#14561486	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1997	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10597353	—	2	1	no	—
9	#11927094	C₂²	1	1	yes	characteristic
10	#11926918	—	2	1	no	—
11	#11926894	—	2	1	no	—
12	#11926746	—	2	2	no	—
13	#11926683	C₄	1	4	yes	—
14	#11926623	—	2	1	no	—
15	#11926617	—	2	2	no	—
16	C₈×D₇₆	—	2	1	no	—
17	#11926597	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
20	#11926512	—	2	1	no	—
21	#11926293	—	2	1	no	—
22	#11926293	—	2	1	no	—
23	#11926291	—	2	1	no	—
24	#11926287	—	2	1	no	—
25	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
26	C₁₉×#423	C₂²	1	1	yes	characteristic
27	C₄×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
28	C₁₉×#343	—	2	1	no	—
29	C₁₉×#343	—	2	1	no	—
30	C₂²×Q₈×C₁₉	C₂×C₄	1	2	yes	—
31	C₂×#64541	—	2	1	no	—
32	C₂×M₁₆×C₁₉	—	2	1	no	—
33	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
34	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
35	C₁₉×C₄⋊Q₈	—	2	1	no	—
36	C₁₉×C₄⋊Q₈	—	2	1	no	—
37	C₁₉×C₄⋊Q₈	—	2	1	no	—
38	C₁₉×C₄⋊Q₈	—	2	1	no	—
39	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
40	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
41	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
42	C₂×#64527	—	2	4	no	—
43	#10597376	—	2	1	no	—
44	#10597376	C₂×C₄	1	2	yes	—
45	C₂²×C₁₉⋊C₈	—	2	1	no	—
46	C₂²×C₁₉⋊C₈	—	2	1	no	—
47	#10597353	—	4	2	no	—
48	#10597353	—	4	2	no	—
49	C₂×#64515	—	2	2	no	—
50	C₂×C₈×D₁₉	—	2	2	no	—
51	#10597321	—	2	1	no	—
52	#10597320	D₄	1	1	yes	characteristic
53	C₄×D₇₆	—	2	1	no	—
54	C₄×D₇₆	D₄	1	2	yes	—
55	#10597315	D₄	1	1	yes	characteristic
56	C₁₉×C₄⋊C₈	—	2	1	no	—
57	C₁₉×C₄⋊C₈	—	2	1	no	—
58	C₁₉×#97	—	2	1	no	—
59	C₄×C₈×C₁₉	—	2	1	no	—
60	#10597292	—	2	4	no	—
61	C₂²⋊(C₁₉⋊C₈)	C₂×C₄	1	2	yes	—
62	(C₂×D₁₉)⋊C₈	—	2	2	no	—
63	(C₂×D₁₉)⋊C₈	—	2	2	no	—
64	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
65	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
66	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
67	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
68	C₄⋊(C₁₉⋊C₈)	D₄	1	1	yes	characteristic
69	C₄⋊(C₁₉⋊C₈)	D₄	1	1	yes	characteristic
70	C₄×C₁₉⋊C₈	—	2	1	no	—
71	C₂×Q₈×C₁₉	—	2	2	no	—
72	C₂×Q₈×C₁₉	—	2	2	no	—
73	C₂×Q₈×C₁₉	—	8	2	no	—
74	C₂²×C₄×C₁₉	—	2	2	no	—
75	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
78	C₂×C₂²⋊D₁₉	—	2	2	no	—
79	#64541	—	4	1	no	—
80	#64541	—	4	1	no	—
81	#64541	—	4	1	no	—
82	#64541	—	4	1	no	—
83	C₂×D₇₆	—	2	1	no	—
84	C₂×D₇₆	—	2	1	no	—
85	C₂×D₇₆	C₂²⋊C₄	1	2	yes	—
86	C₂×C₄×D₁₉	—	2	2	no	—
87	C₂×Dic₃₈	—	2	1	no	—
88	M₁₆×C₁₉	—	2	1	no	—
89	M₁₆×C₁₉	—	2	1	no	—
90	M₁₆×C₁₉	—	2	1	no	—
91	M₁₆×C₁₉	—	4	1	no	—
92	M₁₆×C₁₉	—	4	1	no	—
93	M₁₆×C₁₉	SD₁₆	1	2	yes	complemented
94	C₂×C₈×C₁₉	—	2	1	no	—
95	C₂×C₈×C₁₉	—	2	1	no	—
96	C₂×C₈×C₁₉	—	4	1	no	—
97	C₄⋊C₄×C₁₉	—	4	1	no	—
98	C₄⋊C₄×C₁₉	—	4	1	no	—
99	C₄⋊C₄×C₁₉	—	4	1	no	—
100	C₄⋊C₄×C₁₉	—	4	1	no	—
101	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₁₉	#41	1	1	yes	characteristic
105	#64527	—	8	4	no	—
106	(C₂×D₁₉)⋊C₄	—	2	2	no	—
107	(C₂×D₁₉)⋊C₄	—	2	2	no	—
108	C₄⋊Dic₁₉	—	2	1	no	—
109	C₄⋊Dic₁₉	C₂²⋊C₄	1	2	yes	—
110	C₂×C₁₉⋊C₈	—	2	2	no	—
111	C₂×C₁₉⋊C₈	—	2	1	no	—
112	C₂×C₁₉⋊C₈	—	2	1	no	—
113	C₂×C₁₉⋊C₈	—	4	1	no	—
114	C₂×C₁₉⋊C₈	—	4	1	no	—
115	C₂×C₁₉⋊C₈	—	4	1	no	—
116	C₂×C₁₉⋊C₈	—	8	1	no	—
117	#64515	—	4	4	no	—
118	C₈×D₁₉	—	4	4	no	—
119	#9267	—	19	1	no	maximal, Sylow
120	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
121	C₂²×D₁₉	—	2	2	no	—
122	Q₈×C₁₉	—	4	4	no	—
123	Q₈×C₁₉	—	8	2	no	—
124	C₂×C₄×C₁₉	—	2	1	no	—
125	C₂×C₄×C₁₉	—	2	1	no	—
126	C₂×C₄×C₁₉	—	4	1	no	—
127	C₂×C₄×C₁₉	—	4	2	no	—
128	C₂×C₄×C₁₉	—	8	2	no	—
129	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
132	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
133	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
134	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
135	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
136	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
137	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
138	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
139	C₂²⋊D₁₉	—	4	2	no	—
140	C₂²⋊D₁₉	—	4	2	no	—
141	C₂×Dic₁₉	—	2	2	no	—
142	D₇₆	—	4	2	no	—
143	D₇₆	—	4	1	no	—
144	D₇₆	—	4	1	no	—
145	D₇₆	—	4	1	no	—
146	D₇₆	—	4	1	no	—
147	C₄×D₁₉	—	4	2	no	—
148	C₄×D₁₉	—	4	2	no	—
149	Dic₃₈	—	4	1	no	—
150	Dic₃₈	—	4	1	no	—
151	C₈×C₁₉	—	2	2	no	—
152	C₈×C₁₉	—	4	1	no	—
153	C₈×C₁₉	—	8	1	no	—
154	C₁₉⋊C₈	—	4	2	no	—
155	C₁₉⋊C₈	—	4	1	no	—
156	C₁₉⋊C₈	—	4	1	no	—
157	C₁₉⋊C₈	—	8	1	no	—
158	#3206	—	19	1	no	—
159	#2935	—	19	1	no	—
160	#1997	D₁₉	1	1	yes	characteristic, complemented
161	#1547	—	19	1	no	—
162	C₂×#333	—	19	1	no	—
163	#1301	—	19	1	no	—
164	#1283	—	19	1	no	—
165	C₂²×C₁₉	—	2	1	no	—
166	C₂²×C₁₉	—	2	1	no	—
167	C₂²×C₁₉	#386	1	1	yes	characteristic
168	C₂²×C₁₉	C₄×SD₁₆	1	1	yes	characteristic
169	C₂²×C₁₉	#440	1	1	yes	characteristic
170	C₂×D₁₉	—	4	2	no	—
171	C₂×D₁₉	—	4	2	no	—
172	C₂×D₁₉	—	4	2	no	—
173	C₄×C₁₉	—	2	1	no	—
174	C₄×C₁₉	—	2	1	no	—
175	C₄×C₁₉	—	2	1	no	—
176	C₄×C₁₉	—	8	2	no	—
177	C₄×C₁₉	#450	1	1	yes	characteristic
178	C₄×C₁₉	#452	1	1	yes	characteristic
179	C₄×C₁₉	#461	1	1	yes	characteristic
180	C₄×C₁₉	#463	1	1	yes	characteristic
181	C₄×C₁₉	#471	1	1	yes	characteristic
182	C₄×C₁₉	#473	1	1	yes	characteristic
183	Dic₁₉	—	4	2	no	—
184	C₂×#131	—	38	1	no	—
185	#532	—	19	1	no	—
186	C₂×C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
187	#492	—	38	1	no	—
188	#487	—	38	1	no	—
189	#461	—	38	2	no	—
190	#448	—	19	4	no	—
191	#436	—	38	1	no	—
192	#435	—	38	2	no	—
193	C₈×D₄	—	38	1	no	—
194	#431	—	19	1	no	—
195	#423	C₂×D₁₉	1	1	yes	characteristic
196	C₂×C₄⋊C₈	—	19	1	no	—
197	C₂×C₄⋊C₈	—	19	1	no	—
198	#408	—	38	1	no	—
199	C₄×M₁₆	C₂×D₁₉	1	1	yes	characteristic
200	#343	—	2	1	no	—
201	#343	—	2	1	no	—
202	#333	—	38	1	no	—
203	#333	—	38	1	no	—
204	#332	—	38	1	no	—
205	#331	—	38	1	no	—
206	C₂×C₁₉	—	2	1	no	—
207	C₂×C₁₉	—	2	1	no	—
208	C₂×C₁₉	#1877	1	1	yes	characteristic
209	C₂×C₁₉	#1908	1	1	yes	characteristic
210	C₂×C₁₉	#1914	1	1	yes	characteristic
211	D₁₉	—	8	2	no	—
212	C₂×#41	—	38	1	no	—
213	C₂²×Q₈	C₄×D₁₉	1	2	yes	—
214	C₂×SD₁₆	—	38	4	no	—
215	#131	—	76	2	no	—
216	#131	—	76	2	no	—
217	C₂×M₁₆	—	2	1	no	—
218	C₂×M₁₆	—	38	2	no	—
219	C₂×M₁₆	C₂²⋊D₁₉	1	1	yes	characteristic
220	C₂×M₁₆	C₂²⋊D₁₉	1	1	yes	characteristic
221	C₂²×C₈	—	38	2	no	—
222	C₂²×C₈	—	38	1	no	—
223	C₂²×C₈	—	38	1	no	—
224	C₄⋊Q₈	—	2	1	no	—
225	C₄⋊Q₈	—	2	1	no	—
226	C₄⋊Q₈	—	2	1	no	—
227	C₄⋊Q₈	—	2	1	no	—
228	C₄⋊Q₈	—	19	1	no	—
229	#127	—	19	1	no	—
230	#124	—	38	1	no	—
231	#121	—	19	2	no	—
232	#121	—	38	1	no	—
233	C₄×D₄	—	19	2	no	—
234	C₄×D₄	—	38	1	no	—
235	C₂×C₄⋊C₄	—	2	1	no	—
236	C₂×C₄⋊C₄	—	2	1	no	—
237	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
238	C₄⋊C₈	—	2	1	no	—
239	C₄⋊C₈	—	2	1	no	—
240	C₄⋊C₈	—	19	1	no	—
241	C₄⋊C₈	—	19	1	no	—
242	C₄⋊C₈	—	38	1	no	—
243	C₄⋊C₈	—	38	1	no	—
244	C₄⋊C₈	—	38	1	no	—
245	C₄⋊C₈	—	38	1	no	—
246	#103	—	38	4	no	—
247	C₂²⋊C₈	—	19	2	no	—
248	C₂²⋊C₈	—	38	2	no	—
249	C₂²⋊C₈	—	38	2	no	—
250	#97	—	2	1	no	—
251	C₄×C₈	—	2	1	no	—
252	C₄×C₈	—	38	1	no	—
253	C₁₉	#9267	1	1	yes	characteristic, complemented, Sylow
254	#41	—	76	1	no	—
255	#41	—	76	1	no	—
256	#41	—	76	1	no	—
257	#41	—	76	1	no	—
258	C₂×Q₈	—	2	2	no	—
259	C₂×Q₈	—	2	2	no	—
260	C₂×Q₈	—	8	2	no	—
261	C₂×Q₈	—	38	1	no	—
262	C₂×D₄	—	19	2	no	—
263	C₂×D₄	—	38	2	no	—
264	C₂×D₄	—	38	1	no	—
265	C₂×D₄	—	38	1	no	—
266	C₂²×C₄	—	2	2	no	—
267	C₂²×C₄	—	38	2	no	—
268	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
269	C₂²×C₄	#64541	1	1	yes	characteristic
270	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
271	SD₁₆	—	152	4	no	—
272	M₁₆	—	2	1	no	—
273	M₁₆	—	2	1	no	—
274	M₁₆	—	2	1	no	—
275	M₁₆	—	4	1	no	—
276	M₁₆	—	4	1	no	—
277	M₁₆	—	76	4	no	—
278	M₁₆	#64527	1	2	yes	complemented
279	C₂×C₈	—	2	1	no	—
280	C₂×C₈	—	2	1	no	—
281	C₂×C₈	—	4	1	no	—
282	C₂×C₈	—	38	1	no	—
283	C₂×C₈	—	38	1	no	—
284	C₂×C₈	—	38	2	no	—
285	C₂×C₈	—	76	4	no	—
286	C₂×C₈	—	76	1	no	—
287	C₂×C₈	—	76	1	no	—
288	C₂×C₈	—	76	1	no	—
289	C₂×C₈	—	152	1	no	—
290	C₄⋊C₄	—	4	1	no	—
291	C₄⋊C₄	—	4	1	no	—
292	C₄⋊C₄	—	4	1	no	—
293	C₄⋊C₄	—	4	1	no	—
294	C₄⋊C₄	—	19	2	no	—
295	C₄⋊C₄	—	38	1	no	—
296	C₂²⋊C₄	—	38	2	no	—
297	C₂²⋊C₄	—	38	2	no	—
298	C₄²	D₄×D₁₉	1	1	yes	characteristic
299	C₄²	D₄×D₁₉	1	1	yes	characteristic
300	C₄²	D₄×D₁₉	1	1	yes	characteristic
301	C₄²	#64543	1	1	yes	characteristic
302	C₂³	—	38	2	no	—
303	C₂³	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
304	Q₈	—	4	4	no	—
305	Q₈	—	8	2	no	—
306	Q₈	—	76	1	no	—
307	Q₈	—	76	1	no	—
308	D₄	—	76	2	no	—
309	D₄	—	76	2	no	—
310	D₄	—	76	2	no	—
311	D₄	—	76	1	no	—
312	D₄	—	76	1	no	—
313	D₄	—	76	1	no	—
314	D₄	—	76	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	8	2	no	—
320	C₂×C₄	—	38	2	no	—
321	C₂×C₄	—	76	2	no	—
322	C₂×C₄	—	76	2	no	—
323	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
324	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
325	C₂×C₄	#10597329	1	1	yes	characteristic
326	C₂×C₄	#10597330	1	1	yes	characteristic
327	C₂×C₄	#10597383	1	1	yes	characteristic
328	C₂×C₄	#10597384	1	1	yes	characteristic
329	C₂×C₄	C₂×#64527	1	1	yes	characteristic
330	C₂×C₄	#10597388	1	1	yes	characteristic
331	C₂×C₄	#10597395	1	1	yes	characteristic
332	C₂×C₄	#10597396	1	1	yes	characteristic
333	C₈	—	2	2	no	—
334	C₈	—	4	1	no	—
335	C₈	—	8	1	no	—
336	C₈	—	76	2	no	—
337	C₈	—	76	1	no	—
338	C₈	—	76	1	no	—
339	C₈	—	152	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	76	2	no	—
343	C₂²	—	76	2	no	—
344	C₂²	—	76	2	no	—
345	C₂²	#11926420	1	1	yes	characteristic
346	C₂²	C₄×#64527	1	1	yes	characteristic
347	C₂²	#11926642	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	8	2	no	—
352	C₄	—	76	2	no	—
353	C₄	#11926694	1	1	yes	characteristic
354	C₄	#11926704	1	1	yes	characteristic
355	C₄	#11926746	1	1	yes	characteristic
356	C₄	#11926756	1	1	yes	characteristic
357	C₄	#11926810	1	1	yes	characteristic
358	C₄	#11926818	1	1	yes	characteristic
359	C₂	—	2	1	no	—
360	C₂	—	2	1	no	—
361	C₂	—	152	2	no	—
362	C₂	#14563972	1	1	yes	characteristic
363	C₂	#14564147	1	1	yes	characteristic
364	C₂	#14564179	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11926420
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1997	C₂
socle	C₂²×C₁₉	#386
prime radical	#11927094	C₂²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1997	D₁₉
hypocenter	C₁₉	#9267
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1997

Complements

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

N	H ≅ G/N	type
4. C₂×#11926293	C₂	semidirect
5. #14561530	C₂	semidirect
6. #14561486	C₂	semidirect
7. C₁₉×#1997	C₂	semidirect
93. M₁₆×C₁₉	SD₁₆	semidirect
160. #1997	D₁₉	semidirect
253. C₁₉	#9267	semidirect
278. M₁₆	#64527	semidirect

Maximal subgroups

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

subgroup	index
#14574560	2
#14572381	2
#14562313	2
C₂×#11926293	2
#14561530	2
#14561486	2
C₁₉×#1997	2
#9267	19