Subgroups — #18091221

Order 4864.

686 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14572225	C₂	1	1	yes	maximal, characteristic
2	#14572224	C₂	1	2	yes	maximal
3	C₂×C₁₉⋊#338	C₂	1	1	yes	maximal, characteristic
4	#14563331	C₂	1	2	yes	maximal
5	C₁₉×#1729	C₂	1	1	yes	maximal, characteristic
6	C₂²×#64520	C₂²	1	2	yes	—
7	C₂×#10597372	C₂²	1	2	yes	—
8	#11926587	—	2	4	no	—
9	#11926586	—	2	4	no	—
10	#11926584	—	2	4	no	—
11	#11926582	—	2	2	no	—
12	#11926580	—	2	4	no	—
13	C₁₉⋊#338	C₄	1	4	yes	complemented
14	#11926298	—	2	4	no	—
15	C₂²×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
16	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
17	C₂×C₁₉×#117	C₂²	1	1	yes	characteristic
18	C₁₉×#336	—	2	2	no	—
19	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
20	C₂×M₁₆×C₁₉	—	2	2	no	—
21	C₂×M₁₆×C₁₉	—	2	2	no	—
22	C₂×M₁₆×C₁₉	C₂×C₄	1	2	yes	—
23	C₂×M₁₆×C₁₉	Q₈	1	2	yes	—
24	C₂²×C₈×C₁₉	—	2	2	no	—
25	C₂²×C₈×C₁₉	—	2	1	no	—
26	C₂²×C₈×C₁₉	—	2	1	no	—
27	C₂²×C₈×C₁₉	—	2	2	no	—
28	C₂²×C₈×C₁₉	D₄	1	2	yes	—
29	C₂²×C₈×C₁₉	Q₈	1	2	yes	—
30	C₁₉×#117	—	2	2	no	—
31	C₁₉×#117	—	2	2	no	—
32	C₂×C₄⋊C₄×C₁₉	D₄	1	2	yes	—
33	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
34	C₂×C₄²×C₁₉	—	2	1	no	—
35	C₂×C₂²⋊Dic₁₉	—	2	2	no	—
36	#10597372	—	2	2	no	—
37	#10597372	—	2	2	no	—
38	#10597372	—	2	2	no	—
39	#10597372	—	2	2	no	—
40	C₂×C₄⋊Dic₁₉	D₄	1	4	yes	—
41	C₂×C₄×Dic₁₉	—	2	2	no	—
42	C₂×#64520	—	2	2	no	—
43	C₂×#64520	—	2	2	no	—
44	C₂×#64520	—	2	2	no	—
45	C₂×#64520	—	2	2	no	—
46	C₂×#64520	C₂×C₄	1	4	yes	—
47	C₂×#64520	Q₈	1	4	yes	—
48	C₂²×C₁₉⋊C₈	D₄	1	4	yes	—
49	#10597275	—	2	4	no	—
50	#10597274	—	2	4	no	—
51	#10597265	—	2	8	no	—
52	#10597264	—	2	8	no	—
53	C₂⁴×C₁₉	#41	1	1	yes	characteristic
54	C₂²×C₄×C₁₉	—	2	2	no	—
55	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
63	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	#41	1	2	yes	—
65	C₂²×Dic₁₉	—	2	4	no	—
66	M₁₆×C₁₉	—	2	4	no	—
67	M₁₆×C₁₉	—	2	4	no	—
68	C₂×C₈×C₁₉	—	2	2	no	—
69	C₂×C₈×C₁₉	—	2	2	no	—
70	C₂×C₈×C₁₉	—	2	4	no	—
71	C₂×C₈×C₁₉	—	2	2	no	—
72	C₂×C₈×C₁₉	—	4	1	no	—
73	C₂×C₈×C₁₉	—	4	2	no	—
74	C₂×C₈×C₁₉	—	4	1	no	—
75	C₂×C₈×C₁₉	C₄⋊C₄	1	4	yes	—
76	C₄⋊C₄×C₁₉	—	2	4	no	—
77	C₂²⋊C₄×C₁₉	—	4	2	no	—
78	C₄²×C₁₉	—	4	1	no	—
79	C₄²×C₁₉	—	4	1	no	—
80	C₂²⋊Dic₁₉	—	4	2	no	—
81	C₂²⋊Dic₁₉	—	4	2	no	—
82	C₄⋊Dic₁₉	—	2	4	no	—
83	C₄⋊Dic₁₉	—	2	4	no	—
84	C₄×Dic₁₉	—	4	2	no	—
85	C₄×Dic₁₉	—	4	2	no	—
86	#64520	—	2	4	no	—
87	#64520	—	2	4	no	—
88	#64520	—	2	4	no	—
89	#64520	—	2	4	no	—
90	C₂×C₁₉⋊C₈	—	2	4	no	—
91	C₂×C₁₉⋊C₈	—	2	4	no	—
92	C₂×C₁₉⋊C₈	C₄⋊C₄	1	8	yes	—
93	#11337	—	19	1	no	maximal, Sylow
94	C₂³×C₁₉	—	4	2	no	—
95	C₂³×C₁₉	C₄×Q₈	1	1	yes	characteristic
96	C₂³×C₁₉	#121	1	1	yes	characteristic
97	C₂³×C₁₉	#121	1	2	yes	—
98	C₂³×C₁₉	#125	1	1	yes	characteristic
99	C₂³×C₁₉	#125	1	2	yes	—
100	C₂×C₄×C₁₉	—	4	4	no	—
101	C₂×C₄×C₁₉	—	4	2	no	—
102	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
106	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
107	C₂×C₄×C₁₉	#120	1	2	yes	—
108	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
110	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
111	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
112	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
113	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
114	C₂×C₄×C₁₉	#123	1	2	yes	—
115	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
116	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
117	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
118	C₂×C₄×C₁₉	C₄⋊Q₈	1	2	yes	—
119	C₂×C₄×C₁₉	C₄⋊Q₈	1	2	yes	—
120	C₂×Dic₁₉	—	4	8	no	—
121	C₂×Dic₁₉	—	4	4	no	—
122	C₈×C₁₉	—	2	4	no	—
123	C₈×C₁₉	—	4	2	no	—
124	C₁₉⋊C₈	—	2	8	no	—
125	#2911	—	19	1	no	—
126	#2911	—	19	2	no	—
127	C₂×#338	—	19	1	no	—
128	#1729	—	19	2	no	—
129	#1729	D₁₉	1	1	yes	characteristic
130	C₂²×C₁₉	—	2	4	no	—
131	C₂²×C₁₉	—	2	2	no	—
132	C₂²×C₁₉	—	4	1	no	—
133	C₂²×C₁₉	—	4	1	no	—
134	C₂²×C₁₉	—	4	2	no	—
135	C₂²×C₁₉	#389	1	2	yes	—
136	C₂²×C₁₉	#389	1	1	yes	characteristic
137	C₂²×C₁₉	#392	1	1	yes	characteristic
138	C₂²×C₁₉	#398	1	1	yes	characteristic
139	C₂²×C₁₉	#398	1	2	yes	—
140	C₄×C₁₉	—	8	2	no	—
141	C₄×C₁₉	#384	1	1	yes	characteristic
142	C₄×C₁₉	#385	1	1	yes	characteristic
143	C₄×C₁₉	#385	1	2	yes	—
144	C₄×C₁₉	#395	1	1	yes	characteristic
145	C₄×C₁₉	#395	1	2	yes	—
146	C₄×C₁₉	#399	1	1	yes	characteristic
147	Dic₁₉	—	8	4	no	—
148	C₂²×M₁₆	—	19	2	no	—
149	C₂²×M₁₆	C₂×D₁₉	1	1	yes	characteristic
150	C₂³×C₈	C₂×D₁₉	1	1	yes	characteristic
151	C₂×#117	—	19	2	no	—
152	C₂×#117	C₂×D₁₉	1	1	yes	characteristic
153	#428	—	38	4	no	—
154	#428	—	38	4	no	—
155	#428	—	38	4	no	—
156	#427	—	38	4	no	—
157	#427	—	38	2	no	—
158	#338	—	19	4	no	—
159	#336	—	2	2	no	—
160	#336	—	38	4	no	—
161	C₂×C₁₉	—	2	2	no	—
162	C₂×C₁₉	—	2	1	no	—
163	C₂×C₁₉	—	2	1	no	—
164	C₂×C₁₉	—	2	2	no	—
165	C₂×C₁₉	#1444	1	1	yes	characteristic
166	C₂×C₁₉	#2017	1	2	yes	—
167	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	19	4	no	—
171	C₂×M₁₆	—	19	4	no	—
172	C₂×M₁₆	—	38	2	no	—
173	C₂×M₁₆	—	38	2	no	—
174	C₂×M₁₆	—	38	2	no	—
175	C₂×M₁₆	—	38	2	no	—
176	C₂×M₁₆	Dic₃₈	1	2	yes	—
177	C₂×M₁₆	C₄×D₁₉	1	2	yes	—
178	C₂²×C₈	—	2	1	no	—
179	C₂²×C₈	—	2	2	no	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	—	2	1	no	—
182	C₂²×C₈	—	19	4	no	—
183	C₂²×C₈	Dic₃₈	1	2	yes	—
184	C₂²×C₈	C₂²⋊D₁₉	1	2	yes	—
185	#117	—	2	2	no	—
186	#117	—	2	2	no	—
187	#117	—	38	2	no	—
188	#117	—	38	2	no	—
189	#117	—	38	2	no	—
190	#117	—	38	2	no	—
191	C₂×C₄⋊C₄	—	19	4	no	—
192	C₂×C₄⋊C₄	D₇₆	1	2	yes	—
193	C₂×C₂²⋊C₄	—	2	1	no	—
194	C₂×C₂²⋊C₄	—	38	2	no	—
195	C₂×C₄²	—	2	1	no	—
196	C₂×C₄²	—	38	2	no	—
197	#107	—	38	8	no	—
198	#107	—	38	4	no	—
199	#106	—	38	8	no	—
200	#106	—	38	4	no	—
201	C₁₉	#11337	1	1	yes	characteristic, complemented, Sylow
202	C₂⁴	#64541	1	1	yes	characteristic
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	38	4	no	—
205	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
206	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
208	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
209	C₂²×C₄	#64541	1	1	yes	characteristic
210	C₂²×C₄	D₄×D₁₉	1	2	yes	—
211	C₂²×C₄	D₄×D₁₉	1	2	yes	—
212	C₂²×C₄	#64543	1	2	yes	—
213	C₂²×C₄	Q₈×D₁₉	1	2	yes	—
214	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
215	M₁₆	—	2	4	no	—
216	M₁₆	—	2	4	no	—
217	M₁₆	—	38	4	no	—
218	M₁₆	—	38	4	no	—
219	M₁₆	—	38	4	no	—
220	M₁₆	—	38	4	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	4	1	no	—
226	C₂×C₈	—	4	1	no	—
227	C₂×C₈	—	4	2	no	—
228	C₂×C₈	—	19	8	no	—
229	C₂×C₈	—	38	4	no	—
230	C₂×C₈	—	38	4	no	—
231	C₂×C₈	Dic₁₉⋊C₄	1	4	yes	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	38	4	no	—
234	C₄⋊C₄	—	38	4	no	—
235	C₂²⋊C₄	—	4	2	no	—
236	C₂²⋊C₄	—	76	2	no	—
237	C₂²⋊C₄	—	76	2	no	—
238	C₄²	—	4	1	no	—
239	C₄²	—	4	1	no	—
240	C₄²	—	76	2	no	—
241	C₄²	—	76	2	no	—
242	C₂³	—	4	2	no	—
243	C₂³	C₄×Dic₃₈	1	1	yes	characteristic
244	C₂³	#10597316	1	1	yes	characteristic
245	C₂³	#10597330	1	2	yes	—
246	C₂³	#10597335	1	2	yes	—
247	C₂³	#10597376	1	1	yes	characteristic
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	76	8	no	—
251	C₂×C₄	—	76	4	no	—
252	C₂×C₄	#10597315	1	1	yes	characteristic
253	C₂×C₄	C₄×D₇₆	1	1	yes	characteristic
254	C₂×C₄	#10597324	1	2	yes	—
255	C₂×C₄	#10597324	1	2	yes	—
256	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	2	yes	—
257	C₂×C₄	#10597328	1	2	yes	—
258	C₂×C₄	#10597329	1	2	yes	—
259	C₂×C₄	#10597334	1	2	yes	—
260	C₂×C₄	#10597334	1	2	yes	—
261	C₂×C₄	C₄⋊C₄×D₁₉	1	2	yes	—
262	C₂×C₄	#10597342	1	2	yes	—
263	C₂×C₄	#10597343	1	2	yes	—
264	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
265	C₂×C₄	#10597370	1	1	yes	characteristic
266	C₂×C₄	#10597381	1	1	yes	characteristic
267	C₂×C₄	#10597383	1	1	yes	characteristic
268	C₂×C₄	#10597386	1	1	yes	characteristic
269	C₂×C₄	#10597390	1	1	yes	characteristic
270	C₈	—	2	4	no	—
271	C₈	—	4	2	no	—
272	C₈	—	38	8	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	4	no	—
275	C₂²	—	4	1	no	—
276	C₂²	—	4	1	no	—
277	C₂²	—	4	2	no	—
278	C₂²	#11926441	1	2	yes	—
279	C₂²	#11926442	1	1	yes	characteristic
280	C₂²	#11926453	1	1	yes	characteristic
281	C₂²	#11926474	1	1	yes	characteristic
282	C₂²	#11926478	1	2	yes	—
283	C₄	—	8	2	no	—
284	C₄	—	152	4	no	—
285	C₄	#11926413	1	1	yes	characteristic
286	C₄	#11926416	1	1	yes	characteristic
287	C₄	#11926418	1	2	yes	—
288	C₄	#11926464	1	1	yes	characteristic
289	C₄	#11926465	1	2	yes	—
290	C₄	#11926480	1	1	yes	characteristic
291	C₂	—	2	2	no	—
292	C₂	—	2	1	no	—
293	C₂	—	2	1	no	—
294	C₂	—	2	2	no	—
295	C₂	#14561899	1	1	yes	characteristic
296	C₂	#14564739	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#10597386
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₁₉×#1729	C₂
socle	C₂²×C₁₉	#392
prime radical	C₂⁴×C₁₉	#41
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	#1729	D₁₉
hypocenter	C₁₉	#11337
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1729

Complements

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

N	H ≅ G/N	type
13. C₁₉⋊#338	C₄	semidirect
201. C₁₉	#11337	semidirect

Maximal subgroups

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

subgroup	index
#14572225	2
#14572224	2
C₂×C₁₉⋊#338	2
#14563331	2
C₁₉×#1729	2
#11337	19