Subgroups — #18075704

Order 4864.

454 conjugacy classes of subgroups.

The subgroup lattice has 330 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	#14563656	C₂	1	2	yes	maximal
3	#14562391	C₂	1	1	yes	maximal, characteristic, complemented
4	#14561538	C₂	1	2	yes	maximal, complemented
5	C₄×C₁₉×#106	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	#11926568	C₂²	1	2	yes	—
16	C₂×#10597262	—	2	2	no	—
17	C₄×C₈×D₁₉	—	2	1	no	—
18	#11926304	—	2	2	no	—
19	#11926295	—	2	1	no	—
20	#11926295	—	2	1	no	—
21	C₂×C₁₉×#106	—	2	1	no	—
22	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
23	C₄×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
24	C₁₉×#340	—	2	1	no	—
25	C₂×#64541	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
29	C₂×C₄²×C₁₉	—	2	2	no	—
30	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
31	C₄×C₂²⋊D₁₉	—	2	1	no	—
32	C₄×C₂²⋊D₁₉	—	2	1	no	—
33	#10597372	D₄	1	1	yes	characteristic
34	C₂×#64520	—	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×#106	—	2	4	no	—
44	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
47	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
48	C₁₉×#95	—	2	2	no	—
49	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
50	(C₂×D₁₉)⋊C₈	—	2	1	no	—
51	(C₂×D₁₉)⋊C₈	—	2	1	no	—
52	Dic₁₉⋊C₈	—	2	1	no	—
53	Dic₁₉⋊C₈	—	2	1	no	—
54	C₈×Dic₁₉	—	2	1	no	—
55	C₈×Dic₁₉	—	2	1	no	—
56	#10597262	—	4	4	no	—
57	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
58	C₄⋊(C₁₉⋊C₈)	—	2	2	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	—	4	2	no	—
63	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
66	C₂×C₂²⋊D₁₉	—	2	1	no	—
67	#64541	—	2	2	no	—
68	#64541	SD₁₆	1	4	yes	—
69	C₂×D₇₆	C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×D₁₉	—	2	1	no	—
71	C₂×C₄×D₁₉	—	2	1	no	—
72	C₂×C₄×D₁₉	—	2	1	no	—
73	C₂×Dic₃₈	C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₉	—	2	2	no	—
75	C₂×C₈×C₁₉	—	4	1	no	—
76	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
80	C₄⋊C₄×C₁₉	—	2	4	no	—
81	C₄⋊C₄×C₁₉	—	4	2	no	—
82	C₄²×C₁₉	—	4	2	no	—
83	C₄²×C₁₉	—	4	2	no	—
84	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
87	C₄²×C₁₉	#41	1	1	yes	characteristic
88	C₂²⋊Dic₁₉	—	2	1	no	—
89	(C₂×D₁₉)⋊C₄	—	2	1	no	—
90	(C₂×D₁₉)⋊C₄	—	2	1	no	—
91	C₄⋊Dic₁₉	C₄⋊C₄	1	1	yes	characteristic
92	C₄⋊Dic₁₉	C₄⋊C₄	1	1	yes	characteristic
93	Dic₁₉⋊C₄	—	2	1	no	—
94	Dic₁₉⋊C₄	—	2	1	no	—
95	C₄×Dic₁₉	—	2	1	no	—
96	C₄×Dic₁₉	—	2	1	no	—
97	C₄×Dic₁₉	—	2	1	no	—
98	#64520	—	4	2	no	—
99	#64520	—	4	2	no	—
100	C₂×C₁₉⋊C₈	—	2	1	no	—
101	C₂×C₁₉⋊C₈	—	2	1	no	—
102	C₂×C₁₉⋊C₈	—	4	2	no	—
103	C₈×D₁₉	—	4	2	no	—
104	C₈×D₁₉	—	4	2	no	—
105	#8666	—	19	1	no	maximal, Sylow
106	C₂³×C₁₉	#121	1	1	yes	characteristic
107	C₂²×D₁₉	—	2	1	no	—
108	C₂×C₄×C₁₉	—	2	1	no	—
109	C₂×C₄×C₁₉	—	2	1	no	—
110	C₂×C₄×C₁₉	—	4	1	no	—
111	C₂×C₄×C₁₉	—	4	4	no	—
112	C₂×C₄×C₁₉	—	8	2	no	—
113	C₂×C₄×C₁₉	—	8	2	no	—
114	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
117	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
118	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
119	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
121	C₂×C₄×C₁₉	#135	1	2	yes	—
122	C₂²⋊D₁₉	—	2	4	no	—
123	C₂×Dic₁₉	—	2	1	no	—
124	C₂×Dic₁₉	—	2	1	no	—
125	C₂×Dic₁₉	—	2	1	no	—
126	D₇₆	—	2	1	no	—
127	D₇₆	#106	1	2	yes	complemented
128	C₄×D₁₉	—	2	4	no	—
129	C₄×D₁₉	—	4	2	no	—
130	C₄×D₁₉	—	4	1	no	—
131	C₄×D₁₉	—	4	1	no	—
132	Dic₃₈	—	2	1	no	—
133	Dic₃₈	#106	1	2	yes	complemented
134	C₈×C₁₉	—	2	1	no	—
135	C₈×C₁₉	—	2	1	no	—
136	C₈×C₁₉	—	2	1	no	—
137	C₈×C₁₉	#104	1	2	yes	complemented
138	C₁₉⋊C₈	—	4	1	no	—
139	C₁₉⋊C₈	—	4	1	no	—
140	C₁₉⋊C₈	—	8	2	no	—
141	C₈×#41	—	19	1	no	—
142	#1806	—	19	2	no	—
143	C₄×#106	D₁₉	1	1	yes	characteristic, complemented
144	#1566	—	19	1	no	—
145	#1303	—	19	2	no	—
146	C₂²×C₁₉	—	2	1	no	—
147	C₂²×C₁₉	—	2	1	no	—
148	C₂²×C₁₉	#380	1	1	yes	characteristic
149	C₂²×C₁₉	#465	1	2	yes	—
150	C₂×D₁₉	—	2	2	no	—
151	C₂×D₁₉	—	4	1	no	—
152	C₂×D₁₉	—	4	1	no	—
153	C₄×C₁₉	—	2	1	no	—
154	C₄×C₁₉	—	2	1	no	—
155	C₄×C₁₉	—	2	1	no	—
156	C₄×C₁₉	—	8	4	no	—
157	C₄×C₁₉	C₂×#104	1	1	yes	characteristic
158	C₄×C₁₉	#421	1	1	yes	characteristic
159	C₄×C₁₉	C₂×#106	1	1	yes	characteristic
160	C₄×C₁₉	#427	1	1	yes	characteristic
161	C₄×C₁₉	#465	1	2	yes	—
162	Dic₁₉	—	2	2	no	—
163	Dic₁₉	—	4	1	no	—
164	Dic₁₉	—	4	1	no	—
165	C₄×#41	—	19	1	no	—
166	C₈×Q₈	—	19	1	no	—
167	C₈×Q₈	—	19	1	no	—
168	C₈×D₄	—	19	1	no	—
169	C₈×D₄	—	19	1	no	—
170	C₈×D₄	—	38	1	no	—
171	C₈×D₄	—	38	1	no	—
172	#431	—	19	1	no	—
173	#431	—	38	1	no	—
174	C₂×#106	—	2	1	no	—
175	#423	—	19	2	no	—
176	C₂×#104	—	38	2	no	—
177	C₂×C₄×C₈	—	38	1	no	—
178	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
179	C₄×C₄⋊C₄	C₂×D₁₉	1	2	yes	—
180	#340	—	2	1	no	—
181	#339	—	38	2	no	—
182	#334	—	38	1	no	—
183	#334	—	38	1	no	—
184	C₂×C₁₉	—	2	2	no	—
185	C₂×C₁₉	#1806	1	1	yes	characteristic
186	C₂×C₁₉	#1807	1	1	yes	characteristic
187	C₂×C₁₉	#1814	1	1	yes	characteristic
188	D₁₉	—	4	2	no	—
189	C₂×#41	—	19	1	no	—
190	C₂×M₁₆	—	38	2	no	—
191	C₂²×C₈	—	38	1	no	—
192	C₂²×C₈	—	38	1	no	—
193	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
194	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
195	C₄×Q₈	—	19	1	no	—
196	C₄×Q₈	—	19	1	no	—
197	C₄×D₄	—	19	1	no	—
198	C₄×D₄	—	19	1	no	—
199	C₄×D₄	—	38	1	no	—
200	C₄×D₄	—	38	1	no	—
201	#117	—	19	1	no	—
202	#117	—	38	1	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄²	—	2	2	no	—
205	C₂×C₄²	—	38	1	no	—
206	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
207	#106	—	2	4	no	—
208	C₄⋊C₈	—	19	1	no	—
209	C₄⋊C₈	—	19	1	no	—
210	C₄⋊C₈	—	38	1	no	—
211	C₄⋊C₈	—	38	1	no	—
212	C₄⋊C₈	—	38	2	no	—
213	C₄⋊C₈	—	38	2	no	—
214	#104	—	76	4	no	—
215	C₂²⋊C₈	—	38	1	no	—
216	C₂²⋊C₈	—	38	1	no	—
217	C₂²⋊C₈	—	38	1	no	—
218	C₄×C₈	—	38	1	no	—
219	C₄×C₈	—	38	1	no	—
220	C₄×C₈	—	38	1	no	—
221	C₄×C₈	C₄×D₁₉	1	1	yes	characteristic
222	C₄×C₈	C₄×D₁₉	1	1	yes	characteristic
223	C₄×C₈	D₇₆	1	1	yes	characteristic
224	C₄×C₈	D₇₆	1	1	yes	characteristic
225	#95	—	2	2	no	—
226	C₁₉	#8666	1	1	yes	characteristic, complemented, Sylow
227	#41	—	19	4	no	—
228	#41	—	38	2	no	—
229	C₂×Q₈	—	19	1	no	—
230	C₂×D₄	—	19	1	no	—
231	C₂×D₄	—	38	1	no	—
232	C₂²×C₄	—	4	2	no	—
233	C₂²×C₄	—	38	1	no	—
234	C₂²×C₄	—	38	1	no	—
235	C₂²×C₄	—	38	1	no	—
236	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
237	C₂²×C₄	#64543	1	1	yes	characteristic
238	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
239	M₁₆	—	76	2	no	—
240	M₁₆	—	76	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	4	1	no	—
243	C₂×C₈	—	38	1	no	—
244	C₂×C₈	—	38	1	no	—
245	C₂×C₈	—	76	2	no	—
246	C₂×C₈	—	76	2	no	—
247	C₂×C₈	—	76	2	no	—
248	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
249	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
250	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
251	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₄⋊C₄	—	19	1	no	—
255	C₄⋊C₄	—	19	1	no	—
256	C₄⋊C₄	—	38	1	no	—
257	C₄⋊C₄	—	38	1	no	—
258	C₂²⋊C₄	—	38	1	no	—
259	C₂²⋊C₄	—	38	1	no	—
260	C₂²⋊C₄	—	38	1	no	—
261	C₄²	—	4	2	no	—
262	C₄²	—	4	2	no	—
263	C₄²	—	38	1	no	—
264	C₄²	—	38	1	no	—
265	C₄²	—	38	1	no	—
266	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
267	C₄²	C₂×D₇₆	1	1	yes	characteristic
268	C₄²	Q₈×D₁₉	1	1	yes	characteristic
269	C₄²	#64545	1	1	yes	characteristic
270	C₂³	—	38	1	no	—
271	C₂³	#10597384	1	1	yes	characteristic
272	Q₈	—	19	2	no	—
273	Q₈	—	38	1	no	—
274	D₄	—	19	2	no	—
275	D₄	—	38	4	no	—
276	D₄	—	38	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	8	2	no	—
282	C₂×C₄	—	8	2	no	—
283	C₂×C₄	—	38	1	no	—
284	C₂×C₄	—	38	4	no	—
285	C₂×C₄	—	38	1	no	—
286	C₂×C₄	—	38	1	no	—
287	C₂×C₄	—	76	2	no	—
288	C₂×C₄	—	76	1	no	—
289	C₂×C₄	—	76	1	no	—
290	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
291	C₂×C₄	#10597338	1	1	yes	characteristic
292	C₂×C₄	#10597341	1	1	yes	characteristic
293	C₂×C₄	#10597343	1	1	yes	characteristic
294	C₂×C₄	SD₁₆×D₁₉	1	2	yes	—
295	C₂×C₄	#10597362	1	2	yes	—
296	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
297	C₂×C₄	#10597392	1	1	yes	characteristic
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	—	76	1	no	—
302	C₈	—	76	1	no	—
303	C₈	—	152	2	no	—
304	C₈	#10597262	1	2	yes	complemented
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	38	2	no	—
308	C₂²	—	76	1	no	—
309	C₂²	—	76	1	no	—
310	C₂²	#11926401	1	1	yes	characteristic
311	C₂²	#11926770	1	2	yes	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	8	4	no	—
316	C₄	—	38	2	no	—
317	C₄	—	76	1	no	—
318	C₄	—	76	1	no	—
319	C₄	C₂×#10597262	1	1	yes	characteristic
320	C₄	#11926562	1	1	yes	characteristic
321	C₄	#106×D₁₉	1	1	yes	characteristic
322	C₄	#11926579	1	1	yes	characteristic
323	C₄	#11926774	1	2	yes	—
324	C₂	—	2	2	no	—
325	C₂	—	76	2	no	—
326	C₂	#14563656	1	1	yes	characteristic
327	C₂	#14563657	1	1	yes	characteristic
328	C₂	#14563686	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10597341
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₄×C₁₉×#106	C₂
socle	C₂²×C₁₉	#380
prime radical	C₂×#64541	D₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₄×#106	D₁₉
hypocenter	C₁₉	#8666
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#106

Complements

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

N	H ≅ G/N	type
3. #14562391	C₂	semidirect
4. #14561538	C₂	semidirect
5. C₄×C₁₉×#106	C₂	semidirect
7. C₈×Dic₃₈	C₄	semidirect
8. C₈×Dic₃₈	C₄	semidirect
9. C₈×D₇₆	C₄	semidirect
10. C₈×D₇₆	C₄	semidirect
127. D₇₆	#106	semidirect
133. Dic₃₈	#106	semidirect
137. C₈×C₁₉	#104	semidirect
143. C₄×#106	D₁₉	semidirect
226. C₁₉	#8666	semidirect
304. C₈	#10597262	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
#14563656	2
#14562391	2
#14561538	2
C₄×C₁₉×#106	2
#8666	19