Subgroups — #18075692

Order 4864.

434 conjugacy classes of subgroups.

The subgroup lattice has 336 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	#14563772	C₂	1	1	yes	maximal, characteristic, complemented
3	#14562381	C₂	1	1	yes	maximal, characteristic, complemented
4	#14562061	C₂	1	1	yes	maximal, characteristic
5	#14562059	C₂	1	1	yes	maximal, characteristic
6	#14561532	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1301	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	1	no	—
11	#11926621	—	2	1	no	—
12	#11926621	C₄	1	2	yes	—
13	#11926621	C₄	1	2	yes	—
14	#11926597	C₂²	1	1	yes	characteristic
15	#11926568	C₂²	1	1	yes	characteristic
16	C₂×(C₂×D₁₉)⋊C₈	—	2	1	no	—
17	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
18	#11926308	—	2	1	no	—
19	#11926308	—	2	1	no	—
20	#11926296	—	2	1	no	—
21	#11926294	—	2	1	no	—
22	#11926273	—	2	2	no	—
23	#11926271	—	2	2	no	—
24	C₁₉×#431	C₂²	1	1	yes	characteristic
25	C₁₉×#423	C₂²	1	1	yes	characteristic
26	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	C₂²×D₇₆	Q₈	1	1	yes	characteristic
28	C₂²×C₄×D₁₉	—	2	1	no	—
29	C₂×M₁₆×C₁₉	—	2	1	no	—
30	C₂²×C₈×C₁₉	—	2	1	no	—
31	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
32	C₂×(C₂×D₁₉)⋊C₄	—	2	1	no	—
33	C₂×C₄⋊Dic₁₉	Q₈	1	1	yes	characteristic
34	C₂×#64520	—	2	1	no	—
35	C₂²×C₁₉⋊C₈	D₄	1	1	yes	characteristic
36	C₂²×C₁₉⋊C₈	D₄	1	1	yes	characteristic
37	C₂×#64515	—	2	2	no	—
38	C₂×#64515	—	2	2	no	—
39	C₂×#64515	—	4	2	no	—
40	C₂×#64515	—	4	2	no	—
41	C₄×D₇₆	—	2	1	no	—
42	C₄×D₇₆	—	2	1	no	—
43	C₄×D₇₆	C₂×C₄	1	2	yes	—
44	C₄×D₇₆	D₄	1	2	yes	—
45	C₁₉×C₄⋊C₈	—	2	1	no	—
46	C₁₉×C₄⋊C₈	—	2	1	no	—
47	C₁₉×C₄⋊C₈	—	2	1	no	—
48	C₁₉×C₄⋊C₈	—	2	1	no	—
49	C₁₉×C₄⋊C₈	C₂×C₄	1	2	yes	—
50	C₁₉×C₂²⋊C₈	—	2	1	no	—
51	C₄×C₈×C₁₉	—	2	1	no	—
52	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
53	(C₂×D₁₉)⋊C₈	—	2	2	no	—
54	(C₂×D₁₉)⋊C₈	—	2	2	no	—
55	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
56	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
57	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
58	C₄×C₁₉⋊C₈	—	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	D₄	1	1	yes	characteristic
62	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
63	C₂³×D₁₉	—	2	1	no	—
64	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
67	C₂²×Dic₁₉	—	2	1	no	—
68	C₂×D₇₆	—	2	1	no	—
69	C₂×D₇₆	—	4	2	no	—
70	C₂×D₇₆	C₄⋊C₄	1	2	yes	—
71	C₂×C₄×D₁₉	—	2	2	no	—
72	C₂×C₄×D₁₉	—	2	2	no	—
73	C₂×C₄×D₁₉	—	4	1	no	—
74	C₂×C₄×D₁₉	—	4	1	no	—
75	C₂×C₄×D₁₉	—	4	1	no	—
76	C₂×C₄×D₁₉	—	4	1	no	—
77	M₁₆×C₁₉	—	4	1	no	—
78	M₁₆×C₁₉	—	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	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₁₉	#41	1	1	yes	characteristic
88	(C₂×D₁₉)⋊C₄	—	2	2	no	—
89	(C₂×D₁₉)⋊C₄	—	2	2	no	—
90	C₄⋊Dic₁₉	—	2	1	no	—
91	C₄⋊Dic₁₉	C₄⋊C₄	1	2	yes	—
92	#64520	—	4	1	no	—
93	#64520	—	4	1	no	—
94	C₂×C₁₉⋊C₈	—	2	1	no	—
95	C₂×C₁₉⋊C₈	—	2	1	no	—
96	C₂×C₁₉⋊C₈	—	2	1	no	—
97	C₂×C₁₉⋊C₈	—	2	1	no	—
98	C₂×C₁₉⋊C₈	—	4	1	no	—
99	C₂×C₁₉⋊C₈	—	4	1	no	—
100	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
104	#64515	—	4	4	no	—
105	#64515	—	4	4	no	—
106	#8664	—	19	1	no	maximal, Sylow
107	C₂³×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
108	C₂²×D₁₉	—	2	2	no	—
109	C₂²×D₁₉	—	4	2	no	—
110	C₂²×D₁₉	—	4	2	no	—
111	C₂²×D₁₉	—	4	1	no	—
112	C₂²×D₁₉	—	4	1	no	—
113	C₂×C₄×C₁₉	—	2	2	no	—
114	C₂×C₄×C₁₉	—	4	1	no	—
115	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
118	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
119	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
120	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
121	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
122	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
123	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
124	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₂×Dic₁₉	—	2	2	no	—
126	C₂×Dic₁₉	—	4	1	no	—
127	C₂×Dic₁₉	—	4	1	no	—
128	D₇₆	—	2	4	no	—
129	D₇₆	—	4	2	no	—
130	C₄×D₁₉	—	4	2	no	—
131	C₄×D₁₉	—	4	2	no	—
132	C₄×D₁₉	—	4	2	no	—
133	C₄×D₁₉	—	4	2	no	—
134	C₈×C₁₉	—	4	2	no	—
135	C₈×C₁₉	—	8	1	no	—
136	C₈×C₁₉	—	8	1	no	—
137	C₁₉⋊C₈	—	2	1	no	—
138	C₁₉⋊C₈	—	2	1	no	—
139	C₁₉⋊C₈	—	2	1	no	—
140	C₁₉⋊C₈	—	2	1	no	—
141	C₁₉⋊C₈	—	8	1	no	—
142	C₁₉⋊C₈	—	8	1	no	—
143	C₂×#436	—	19	1	no	—
144	#1837	—	19	1	no	—
145	#1564	—	19	1	no	—
146	#1486	—	19	1	no	—
147	#1486	—	19	1	no	—
148	#1301	—	19	1	no	—
149	#1301	D₁₉	1	1	yes	characteristic, complemented
150	C₂²×C₁₉	—	2	1	no	—
151	C₂²×C₁₉	—	2	1	no	—
152	C₂²×C₁₉	#380	1	1	yes	characteristic
153	C₂²×C₁₉	#478	1	1	yes	characteristic
154	C₂²×C₁₉	#478	1	1	yes	characteristic
155	C₂×D₁₉	—	4	2	no	—
156	C₂×D₁₉	—	4	2	no	—
157	C₂×D₁₉	—	4	2	no	—
158	C₂×D₁₉	—	8	2	no	—
159	C₂×D₁₉	—	8	2	no	—
160	C₄×C₁₉	—	2	1	no	—
161	C₄×C₁₉	—	2	1	no	—
162	C₄×C₁₉	—	2	1	no	—
163	C₄×C₁₉	#416	1	1	yes	characteristic
164	C₄×C₁₉	#418	1	1	yes	characteristic
165	C₄×C₁₉	#430	1	2	yes	—
166	C₄×C₁₉	#463	1	1	yes	characteristic
167	C₄×C₁₉	#463	1	1	yes	characteristic
168	Dic₁₉	—	4	2	no	—
169	C₂²×M₁₆	—	38	1	no	—
170	C₂×C₄×D₄	—	19	1	no	—
171	#436	—	19	2	no	—
172	#436	—	19	2	no	—
173	#436	—	38	1	no	—
174	#436	—	38	1	no	—
175	#431	—	19	1	no	—
176	#431	C₂×D₁₉	1	1	yes	characteristic
177	#423	—	19	1	no	—
178	#423	C₂×D₁₉	1	1	yes	characteristic
179	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
180	C₂×C₂²⋊C₈	—	38	1	no	—
181	C₂×C₄×C₈	—	19	1	no	—
182	#341	—	38	1	no	—
183	#341	—	38	1	no	—
184	#335	—	38	1	no	—
185	#334	—	38	1	no	—
186	D₄⋊C₈	—	38	2	no	—
187	D₄⋊C₈	—	38	2	no	—
188	C₂×C₁₉	—	2	1	no	—
189	C₂×C₁₉	—	2	1	no	—
190	C₂×C₁₉	#1804	1	1	yes	characteristic
191	C₂×C₁₉	#1811	1	1	yes	characteristic
192	C₂×C₁₉	#1814	1	1	yes	characteristic
193	D₁₉	—	8	2	no	—
194	C₂²×D₄	—	19	1	no	—
195	C₂³×C₄	—	38	1	no	—
196	C₂×M₁₆	—	2	1	no	—
197	C₂×M₁₆	—	38	2	no	—
198	C₂×M₁₆	—	38	2	no	—
199	C₂×M₁₆	—	38	1	no	—
200	C₂×M₁₆	—	76	2	no	—
201	C₂×M₁₆	—	76	2	no	—
202	C₂²×C₈	—	2	1	no	—
203	C₂²×C₈	—	19	1	no	—
204	C₂²×C₈	—	19	1	no	—
205	C₄×D₄	—	19	2	no	—
206	C₄×D₄	—	19	2	no	—
207	C₄×D₄	—	38	1	no	—
208	C₄×D₄	—	38	1	no	—
209	C₂×C₄⋊C₄	—	19	1	no	—
210	C₂×C₂²⋊C₄	—	38	1	no	—
211	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
212	C₄⋊C₈	—	2	1	no	—
213	C₄⋊C₈	—	2	1	no	—
214	C₄⋊C₈	—	2	1	no	—
215	C₄⋊C₈	—	2	1	no	—
216	C₄⋊C₈	—	38	1	no	—
217	C₄⋊C₈	—	38	1	no	—
218	C₄⋊C₈	—	38	1	no	—
219	C₄⋊C₈	C₄×D₁₉	1	2	yes	—
220	C₂²⋊C₈	—	2	1	no	—
221	C₂²⋊C₈	—	38	2	no	—
222	C₂²⋊C₈	—	38	2	no	—
223	C₂²⋊C₈	—	38	1	no	—
224	C₄×C₈	—	2	1	no	—
225	C₄×C₈	—	19	1	no	—
226	C₄×C₈	—	19	1	no	—
227	C₄×C₈	—	19	1	no	—
228	C₄×C₈	—	19	1	no	—
229	C₄×C₈	—	38	1	no	—
230	C₁₉	#8664	1	1	yes	characteristic, complemented, Sylow
231	C₂⁴	—	38	1	no	—
232	C₂×D₄	—	19	2	no	—
233	C₂×D₄	—	38	1	no	—
234	C₂×D₄	—	76	2	no	—
235	C₂²×C₄	—	38	1	no	—
236	C₂²×C₄	—	38	2	no	—
237	C₂²×C₄	—	38	2	no	—
238	C₂²×C₄	—	76	1	no	—
239	C₂²×C₄	—	76	1	no	—
240	C₂²×C₄	—	76	1	no	—
241	C₂²×C₄	—	76	1	no	—
242	C₂²×C₄	#64541	1	1	yes	characteristic
243	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
244	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
245	M₁₆	—	4	1	no	—
246	M₁₆	—	4	1	no	—
247	M₁₆	—	76	4	no	—
248	M₁₆	—	76	4	no	—
249	M₁₆	—	76	1	no	—
250	M₁₆	—	76	1	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	4	1	no	—
253	C₂×C₈	—	4	1	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	19	1	no	—
257	C₂×C₈	—	19	1	no	—
258	C₂×C₈	—	19	1	no	—
259	C₂×C₈	—	19	1	no	—
260	C₂×C₈	—	38	1	no	—
261	C₂×C₈	—	38	1	no	—
262	C₂×C₈	—	38	1	no	—
263	C₂×C₈	—	38	1	no	—
264	C₂×C₈	—	76	1	no	—
265	C₂×C₈	—	76	1	no	—
266	C₄⋊C₄	—	19	2	no	—
267	C₄⋊C₄	—	38	1	no	—
268	C₂²⋊C₄	—	38	2	no	—
269	C₂²⋊C₄	—	38	2	no	—
270	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
271	C₄²	#64541	1	1	yes	characteristic
272	C₄²	D₄×D₁₉	1	1	yes	characteristic
273	C₄²	D₄×D₁₉	1	1	yes	characteristic
274	C₂³	—	38	2	no	—
275	C₂³	—	76	2	no	—
276	C₂³	—	76	2	no	—
277	C₂³	—	76	1	no	—
278	C₂³	—	76	1	no	—
279	C₂³	#10597342	1	1	yes	characteristic
280	D₄	—	38	4	no	—
281	D₄	—	76	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	4	1	no	—
284	C₂×C₄	—	38	2	no	—
285	C₂×C₄	—	76	1	no	—
286	C₂×C₄	—	76	1	no	—
287	C₂×C₄	—	76	2	no	—
288	C₂×C₄	—	76	2	no	—
289	C₂×C₄	—	76	2	no	—
290	C₂×C₄	—	76	2	no	—
291	C₂×C₄	#10597318	1	1	yes	characteristic
292	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
293	C₂×C₄	#10597330	1	1	yes	characteristic
294	C₂×C₄	#10597330	1	1	yes	characteristic
295	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
296	C₂×C₄	#10597342	1	1	yes	characteristic
297	C₂×C₄	#10597357	1	1	yes	characteristic
298	C₂×C₄	#10597360	1	1	yes	characteristic
299	C₂×C₄	#10597362	1	1	yes	characteristic
300	C₂×C₄	#10597365	1	1	yes	characteristic
301	C₈	—	4	2	no	—
302	C₈	—	8	1	no	—
303	C₈	—	8	1	no	—
304	C₈	—	38	1	no	—
305	C₈	—	38	1	no	—
306	C₈	—	38	1	no	—
307	C₈	—	38	1	no	—
308	C₈	—	152	1	no	—
309	C₈	—	152	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	76	2	no	—
313	C₂²	—	76	2	no	—
314	C₂²	—	76	2	no	—
315	C₂²	—	152	2	no	—
316	C₂²	—	152	2	no	—
317	C₂²	#11926403	1	1	yes	characteristic
318	C₂²	#11926843	1	1	yes	characteristic
319	C₂²	#11926847	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	76	2	no	—
324	C₄	#11926539	1	1	yes	characteristic
325	C₄	#11926547	1	1	yes	characteristic
326	C₄	#11926591	1	2	yes	—
327	C₄	#11926758	1	1	yes	characteristic
328	C₄	#11926762	1	1	yes	characteristic
329	C₂	—	2	1	no	—
330	C₂	—	2	1	no	—
331	C₂	—	152	2	no	—
332	C₂	#14563644	1	1	yes	characteristic
333	C₂	#14563676	1	1	yes	characteristic
334	C₂	#14563688	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10597330
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1301	C₂
socle	C₂²×C₁₉	#380
prime radical	C₂²×D₇₆	Q₈
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1301	D₁₉
hypocenter	C₁₉	#8664
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1301

Complements

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

N	H ≅ G/N	type
2. #14563772	C₂	semidirect
3. #14562381	C₂	semidirect
6. #14561532	C₂	semidirect
7. C₁₉×#1301	C₂	semidirect
149. #1301	D₁₉	semidirect
230. C₁₉	#8664	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
#14563772	2
#14562381	2
#14562061	2
#14562059	2
#14561532	2
C₁₉×#1301	2
#8664	19