Subgroups — #18075705

Order 4864.

454 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#64541	C₂	1	1	yes	maximal, characteristic
2	#14563652	C₂	1	2	yes	maximal
3	C₄×#10597275	C₂	1	1	yes	maximal, characteristic, complemented
4	#14561536	C₂	1	2	yes	maximal, complemented
5	C₁₉×#1567	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#64541	C₂²	1	1	yes	characteristic
7	C₈×Dic₃₈	C₄	1	1	yes	characteristic, complemented
8	C₈×Dic₃₈	C₄	1	1	yes	characteristic, complemented
9	C₈×D₇₆	C₄	1	1	yes	characteristic, complemented
10	C₈×D₇₆	C₄	1	1	yes	characteristic, complemented
11	C₈×C₂²⋊D₁₉	—	2	1	no	—
12	C₈×C₂²⋊D₁₉	—	2	1	no	—
13	#11926597	C₂²	1	1	yes	characteristic
14	#11926595	—	2	1	no	—
15	C₂×#10597275	—	2	1	no	—
16	C₂×#10597282	—	2	2	no	—
17	C₄×C₈×D₁₉	—	2	1	no	—
18	C₄×C₄⋊Dic₁₉	C₂²	1	2	yes	—
19	#11926307	—	2	1	no	—
20	#11926305	—	2	2	no	—
21	C₁₉×#423	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
23	C₁₉×#335	—	2	1	no	—
24	C₁₉×#335	—	2	1	no	—
25	C₂×#64541	D₄	1	1	yes	characteristic
26	C₂×M₁₆×C₁₉	—	2	2	no	—
27	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
30	C₄×C₂²⋊D₁₉	—	2	1	no	—
31	C₄×C₂²⋊D₁₉	—	2	1	no	—
32	#10597372	D₄	1	1	yes	characteristic
33	C₂×C₄⋊Dic₁₉	—	2	2	no	—
34	C₂×C₄×Dic₁₉	—	2	2	no	—
35	C₂×C₈×D₁₉	—	2	1	no	—
36	C₂×C₈×D₁₉	—	2	1	no	—
37	C₄×D₇₆	C₂×C₄	1	1	yes	characteristic
38	C₄×D₇₆	Q₈	1	1	yes	characteristic
39	#10597318	—	2	1	no	—
40	C₄²×D₁₉	—	2	1	no	—
41	C₄×Dic₃₈	C₂×C₄	1	1	yes	characteristic
42	C₄×Dic₃₈	Q₈	1	1	yes	characteristic
43	C₁₉×C₄⋊C₈	—	2	2	no	—
44	C₁₉×C₄⋊C₈	—	2	2	no	—
45	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
48	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
49	C₁₉⋊#95	—	2	2	no	—
50	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
51	#10597282	—	4	4	no	—
52	(C₂×D₁₉)⋊C₈	—	2	1	no	—
53	(C₂×D₁₉)⋊C₈	—	2	1	no	—
54	#10597275	—	2	4	no	—
55	Dic₁₉⋊C₈	—	2	1	no	—
56	Dic₁₉⋊C₈	—	2	1	no	—
57	C₈×Dic₁₉	—	2	1	no	—
58	C₈×Dic₁₉	—	2	1	no	—
59	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	1	yes	characteristic
60	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	1	yes	characteristic
61	C₄×C₁₉⋊C₈	—	2	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	C₂×C₂²⋊D₁₉	—	2	1	no	—
66	C₂²×Dic₁₉	—	4	2	no	—
67	#64541	—	2	1	no	—
68	#64541	—	2	1	no	—
69	#64541	D₈	1	2	yes	—
70	#64541	Q₁₆	1	2	yes	—
71	C₂×D₇₆	C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×D₁₉	—	2	1	no	—
73	C₂×C₄×D₁₉	—	2	1	no	—
74	C₂×C₄×D₁₉	—	2	1	no	—
75	C₂×Dic₃₈	C₄⋊C₄	1	1	yes	characteristic
76	M₁₆×C₁₉	—	4	2	no	—
77	M₁₆×C₁₉	—	4	2	no	—
78	C₂×C₈×C₁₉	—	2	2	no	—
79	C₂×C₈×C₁₉	—	4	1	no	—
80	C₂×C₈×C₁₉	—	4	2	no	—
81	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
85	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
88	C₄²×C₁₉	#41	1	1	yes	characteristic
89	C₂²⋊Dic₁₉	—	2	1	no	—
90	(C₂×D₁₉)⋊C₄	—	2	1	no	—
91	(C₂×D₁₉)⋊C₄	—	2	1	no	—
92	C₄⋊Dic₁₉	—	2	4	no	—
93	C₄⋊Dic₁₉	—	4	2	no	—
94	C₄⋊Dic₁₉	C₄⋊C₄	1	1	yes	characteristic
95	C₄⋊Dic₁₉	C₄⋊C₄	1	1	yes	characteristic
96	Dic₁₉⋊C₄	—	2	1	no	—
97	Dic₁₉⋊C₄	—	2	1	no	—
98	C₄×Dic₁₉	—	2	1	no	—
99	C₄×Dic₁₉	—	2	1	no	—
100	C₄×Dic₁₉	—	2	1	no	—
101	C₄×Dic₁₉	—	4	2	no	—
102	C₄×Dic₁₉	—	4	2	no	—
103	C₂×C₁₉⋊C₈	—	2	1	no	—
104	C₂×C₁₉⋊C₈	—	2	1	no	—
105	C₈×D₁₉	—	4	2	no	—
106	C₈×D₁₉	—	4	2	no	—
107	#8667	—	19	1	no	maximal, Sylow
108	C₂³×C₁₉	#121	1	1	yes	characteristic
109	C₂²×D₁₉	—	2	1	no	—
110	C₂×C₄×C₁₉	—	2	1	no	—
111	C₂×C₄×C₁₉	—	2	1	no	—
112	C₂×C₄×C₁₉	—	4	1	no	—
113	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
116	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
117	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
118	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
119	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
121	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
122	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
123	C₂²⋊D₁₉	—	2	2	no	—
124	C₂²⋊D₁₉	—	2	2	no	—
125	C₂×Dic₁₉	—	2	1	no	—
126	C₂×Dic₁₉	—	2	1	no	—
127	C₂×Dic₁₉	—	2	1	no	—
128	C₂×Dic₁₉	—	4	4	no	—
129	C₂×Dic₁₉	—	8	2	no	—
130	C₂×Dic₁₉	—	8	2	no	—
131	D₇₆	—	2	1	no	—
132	D₇₆	#107	1	2	yes	complemented
133	C₄×D₁₉	—	2	2	no	—
134	C₄×D₁₉	—	2	2	no	—
135	C₄×D₁₉	—	4	2	no	—
136	C₄×D₁₉	—	4	1	no	—
137	C₄×D₁₉	—	4	1	no	—
138	Dic₃₈	—	2	1	no	—
139	Dic₃₈	#107	1	2	yes	complemented
140	C₈×C₁₉	—	2	1	no	—
141	C₈×C₁₉	—	2	1	no	—
142	C₈×C₁₉	—	2	1	no	—
143	C₈×C₁₉	—	8	2	no	—
144	C₈×C₁₉	#104	1	2	yes	complemented
145	C₁₉⋊C₈	—	4	1	no	—
146	C₁₉⋊C₈	—	4	1	no	—
147	C₈×#41	—	19	1	no	—
148	#1806	—	19	2	no	—
149	C₄×#107	—	19	1	no	—
150	#1567	D₁₉	1	1	yes	characteristic, complemented
151	#1303	—	19	2	no	—
152	C₂²×C₁₉	—	2	1	no	—
153	C₂²×C₁₉	—	2	1	no	—
154	C₂²×C₁₉	#380	1	1	yes	characteristic
155	C₂²×C₁₉	#466	1	1	yes	characteristic
156	C₂²×C₁₉	#467	1	1	yes	characteristic
157	C₂×D₁₉	—	2	2	no	—
158	C₂×D₁₉	—	4	1	no	—
159	C₂×D₁₉	—	4	1	no	—
160	C₄×C₁₉	—	2	1	no	—
161	C₄×C₁₉	—	2	1	no	—
162	C₄×C₁₉	—	2	1	no	—
163	C₄×C₁₉	C₂×#104	1	1	yes	characteristic
164	C₄×C₁₉	#421	1	1	yes	characteristic
165	C₄×C₁₉	C₂×#107	1	1	yes	characteristic
166	C₄×C₁₉	#427	1	1	yes	characteristic
167	C₄×C₁₉	#466	1	1	yes	characteristic
168	C₄×C₁₉	#467	1	1	yes	characteristic
169	Dic₁₉	—	2	2	no	—
170	Dic₁₉	—	4	1	no	—
171	Dic₁₉	—	4	1	no	—
172	Dic₁₉	—	8	4	no	—
173	C₄×#41	—	19	1	no	—
174	C₈×Q₈	—	19	1	no	—
175	C₈×Q₈	—	19	1	no	—
176	C₈×D₄	—	19	1	no	—
177	C₈×D₄	—	19	1	no	—
178	C₈×D₄	—	38	1	no	—
179	C₈×D₄	—	38	1	no	—
180	#431	—	19	1	no	—
181	#431	—	38	1	no	—
182	C₂×#107	—	38	1	no	—
183	#423	C₂×D₁₉	1	2	yes	—
184	C₂×#104	—	38	2	no	—
185	C₂×C₄×C₈	—	38	1	no	—
186	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
187	C₄×C₄⋊C₄	—	19	2	no	—
188	#340	—	38	1	no	—
189	#339	—	38	2	no	—
190	#335	—	2	1	no	—
191	#335	—	2	1	no	—
192	C₂×C₁₉	—	2	1	no	—
193	C₂×C₁₉	—	2	1	no	—
194	C₂×C₁₉	#1806	1	1	yes	characteristic
195	C₂×C₁₉	#1808	1	1	yes	characteristic
196	C₂×C₁₉	#1814	1	1	yes	characteristic
197	D₁₉	—	4	2	no	—
198	C₂×#41	—	19	1	no	—
199	C₂×M₁₆	—	2	2	no	—
200	C₂²×C₈	—	38	1	no	—
201	C₂²×C₈	—	38	1	no	—
202	C₂²×C₈	D₇₆	1	1	yes	characteristic
203	C₂²×C₈	D₇₆	1	1	yes	characteristic
204	C₄×Q₈	—	19	1	no	—
205	C₄×Q₈	—	19	1	no	—
206	C₄×D₄	—	19	1	no	—
207	C₄×D₄	—	19	1	no	—
208	C₄×D₄	—	38	1	no	—
209	C₄×D₄	—	38	1	no	—
210	#117	—	19	1	no	—
211	#117	—	38	1	no	—
212	C₂×C₄⋊C₄	—	38	2	no	—
213	C₂×C₄²	—	38	1	no	—
214	C₂×C₄²	—	38	2	no	—
215	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
216	#107	—	38	4	no	—
217	C₄⋊C₈	—	2	2	no	—
218	C₄⋊C₈	—	2	2	no	—
219	C₄⋊C₈	—	19	1	no	—
220	C₄⋊C₈	—	19	1	no	—
221	C₄⋊C₈	—	38	1	no	—
222	C₄⋊C₈	—	38	1	no	—
223	#104	—	76	4	no	—
224	C₂²⋊C₈	—	38	1	no	—
225	C₂²⋊C₈	—	38	1	no	—
226	C₂²⋊C₈	—	38	1	no	—
227	C₄×C₈	—	38	1	no	—
228	C₄×C₈	—	38	1	no	—
229	C₄×C₈	—	38	1	no	—
230	C₄×C₈	C₄×D₁₉	1	1	yes	characteristic
231	C₄×C₈	C₄×D₁₉	1	1	yes	characteristic
232	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
233	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
234	#95	—	38	2	no	—
235	C₁₉	#8667	1	1	yes	characteristic, complemented, Sylow
236	#41	—	19	2	no	—
237	#41	—	19	2	no	—
238	#41	—	38	1	no	—
239	#41	—	38	1	no	—
240	C₂×Q₈	—	19	1	no	—
241	C₂×D₄	—	19	1	no	—
242	C₂×D₄	—	38	1	no	—
243	C₂²×C₄	—	38	1	no	—
244	C₂²×C₄	—	38	1	no	—
245	C₂²×C₄	—	38	1	no	—
246	C₂²×C₄	—	76	2	no	—
247	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
248	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
249	C₂²×C₄	#64545	1	1	yes	characteristic
250	M₁₆	—	4	2	no	—
251	M₁₆	—	4	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	4	1	no	—
254	C₂×C₈	—	4	2	no	—
255	C₂×C₈	—	38	1	no	—
256	C₂×C₈	—	38	1	no	—
257	C₂×C₈	—	76	2	no	—
258	C₂×C₈	—	76	2	no	—
259	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
260	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
261	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
262	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
263	C₄⋊C₄	—	19	1	no	—
264	C₄⋊C₄	—	19	1	no	—
265	C₄⋊C₄	—	38	4	no	—
266	C₄⋊C₄	—	38	1	no	—
267	C₄⋊C₄	—	38	1	no	—
268	C₄⋊C₄	—	76	2	no	—
269	C₂²⋊C₄	—	38	1	no	—
270	C₂²⋊C₄	—	38	1	no	—
271	C₂²⋊C₄	—	38	1	no	—
272	C₄²	—	38	1	no	—
273	C₄²	—	38	1	no	—
274	C₄²	—	38	1	no	—
275	C₄²	—	76	2	no	—
276	C₄²	—	76	2	no	—
277	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
278	C₄²	#64543	1	1	yes	characteristic
279	C₄²	Q₈×D₁₉	1	1	yes	characteristic
280	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
281	C₂³	—	38	1	no	—
282	C₂³	#10597341	1	1	yes	characteristic
283	Q₈	—	19	2	no	—
284	Q₈	—	38	1	no	—
285	D₄	—	19	2	no	—
286	D₄	—	38	2	no	—
287	D₄	—	38	2	no	—
288	D₄	—	38	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	38	1	no	—
293	C₂×C₄	—	38	2	no	—
294	C₂×C₄	—	38	2	no	—
295	C₂×C₄	—	38	1	no	—
296	C₂×C₄	—	38	1	no	—
297	C₂×C₄	—	76	2	no	—
298	C₂×C₄	—	76	1	no	—
299	C₂×C₄	—	76	1	no	—
300	C₂×C₄	—	76	4	no	—
301	C₂×C₄	—	152	2	no	—
302	C₂×C₄	—	152	2	no	—
303	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
304	C₂×C₄	#10597338	1	1	yes	characteristic
305	C₂×C₄	#10597343	1	1	yes	characteristic
306	C₂×C₄	D₈×D₁₉	1	1	yes	characteristic
307	C₂×C₄	#10597358	1	1	yes	characteristic
308	C₂×C₄	Q₁₆×D₁₉	1	1	yes	characteristic
309	C₂×C₄	#10597365	1	1	yes	characteristic
310	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
311	C₂×C₄	#10597384	1	1	yes	characteristic
312	C₂×C₄	#10597392	1	1	yes	characteristic
313	C₈	—	2	1	no	—
314	C₈	—	2	1	no	—
315	C₈	—	2	1	no	—
316	C₈	—	8	2	no	—
317	C₈	—	76	1	no	—
318	C₈	—	76	1	no	—
319	C₈	#10597282	1	2	yes	complemented
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	38	2	no	—
323	C₂²	—	76	1	no	—
324	C₂²	—	76	1	no	—
325	C₂²	#11926401	1	1	yes	characteristic
326	C₂²	#11926781	1	1	yes	characteristic
327	C₂²	#11926786	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	38	2	no	—
332	C₄	—	76	1	no	—
333	C₄	—	76	1	no	—
334	C₄	—	152	4	no	—
335	C₄	C₂×#10597282	1	1	yes	characteristic
336	C₄	#11926560	1	1	yes	characteristic
337	C₄	#107×D₁₉	1	1	yes	characteristic
338	C₄	#11926581	1	1	yes	characteristic
339	C₄	#11926777	1	1	yes	characteristic
340	C₄	#11926782	1	1	yes	characteristic
341	C₂	—	2	1	no	—
342	C₂	—	2	1	no	—
343	C₂	—	76	2	no	—
344	C₂	#14563652	1	1	yes	characteristic
345	C₂	#14563661	1	1	yes	characteristic
346	C₂	#14563690	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10597384
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1567	C₂
socle	C₂²×C₁₉	#380
prime radical	C₂×#64541	D₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1567	D₁₉
hypocenter	C₁₉	#8667
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

Complements

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

N	H ≅ G/N	type
3. C₄×#10597275	C₂	semidirect
4. #14561536	C₂	semidirect
5. C₁₉×#1567	C₂	semidirect
7. C₈×Dic₃₈	C₄	semidirect
8. C₈×Dic₃₈	C₄	semidirect
9. C₈×D₇₆	C₄	semidirect
10. C₈×D₇₆	C₄	semidirect
132. D₇₆	#107	semidirect
139. Dic₃₈	#107	semidirect
144. C₈×C₁₉	#104	semidirect
150. #1567	D₁₉	semidirect
235. C₁₉	#8667	semidirect
319. C₈	#10597282	semidirect

Maximal subgroups

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

subgroup	index
C₈×#64541	2
#14563652	2
C₄×#10597275	2
#14561536	2
C₁₉×#1567	2
#8667	19