Subgroups — #18085317

Order 4864.

670 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926923	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11926890	C₂	1	1	yes	maximal, characteristic
3	#14564597	C₂	1	1	yes	maximal, characteristic, complemented
4	#14564596	C₂	1	1	yes	maximal, characteristic, complemented
5	#14564442	C₂	1	1	yes	maximal, characteristic
6	C₁₉⋊#1273	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₉×#486	C₂	1	1	yes	maximal, characteristic
8	C₂²×#64528	—	2	2	no	—
9	C₂×#10597315	C₂²	1	1	yes	characteristic
10	#11926923	—	2	2	no	—
11	#11926923	—	2	2	no	—
12	#11926890	—	2	2	no	—
13	#11926890	—	2	2	no	—
14	C₂×#10597289	—	2	2	no	—
15	C₂×#10597289	—	2	2	no	—
16	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
18	#11926414	C₂²	1	1	yes	characteristic
19	#11926307	—	2	2	no	—
20	#11926306	—	2	2	no	—
21	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
22	C₂×C₁₉×#127	C₂²	1	1	yes	characteristic
23	C₁₉×#486	—	2	2	no	—
24	C₁₉×#486	—	2	2	no	—
25	C₂×C₁₉×#102	—	2	2	no	—
26	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
27	C₂²×Q₈×C₁₉	—	2	1	no	—
28	C₂²×D₄×C₁₉	—	2	1	no	—
29	C₂²×D₄×C₁₉	—	2	1	no	—
30	C₂²×D₄×C₁₉	—	2	1	no	—
31	C₂²×Dic₃₈	—	2	1	no	—
32	C₂²×C₈×C₁₉	D₄	1	2	yes	—
33	C₁₉×C₄⋊Q₈	—	2	1	no	—
34	C₁₉×C₄⋊Q₈	—	2	1	no	—
35	C₁₉×C₄⋊Q₈	—	2	1	no	—
36	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
37	C₁₉×#127	—	2	1	no	—
38	C₁₉×#127	—	2	1	no	—
39	C₁₉×#127	—	2	1	no	—
40	C₁₉×#127	D₄	1	2	yes	—
41	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
42	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
43	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
44	C₂×#64528	—	2	8	no	—
45	C₂×#64528	—	4	4	no	—
46	C₂×#64528	—	4	4	no	—
47	C₂×C₄⋊Dic₁₉	—	2	1	no	—
48	C₂×C₄⋊Dic₁₉	—	2	1	no	—
49	C₂×C₄⋊Dic₁₉	—	2	1	no	—
50	C₂×C₄⋊Dic₁₉	—	2	1	no	—
51	C₂×C₄⋊Dic₁₉	—	2	1	no	—
52	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
53	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
54	#10597315	—	2	1	no	—
55	#10597315	—	2	1	no	—
56	#10597315	—	2	1	no	—
57	#10597315	D₄	1	2	yes	—
58	C₁₉×#102	—	2	8	no	—
59	C₄×C₈×C₁₉	—	2	1	no	—
60	C₄×C₈×C₁₉	—	2	1	no	—
61	#10597289	—	2	8	no	—
62	#10597289	—	4	4	no	—
63	C₄×C₁₉⋊C₈	—	2	1	no	—
64	C₄×C₁₉⋊C₈	—	2	1	no	—
65	C₄×C₁₉⋊C₈	—	2	1	no	—
66	C₄×C₁₉⋊C₈	—	2	1	no	—
67	C₂⁴×C₁₉	—	4	1	no	—
68	C₂×Q₈×C₁₉	—	2	2	no	—
69	C₂×Q₈×C₁₉	—	4	1	no	—
70	C₂×Q₈×C₁₉	—	4	2	no	—
71	C₂×Q₈×C₁₉	—	4	2	no	—
72	C₂×D₄×C₁₉	—	2	2	no	—
73	C₂×D₄×C₁₉	—	2	2	no	—
74	C₂×D₄×C₁₉	—	2	2	no	—
75	C₂×D₄×C₁₉	—	4	1	no	—
76	C₂×D₄×C₁₉	—	4	1	no	—
77	C₂×D₄×C₁₉	—	4	1	no	—
78	C₂×D₄×C₁₉	—	4	2	no	—
79	C₂×D₄×C₁₉	—	4	2	no	—
80	C₂×D₄×C₁₉	—	4	2	no	—
81	C₂×D₄×C₁₉	—	4	2	no	—
82	C₂×D₄×C₁₉	—	4	2	no	—
83	C₂×D₄×C₁₉	—	4	2	no	—
84	C₂²×C₄×C₁₉	—	4	1	no	—
85	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
88	C₂²×Dic₁₉	—	4	1	no	—
89	C₂²×Dic₁₉	—	4	1	no	—
90	C₂×Dic₃₈	—	2	2	no	—
91	C₂×Dic₃₈	—	4	2	no	—
92	C₂×Dic₃₈	—	4	2	no	—
93	C₂×Dic₃₈	—	4	1	no	—
94	C₂×C₈×C₁₉	—	2	2	no	—
95	C₂×C₈×C₁₉	—	2	2	no	—
96	C₂×C₈×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	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	#64528	—	4	16	no	—
106	C₄⋊Dic₁₉	—	2	2	no	—
107	C₄⋊Dic₁₉	—	2	2	no	—
108	C₄⋊Dic₁₉	—	4	1	no	—
109	C₄⋊Dic₁₉	—	4	1	no	—
110	C₄⋊Dic₁₉	—	4	2	no	—
111	C₄⋊Dic₁₉	—	4	2	no	—
112	C₄⋊Dic₁₉	—	4	2	no	—
113	C₂×C₁₉⋊C₈	—	2	2	no	—
114	C₂×C₁₉⋊C₈	—	2	2	no	—
115	C₂×C₁₉⋊C₈	—	2	2	no	—
116	C₂×C₁₉⋊C₈	—	2	2	no	—
117	C₂×C₁₉⋊C₈	—	2	2	no	—
118	C₂×C₁₉⋊C₈	D₈	1	4	yes	—
119	#10306	—	19	1	no	maximal, Sylow
120	C₂³×C₁₉	—	4	2	no	—
121	C₂³×C₁₉	—	8	1	no	—
122	C₂³×C₁₉	—	8	1	no	—
123	C₂³×C₁₉	—	8	1	no	—
124	C₂³×C₁₉	—	8	1	no	—
125	C₂³×C₁₉	—	8	1	no	—
126	C₂³×C₁₉	—	8	1	no	—
127	C₂³×C₁₉	#127	1	1	yes	characteristic
128	Q₈×C₁₉	—	4	4	no	—
129	Q₈×C₁₉	—	8	2	no	—
130	D₄×C₁₉	—	4	4	no	—
131	D₄×C₁₉	—	4	4	no	—
132	D₄×C₁₉	—	4	4	no	—
133	D₄×C₁₉	—	8	2	no	—
134	D₄×C₁₉	—	8	2	no	—
135	D₄×C₁₉	—	8	2	no	—
136	C₂×C₄×C₁₉	—	4	2	no	—
137	C₂×C₄×C₁₉	—	8	1	no	—
138	C₂×C₄×C₁₉	—	8	1	no	—
139	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
140	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
141	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
142	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
143	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
144	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
145	C₂×C₄×C₁₉	C₂×D₈	1	2	yes	—
146	C₂×C₄×C₁₉	#135	1	2	yes	—
147	C₂×C₄×C₁₉	#135	1	2	yes	—
148	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
149	C₂×C₄×C₁₉	#137	1	2	yes	—
150	C₂×C₄×C₁₉	#137	1	2	yes	—
151	C₂×Dic₁₉	—	4	2	no	—
152	C₂×Dic₁₉	—	8	1	no	—
153	C₂×Dic₁₉	—	8	1	no	—
154	C₂×Dic₁₉	—	8	1	no	—
155	C₂×Dic₁₉	—	8	1	no	—
156	C₂×Dic₁₉	—	8	1	no	—
157	Dic₃₈	—	4	4	no	—
158	Dic₃₈	—	8	2	no	—
159	C₈×C₁₉	—	4	2	no	—
160	C₁₉⋊C₈	—	2	4	no	—
161	C₁₉⋊C₈	—	4	2	no	—
162	C₂×#494	—	19	1	no	—
163	C₂×#486	—	19	1	no	—
164	C₂×#486	D₁₉	1	1	yes	characteristic
165	#1990	—	19	1	no	—
166	#1990	—	19	1	no	—
167	#1957	—	19	1	no	—
168	#1273	—	19	1	no	—
169	C₂²×C₁₉	—	8	2	no	—
170	C₂²×C₁₉	—	8	2	no	—
171	C₂²×C₁₉	—	8	2	no	—
172	C₂²×C₁₉	—	16	1	no	—
173	C₂²×C₁₉	—	16	1	no	—
174	C₂²×C₁₉	—	16	1	no	—
175	C₂²×C₁₉	—	16	1	no	—
176	C₂²×C₁₉	#392	1	1	yes	characteristic
177	C₂²×C₁₉	#493	1	1	yes	characteristic
178	C₂²×C₁₉	#495	1	1	yes	characteristic
179	C₂²×C₁₉	#495	1	1	yes	characteristic
180	C₂²×C₁₉	#496	1	1	yes	characteristic
181	C₂²×C₁₉	#497	1	1	yes	characteristic
182	C₂²×C₁₉	#497	1	1	yes	characteristic
183	C₄×C₁₉	—	8	2	no	—
184	C₄×C₁₉	#447	1	2	yes	—
185	C₄×C₁₉	#452	1	2	yes	—
186	C₄×C₁₉	#452	1	2	yes	—
187	C₄×C₁₉	#459	1	2	yes	—
188	C₄×C₁₉	#464	1	2	yes	—
189	C₄×C₁₉	#464	1	2	yes	—
190	Dic₁₉	—	8	2	no	—
191	Dic₁₉	—	16	1	no	—
192	C₂²×Q₁₆	—	38	2	no	—
193	C₂×C₄⋊Q₈	—	19	1	no	—
194	C₂×C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
195	C₂×#127	C₂×D₁₉	1	1	yes	characteristic
196	#494	—	38	2	no	—
197	#494	—	38	2	no	—
198	#486	—	2	2	no	—
199	#486	—	2	2	no	—
200	#486	—	38	2	no	—
201	#486	—	38	2	no	—
202	C₂×#102	—	2	2	no	—
203	C₂×#102	—	38	2	no	—
204	C₂×#102	—	38	2	no	—
205	C₂×C₄×C₈	—	19	1	no	—
206	C₂×C₄×C₈	—	19	1	no	—
207	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
208	#384	—	19	1	no	—
209	#340	—	38	2	no	—
210	#340	—	38	2	no	—
211	C₂×C₁₉	—	16	2	no	—
212	C₂×C₁₉	#1709	1	1	yes	characteristic
213	C₂×C₁₉	#1720	1	1	yes	characteristic
214	C₂×C₁₉	#1720	1	1	yes	characteristic
215	C₂×C₁₉	#1722	1	1	yes	characteristic
216	C₂×C₁₉	#2011	1	1	yes	characteristic
217	C₂×C₁₉	#2015	1	1	yes	characteristic
218	C₂×C₁₉	#2015	1	1	yes	characteristic
219	C₂²×Q₈	—	2	1	no	—
220	C₂²×Q₈	—	38	1	no	—
221	C₂²×D₄	—	2	1	no	—
222	C₂²×D₄	—	2	1	no	—
223	C₂²×D₄	—	2	1	no	—
224	C₂×Q₁₆	—	38	8	no	—
225	C₂×Q₁₆	—	76	4	no	—
226	C₂×Q₁₆	—	76	4	no	—
227	C₂²×C₈	—	19	2	no	—
228	C₂²×C₈	—	19	2	no	—
229	C₂²×C₈	C₂²⋊D₁₉	1	2	yes	—
230	C₄⋊Q₈	—	2	1	no	—
231	C₄⋊Q₈	—	2	1	no	—
232	C₄⋊Q₈	—	2	1	no	—
233	C₄⋊Q₈	—	19	2	no	—
234	C₄⋊Q₈	—	38	1	no	—
235	C₄⋊Q₈	—	38	1	no	—
236	C₄⋊Q₈	—	38	1	no	—
237	C₄⋊Q₈	C₂²⋊D₁₉	1	2	yes	—
238	#127	—	2	1	no	—
239	#127	—	2	1	no	—
240	#127	—	2	1	no	—
241	#127	C₂²⋊D₁₉	1	2	yes	—
242	C₂×C₄⋊C₄	—	2	1	no	—
243	C₂×C₄⋊C₄	—	2	1	no	—
244	C₂×C₄⋊C₄	—	38	1	no	—
245	C₂×C₄⋊C₄	—	38	1	no	—
246	C₂×C₄⋊C₄	—	38	1	no	—
247	C₂×C₄⋊C₄	—	38	1	no	—
248	C₂×C₄⋊C₄	—	38	1	no	—
249	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
250	#102	—	2	8	no	—
251	#102	—	38	8	no	—
252	#102	—	76	4	no	—
253	C₄×C₈	—	2	1	no	—
254	C₄×C₈	—	2	1	no	—
255	C₄×C₈	—	38	1	no	—
256	C₄×C₈	—	38	1	no	—
257	C₄×C₈	—	38	1	no	—
258	C₄×C₈	—	38	1	no	—
259	C₁₉	#10306	1	1	yes	characteristic, complemented, Sylow
260	C₂⁴	—	4	1	no	—
261	C₂×Q₈	—	2	2	no	—
262	C₂×Q₈	—	4	1	no	—
263	C₂×Q₈	—	4	2	no	—
264	C₂×Q₈	—	4	2	no	—
265	C₂×Q₈	—	38	2	no	—
266	C₂×Q₈	—	76	1	no	—
267	C₂×Q₈	—	76	2	no	—
268	C₂×Q₈	—	76	2	no	—
269	C₂×D₄	—	2	2	no	—
270	C₂×D₄	—	2	2	no	—
271	C₂×D₄	—	2	2	no	—
272	C₂×D₄	—	4	1	no	—
273	C₂×D₄	—	4	1	no	—
274	C₂×D₄	—	4	1	no	—
275	C₂×D₄	—	4	2	no	—
276	C₂×D₄	—	4	2	no	—
277	C₂×D₄	—	4	2	no	—
278	C₂×D₄	—	4	2	no	—
279	C₂×D₄	—	4	2	no	—
280	C₂×D₄	—	4	2	no	—
281	C₂²×C₄	—	4	1	no	—
282	C₂²×C₄	—	76	1	no	—
283	C₂²×C₄	—	76	1	no	—
284	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
285	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
286	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
287	Q₁₆	—	76	16	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	19	4	no	—
292	C₂×C₈	—	38	2	no	—
293	C₂×C₈	—	38	2	no	—
294	C₂×C₈	—	38	2	no	—
295	C₂×C₈	—	38	2	no	—
296	C₂×C₈	—	38	2	no	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	4	1	no	—
300	C₄⋊C₄	—	4	1	no	—
301	C₄⋊C₄	—	38	2	no	—
302	C₄⋊C₄	—	38	2	no	—
303	C₄⋊C₄	—	76	1	no	—
304	C₄⋊C₄	—	76	1	no	—
305	C₄⋊C₄	—	76	2	no	—
306	C₄⋊C₄	—	76	2	no	—
307	C₄⋊C₄	—	76	2	no	—
308	C₄²	D₄×D₁₉	1	1	yes	characteristic
309	C₄²	#64543	1	1	yes	characteristic
310	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
311	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
312	C₂³	—	4	2	no	—
313	C₂³	—	8	1	no	—
314	C₂³	—	8	1	no	—
315	C₂³	—	8	1	no	—
316	C₂³	—	8	1	no	—
317	C₂³	—	8	1	no	—
318	C₂³	—	8	1	no	—
319	C₂³	#10597386	1	1	yes	characteristic
320	Q₈	—	4	4	no	—
321	Q₈	—	8	2	no	—
322	Q₈	—	76	4	no	—
323	Q₈	—	152	2	no	—
324	D₄	—	4	4	no	—
325	D₄	—	4	4	no	—
326	D₄	—	4	4	no	—
327	D₄	—	8	2	no	—
328	D₄	—	8	2	no	—
329	D₄	—	8	2	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	8	1	no	—
332	C₂×C₄	—	8	1	no	—
333	C₂×C₄	—	76	2	no	—
334	C₂×C₄	—	152	1	no	—
335	C₂×C₄	—	152	1	no	—
336	C₂×C₄	—	152	1	no	—
337	C₂×C₄	—	152	1	no	—
338	C₂×C₄	—	152	1	no	—
339	C₂×C₄	D₈×D₁₉	1	2	yes	—
340	C₂×C₄	#10597357	1	2	yes	—
341	C₂×C₄	#10597361	1	2	yes	—
342	C₂×C₄	#10597362	1	2	yes	—
343	C₂×C₄	#10597383	1	1	yes	characteristic
344	C₂×C₄	#10597383	1	1	yes	characteristic
345	C₂×C₄	#10597384	1	1	yes	characteristic
346	C₂×C₄	#10597385	1	1	yes	characteristic
347	C₂×C₄	#10597385	1	1	yes	characteristic
348	C₂×C₄	#10597396	1	2	yes	—
349	C₂×C₄	#10597397	1	2	yes	—
350	C₂×C₄	#10597399	1	1	yes	characteristic
351	C₈	—	4	2	no	—
352	C₈	—	38	4	no	—
353	C₈	—	76	2	no	—
354	C₂²	—	8	2	no	—
355	C₂²	—	8	2	no	—
356	C₂²	—	8	2	no	—
357	C₂²	—	16	1	no	—
358	C₂²	—	16	1	no	—
359	C₂²	—	16	1	no	—
360	C₂²	—	16	1	no	—
361	C₂²	#11926452	1	1	yes	characteristic
362	C₂²	#11926919	1	1	yes	characteristic
363	C₂²	#11926925	1	1	yes	characteristic
364	C₂²	#11926926	1	1	yes	characteristic
365	C₂²	#11926932	1	1	yes	characteristic
366	C₂²	#11926935	1	1	yes	characteristic
367	C₂²	#11926939	1	1	yes	characteristic
368	C₄	—	8	2	no	—
369	C₄	—	304	1	no	—
370	C₄	—	152	2	no	—
371	C₄	#11926676	1	2	yes	—
372	C₄	#11926703	1	2	yes	—
373	C₄	#11926706	1	2	yes	—
374	C₄	#11926737	1	2	yes	—
375	C₄	#11926763	1	2	yes	—
376	C₄	#11926764	1	2	yes	—
377	C₂	—	16	2	no	—
378	C₂	#14563239	1	1	yes	characteristic
379	C₂	#14563296	1	1	yes	characteristic
380	C₂	#14563297	1	1	yes	characteristic
381	C₂	#14563308	1	1	yes	characteristic
382	C₂	#14564704	1	1	yes	characteristic
383	C₂	#14564727	1	1	yes	characteristic
384	C₂	#14564729	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10597386
commutator, cosocle	C₂×C₄²×C₁₉	C₂³
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₁₉×#486	C₂
socle	C₂³×C₁₉	#127
prime radical	C₂×C₁₉×#127	C₂²
hypercenter	C₂×#486	D₁₉
hypocenter	C₁₉	#10306
LCS	G ▷ C₂×C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#486

Complements

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

N	H ≅ G/N	type
1. C₂×#11926923	C₂	semidirect
3. #14564597	C₂	semidirect
4. #14564596	C₂	semidirect
6. C₁₉⋊#1273	C₂	semidirect
259. C₁₉	#10306	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926923	2
C₂×#11926890	2
#14564597	2
#14564596	2
#14564442	2
C₁₉⋊#1273	2
C₂×C₁₉×#486	2
#10306	19