Subgroups — #18080929

Order 4864.

454 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14564562	C₂	1	1	yes	maximal, characteristic
2	#14564441	C₂	1	1	yes	maximal, characteristic
3	#14564398	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11926287	C₂	1	1	yes	maximal, characteristic
5	#14561519	C₂	1	1	yes	maximal, characteristic, complemented
6	#14561470	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1770	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10597320	C₂²	1	1	yes	characteristic
9	C₂×#10597265	—	2	1	no	—
10	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×#10597278	—	2	1	no	—
13	C₂×#10597278	—	2	1	no	—
14	C₂×#10597266	—	2	1	no	—
15	C₂×#10597266	—	2	1	no	—
16	#11926414	C₂²	1	1	yes	characteristic
17	#11926306	—	2	1	no	—
18	#11926287	C₄	1	4	yes	complemented
19	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₁₉×#97	—	2	1	no	—
21	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
23	C₁₉×#336	—	2	1	no	—
24	C₂²×D₇₆	—	2	1	no	—
25	C₂²×D₇₆	—	2	1	no	—
26	C₂²×D₇₆	—	2	1	no	—
27	C₂²×C₈×C₁₉	—	2	1	no	—
28	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₉	—	2	1	no	—
33	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
34	C₂×C₄⋊Dic₁₉	—	2	1	no	—
35	C₂×C₄⋊Dic₁₉	—	2	1	no	—
36	C₂×C₄⋊Dic₁₉	—	2	1	no	—
37	C₂²×C₁₉⋊C₈	—	2	1	no	—
38	C₂²×C₁₉⋊C₈	—	2	1	no	—
39	#10597320	—	2	1	no	—
40	#10597320	—	2	1	no	—
41	#10597320	—	2	1	no	—
42	#10597320	C₂×C₄	1	2	yes	—
43	C₁₉×C₄⋊C₈	—	2	1	no	—
44	C₁₉×C₄⋊C₈	C₂×C₄	1	2	yes	—
45	C₁₉×#97	—	4	2	no	—
46	C₄×C₈×C₁₉	—	2	1	no	—
47	C₄×C₈×C₁₉	—	2	1	no	—
48	C₁₉×#95	—	2	1	no	—
49	#10597278	—	4	2	no	—
50	#10597278	—	4	2	no	—
51	#10597266	—	4	2	no	—
52	#10597266	—	4	2	no	—
53	#10597265	—	4	2	no	—
54	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
55	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
56	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
57	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	2	yes	—
58	C₂³×D₁₉	—	4	1	no	—
59	C₂²×C₄×C₁₉	—	2	1	no	—
60	C₂²×C₄×C₁₉	—	2	1	no	—
61	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
64	C₂²×Dic₁₉	—	4	1	no	—
65	C₂×D₇₆	—	2	2	no	—
66	C₂×D₇₆	—	2	2	no	—
67	C₂×D₇₆	—	2	2	no	—
68	C₂×D₇₆	—	4	1	no	—
69	C₂×D₇₆	—	4	1	no	—
70	C₂×D₇₆	—	4	1	no	—
71	C₂×D₇₆	—	4	1	no	—
72	C₂×D₇₆	—	4	1	no	—
73	C₂×D₇₆	—	4	1	no	—
74	C₂×D₇₆	—	4	1	no	—
75	C₂×D₇₆	—	4	1	no	—
76	C₂×D₇₆	—	4	1	no	—
77	C₂×D₇₆	—	4	1	no	—
78	C₂×D₇₆	—	4	1	no	—
79	C₂×D₇₆	—	8	1	no	—
80	C₂×D₇₆	—	8	1	no	—
81	C₂×C₈×C₁₉	—	2	1	no	—
82	C₂×C₈×C₁₉	—	2	1	no	—
83	C₂×C₈×C₁₉	—	2	2	no	—
84	C₂×C₈×C₁₉	—	2	1	no	—
85	C₂×C₈×C₁₉	—	2	1	no	—
86	C₂×C₈×C₁₉	—	2	1	no	—
87	C₂×C₈×C₁₉	—	4	1	no	—
88	C₂×C₈×C₁₉	—	4	1	no	—
89	C₂×C₈×C₁₉	SD₁₆	1	2	yes	—
90	C₄⋊C₄×C₁₉	—	2	1	no	—
91	C₄⋊C₄×C₁₉	—	2	1	no	—
92	C₄⋊C₄×C₁₉	—	2	1	no	—
93	C₄⋊C₄×C₁₉	—	2	1	no	—
94	C₄²×C₁₉	—	4	1	no	—
95	C₄²×C₁₉	—	4	1	no	—
96	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₁₉	#41	1	1	yes	characteristic
98	C₄²×C₁₉	#41	1	1	yes	characteristic
99	C₄²×C₁₉	#41	1	1	yes	characteristic
100	C₄⋊Dic₁₉	—	4	1	no	—
101	C₄⋊Dic₁₉	—	4	1	no	—
102	C₄⋊Dic₁₉	—	4	1	no	—
103	C₄⋊Dic₁₉	—	4	1	no	—
104	C₄⋊Dic₁₉	—	8	1	no	—
105	C₂×C₁₉⋊C₈	—	2	2	no	—
106	C₂×C₁₉⋊C₈	—	4	1	no	—
107	C₂×C₁₉⋊C₈	—	4	1	no	—
108	C₂×C₁₉⋊C₈	—	4	1	no	—
109	C₂×C₁₉⋊C₈	—	4	1	no	—
110	C₂×C₁₉⋊C₈	—	4	1	no	—
111	#9572	—	19	1	no	maximal, Sylow
112	C₂³×C₁₉	#121	1	1	yes	characteristic
113	C₂²×D₁₉	—	4	2	no	—
114	C₂²×D₁₉	—	8	1	no	—
115	C₂²×D₁₉	—	8	1	no	—
116	C₂²×D₁₉	—	8	1	no	—
117	C₂²×D₁₉	—	8	1	no	—
118	C₂²×D₁₉	—	8	1	no	—
119	C₂²×D₁₉	—	8	1	no	—
120	C₂×C₄×C₁₉	—	2	1	no	—
121	C₂×C₄×C₁₉	—	2	1	no	—
122	C₂×C₄×C₁₉	—	4	1	no	—
123	C₂×C₄×C₁₉	—	4	1	no	—
124	C₂×C₄×C₁₉	—	4	1	no	—
125	C₂×C₄×C₁₉	—	4	1	no	—
126	C₂×C₄×C₁₉	—	4	1	no	—
127	C₂×C₄×C₁₉	—	4	1	no	—
128	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
129	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
132	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
133	C₂×C₄×C₁₉	#126	1	1	yes	characteristic
134	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
135	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
136	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
137	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
138	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
139	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
140	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
142	C₂×Dic₁₉	—	8	1	no	—
143	C₂×Dic₁₉	—	8	1	no	—
144	C₂×Dic₁₉	—	8	1	no	—
145	D₇₆	—	4	2	no	—
146	D₇₆	—	4	2	no	—
147	D₇₆	—	4	2	no	—
148	D₇₆	—	4	2	no	—
149	D₇₆	—	8	2	no	—
150	D₇₆	—	8	2	no	—
151	D₇₆	—	8	1	no	—
152	D₇₆	—	8	1	no	—
153	D₇₆	—	8	1	no	—
154	D₇₆	—	8	1	no	—
155	C₈×C₁₉	—	2	2	no	—
156	C₈×C₁₉	—	4	2	no	—
157	C₈×C₁₉	—	4	1	no	—
158	C₁₉⋊C₈	—	4	2	no	—
159	C₁₉⋊C₈	—	8	1	no	—
160	#1982	—	19	1	no	—
161	#1957	—	19	1	no	—
162	#1948	—	19	1	no	—
163	#1770	D₁₉	1	1	yes	characteristic, complemented
164	C₂×#331	—	19	1	no	—
165	#1296	—	19	1	no	—
166	#1277	—	19	1	no	—
167	C₂²×C₁₉	#388	1	1	yes	characteristic
168	C₂²×C₁₉	#460	1	1	yes	characteristic
169	C₂²×C₁₉	#463	1	1	yes	characteristic
170	C₂²×C₁₉	#468	1	1	yes	characteristic
171	C₂²×C₁₉	#469	1	1	yes	characteristic
172	C₂²×C₁₉	#471	1	1	yes	characteristic
173	C₂²×C₁₉	#472	1	1	yes	characteristic
174	C₂×D₁₉	—	8	2	no	—
175	C₂×D₁₉	—	8	2	no	—
176	C₂×D₁₉	—	8	2	no	—
177	C₂×D₁₉	—	16	1	no	—
178	C₂×D₁₉	—	16	1	no	—
179	C₂×D₁₉	—	16	1	no	—
180	C₂×D₁₉	—	16	1	no	—
181	C₄×C₁₉	—	2	1	no	—
182	C₄×C₁₉	—	2	1	no	—
183	C₄×C₁₉	—	8	1	no	—
184	C₄×C₁₉	—	8	1	no	—
185	C₄×C₁₉	C₄×SD₁₆	1	1	yes	characteristic
186	C₄×C₁₉	#440	1	1	yes	characteristic
187	C₄×C₁₉	#442	1	1	yes	characteristic
188	C₄×C₁₉	#442	1	1	yes	characteristic
189	C₄×C₁₉	#481	1	1	yes	characteristic
190	C₄×C₁₉	#482	1	1	yes	characteristic
191	C₄×C₁₉	#489	1	1	yes	characteristic
192	C₄×C₁₉	#490	1	1	yes	characteristic
193	Dic₁₉	—	16	1	no	—
194	C₂×#127	—	19	1	no	—
195	C₂×#106	—	38	1	no	—
196	C₂×C₄⋊C₈	—	19	1	no	—
197	C₂×C₄⋊C₈	—	19	1	no	—
198	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
199	C₂×#102	—	38	1	no	—
200	C₂×#102	—	38	1	no	—
201	C₂×#102	—	38	1	no	—
202	C₂×#102	—	38	1	no	—
203	C₂×#97	—	2	1	no	—
204	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
205	#384	—	19	1	no	—
206	C₄×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
207	#340	—	38	1	no	—
208	#336	—	2	1	no	—
209	#331	—	19	4	no	—
210	C₂×C₁₉	#1686	1	1	yes	characteristic
211	C₂×C₁₉	#1688	1	1	yes	characteristic
212	C₂×C₁₉	#1697	1	1	yes	characteristic
213	C₂×C₁₉	#1701	1	1	yes	characteristic
214	C₂×C₁₉	#1926	1	1	yes	characteristic
215	C₂×C₁₉	#1931	1	1	yes	characteristic
216	C₂×C₁₉	#1936	1	1	yes	characteristic
217	D₁₉	—	16	2	no	—
218	C₂²×D₄	—	38	1	no	—
219	C₂²×D₄	—	38	1	no	—
220	C₂²×D₄	—	38	1	no	—
221	C₂²×C₈	—	2	1	no	—
222	C₂²×C₈	—	38	1	no	—
223	C₂²×C₈	—	38	1	no	—
224	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
225	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
226	#127	—	19	2	no	—
227	#127	—	38	1	no	—
228	#127	—	38	1	no	—
229	#127	—	38	1	no	—
230	C₂×C₄⋊C₄	—	38	1	no	—
231	C₂×C₄⋊C₄	—	38	1	no	—
232	C₂×C₄⋊C₄	—	38	1	no	—
233	C₂×C₄⋊C₄	D₇₆	1	1	yes	characteristic
234	C₂×C₄⋊C₄	D₇₆	1	1	yes	characteristic
235	C₂×C₄²	—	2	1	no	—
236	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
237	#106	—	76	2	no	—
238	C₄⋊C₈	—	2	1	no	—
239	C₄⋊C₈	—	19	2	no	—
240	C₄⋊C₈	—	38	1	no	—
241	C₄⋊C₈	—	38	1	no	—
242	C₄⋊C₈	—	38	1	no	—
243	C₄⋊C₈	C₄×D₁₉	1	2	yes	—
244	#102	—	76	2	no	—
245	#102	—	76	2	no	—
246	#102	—	76	2	no	—
247	#102	—	76	2	no	—
248	#97	—	4	2	no	—
249	C₄×C₈	—	2	1	no	—
250	C₄×C₈	—	2	1	no	—
251	#95	—	2	1	no	—
252	C₁₉	#9572	1	1	yes	characteristic, complemented, Sylow
253	C₂⁴	—	76	1	no	—
254	C₂×D₄	—	38	2	no	—
255	C₂×D₄	—	38	2	no	—
256	C₂×D₄	—	38	2	no	—
257	C₂×D₄	—	76	1	no	—
258	C₂×D₄	—	76	1	no	—
259	C₂×D₄	—	76	1	no	—
260	C₂×D₄	—	76	1	no	—
261	C₂×D₄	—	76	1	no	—
262	C₂×D₄	—	76	1	no	—
263	C₂×D₄	—	76	1	no	—
264	C₂×D₄	—	76	1	no	—
265	C₂×D₄	—	76	1	no	—
266	C₂×D₄	—	76	1	no	—
267	C₂×D₄	—	76	1	no	—
268	C₂×D₄	—	152	1	no	—
269	C₂×D₄	—	152	1	no	—
270	C₂²×C₄	—	2	1	no	—
271	C₂²×C₄	—	2	1	no	—
272	C₂²×C₄	—	76	1	no	—
273	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
274	C₂²×C₄	#64541	1	1	yes	characteristic
275	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
276	C₂×C₈	—	2	1	no	—
277	C₂×C₈	—	2	1	no	—
278	C₂×C₈	—	2	1	no	—
279	C₂×C₈	—	2	1	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	2	1	no	—
282	C₂×C₈	—	4	1	no	—
283	C₂×C₈	—	4	1	no	—
284	C₂×C₈	—	38	2	no	—
285	C₂×C₈	—	76	1	no	—
286	C₂×C₈	—	76	1	no	—
287	C₂×C₈	—	76	1	no	—
288	C₂×C₈	—	76	1	no	—
289	C₂×C₈	—	76	1	no	—
290	C₂×C₈	#64527	1	2	yes	—
291	C₄⋊C₄	—	2	1	no	—
292	C₄⋊C₄	—	2	1	no	—
293	C₄⋊C₄	—	2	1	no	—
294	C₄⋊C₄	—	2	1	no	—
295	C₄⋊C₄	—	76	1	no	—
296	C₄⋊C₄	—	76	1	no	—
297	C₄⋊C₄	—	76	1	no	—
298	C₄⋊C₄	—	76	1	no	—
299	C₄⋊C₄	—	152	1	no	—
300	C₄²	—	4	1	no	—
301	C₄²	—	4	1	no	—
302	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
303	C₄²	#64541	1	1	yes	characteristic
304	C₄²	#64541	1	1	yes	characteristic
305	C₄²	#64541	1	1	yes	characteristic
306	C₂³	—	76	2	no	—
307	C₂³	—	152	1	no	—
308	C₂³	—	152	1	no	—
309	C₂³	—	152	1	no	—
310	C₂³	—	152	1	no	—
311	C₂³	—	152	1	no	—
312	C₂³	—	152	1	no	—
313	C₂³	#10597376	1	1	yes	characteristic
314	D₄	—	76	2	no	—
315	D₄	—	76	2	no	—
316	D₄	—	76	2	no	—
317	D₄	—	76	2	no	—
318	D₄	—	152	2	no	—
319	D₄	—	152	2	no	—
320	D₄	—	152	1	no	—
321	D₄	—	152	1	no	—
322	D₄	—	152	1	no	—
323	D₄	—	152	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	4	1	no	—
327	C₂×C₄	—	4	1	no	—
328	C₂×C₄	—	4	1	no	—
329	C₂×C₄	—	4	1	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	152	1	no	—
333	C₂×C₄	—	152	1	no	—
334	C₂×C₄	—	152	1	no	—
335	C₂×C₄	#10597318	1	1	yes	characteristic
336	C₂×C₄	C₄×D₇₆	1	1	yes	characteristic
337	C₂×C₄	#10597321	1	1	yes	characteristic
338	C₂×C₄	#10597322	1	1	yes	characteristic
339	C₂×C₄	#10597354	1	1	yes	characteristic
340	C₂×C₄	#10597354	1	1	yes	characteristic
341	C₂×C₄	#10597354	1	1	yes	characteristic
342	C₂×C₄	#10597355	1	1	yes	characteristic
343	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
344	C₂×C₄	#10597375	1	1	yes	characteristic
345	C₂×C₄	#10597378	1	1	yes	characteristic
346	C₂×C₄	C₂×#64527	1	1	yes	characteristic
347	C₂×C₄	#10597395	1	1	yes	characteristic
348	C₂×C₄	#10597396	1	1	yes	characteristic
349	C₈	—	2	2	no	—
350	C₈	—	4	1	no	—
351	C₈	—	4	2	no	—
352	C₈	—	76	2	no	—
353	C₈	—	152	1	no	—
354	C₂²	—	304	1	no	—
355	C₂²	—	304	1	no	—
356	C₂²	—	304	1	no	—
357	C₂²	—	304	1	no	—
358	C₂²	—	152	2	no	—
359	C₂²	—	152	2	no	—
360	C₂²	—	152	2	no	—
361	C₂²	#11926435	1	1	yes	characteristic
362	C₂²	#11926744	1	1	yes	characteristic
363	C₂²	#11926759	1	1	yes	characteristic
364	C₂²	#11926790	1	1	yes	characteristic
365	C₂²	#11926797	1	1	yes	characteristic
366	C₂²	#11926807	1	1	yes	characteristic
367	C₂²	#11926814	1	1	yes	characteristic
368	C₄	—	2	1	no	—
369	C₄	—	2	1	no	—
370	C₄	—	8	1	no	—
371	C₄	—	8	1	no	—
372	C₄	—	304	1	no	—
373	C₄	C₄×#64527	1	1	yes	characteristic
374	C₄	#11926646	1	1	yes	characteristic
375	C₄	#11926654	1	1	yes	characteristic
376	C₄	#11926657	1	1	yes	characteristic
377	C₄	#11926862	1	1	yes	characteristic
378	C₄	#11926868	1	1	yes	characteristic
379	C₄	#11926904	1	1	yes	characteristic
380	C₄	#11926909	1	1	yes	characteristic
381	C₂	—	304	2	no	—
382	C₂	#14563107	1	1	yes	characteristic
383	C₂	#14563119	1	1	yes	characteristic
384	C₂	#14563168	1	1	yes	characteristic
385	C₂	#14563196	1	1	yes	characteristic
386	C₂	#14564262	1	1	yes	characteristic
387	C₂	#14564297	1	1	yes	characteristic
388	C₂	#14564332	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10597376
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1770	C₂
socle	C₂³×C₁₉	#121
prime radical	C₂×#10597320	C₂²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1770	D₁₉
hypocenter	C₁₉	#9572
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1770

Complements

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

N	H ≅ G/N	type
3. #14564398	C₂	semidirect
5. #14561519	C₂	semidirect
6. #14561470	C₂	semidirect
7. C₁₉×#1770	C₂	semidirect
18. #11926287	C₄	semidirect
163. #1770	D₁₉	semidirect
252. C₁₉	#9572	semidirect

Maximal subgroups

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

subgroup	index
#14564562	2
#14564441	2
#14564398	2
C₂×#11926287	2
#14561519	2
#14561470	2
C₁₉×#1770	2
#9572	19