Subgroups — #18081064

Order 4864.

318 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14564580	C₂	1	1	yes	maximal, characteristic
2	#14564411	C₂	1	1	yes	maximal, characteristic
3	#14562090	C₂	1	1	yes	maximal, characteristic
4	#14561925	C₂	1	1	yes	maximal, characteristic
5	#14561537	C₂	1	1	yes	maximal, characteristic
6	#14561474	C₂	1	1	yes	maximal, characteristic
7	C₁₉×#1542	C₂	1	1	yes	maximal, characteristic
8	C₂×#10597276	—	2	1	no	—
9	#11926568	C₂²	1	1	yes	characteristic
10	C₄×#64520	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
12	C₄×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
13	#11926309	—	2	1	no	—
14	#11926303	—	2	1	no	—
15	#11926303	—	2	1	no	—
16	Q₈⋊(C₁₉⋊C₈)	—	2	1	no	—
17	Q₈⋊(C₁₉⋊C₈)	—	2	1	no	—
18	#11926266	—	2	1	no	—
19	#11926265	—	2	1	no	—
20	C₁₉×#533	C₂²	1	1	yes	characteristic
21	C₁₉×#431	C₂²	1	1	yes	characteristic
22	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₁₉×#333	C₄	1	1	yes	characteristic, complemented
24	C₁₉×#333	C₄	1	1	yes	characteristic, complemented
25	C₁₉×#332	C₄	1	1	yes	characteristic, complemented
26	C₁₉×#332	C₄	1	1	yes	characteristic, complemented
27	C₂²×C₈×C₁₉	—	2	1	no	—
28	C₁₉×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
29	C₁₉×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
30	C₁₉×#125	—	2	1	no	—
31	C₁₉×C₂²⋊Q₈	—	2	1	no	—
32	C₁₉×C₂²⋊Q₈	—	2	1	no	—
33	C₄×Q₈×C₁₉	—	2	1	no	—
34	C₄×Q₈×C₁₉	—	2	1	no	—
35	C₁₉×#117	—	2	1	no	—
36	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
37	C₂×C₄⋊Dic₁₉	—	2	1	no	—
38	C₂×C₄×Dic₁₉	—	2	1	no	—
39	C₂×#64520	—	2	1	no	—
40	C₂×#64520	D₄	1	1	yes	characteristic
41	C₂×#64520	D₄	1	1	yes	characteristic
42	C₂²×C₁₉⋊C₈	—	2	1	no	—
43	C₁₉×C₄⋊C₈	—	2	1	no	—
44	C₁₉×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
45	C₁₉×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
46	C₁₉×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
47	C₁₉×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
48	C₁₉×C₂²⋊C₈	—	2	1	no	—
49	C₄×C₈×C₁₉	—	2	1	no	—
50	C₁₉⋊#95	—	2	1	no	—
51	#10597276	—	4	2	no	—
52	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
53	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
54	#10597260	—	2	1	no	—
55	C₄×C₁₉⋊C₈	—	2	1	no	—
56	C₄×C₁₉⋊C₈	—	2	1	no	—
57	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
58	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
59	C₂×Q₈×C₁₉	—	2	1	no	—
60	C₂×Q₈×C₁₉	—	2	1	no	—
61	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
64	C₂²×Dic₁₉	—	4	1	no	—
65	C₂×C₈×C₁₉	—	2	1	no	—
66	C₂×C₈×C₁₉	—	2	1	no	—
67	C₂×C₈×C₁₉	—	2	1	no	—
68	C₂×C₈×C₁₉	—	2	1	no	—
69	C₂×C₈×C₁₉	—	4	1	no	—
70	C₂×C₈×C₁₉	—	4	1	no	—
71	C₄⋊C₄×C₁₉	—	2	1	no	—
72	C₄⋊C₄×C₁₉	—	2	1	no	—
73	C₄⋊C₄×C₁₉	—	2	1	no	—
74	C₄⋊C₄×C₁₉	—	2	1	no	—
75	C₄⋊C₄×C₁₉	—	4	1	no	—
76	C₄⋊C₄×C₁₉	—	4	1	no	—
77	C₂²⋊C₄×C₁₉	—	4	1	no	—
78	C₄²×C₁₉	—	4	1	no	—
79	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₁₉	#41	1	1	yes	characteristic
82	C₄²×C₁₉	#41	1	1	yes	characteristic
83	C₄⋊Dic₁₉	—	2	2	no	—
84	C₄⋊Dic₁₉	—	4	1	no	—
85	C₄×Dic₁₉	—	4	1	no	—
86	C₄×Dic₁₉	—	4	1	no	—
87	#64520	—	2	1	no	—
88	#64520	—	2	1	no	—
89	#64520	—	2	1	no	—
90	#64520	—	2	1	no	—
91	#64520	—	4	1	no	—
92	#64520	—	4	1	no	—
93	C₂×C₁₉⋊C₈	—	2	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	—	4	2	no	—
98	C₂×C₁₉⋊C₈	—	4	1	no	—
99	#9591	—	19	1	no	maximal, Sylow
100	C₂³×C₁₉	#124	1	1	yes	characteristic
101	Q₈×C₁₉	—	4	1	no	—
102	Q₈×C₁₉	—	4	1	no	—
103	Q₈×C₁₉	—	8	1	no	—
104	C₂×C₄×C₁₉	—	2	1	no	—
105	C₂×C₄×C₁₉	—	2	1	no	—
106	C₂×C₄×C₁₉	—	4	1	no	—
107	C₂×C₄×C₁₉	—	4	1	no	—
108	C₂×C₄×C₁₉	—	4	1	no	—
109	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
110	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
113	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
114	C₂×C₄×C₁₉	#126	1	1	yes	characteristic
115	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
116	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
117	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
118	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
119	C₂×Dic₁₉	—	4	2	no	—
120	C₂×Dic₁₉	—	8	1	no	—
121	C₂×Dic₁₉	—	8	1	no	—
122	C₈×C₁₉	—	4	1	no	—
123	C₈×C₁₉	—	4	1	no	—
124	C₈×C₁₉	—	4	1	no	—
125	C₈×C₁₉	—	4	1	no	—
126	C₁₉⋊C₈	—	2	2	no	—
127	C₁₉⋊C₈	—	4	1	no	—
128	C₁₉⋊C₈	—	4	1	no	—
129	C₁₉⋊C₈	—	4	1	no	—
130	C₁₉⋊C₈	—	8	1	no	—
131	#1986	—	19	1	no	—
132	#1950	—	19	1	no	—
133	#1542	D₁₉	1	1	yes	characteristic
134	#1493	—	19	1	no	—
135	#1451	—	19	1	no	—
136	#1303	—	19	1	no	—
137	#1279	—	19	1	no	—
138	C₂²×C₁₉	—	2	1	no	—
139	C₂²×C₁₉	—	2	1	no	—
140	C₂²×C₁₉	#388	1	1	yes	characteristic
141	C₂²×C₁₉	#489	1	1	yes	characteristic
142	C₂²×C₁₉	#491	1	1	yes	characteristic
143	C₄×C₁₉	—	2	1	no	—
144	C₄×C₁₉	—	2	1	no	—
145	C₄×C₁₉	—	2	1	no	—
146	C₄×C₁₉	—	8	1	no	—
147	C₄×C₁₉	—	8	1	no	—
148	C₄×C₁₉	#440	1	1	yes	characteristic
149	C₄×C₁₉	#441	1	1	yes	characteristic
150	C₄×C₁₉	#443	1	1	yes	characteristic
151	C₄×C₁₉	#444	1	1	yes	characteristic
152	C₄×C₁₉	#468	1	1	yes	characteristic
153	C₄×C₁₉	#470	1	1	yes	characteristic
154	Dic₁₉	—	8	2	no	—
155	#533	C₂×D₁₉	1	1	yes	characteristic
156	#431	C₂×D₁₉	1	1	yes	characteristic
157	C₂×#108	—	38	1	no	—
158	#423	—	19	1	no	—
159	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
160	C₄×M₁₆	—	19	1	no	—
161	C₂×C₄×C₈	—	19	1	no	—
162	C₄×C₄⋊C₄	—	19	1	no	—
163	#341	—	38	1	no	—
164	#339	—	38	1	no	—
165	#339	—	38	1	no	—
166	#333	Dic₁₉	1	1	yes	characteristic, complemented
167	#333	Dic₁₉	1	1	yes	characteristic, complemented
168	#332	Dic₁₉	1	1	yes	characteristic, complemented
169	#332	Dic₁₉	1	1	yes	characteristic, complemented
170	Q₈⋊C₈	—	38	1	no	—
171	Q₈⋊C₈	—	38	1	no	—
172	#322	—	38	1	no	—
173	#322	—	38	1	no	—
174	C₂×C₁₉	—	2	1	no	—
175	C₂×C₁₉	—	2	1	no	—
176	C₂×C₁₉	#1929	1	1	yes	characteristic
177	C₂×C₁₉	#1932	1	1	yes	characteristic
178	C₂×C₁₉	#1935	1	1	yes	characteristic
179	C₂×M₁₆	—	19	1	no	—
180	C₂×M₁₆	—	19	1	no	—
181	C₂×M₁₆	—	38	1	no	—
182	C₂²×C₈	—	2	1	no	—
183	C₂²×C₈	—	38	1	no	—
184	C₄⋊Q₈	C₂×Dic₁₉	1	1	yes	characteristic
185	C₄⋊Q₈	C₂×Dic₁₉	1	1	yes	characteristic
186	#125	—	2	1	no	—
187	C₂²⋊Q₈	—	2	1	no	—
188	C₂²⋊Q₈	—	2	1	no	—
189	C₄×Q₈	—	2	1	no	—
190	C₄×Q₈	—	2	1	no	—
191	#117	—	2	1	no	—
192	C₂×C₄⋊C₄	—	38	1	no	—
193	C₂×C₄²	—	38	1	no	—
194	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
195	#108	—	76	2	no	—
196	C₄⋊C₈	—	2	1	no	—
197	C₄⋊C₈	—	38	1	no	—
198	C₄⋊C₈	—	38	1	no	—
199	C₄⋊C₈	C₂×Dic₁₉	1	1	yes	characteristic
200	C₄⋊C₈	C₂×Dic₁₉	1	1	yes	characteristic
201	C₄⋊C₈	C₂×Dic₁₉	1	1	yes	characteristic
202	C₄⋊C₈	C₂×Dic₁₉	1	1	yes	characteristic
203	C₂²⋊C₈	—	2	1	no	—
204	#97	—	38	1	no	—
205	C₄×C₈	—	2	1	no	—
206	C₄×C₈	—	19	1	no	—
207	C₄×C₈	—	19	1	no	—
208	C₄×C₈	—	38	1	no	—
209	C₄×C₈	—	38	1	no	—
210	#95	—	38	1	no	—
211	C₁₉	#9591	1	1	yes	characteristic, complemented, Sylow
212	C₂×Q₈	—	2	1	no	—
213	C₂×Q₈	—	2	1	no	—
214	C₂²×C₄	—	76	1	no	—
215	C₂²×C₄	#64541	1	1	yes	characteristic
216	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
217	C₂²×C₄	#64543	1	1	yes	characteristic
218	M₁₆	—	38	1	no	—
219	M₁₆	—	38	1	no	—
220	M₁₆	—	38	1	no	—
221	M₁₆	—	38	1	no	—
222	M₁₆	—	76	1	no	—
223	M₁₆	—	76	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	4	1	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	—	38	1	no	—
231	C₂×C₈	—	38	1	no	—
232	C₂×C₈	—	38	1	no	—
233	C₂×C₈	—	38	1	no	—
234	C₂×C₈	—	76	2	no	—
235	C₂×C₈	—	76	1	no	—
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	2	1	no	—
238	C₄⋊C₄	—	2	1	no	—
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	—	4	1	no	—
241	C₄⋊C₄	—	4	1	no	—
242	C₄⋊C₄	—	38	2	no	—
243	C₄⋊C₄	—	76	1	no	—
244	C₂²⋊C₄	—	4	1	no	—
245	C₄²	—	4	1	no	—
246	C₄²	—	76	1	no	—
247	C₄²	—	76	1	no	—
248	C₄²	#64541	1	1	yes	characteristic
249	C₄²	D₄×D₁₉	1	1	yes	characteristic
250	C₄²	#64543	1	1	yes	characteristic
251	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
252	C₂³	#10597331	1	1	yes	characteristic
253	Q₈	—	4	1	no	—
254	Q₈	—	4	1	no	—
255	Q₈	—	8	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	76	2	no	—
262	C₂×C₄	—	152	1	no	—
263	C₂×C₄	—	152	1	no	—
264	C₂×C₄	#10597325	1	1	yes	characteristic
265	C₂×C₄	#10597329	1	1	yes	characteristic
266	C₂×C₄	#10597330	1	1	yes	characteristic
267	C₂×C₄	#10597360	1	1	yes	characteristic
268	C₂×C₄	#10597361	1	1	yes	characteristic
269	C₂×C₄	#10597364	1	1	yes	characteristic
270	C₂×C₄	#10597364	1	1	yes	characteristic
271	C₂×C₄	#10597372	1	1	yes	characteristic
272	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
273	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
274	C₈	—	4	1	no	—
275	C₈	—	4	1	no	—
276	C₈	—	4	1	no	—
277	C₈	—	4	1	no	—
278	C₈	—	38	2	no	—
279	C₈	—	76	1	no	—
280	C₈	—	76	1	no	—
281	C₈	—	76	1	no	—
282	C₈	—	152	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	#11926437	1	1	yes	characteristic
286	C₂²	#11926905	1	1	yes	characteristic
287	C₂²	#11926911	1	1	yes	characteristic
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	2	1	no	—
291	C₄	—	8	1	no	—
292	C₄	—	8	1	no	—
293	C₄	—	152	2	no	—
294	C₄	#11926645	1	1	yes	characteristic
295	C₄	#11926651	1	1	yes	characteristic
296	C₄	#11926661	1	1	yes	characteristic
297	C₄	#11926666	1	1	yes	characteristic
298	C₄	#11926793	1	1	yes	characteristic
299	C₄	#11926801	1	1	yes	characteristic
300	C₂	—	2	1	no	—
301	C₂	—	2	1	no	—
302	C₂	#14564285	1	1	yes	characteristic
303	C₂	#14564306	1	1	yes	characteristic
304	C₂	#14564327	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10597330
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1542	C₂
socle	C₂²×C₁₉	#388
prime radical	C₂³×C₁₉	#124
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1542	D₁₉
hypocenter	C₁₉	#9591
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1542

Complements

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

N	H ≅ G/N	type
23. C₁₉×#333	C₄	semidirect
24. C₁₉×#333	C₄	semidirect
25. C₁₉×#332	C₄	semidirect
26. C₁₉×#332	C₄	semidirect
166. #333	Dic₁₉	semidirect
167. #333	Dic₁₉	semidirect
168. #332	Dic₁₉	semidirect
169. #332	Dic₁₉	semidirect
211. C₁₉	#9591	semidirect

Maximal subgroups

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

subgroup	index
#14564580	2
#14564411	2
#14562090	2
#14561925	2
#14561537	2
#14561474	2
C₁₉×#1542	2
#9591	19