Subgroups — #18071021

Order 4864.

562 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂²⋊(C₁₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	#14563329	C₂	1	1	yes	maximal, characteristic
3	#14561461	C₂	1	4	yes	maximal, complemented
4	C₁₉×#2857	C₂	1	1	yes	maximal, characteristic
5	C₂²×#64520	—	2	1	no	—
6	C₂×C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
7	C₂×C₄×C₁₉⋊C₈	—	2	1	no	—
8	#11926299	—	2	1	no	—
9	#11926299	C₂²	1	2	yes	—
10	#11926299	C₂²	1	2	yes	—
11	C₂×C₁₉×C₄⋊Q₈	C₄	1	1	yes	characteristic
12	C₂×C₁₉×C₂²⋊Q₈	C₄	1	2	yes	—
13	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
14	C₁₉×#391	C₂²	1	2	yes	—
15	C₁₉×#389	—	2	1	no	—
16	C₁₉×#389	—	2	1	no	—
17	C₁₉×#384	C₄	1	1	yes	characteristic
18	C₁₉×#380	—	2	1	no	—
19	C₁₉×#380	—	2	1	no	—
20	C₂²×Q₈×C₁₉	C₂×C₄	1	2	yes	—
21	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
24	C₁₉×C₄⋊Q₈	—	4	2	no	—
25	C₁₉×C₂²⋊Q₈	—	4	4	no	—
26	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
27	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
28	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
29	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
30	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
31	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
32	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
33	C₂×C₄²×C₁₉	—	4	2	no	—
34	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
38	C₂×#64520	—	2	4	no	—
39	C₂×#64520	—	4	2	no	—
40	C₂×#64520	—	4	2	no	—
41	C₂²×C₁₉⋊C₈	—	2	2	no	—
42	C₂²×C₁₉⋊C₈	—	2	2	no	—
43	C₁₉×#95	—	2	2	no	—
44	C₁₉×#95	—	2	2	no	—
45	C₂²⋊(C₁₉⋊C₈)	—	2	4	no	—
46	C₄×C₁₉⋊C₈	—	4	2	no	—
47	C₂⁴×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×Q₈×C₁₉	—	2	4	no	—
49	C₂×Q₈×C₁₉	—	4	2	no	—
50	C₂×Q₈×C₁₉	—	4	2	no	—
51	C₂²×C₄×C₁₉	—	2	2	no	—
52	C₂²×C₄×C₁₉	—	2	2	no	—
53	C₂²×C₄×C₁₉	—	2	2	no	—
54	C₂²×C₄×C₁₉	—	2	2	no	—
55	C₂²×C₄×C₁₉	—	2	2	no	—
56	C₂²×C₄×C₁₉	—	2	2	no	—
57	C₂²×C₄×C₁₉	—	2	2	no	—
58	C₂²×C₄×C₁₉	—	2	2	no	—
59	C₂²×C₄×C₁₉	—	4	1	no	—
60	C₂²×C₄×C₁₉	—	4	1	no	—
61	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	SD₁₆	1	2	yes	—
65	C₂²×C₄×C₁₉	Q₁₆	1	2	yes	—
66	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
69	C₄⋊C₄×C₁₉	—	4	2	no	—
70	C₄⋊C₄×C₁₉	—	4	2	no	—
71	C₄⋊C₄×C₁₉	—	4	2	no	—
72	C₄⋊C₄×C₁₉	—	4	4	no	—
73	C₄⋊C₄×C₁₉	—	4	2	no	—
74	C₄⋊C₄×C₁₉	—	4	2	no	—
75	C₂²⋊C₄×C₁₉	—	4	4	no	—
76	C₄²×C₁₉	—	2	2	no	—
77	C₄²×C₁₉	—	2	2	no	—
78	C₄²×C₁₉	—	4	1	no	—
79	C₄²×C₁₉	—	4	1	no	—
80	#64520	—	4	8	no	—
81	C₂×C₁₉⋊C₈	—	2	4	no	—
82	C₂×C₁₉⋊C₈	—	2	4	no	—
83	C₂×C₁₉⋊C₈	—	4	2	no	—
84	C₂×C₁₉⋊C₈	—	4	2	no	—
85	C₂×C₁₉⋊C₈	—	4	2	no	—
86	C₂×C₁₉⋊C₈	—	4	2	no	—
87	#7687	—	19	1	no	maximal, Sylow
88	C₂³×C₁₉	—	2	1	no	—
89	C₂³×C₁₉	—	2	1	no	—
90	C₂³×C₁₉	—	2	1	no	—
91	C₂³×C₁₉	—	2	1	no	—
92	C₂³×C₁₉	—	2	1	no	—
93	C₂³×C₁₉	—	2	1	no	—
94	C₂³×C₁₉	#103	1	2	yes	—
95	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	Q₈×C₁₉	—	8	4	no	—
97	C₂×C₄×C₁₉	—	2	1	no	—
98	C₂×C₄×C₁₉	—	2	1	no	—
99	C₂×C₄×C₁₉	—	2	1	no	—
100	C₂×C₄×C₁₉	—	2	1	no	—
101	C₂×C₄×C₁₉	—	2	1	no	—
102	C₂×C₄×C₁₉	—	2	1	no	—
103	C₂×C₄×C₁₉	—	2	1	no	—
104	C₂×C₄×C₁₉	—	2	1	no	—
105	C₂×C₄×C₁₉	—	2	1	no	—
106	C₂×C₄×C₁₉	—	2	1	no	—
107	C₂×C₄×C₁₉	—	2	4	no	—
108	C₂×C₄×C₁₉	—	2	4	no	—
109	C₂×C₄×C₁₉	—	4	4	no	—
110	C₂×C₄×C₁₉	—	4	4	no	—
111	C₂×C₄×C₁₉	—	4	2	no	—
112	C₂×C₄×C₁₉	—	4	2	no	—
113	C₂×C₄×C₁₉	—	4	2	no	—
114	C₂×C₄×C₁₉	—	4	2	no	—
115	C₂×C₄×C₁₉	—	4	2	no	—
116	C₂×C₄×C₁₉	—	4	2	no	—
117	C₂×C₄×C₁₉	#99	1	2	yes	—
118	C₂×C₄×C₁₉	#100	1	2	yes	—
119	C₂×C₄×C₁₉	#103	1	2	yes	—
120	C₂×C₄×C₁₉	#104	1	2	yes	—
121	C₂×C₄×C₁₉	#104	1	2	yes	—
122	C₂×C₄×C₁₉	#131	1	2	yes	—
123	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
125	C₂×C₄×C₁₉	#137	1	2	yes	—
126	C₁₉⋊C₈	—	4	4	no	—
127	C₁₉⋊C₈	—	4	4	no	—
128	#2857	D₁₉	1	1	yes	characteristic
129	C₄×C₂²⋊C₈	—	19	1	no	—
130	#1728	—	19	1	no	—
131	#1272	—	19	4	no	—
132	C₂²×C₁₉	—	2	2	no	—
133	C₂²×C₁₉	—	2	2	no	—
134	C₂²×C₁₉	—	2	2	no	—
135	C₂²×C₁₉	—	4	1	no	—
136	C₂²×C₁₉	—	4	1	no	—
137	C₂²×C₁₉	—	4	1	no	—
138	C₂²×C₁₉	—	4	1	no	—
139	C₂²×C₁₉	#408	1	1	yes	characteristic
140	C₂²×C₁₉	C₂×#99	1	1	yes	characteristic
141	C₂²×C₁₉	C₂×#100	1	1	yes	characteristic
142	C₂²×C₁₉	C₂×#103	1	1	yes	characteristic
143	C₂²×C₁₉	#419	1	1	yes	characteristic
144	C₂²×C₁₉	C₂×#104	1	1	yes	characteristic
145	C₂²×C₁₉	C₂×#104	1	1	yes	characteristic
146	C₄×C₁₉	—	2	2	no	—
147	C₄×C₁₉	—	2	2	no	—
148	C₄×C₁₉	—	2	1	no	—
149	C₄×C₁₉	—	2	1	no	—
150	C₄×C₁₉	—	2	2	no	—
151	C₄×C₁₉	—	2	2	no	—
152	C₄×C₁₉	—	8	2	no	—
153	C₄×C₁₉	—	8	2	no	—
154	C₄×C₁₉	#328	1	4	yes	—
155	C₂²×M₁₆	—	38	1	no	—
156	C₂×C₄⋊Q₈	Dic₁₉	1	1	yes	characteristic
157	C₂×C₂²⋊Q₈	Dic₁₉	1	2	yes	—
158	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
159	C₂×C₂²⋊C₈	—	38	1	no	—
160	C₂×C₄×C₈	—	38	1	no	—
161	#391	C₂×D₁₉	1	2	yes	—
162	#389	—	2	1	no	—
163	#389	—	2	1	no	—
164	#384	Dic₁₉	1	1	yes	characteristic
165	#380	—	2	1	no	—
166	#380	—	2	1	no	—
167	#336	—	19	2	no	—
168	#336	—	19	2	no	—
169	#336	—	38	1	no	—
170	C₂×C₁₉	—	4	2	no	—
171	C₂×C₁₉	#1471	1	1	yes	characteristic
172	C₂×C₁₉	#1485	1	1	yes	characteristic
173	C₂×C₁₉	#1493	1	1	yes	characteristic
174	C₂×C₁₉	C₂×#328	1	1	yes	characteristic
175	C₂×C₁₉	#1508	1	1	yes	characteristic
176	C₂×C₁₉	#1525	1	1	yes	characteristic
177	C₂×C₁₉	#1548	1	1	yes	characteristic
178	C₂²×Q₈	C₂×Dic₁₉	1	2	yes	—
179	C₂³×C₄	C₂×Dic₁₉	1	1	yes	characteristic
180	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
181	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
182	C₂×M₁₆	—	38	4	no	—
183	C₂×M₁₆	—	76	2	no	—
184	C₂×M₁₆	—	76	2	no	—
185	C₂²×C₈	—	38	2	no	—
186	C₂²×C₈	—	38	2	no	—
187	C₄⋊Q₈	—	4	2	no	—
188	C₂²⋊Q₈	—	4	4	no	—
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₄⋊C₄	—	2	1	no	—
191	C₂×C₄⋊C₄	—	2	1	no	—
192	C₂×C₄⋊C₄	—	2	1	no	—
193	C₂×C₄⋊C₄	—	2	1	no	—
194	C₂×C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
195	C₂×C₂²⋊C₄	—	2	2	no	—
196	C₂×C₄²	—	4	2	no	—
197	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
198	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
199	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
200	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
201	C₂²⋊C₈	—	38	4	no	—
202	C₄×C₈	—	76	2	no	—
203	#95	—	2	2	no	—
204	#95	—	2	2	no	—
205	C₁₉	#7687	1	1	yes	characteristic, complemented, Sylow
206	C₂⁴	C₂²⋊Dic₁₉	1	1	yes	characteristic
207	C₂×Q₈	—	2	4	no	—
208	C₂×Q₈	—	4	2	no	—
209	C₂×Q₈	—	4	2	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	4	1	no	—
219	C₂²×C₄	—	4	1	no	—
220	C₂²×C₄	#64527	1	2	yes	—
221	C₂²×C₄	#64528	1	2	yes	—
222	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
223	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
224	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
225	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
226	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
228	M₁₆	—	76	8	no	—
229	C₂×C₈	—	38	4	no	—
230	C₂×C₈	—	38	4	no	—
231	C₂×C₈	—	76	2	no	—
232	C₂×C₈	—	76	2	no	—
233	C₂×C₈	—	76	2	no	—
234	C₂×C₈	—	76	2	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₄⋊C₄	—	4	4	no	—
238	C₄⋊C₄	—	4	2	no	—
239	C₄⋊C₄	—	4	2	no	—
240	C₄⋊C₄	—	4	2	no	—
241	C₂²⋊C₄	—	4	4	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	4	1	no	—
245	C₄²	—	4	1	no	—
246	C₂³	—	2	1	no	—
247	C₂³	—	2	1	no	—
248	C₂³	—	2	1	no	—
249	C₂³	—	2	1	no	—
250	C₂³	—	2	1	no	—
251	C₂³	—	2	1	no	—
252	C₂³	#10597292	1	2	yes	—
253	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
254	Q₈	—	8	4	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	#10597290	1	2	yes	—
276	C₂×C₄	#10597291	1	2	yes	—
277	C₂×C₄	#10597292	1	2	yes	—
278	C₂×C₄	#10597294	1	2	yes	—
279	C₂×C₄	#10597294	1	2	yes	—
280	C₂×C₄	C₂×#64527	1	1	yes	characteristic
281	C₂×C₄	#10597388	1	2	yes	—
282	C₂×C₄	C₂×#64528	1	1	yes	characteristic
283	C₂×C₄	#10597394	1	2	yes	—
284	C₈	—	76	4	no	—
285	C₈	—	76	4	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	4	1	no	—
290	C₂²	—	4	1	no	—
291	C₂²	—	4	1	no	—
292	C₂²	—	4	1	no	—
293	C₂²	#11926511	1	1	yes	characteristic
294	C₂²	C₂×#10597291	1	1	yes	characteristic
295	C₂²	C₂×#10597290	1	1	yes	characteristic
296	C₂²	C₂×#10597292	1	1	yes	characteristic
297	C₂²	#11926553	1	1	yes	characteristic
298	C₂²	C₂×#10597294	1	1	yes	characteristic
299	C₂²	C₂×#10597294	1	1	yes	characteristic
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	2	no	—
303	C₄	—	2	2	no	—
304	C₄	—	2	2	no	—
305	C₄	—	2	2	no	—
306	C₄	—	8	2	no	—
307	C₄	—	8	2	no	—
308	C₄	#11926281	1	4	yes	—
309	C₂	—	4	2	no	—
310	C₂	#14561995	1	1	yes	characteristic
311	C₂	#14562056	1	1	yes	characteristic
312	C₂	#14562090	1	1	yes	characteristic
313	C₂	C₂×#11926281	1	1	yes	characteristic
314	C₂	#14562155	1	1	yes	characteristic
315	C₂	#14562227	1	1	yes	characteristic
316	C₂	#14562316	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₁₉
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#2857	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#2857	D₁₉
hypocenter	C₁₉	#7687
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #2857

Complements

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

N	H ≅ G/N	type
3. #14561461	C₂	semidirect
205. C₁₉	#7687	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₂²⋊(C₁₉⋊C₈)	2
#14563329	2
#14561461	2
C₁₉×#2857	2
#7687	19