Subgroups — #18079395

Order 4864.

496 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14573444	C₂	1	1	yes	maximal, characteristic
2	#14572744	C₂	1	1	yes	maximal, characteristic
3	#14563656	C₂	1	1	yes	maximal, characteristic
4	#14562320	C₂	1	1	yes	maximal, characteristic, complemented
5	#14561538	C₂	1	1	yes	maximal, characteristic, complemented
6	#14561510	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1981	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#64541	C₂²	1	1	yes	characteristic
9	#11926836	C₄	1	2	yes	complemented
10	#11926831	C₄	1	2	yes	complemented
11	#11926826	—	2	1	no	—
12	#11926821	—	2	1	no	—
13	#11926668	—	2	1	no	—
14	#11926609	—	2	1	no	—
15	C₈×C₂²⋊D₁₉	—	2	2	no	—
16	#11926597	C₂²	1	1	yes	characteristic
17	#11926596	—	2	1	no	—
18	#11926584	—	2	1	no	—
19	#11926568	C₂²	1	1	yes	characteristic
20	#11926568	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×D₁₉	—	2	1	no	—
22	C₂×#10597262	—	2	1	no	—
23	#11926543	—	2	1	no	—
24	#11926304	—	2	1	no	—
25	#11926290	—	2	1	no	—
26	#11926290	—	2	1	no	—
27	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
28	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
29	C₂×C₁₉×#103	—	2	2	no	—
30	C₄×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
31	C₂²×Q₈×C₁₉	—	2	1	no	—
32	C₂×#64541	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₉	D₄	1	2	yes	—
34	C₁₉×C₄⋊Q₈	—	2	1	no	—
35	C₁₉×C₄⋊Q₈	—	2	1	no	—
36	C₁₉×C₄⋊Q₈	—	2	1	no	—
37	C₁₉×C₄⋊Q₈	C₂×C₄	1	2	yes	—
38	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
39	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
40	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
41	C₂×C₄²×C₁₉	—	2	1	no	—
42	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
43	C₄×C₂²⋊D₁₉	—	2	1	no	—
44	C₄×C₂²⋊D₁₉	—	2	1	no	—
45	#10597372	D₄	1	1	yes	characteristic
46	C₂×#64520	—	2	1	no	—
47	C₂×#64520	—	2	1	no	—
48	C₂×C₈×D₁₉	—	2	2	no	—
49	C₄×D₇₆	C₂×C₄	1	1	yes	characteristic
50	C₄×D₇₆	D₄	1	1	yes	characteristic
51	#10597318	—	2	1	no	—
52	C₄²×D₁₉	—	2	1	no	—
53	C₄×Dic₃₈	C₂×C₄	1	1	yes	characteristic
54	C₄×Dic₃₈	D₄	1	1	yes	characteristic
55	C₁₉×C₄⋊C₈	—	2	1	no	—
56	C₁₉×C₄⋊C₈	—	2	1	no	—
57	C₁₉×#103	—	4	4	no	—
58	C₁₉×#95	—	2	1	no	—
59	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
60	(C₂×D₁₉)⋊C₈	—	2	2	no	—
61	Dic₁₉⋊C₈	—	2	2	no	—
62	C₈×Dic₁₉	—	2	2	no	—
63	#10597267	—	2	2	no	—
64	#10597266	—	2	2	no	—
65	#10597265	—	2	2	no	—
66	#10597264	—	2	2	no	—
67	#10597262	—	4	2	no	—
68	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
69	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
70	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
71	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	2	yes	—
72	C₄⋊(C₁₉⋊C₈)	D₄	1	1	yes	characteristic
73	C₄⋊(C₁₉⋊C₈)	D₄	1	1	yes	characteristic
74	C₄×C₁₉⋊C₈	—	2	1	no	—
75	C₂×Q₈×C₁₉	—	2	2	no	—
76	C₂×Q₈×C₁₉	—	4	2	no	—
77	C₂×Q₈×C₁₉	—	4	2	no	—
78	C₂×Q₈×C₁₉	—	4	1	no	—
79	C₂²×C₄×C₁₉	—	4	1	no	—
80	C₂²×C₄×C₁₉	—	4	1	no	—
81	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
84	C₂×C₂²⋊D₁₉	—	2	1	no	—
85	#64541	—	2	1	no	—
86	#64541	—	2	1	no	—
87	#64541	SD₁₆	1	2	yes	—
88	#64541	Q₁₆	1	2	yes	—
89	C₂×D₇₆	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×D₁₉	—	2	1	no	—
91	C₂×C₄×D₁₉	—	2	1	no	—
92	C₂×C₄×D₁₉	—	2	1	no	—
93	C₂×Dic₃₈	C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₈×C₁₉	—	2	2	no	—
95	C₂×C₈×C₁₉	—	4	2	no	—
96	C₄⋊C₄×C₁₉	—	2	2	no	—
97	C₄⋊C₄×C₁₉	—	2	2	no	—
98	C₄⋊C₄×C₁₉	—	2	2	no	—
99	C₄⋊C₄×C₁₉	—	4	1	no	—
100	C₄⋊C₄×C₁₉	—	4	1	no	—
101	C₄⋊C₄×C₁₉	—	4	1	no	—
102	C₄²×C₁₉	—	4	1	no	—
103	C₄²×C₁₉	—	4	1	no	—
104	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₁₉	#41	1	1	yes	characteristic
108	C₂²⋊Dic₁₉	—	2	1	no	—
109	(C₂×D₁₉)⋊C₄	—	2	1	no	—
110	(C₂×D₁₉)⋊C₄	—	2	1	no	—
111	C₄⋊Dic₁₉	C₂²⋊C₄	1	1	yes	characteristic
112	C₄⋊Dic₁₉	C₂²⋊C₄	1	1	yes	characteristic
113	Dic₁₉⋊C₄	—	2	1	no	—
114	Dic₁₉⋊C₄	—	2	1	no	—
115	C₄×Dic₁₉	—	2	1	no	—
116	C₄×Dic₁₉	—	2	1	no	—
117	C₄×Dic₁₉	—	2	1	no	—
118	#64520	—	4	1	no	—
119	#64520	—	4	1	no	—
120	#64520	—	4	1	no	—
121	#64520	—	4	1	no	—
122	C₂×C₁₉⋊C₈	—	2	2	no	—
123	C₂×C₁₉⋊C₈	—	2	1	no	—
124	C₂×C₁₉⋊C₈	—	2	1	no	—
125	C₂×C₁₉⋊C₈	—	4	1	no	—
126	C₈×D₁₉	—	8	2	no	—
127	#9317	—	19	1	no	maximal, Sylow
128	C₂³×C₁₉	#121	1	1	yes	characteristic
129	C₂²×D₁₉	—	2	1	no	—
130	Q₈×C₁₉	—	4	4	no	—
131	Q₈×C₁₉	—	8	1	no	—
132	Q₈×C₁₉	—	8	1	no	—
133	C₂×C₄×C₁₉	—	2	1	no	—
134	C₂×C₄×C₁₉	—	2	1	no	—
135	C₂×C₄×C₁₉	—	4	1	no	—
136	C₂×C₄×C₁₉	—	4	2	no	—
137	C₂×C₄×C₁₉	—	4	2	no	—
138	C₂×C₄×C₁₉	—	8	1	no	—
139	C₂×C₄×C₁₉	—	8	2	no	—
140	C₂×C₄×C₁₉	—	8	1	no	—
141	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
144	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
145	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
146	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
147	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
148	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
149	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
150	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
151	C₂²⋊D₁₉	—	2	2	no	—
152	C₂²⋊D₁₉	—	2	2	no	—
153	C₂×Dic₁₉	—	2	1	no	—
154	C₂×Dic₁₉	—	2	1	no	—
155	C₂×Dic₁₉	—	2	1	no	—
156	D₇₆	—	2	1	no	—
157	D₇₆	#103	1	2	yes	complemented
158	C₄×D₁₉	—	2	2	no	—
159	C₄×D₁₉	—	2	2	no	—
160	C₄×D₁₉	—	4	1	no	—
161	C₄×D₁₉	—	4	1	no	—
162	C₄×D₁₉	—	8	1	no	—
163	Dic₃₈	—	2	1	no	—
164	Dic₃₈	#103	1	2	yes	complemented
165	C₈×C₁₉	—	4	2	no	—
166	C₁₉⋊C₈	—	4	2	no	—
167	C₁₉⋊C₈	—	4	1	no	—
168	C₁₉⋊C₈	—	4	1	no	—
169	C₁₉⋊C₈	—	8	1	no	—
170	#3079	—	19	1	no	—
171	#2988	—	19	1	no	—
172	#1981	D₁₉	1	1	yes	characteristic, complemented
173	#1806	—	19	1	no	—
174	#1549	—	19	1	no	—
175	#1303	—	19	1	no	—
176	#1292	—	19	1	no	—
177	C₂²×C₁₉	—	2	1	no	—
178	C₂²×C₁₉	—	2	1	no	—
179	C₂²×C₁₉	#386	1	1	yes	characteristic
180	C₂²×C₁₉	#465	1	1	yes	characteristic
181	C₂²×C₁₉	#467	1	1	yes	characteristic
182	C₂×D₁₉	—	2	2	no	—
183	C₂×D₁₉	—	4	1	no	—
184	C₂×D₁₉	—	4	1	no	—
185	C₄×C₁₉	—	2	1	no	—
186	C₄×C₁₉	—	2	1	no	—
187	C₄×C₁₉	—	2	1	no	—
188	C₄×C₁₉	—	8	2	no	—
189	C₄×C₁₉	—	8	2	no	—
190	C₄×C₁₉	C₂×#103	1	1	yes	characteristic
191	C₄×C₁₉	#416	1	1	yes	characteristic
192	C₄×C₁₉	#453	1	1	yes	characteristic
193	C₄×C₁₉	#454	1	1	yes	characteristic
194	C₄×C₁₉	#465	1	1	yes	characteristic
195	C₄×C₁₉	#467	1	1	yes	characteristic
196	Dic₁₉	—	2	2	no	—
197	Dic₁₉	—	4	1	no	—
198	Dic₁₉	—	4	1	no	—
199	C₂×C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
200	C₄×#41	—	19	1	no	—
201	#477	—	19	2	no	—
202	#476	—	19	2	no	—
203	#475	—	38	1	no	—
204	#474	—	38	1	no	—
205	C₈×Q₈	—	38	1	no	—
206	C₈×D₄	—	38	1	no	—
207	C₈×D₄	—	38	2	no	—
208	#431	—	19	1	no	—
209	#431	—	38	1	no	—
210	#428	—	38	1	no	—
211	#423	—	19	1	no	—
212	#423	—	19	1	no	—
213	C₂×C₄⋊C₈	—	38	1	no	—
214	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
215	C₂×#104	—	38	1	no	—
216	#417	—	38	1	no	—
217	C₂×#103	—	2	2	no	—
218	C₄×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
219	#339	—	38	1	no	—
220	#332	—	38	1	no	—
221	#332	—	38	1	no	—
222	C₂×C₁₉	—	2	1	no	—
223	C₂×C₁₉	—	2	1	no	—
224	C₂×C₁₉	#1883	1	1	yes	characteristic
225	C₂×C₁₉	#1892	1	1	yes	characteristic
226	C₂×C₁₉	#1902	1	1	yes	characteristic
227	D₁₉	—	4	2	no	—
228	C₂×#41	—	19	1	no	—
229	C₂²×Q₈	—	2	1	no	—
230	C₂×M₁₆	—	38	1	no	—
231	C₂×M₁₆	—	38	1	no	—
232	C₂²×C₈	—	38	2	no	—
233	C₂²×C₈	C₂²⋊D₁₉	1	2	yes	—
234	C₄⋊Q₈	—	2	1	no	—
235	C₄⋊Q₈	—	2	1	no	—
236	C₄⋊Q₈	—	2	1	no	—
237	C₄⋊Q₈	C₄×D₁₉	1	2	yes	—
238	C₄×Q₈	—	19	1	no	—
239	C₄×Q₈	—	19	1	no	—
240	C₄×D₄	—	19	1	no	—
241	C₄×D₄	—	19	1	no	—
242	C₄×D₄	—	38	1	no	—
243	C₄×D₄	—	38	1	no	—
244	#117	—	19	1	no	—
245	#117	—	38	1	no	—
246	C₂×C₄⋊C₄	—	2	1	no	—
247	C₂×C₄⋊C₄	—	2	1	no	—
248	C₂×C₄⋊C₄	—	2	1	no	—
249	C₂×C₄²	—	2	1	no	—
250	C₂×C₄²	—	38	1	no	—
251	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
252	#107	—	38	2	no	—
253	#106	—	38	2	no	—
254	C₄⋊C₈	—	2	1	no	—
255	C₄⋊C₈	—	2	1	no	—
256	C₄⋊C₈	—	19	2	no	—
257	C₄⋊C₈	—	19	1	no	—
258	C₄⋊C₈	—	19	1	no	—
259	C₄⋊C₈	—	38	2	no	—
260	C₄⋊C₈	—	38	1	no	—
261	C₄⋊C₈	—	38	1	no	—
262	C₄⋊C₈	—	38	1	no	—
263	#104	—	76	2	no	—
264	#103	—	4	4	no	—
265	#103	—	38	2	no	—
266	#102	—	38	2	no	—
267	C₂²⋊C₈	—	38	2	no	—
268	C₂²⋊C₈	—	38	1	no	—
269	C₄×C₈	—	38	2	no	—
270	C₄×C₈	—	38	1	no	—
271	#95	—	2	1	no	—
272	C₁₉	#9317	1	1	yes	characteristic, complemented, Sylow
273	#41	—	19	2	no	—
274	#41	—	19	2	no	—
275	#41	—	38	1	no	—
276	#41	—	38	1	no	—
277	C₂×Q₈	—	2	2	no	—
278	C₂×Q₈	—	4	2	no	—
279	C₂×Q₈	—	4	2	no	—
280	C₂×Q₈	—	4	1	no	—
281	C₂×Q₈	—	19	1	no	—
282	C₂×D₄	—	19	1	no	—
283	C₂×D₄	—	38	1	no	—
284	C₂²×C₄	—	4	1	no	—
285	C₂²×C₄	—	4	1	no	—
286	C₂²×C₄	—	38	1	no	—
287	C₂²×C₄	—	38	1	no	—
288	C₂²×C₄	—	38	1	no	—
289	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
290	C₂²×C₄	#64543	1	1	yes	characteristic
291	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
292	M₁₆	—	76	1	no	—
293	M₁₆	—	76	1	no	—
294	M₁₆	—	76	1	no	—
295	M₁₆	—	76	1	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	—	4	2	no	—
298	C₂×C₈	—	38	1	no	—
299	C₂×C₈	—	38	1	no	—
300	C₂×C₈	—	38	2	no	—
301	C₂×C₈	—	76	1	no	—
302	C₂×C₈	—	152	2	no	—
303	C₄⋊C₄	—	2	2	no	—
304	C₄⋊C₄	—	2	2	no	—
305	C₄⋊C₄	—	2	2	no	—
306	C₄⋊C₄	—	4	1	no	—
307	C₄⋊C₄	—	4	1	no	—
308	C₄⋊C₄	—	4	1	no	—
309	C₄⋊C₄	—	19	1	no	—
310	C₄⋊C₄	—	19	1	no	—
311	C₄⋊C₄	—	38	1	no	—
312	C₄⋊C₄	—	38	1	no	—
313	C₂²⋊C₄	—	38	1	no	—
314	C₂²⋊C₄	—	38	1	no	—
315	C₂²⋊C₄	—	38	1	no	—
316	C₄²	—	4	1	no	—
317	C₄²	—	4	1	no	—
318	C₄²	—	38	1	no	—
319	C₄²	—	38	1	no	—
320	C₄²	—	38	1	no	—
321	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
322	C₄²	#64541	1	1	yes	characteristic
323	C₄²	D₄×D₁₉	1	1	yes	characteristic
324	C₄²	D₄×D₁₉	1	1	yes	characteristic
325	C₂³	—	38	1	no	—
326	C₂³	#10597384	1	1	yes	characteristic
327	Q₈	—	4	4	no	—
328	Q₈	—	8	1	no	—
329	Q₈	—	8	1	no	—
330	Q₈	—	19	2	no	—
331	Q₈	—	38	1	no	—
332	D₄	—	19	2	no	—
333	D₄	—	38	2	no	—
334	D₄	—	38	2	no	—
335	D₄	—	38	1	no	—
336	C₂×C₄	—	2	1	no	—
337	C₂×C₄	—	2	1	no	—
338	C₂×C₄	—	4	1	no	—
339	C₂×C₄	—	4	2	no	—
340	C₂×C₄	—	4	2	no	—
341	C₂×C₄	—	8	2	no	—
342	C₂×C₄	—	8	1	no	—
343	C₂×C₄	—	8	1	no	—
344	C₂×C₄	—	38	1	no	—
345	C₂×C₄	—	38	2	no	—
346	C₂×C₄	—	38	2	no	—
347	C₂×C₄	—	38	1	no	—
348	C₂×C₄	—	38	1	no	—
349	C₂×C₄	—	76	1	no	—
350	C₂×C₄	—	76	1	no	—
351	C₂×C₄	—	152	1	no	—
352	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
353	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
354	C₂×C₄	#10597329	1	1	yes	characteristic
355	C₂×C₄	#10597330	1	1	yes	characteristic
356	C₂×C₄	SD₁₆×D₁₉	1	1	yes	characteristic
357	C₂×C₄	#10597362	1	1	yes	characteristic
358	C₂×C₄	Q₁₆×D₁₉	1	1	yes	characteristic
359	C₂×C₄	#10597365	1	1	yes	characteristic
360	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
361	C₂×C₄	#10597383	1	1	yes	characteristic
362	C₈	—	4	2	no	—
363	C₈	—	76	2	no	—
364	C₈	—	76	1	no	—
365	C₈	—	76	1	no	—
366	C₈	—	152	1	no	—
367	C₂²	—	2	1	no	—
368	C₂²	—	2	1	no	—
369	C₂²	—	38	2	no	—
370	C₂²	—	76	1	no	—
371	C₂²	—	76	1	no	—
372	C₂²	#11926420	1	1	yes	characteristic
373	C₂²	#11926770	1	1	yes	characteristic
374	C₂²	#11926782	1	1	yes	characteristic
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	2	1	no	—
378	C₄	—	8	2	no	—
379	C₄	—	8	2	no	—
380	C₄	—	38	2	no	—
381	C₄	—	76	1	no	—
382	C₄	—	76	1	no	—
383	C₄	#103×D₁₉	1	1	yes	characteristic
384	C₄	#11926539	1	1	yes	characteristic
385	C₄	#11926711	1	1	yes	characteristic
386	C₄	#11926716	1	1	yes	characteristic
387	C₄	#11926776	1	1	yes	characteristic
388	C₄	#11926784	1	1	yes	characteristic
389	C₂	—	2	1	no	—
390	C₂	—	2	1	no	—
391	C₂	—	76	2	no	—
392	C₂	#14564014	1	1	yes	characteristic
393	C₂	#14564055	1	1	yes	characteristic
394	C₂	#14564105	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11926420
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1981	C₂
socle	C₂²×C₁₉	#386
prime radical	C₂×#64541	D₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1981	D₁₉
hypocenter	C₁₉	#9317
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1981

Complements

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

N	H ≅ G/N	type
4. #14562320	C₂	semidirect
5. #14561538	C₂	semidirect
6. #14561510	C₂	semidirect
7. C₁₉×#1981	C₂	semidirect
9. #11926836	C₄	semidirect
10. #11926831	C₄	semidirect
157. D₇₆	#103	semidirect
164. Dic₃₈	#103	semidirect
172. #1981	D₁₉	semidirect
272. C₁₉	#9317	semidirect

Maximal subgroups

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

subgroup	index
#14573444	2
#14572744	2
#14563656	2
#14562320	2
#14561538	2
#14561510	2
C₁₉×#1981	2
#9317	19