Subgroups — #18088360

Order 4864.

950 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#11926410	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926306	C₂	1	4	yes	maximal
4	C₂×C₁₉×#382	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₁₉⋊C₈	C₂²	1	2	yes	—
6	C₂²×C₄⋊Dic₁₉	C₂²	1	2	yes	—
7	C₂×C₄×C₁₉⋊C₈	—	2	2	no	—
8	C₂×C₄×C₁₉⋊C₈	—	2	2	no	—
9	C₂×C₄×C₁₉⋊C₈	C₄	1	4	yes	complemented
10	#11926410	—	2	2	no	—
11	#11926410	C₄	1	4	yes	complemented
12	C₂×C₁₉⋊#95	—	2	2	no	—
13	#11926306	—	2	8	no	—
14	C₂²×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
15	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
16	C₁₉×#382	—	2	2	no	—
17	C₁₉×#382	C₄	1	4	yes	complemented
18	C₂×C₁₉×#95	—	2	2	no	—
19	C₂³×C₄×C₁₉	—	2	2	no	—
20	C₂³×C₄×C₁₉	D₄	1	2	yes	—
21	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
22	C₂³×Dic₁₉	—	2	2	no	—
23	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
24	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
25	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
26	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₁₉	D₄	1	2	yes	—
31	C₂×C₄²×C₁₉	D₄	1	2	yes	—
32	C₂×C₄²×C₁₉	Q₈	1	2	yes	—
33	C₂×C₄⋊Dic₁₉	—	2	4	no	—
34	C₂×C₄⋊Dic₁₉	—	2	4	no	—
35	C₂×C₄⋊Dic₁₉	—	2	2	no	—
36	C₂×C₄⋊Dic₁₉	C₂×C₄	1	4	yes	—
37	C₂²×C₁₉⋊C₈	—	2	2	no	—
38	C₂²×C₁₉⋊C₈	—	2	2	no	—
39	C₂²×C₁₉⋊C₈	—	2	2	no	—
40	C₂²×C₁₉⋊C₈	—	2	2	no	—
41	C₂²×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
42	C₂²×C₁₉⋊C₈	D₄	1	4	yes	—
43	C₂²×C₁₉⋊C₈	Q₈	1	4	yes	—
44	C₁₉×#95	—	2	8	no	—
45	C₁₉⋊#95	—	2	8	no	—
46	C₄×C₁₉⋊C₈	—	2	8	no	—
47	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₉	—	2	8	no	—
49	C₂²×C₄×C₁₉	—	2	4	no	—
50	C₂²×C₄×C₁₉	—	4	2	no	—
51	C₂²×C₄×C₁₉	—	4	2	no	—
52	C₂²×C₄×C₁₉	—	4	8	no	—
53	C₂²×C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
54	C₂²×C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
55	C₂²×C₄×C₁₉	C₄⋊C₄	1	4	yes	—
56	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
64	C₂²×C₄×C₁₉	#41	1	2	yes	—
65	C₂²×C₄×C₁₉	#41	1	2	yes	—
66	C₂²×Dic₁₉	—	2	4	no	—
67	C₂²×Dic₁₉	—	4	2	no	—
68	C₂²×Dic₁₉	—	4	2	no	—
69	C₂²×Dic₁₉	—	4	8	no	—
70	C₄⋊C₄×C₁₉	—	2	8	no	—
71	C₄⋊C₄×C₁₉	—	4	4	no	—
72	C₄²×C₁₉	C₄²	1	4	yes	—
73	C₄²×C₁₉	C₂²⋊C₄	1	4	yes	—
74	C₄²×C₁₉	C₄⋊C₄	1	4	yes	—
75	C₄²×C₁₉	C₄⋊C₄	1	4	yes	—
76	C₄⋊Dic₁₉	—	2	8	no	—
77	C₄⋊Dic₁₉	—	4	4	no	—
78	C₂×C₁₉⋊C₈	—	2	4	no	—
79	C₂×C₁₉⋊C₈	—	2	4	no	—
80	C₂×C₁₉⋊C₈	—	2	4	no	—
81	C₂×C₁₉⋊C₈	—	2	4	no	—
82	C₂×C₁₉⋊C₈	—	2	4	no	—
83	C₂×C₁₉⋊C₈	—	2	4	no	—
84	C₂×C₁₉⋊C₈	C₄⋊C₄	1	8	yes	—
85	#10842	—	19	1	no	maximal, Sylow
86	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂³×C₁₉	#124	1	1	yes	characteristic
90	C₂³×C₁₉	#124	1	1	yes	characteristic
91	C₂³×C₁₉	#127	1	1	yes	characteristic
92	C₂³×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
93	C₂³×C₁₉	#135	1	8	yes	—
94	C₂×C₄×C₁₉	—	2	8	no	—
95	C₂×C₄×C₁₉	—	2	8	no	—
96	C₂×C₄×C₁₉	—	2	8	no	—
97	C₂×C₄×C₁₉	—	4	4	no	—
98	C₂×C₄×C₁₉	—	4	8	no	—
99	C₂×C₄×C₁₉	—	8	4	no	—
100	C₂×C₄×C₁₉	—	8	4	no	—
101	C₂×C₄×C₁₉	#95	1	8	yes	—
102	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
103	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
105	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₉	#117	1	2	yes	—
108	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
111	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
112	C₂×C₄×C₁₉	#121	1	2	yes	—
113	C₂×C₄×C₁₉	#121	1	2	yes	—
114	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
115	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
116	C₂×C₄×C₁₉	#125	1	2	yes	—
117	C₂×C₄×C₁₉	C₄⋊Q₈	1	2	yes	—
118	C₂×Dic₁₉	—	4	4	no	—
119	C₂×Dic₁₉	—	4	8	no	—
120	C₂×Dic₁₉	—	8	4	no	—
121	C₂×Dic₁₉	—	8	4	no	—
122	C₁₉⋊C₈	—	2	8	no	—
123	C₂²×C₄×C₈	—	19	1	no	—
124	C₂×#382	—	19	1	no	—
125	C₂×#382	D₁₉	1	1	yes	characteristic
126	C₂×#340	—	19	4	no	—
127	C₂²×C₁₉	C₂×#95	1	1	yes	characteristic
128	C₂²×C₁₉	#382	1	1	yes	characteristic
129	C₂²×C₁₉	#384	1	1	yes	characteristic
130	C₂²×C₁₉	#390	1	1	yes	characteristic
131	C₂²×C₁₉	#390	1	1	yes	characteristic
132	C₂²×C₁₉	#391	1	1	yes	characteristic
133	C₂²×C₁₉	#391	1	1	yes	characteristic
134	C₂²×C₁₉	#416	1	4	yes	—
135	C₂²×C₁₉	#416	1	4	yes	—
136	C₂²×C₁₉	#427	1	4	yes	—
137	C₂²×C₁₉	#488	1	4	yes	—
138	C₂²×C₁₉	#488	1	4	yes	—
139	C₂²×C₁₉	#495	1	4	yes	—
140	C₂²×C₁₉	#499	1	4	yes	—
141	C₄×C₁₉	—	2	8	no	—
142	C₄×C₁₉	—	8	4	no	—
143	C₄×C₁₉	C₄×C₄⋊C₄	1	2	yes	—
144	C₄×C₁₉	#380	1	2	yes	—
145	C₄×C₁₉	#389	1	2	yes	—
146	C₄×C₁₉	#389	1	2	yes	—
147	Dic₁₉	—	8	4	no	—
148	C₂³×C₈	—	19	2	no	—
149	C₂²×C₄⋊C₄	—	19	2	no	—
150	C₂²×C₄⋊C₄	C₂×D₁₉	1	2	yes	—
151	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
152	C₂×C₄×C₈	—	19	4	no	—
153	C₂×C₄×C₈	—	38	2	no	—
154	C₂×C₄×C₈	—	38	2	no	—
155	#382	—	2	2	no	—
156	#382	—	19	4	no	—
157	#382	—	38	2	no	—
158	#382	Dic₁₉	1	4	yes	complemented
159	C₂×#95	—	2	2	no	—
160	C₂×#95	—	38	2	no	—
161	#340	—	38	8	no	—
162	C₂×C₁₉	C₄⋊#95	1	1	yes	characteristic
163	C₂×C₁₉	#1735	1	2	yes	—
164	C₂×C₁₉	#1835	1	2	yes	—
165	C₂×C₁₉	#1846	1	2	yes	—
166	C₂×C₁₉	#1958	1	2	yes	—
167	C₂×C₁₉	#1958	1	2	yes	—
168	C₂×C₁₉	#1991	1	2	yes	—
169	C₂×C₁₉	#1991	1	2	yes	—
170	C₂³×C₄	—	2	2	no	—
171	C₂³×C₄	—	38	2	no	—
172	C₂³×C₄	C₂²⋊D₁₉	1	2	yes	—
173	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
174	C₂²×C₈	—	19	4	no	—
175	C₂²×C₈	—	19	4	no	—
176	C₂²×C₈	—	19	4	no	—
177	C₂²×C₈	—	38	2	no	—
178	C₂²×C₈	—	38	2	no	—
179	C₂²×C₈	—	38	2	no	—
180	C₂²×C₈	—	38	2	no	—
181	C₂×C₄⋊C₄	—	2	4	no	—
182	C₂×C₄⋊C₄	—	2	4	no	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄⋊C₄	—	19	4	no	—
185	C₂×C₄⋊C₄	—	38	4	no	—
186	C₂×C₄⋊C₄	—	38	4	no	—
187	C₂×C₄⋊C₄	—	38	2	no	—
188	C₂×C₄⋊C₄	C₂×Dic₁₉	1	4	yes	—
189	C₂×C₄²	Dic₃₈	1	2	yes	—
190	C₂×C₄²	C₄×D₁₉	1	2	yes	—
191	C₂×C₄²	C₄×D₁₉	1	2	yes	—
192	C₂×C₄²	D₇₆	1	2	yes	—
193	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
194	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
195	C₄×C₈	—	38	8	no	—
196	#95	—	2	8	no	—
197	#95	—	38	8	no	—
198	C₁₉	#10842	1	1	yes	characteristic, complemented, Sylow
199	C₂⁴	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
200	C₂²×C₄	—	2	8	no	—
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	—	4	2	no	—
203	C₂²×C₄	—	4	2	no	—
204	C₂²×C₄	—	4	8	no	—
205	C₂²×C₄	—	38	4	no	—
206	C₂²×C₄	—	76	2	no	—
207	C₂²×C₄	—	76	2	no	—
208	C₂²×C₄	—	76	8	no	—
209	C₂²×C₄	Dic₁₉⋊C₄	1	4	yes	—
210	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	4	yes	—
211	C₂²×C₄	C₂²⋊Dic₁₉	1	4	yes	—
212	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
214	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
216	C₂²×C₄	D₄×D₁₉	1	2	yes	—
217	C₂²×C₄	#64543	1	2	yes	—
218	C₂²×C₄	Q₈×D₁₉	1	2	yes	—
219	C₂²×C₄	#64545	1	2	yes	—
220	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
222	C₂×C₈	—	19	8	no	—
223	C₂×C₈	—	38	4	no	—
224	C₂×C₈	—	38	4	no	—
225	C₂×C₈	—	38	4	no	—
226	C₂×C₈	—	38	4	no	—
227	C₂×C₈	—	38	4	no	—
228	C₂×C₈	—	38	4	no	—
229	C₄⋊C₄	—	2	8	no	—
230	C₄⋊C₄	—	4	4	no	—
231	C₄⋊C₄	—	38	8	no	—
232	C₄⋊C₄	—	76	4	no	—
233	C₄²	C₄×Dic₁₉	1	4	yes	—
234	C₄²	Dic₁₉⋊C₄	1	4	yes	—
235	C₄²	C₄⋊Dic₁₉	1	4	yes	—
236	C₄²	(C₂×D₁₉)⋊C₄	1	4	yes	—
237	C₂³	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
238	C₂³	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
239	C₂³	#10597382	1	1	yes	characteristic
240	C₂³	#10597386	1	1	yes	characteristic
241	C₂³	#10597390	1	1	yes	characteristic
242	C₂³	#10597393	1	1	yes	characteristic
243	C₂³	#10597396	1	8	yes	—
244	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
245	C₂×C₄	—	2	8	no	—
246	C₂×C₄	—	2	8	no	—
247	C₂×C₄	—	2	8	no	—
248	C₂×C₄	—	4	8	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	8	4	no	—
251	C₂×C₄	—	8	4	no	—
252	C₂×C₄	—	76	8	no	—
253	C₂×C₄	—	76	4	no	—
254	C₂×C₄	—	152	4	no	—
255	C₂×C₄	—	152	4	no	—
256	C₂×C₄	C₁₉⋊#95	1	8	yes	—
257	C₂×C₄	#10597333	1	2	yes	—
258	C₂×C₄	#10597334	1	2	yes	—
259	C₂×C₄	#10597336	1	2	yes	—
260	C₂×C₄	C₄⋊C₄×D₁₉	1	2	yes	—
261	C₂×C₄	#10597338	1	2	yes	—
262	C₂×C₄	#10597339	1	2	yes	—
263	C₂×C₄	#10597341	1	2	yes	—
264	C₂×C₄	#10597343	1	2	yes	—
265	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
266	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
267	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
268	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
269	C₂×C₄	D₄×Dic₁₉	1	2	yes	—
270	C₂×C₄	#10597384	1	2	yes	—
271	C₂×C₄	Q₈×Dic₁₉	1	2	yes	—
272	C₂×C₄	#10597392	1	2	yes	—
273	C₈	—	38	8	no	—
274	C₂²	C₂×C₁₉⋊#95	1	1	yes	characteristic
275	C₂²	#11926409	1	1	yes	characteristic
276	C₂²	#11926413	1	1	yes	characteristic
277	C₂²	#11926445	1	1	yes	characteristic
278	C₂²	#11926447	1	1	yes	characteristic
279	C₂²	#11926449	1	1	yes	characteristic
280	C₂²	#11926451	1	1	yes	characteristic
281	C₂²	#11926538	1	4	yes	—
282	C₂²	#11926540	1	4	yes	—
283	C₂²	#11926580	1	4	yes	—
284	C₂²	#11926898	1	4	yes	—
285	C₂²	#11926901	1	4	yes	—
286	C₂²	#11926926	1	4	yes	—
287	C₂²	#11926943	1	4	yes	—
288	C₄	—	2	8	no	—
289	C₄	—	8	4	no	—
290	C₄	—	152	4	no	—
291	C₄	C₄⋊C₄×Dic₁₉	1	2	yes	—
292	C₄	#11926401	1	2	yes	—
293	C₄	#11926440	1	2	yes	—
294	C₄	#11926443	1	2	yes	—
295	C₂	#14561864	1	1	yes	characteristic
296	C₂	#14563351	1	2	yes	—
297	C₂	#14563765	1	2	yes	—
298	C₂	#14563797	1	2	yes	—
299	C₂	#14564444	1	2	yes	—
300	C₂	#14564447	1	2	yes	—
301	C₂	#14564599	1	2	yes	—
302	C₂	#14564601	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₁₉
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₂×C₁₉×#382	C₂
socle, prime radical	C₂⁴×C₁₉	C₂×D₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂×#382	D₁₉
hypocenter	C₁₉	#10842
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂×#382

Complements

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

N	H ≅ G/N	type
9. C₂×C₄×C₁₉⋊C₈	C₄	semidirect
11. #11926410	C₄	semidirect
17. C₁₉×#382	C₄	semidirect
158. #382	Dic₁₉	semidirect
198. C₁₉	#10842	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₁₉⋊C₈	2
C₂×#11926410	2
C₂×#11926306	2
C₂×C₁₉×#382	2
#10842	19