Subgroups — #18075652

Order 4864.

494 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	C₂×#11926621	C₂	1	1	yes	maximal, characteristic
2	#14563762	C₂	1	1	yes	maximal, characteristic, complemented
3	#14563636	C₂	1	1	yes	maximal, characteristic
4	#14563632	C₂	1	1	yes	maximal, characteristic
5	C₂×#11926294	C₂	1	1	yes	maximal, characteristic, complemented
6	#14561519	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1296	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#64515	—	2	1	no	—
9	C₂×C₄×D₇₆	C₂²	1	1	yes	characteristic
10	#11926621	—	2	2	no	—
11	#11926621	C₄	1	4	yes	complemented
12	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×#10597278	—	2	1	no	—
14	C₂×#10597266	—	2	1	no	—
15	C₂×(C₂×D₁₉)⋊C₈	—	2	1	no	—
16	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
17	#11926414	C₂²	1	1	yes	characteristic
18	#11926307	—	2	1	no	—
19	#11926306	—	2	1	no	—
20	#11926306	—	2	1	no	—
21	#11926306	—	2	1	no	—
22	#11926294	—	2	2	no	—
23	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₁₉×#384	C₂²	1	1	yes	characteristic
26	C₂²×D₇₆	D₄	1	1	yes	characteristic
27	C₂²×C₄×D₁₉	—	2	1	no	—
28	C₂²×C₈×C₁₉	—	2	1	no	—
29	C₂²×C₈×C₁₉	—	2	1	no	—
30	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
31	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
32	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
33	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
34	C₂×(C₂×D₁₉)⋊C₄	—	2	1	no	—
35	C₂×C₄⋊Dic₁₉	—	2	1	no	—
36	C₂×C₄⋊Dic₁₉	—	2	1	no	—
37	C₂×C₄⋊Dic₁₉	—	2	1	no	—
38	C₂×C₄⋊Dic₁₉	D₄	1	1	yes	characteristic
39	C₂²×C₁₉⋊C₈	—	2	1	no	—
40	C₂²×C₁₉⋊C₈	D₄	1	1	yes	characteristic
41	C₂²×C₁₉⋊C₈	D₄	1	1	yes	characteristic
42	C₂×#64515	—	2	4	no	—
43	C₂×#64515	—	4	2	no	—
44	C₂×#64515	—	4	2	no	—
45	C₄×D₇₆	—	2	1	no	—
46	C₄×D₇₆	—	2	1	no	—
47	C₄×D₇₆	C₂×C₄	1	2	yes	—
48	C₄×D₇₆	Q₈	1	2	yes	—
49	C₁₉×C₄⋊C₈	—	2	1	no	—
50	C₁₉×C₄⋊C₈	—	2	1	no	—
51	C₁₉×C₄⋊C₈	—	2	1	no	—
52	C₁₉×C₄⋊C₈	C₂×C₄	1	2	yes	—
53	#10597278	—	4	2	no	—
54	(C₂×D₁₉)⋊C₈	—	2	4	no	—
55	#10597266	—	4	2	no	—
56	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
57	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
58	C₄×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
59	C₄×C₁₉⋊C₈	D₄	1	2	yes	—
60	C₂³×D₁₉	—	2	1	no	—
61	C₂²×C₄×C₁₉	—	4	1	no	—
62	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
65	C₂²×Dic₁₉	—	2	1	no	—
66	C₂²×Dic₁₉	—	4	1	no	—
67	C₂×D₇₆	—	2	1	no	—
68	C₂×D₇₆	—	2	1	no	—
69	C₂×D₇₆	—	2	1	no	—
70	C₂×D₇₆	—	2	1	no	—
71	C₂×D₇₆	—	2	1	no	—
72	C₂×D₇₆	C₄⋊C₄	1	2	yes	—
73	C₂×C₄×D₁₉	—	2	2	no	—
74	C₂×C₄×D₁₉	—	2	2	no	—
75	C₂×C₄×D₁₉	—	4	1	no	—
76	C₂×C₄×D₁₉	—	4	1	no	—
77	C₂×C₄×D₁₉	—	4	1	no	—
78	C₂×C₄×D₁₉	—	4	1	no	—
79	C₂×C₈×C₁₉	—	2	2	no	—
80	C₂×C₈×C₁₉	—	4	1	no	—
81	C₂×C₈×C₁₉	—	4	1	no	—
82	C₂×C₈×C₁₉	—	4	1	no	—
83	C₂×C₈×C₁₉	—	4	1	no	—
84	C₂×C₈×C₁₉	—	4	1	no	—
85	C₄⋊C₄×C₁₉	—	4	1	no	—
86	C₄⋊C₄×C₁₉	—	4	1	no	—
87	C₄⋊C₄×C₁₉	—	4	1	no	—
88	C₄⋊C₄×C₁₉	—	4	1	no	—
89	C₄⋊C₄×C₁₉	—	8	1	no	—
90	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
93	C₄²×C₁₉	#41	1	1	yes	characteristic
94	(C₂×D₁₉)⋊C₄	—	2	2	no	—
95	(C₂×D₁₉)⋊C₄	—	2	2	no	—
96	C₄⋊Dic₁₉	—	2	1	no	—
97	C₄⋊Dic₁₉	—	4	1	no	—
98	C₄⋊Dic₁₉	—	4	1	no	—
99	C₄⋊Dic₁₉	—	4	1	no	—
100	C₄⋊Dic₁₉	—	4	1	no	—
101	C₄⋊Dic₁₉	—	8	1	no	—
102	C₄⋊Dic₁₉	C₄⋊C₄	1	2	yes	—
103	C₂×C₁₉⋊C₈	—	2	1	no	—
104	C₂×C₁₉⋊C₈	—	2	1	no	—
105	C₂×C₁₉⋊C₈	—	2	1	no	—
106	C₂×C₁₉⋊C₈	—	2	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	C₂²⋊C₄	1	2	yes	—
111	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	2	yes	—
112	#64515	—	4	8	no	—
113	#8656	—	19	1	no	maximal, Sylow
114	C₂³×C₁₉	#121	1	1	yes	characteristic
115	C₂²×D₁₉	—	2	2	no	—
116	C₂²×D₁₉	—	4	1	no	—
117	C₂²×D₁₉	—	4	1	no	—
118	C₂²×D₁₉	—	4	1	no	—
119	C₂²×D₁₉	—	4	1	no	—
120	C₂²×D₁₉	—	4	1	no	—
121	C₂²×D₁₉	—	4	1	no	—
122	C₂×C₄×C₁₉	—	2	1	no	—
123	C₂×C₄×C₁₉	—	2	1	no	—
124	C₂×C₄×C₁₉	—	8	1	no	—
125	C₂×C₄×C₁₉	—	8	1	no	—
126	C₂×C₄×C₁₉	—	8	1	no	—
127	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
130	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
131	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
132	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
133	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
134	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
135	C₂×C₄×C₁₉	AGL(1,ℤ/8)	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	#137	1	1	yes	characteristic
140	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
141	C₂×Dic₁₉	—	2	2	no	—
142	C₂×Dic₁₉	—	4	1	no	—
143	C₂×Dic₁₉	—	4	1	no	—
144	C₂×Dic₁₉	—	8	1	no	—
145	C₂×Dic₁₉	—	8	1	no	—
146	C₂×Dic₁₉	—	8	1	no	—
147	D₇₆	—	2	2	no	—
148	D₇₆	—	2	2	no	—
149	D₇₆	—	4	1	no	—
150	D₇₆	—	4	1	no	—
151	C₄×D₁₉	—	4	2	no	—
152	C₄×D₁₉	—	4	2	no	—
153	C₄×D₁₉	—	4	2	no	—
154	C₄×D₁₉	—	4	2	no	—
155	C₈×C₁₉	—	4	2	no	—
156	C₈×C₁₉	—	8	1	no	—
157	C₁₉⋊C₈	—	2	2	no	—
158	C₁₉⋊C₈	—	2	2	no	—
159	C₁₉⋊C₈	—	8	1	no	—
160	C₂×#436	—	19	1	no	—
161	#1834	—	19	1	no	—
162	#1802	—	19	1	no	—
163	#1802	—	19	1	no	—
164	C₂×#334	—	19	1	no	—
165	#1296	—	19	1	no	—
166	#1296	D₁₉	1	1	yes	characteristic, complemented
167	C₂²×C₁₉	#380	1	1	yes	characteristic
168	C₂²×C₁₉	#463	1	1	yes	characteristic
169	C₂²×C₁₉	#463	1	1	yes	characteristic
170	C₂²×C₁₉	#469	1	1	yes	characteristic
171	C₂²×C₁₉	#470	1	1	yes	characteristic
172	C₂²×C₁₉	#471	1	1	yes	characteristic
173	C₂²×C₁₉	#471	1	1	yes	characteristic
174	C₂×D₁₉	—	4	2	no	—
175	C₂×D₁₉	—	4	2	no	—
176	C₂×D₁₉	—	4	2	no	—
177	C₂×D₁₉	—	8	1	no	—
178	C₂×D₁₉	—	8	1	no	—
179	C₂×D₁₉	—	8	1	no	—
180	C₂×D₁₉	—	8	1	no	—
181	C₄×C₁₉	—	2	1	no	—
182	C₄×C₁₉	—	2	1	no	—
183	C₄×C₁₉	—	16	1	no	—
184	C₄×C₁₉	#416	1	1	yes	characteristic
185	C₄×C₁₉	#418	1	1	yes	characteristic
186	C₄×C₁₉	#428	1	1	yes	characteristic
187	C₄×C₁₉	#428	1	1	yes	characteristic
188	C₄×C₁₉	#468	1	1	yes	characteristic
189	C₄×C₁₉	#468	1	1	yes	characteristic
190	C₄×C₁₉	#478	1	1	yes	characteristic
191	C₄×C₁₉	#478	1	1	yes	characteristic
192	Dic₁₉	—	4	2	no	—
193	Dic₁₉	—	16	1	no	—
194	C₂²×M₁₆	—	38	1	no	—
195	C₂×C₄×D₄	—	19	1	no	—
196	#436	—	19	4	no	—
197	#436	—	38	2	no	—
198	C₂×C₄⋊C₈	—	19	1	no	—
199	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
200	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
201	C₂×#102	—	38	1	no	—
202	C₂×#102	—	38	1	no	—
203	C₂×C₂²⋊C₈	—	38	1	no	—
204	C₂×C₄×C₈	—	19	1	no	—
205	#384	—	19	1	no	—
206	#384	C₂×D₁₉	1	1	yes	characteristic
207	#340	—	38	1	no	—
208	#340	—	38	1	no	—
209	#340	—	38	1	no	—
210	#340	—	38	1	no	—
211	#334	—	38	2	no	—
212	C₂×C₁₉	#1701	1	1	yes	characteristic
213	C₂×C₁₉	#1701	1	1	yes	characteristic
214	C₂×C₁₉	#1703	1	1	yes	characteristic
215	C₂×C₁₉	#1703	1	1	yes	characteristic
216	C₂×C₁₉	#1804	1	1	yes	characteristic
217	C₂×C₁₉	#1809	1	1	yes	characteristic
218	C₂×C₁₉	#1815	1	1	yes	characteristic
219	D₁₉	—	8	2	no	—
220	C₂²×D₄	—	19	1	no	—
221	C₂³×C₄	—	38	1	no	—
222	C₂×M₁₆	—	38	4	no	—
223	C₂×M₁₆	—	76	2	no	—
224	C₂×M₁₆	—	76	2	no	—
225	C₂²×C₈	—	2	1	no	—
226	C₂²×C₈	—	2	1	no	—
227	C₂²×C₈	—	19	1	no	—
228	C₂²×C₈	—	19	1	no	—
229	C₂²×C₈	—	38	1	no	—
230	C₄×D₄	—	19	2	no	—
231	C₄×D₄	—	19	2	no	—
232	C₄×D₄	—	38	1	no	—
233	C₄×D₄	—	38	1	no	—
234	C₂×C₄⋊C₄	—	2	1	no	—
235	C₂×C₄⋊C₄	—	2	1	no	—
236	C₂×C₄⋊C₄	—	2	1	no	—
237	C₂×C₄⋊C₄	—	19	1	no	—
238	C₂×C₄⋊C₄	—	38	1	no	—
239	C₂×C₄⋊C₄	—	38	1	no	—
240	C₂×C₄⋊C₄	—	38	1	no	—
241	C₂×C₂²⋊C₄	—	38	1	no	—
242	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
243	C₄⋊C₈	—	2	1	no	—
244	C₄⋊C₈	—	2	1	no	—
245	C₄⋊C₈	—	2	1	no	—
246	C₄⋊C₈	—	38	1	no	—
247	C₄⋊C₈	—	38	1	no	—
248	C₄⋊C₈	C₄×D₁₉	1	2	yes	—
249	#102	—	76	2	no	—
250	#102	—	76	2	no	—
251	C₂²⋊C₈	—	38	4	no	—
252	C₄×C₈	—	19	2	no	—
253	C₄×C₈	—	19	2	no	—
254	C₁₉	#8656	1	1	yes	characteristic, complemented, Sylow
255	C₂⁴	—	38	1	no	—
256	C₂×D₄	—	19	2	no	—
257	C₂×D₄	—	38	1	no	—
258	C₂×D₄	—	38	1	no	—
259	C₂×D₄	—	38	1	no	—
260	C₂×D₄	—	38	1	no	—
261	C₂×D₄	—	38	1	no	—
262	C₂²×C₄	—	4	1	no	—
263	C₂²×C₄	—	38	1	no	—
264	C₂²×C₄	—	38	2	no	—
265	C₂²×C₄	—	38	2	no	—
266	C₂²×C₄	—	76	1	no	—
267	C₂²×C₄	—	76	1	no	—
268	C₂²×C₄	—	76	1	no	—
269	C₂²×C₄	—	76	1	no	—
270	C₂²×C₄	—	76	1	no	—
271	C₂²×C₄	#64541	1	1	yes	characteristic
272	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
273	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
274	M₁₆	—	76	8	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	4	1	no	—
277	C₂×C₈	—	4	1	no	—
278	C₂×C₈	—	4	1	no	—
279	C₂×C₈	—	4	1	no	—
280	C₂×C₈	—	4	1	no	—
281	C₂×C₈	—	19	2	no	—
282	C₂×C₈	—	19	2	no	—
283	C₂×C₈	—	38	1	no	—
284	C₂×C₈	—	38	1	no	—
285	C₂×C₈	—	38	1	no	—
286	C₂×C₈	—	38	1	no	—
287	C₂×C₈	—	76	1	no	—
288	C₂×C₈	—	76	1	no	—
289	C₂×C₈	—	76	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	—	8	1	no	—
295	C₄⋊C₄	—	19	2	no	—
296	C₄⋊C₄	—	38	1	no	—
297	C₄⋊C₄	—	76	1	no	—
298	C₄⋊C₄	—	76	1	no	—
299	C₄⋊C₄	—	76	1	no	—
300	C₄⋊C₄	—	76	1	no	—
301	C₄⋊C₄	—	152	1	no	—
302	C₂²⋊C₄	—	38	2	no	—
303	C₂²⋊C₄	—	38	2	no	—
304	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
305	C₄²	#64541	1	1	yes	characteristic
306	C₄²	D₄×D₁₉	1	1	yes	characteristic
307	C₄²	Q₈×D₁₉	1	1	yes	characteristic
308	C₂³	—	38	2	no	—
309	C₂³	—	76	1	no	—
310	C₂³	—	76	1	no	—
311	C₂³	—	76	1	no	—
312	C₂³	—	76	1	no	—
313	C₂³	—	76	1	no	—
314	C₂³	—	76	1	no	—
315	C₂³	#10597330	1	1	yes	characteristic
316	D₄	—	38	2	no	—
317	D₄	—	38	2	no	—
318	D₄	—	76	1	no	—
319	D₄	—	76	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	8	1	no	—
323	C₂×C₄	—	8	1	no	—
324	C₂×C₄	—	8	1	no	—
325	C₂×C₄	—	38	2	no	—
326	C₂×C₄	—	76	1	no	—
327	C₂×C₄	—	76	1	no	—
328	C₂×C₄	—	76	2	no	—
329	C₂×C₄	—	76	2	no	—
330	C₂×C₄	—	76	2	no	—
331	C₂×C₄	—	76	2	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	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
337	C₂×C₄	#10597330	1	1	yes	characteristic
338	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
339	C₂×C₄	#10597342	1	1	yes	characteristic
340	C₂×C₄	#10597342	1	1	yes	characteristic
341	C₂×C₄	#10597357	1	1	yes	characteristic
342	C₂×C₄	#10597357	1	1	yes	characteristic
343	C₂×C₄	#10597360	1	1	yes	characteristic
344	C₂×C₄	#10597360	1	1	yes	characteristic
345	C₂×C₄	#10597361	1	1	yes	characteristic
346	C₂×C₄	#10597362	1	1	yes	characteristic
347	C₂×C₄	#10597364	1	1	yes	characteristic
348	C₂×C₄	#10597365	1	1	yes	characteristic
349	C₈	—	4	2	no	—
350	C₈	—	8	1	no	—
351	C₈	—	38	2	no	—
352	C₈	—	38	2	no	—
353	C₈	—	152	1	no	—
354	C₂²	—	76	2	no	—
355	C₂²	—	76	2	no	—
356	C₂²	—	76	2	no	—
357	C₂²	—	152	1	no	—
358	C₂²	—	152	1	no	—
359	C₂²	—	152	1	no	—
360	C₂²	—	152	1	no	—
361	C₂²	#11926403	1	1	yes	characteristic
362	C₂²	#11926758	1	1	yes	characteristic
363	C₂²	#11926762	1	1	yes	characteristic
364	C₂²	#11926796	1	1	yes	characteristic
365	C₂²	#11926801	1	1	yes	characteristic
366	C₂²	#11926806	1	1	yes	characteristic
367	C₂²	#11926810	1	1	yes	characteristic
368	C₄	—	2	1	no	—
369	C₄	—	2	1	no	—
370	C₄	—	16	1	no	—
371	C₄	—	304	1	no	—
372	C₄	—	76	2	no	—
373	C₄	#11926539	1	1	yes	characteristic
374	C₄	#11926547	1	1	yes	characteristic
375	C₄	#11926583	1	1	yes	characteristic
376	C₄	#11926585	1	1	yes	characteristic
377	C₄	#11926789	1	1	yes	characteristic
378	C₄	#11926793	1	1	yes	characteristic
379	C₄	#11926843	1	1	yes	characteristic
380	C₄	#11926847	1	1	yes	characteristic
381	C₂	—	152	2	no	—
382	C₂	#14563195	1	1	yes	characteristic
383	C₂	#14563199	1	1	yes	characteristic
384	C₂	#14563209	1	1	yes	characteristic
385	C₂	#14563213	1	1	yes	characteristic
386	C₂	#14563644	1	1	yes	characteristic
387	C₂	#14563667	1	1	yes	characteristic
388	C₂	#14563693	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10597330
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1296	C₂
socle	C₂³×C₁₉	#121
prime radical	C₂²×D₇₆	D₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1296	D₁₉
hypocenter	C₁₉	#8656
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1296

Complements

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

N	H ≅ G/N	type
2. #14563762	C₂	semidirect
5. C₂×#11926294	C₂	semidirect
6. #14561519	C₂	semidirect
7. C₁₉×#1296	C₂	semidirect
11. #11926621	C₄	semidirect
166. #1296	D₁₉	semidirect
254. C₁₉	#8656	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926621	2
#14563762	2
#14563636	2
#14563632	2
C₂×#11926294	2
#14561519	2
C₁₉×#1296	2
#8656	19