Subgroups — #18091224

Order 4864.

702 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	C₂×#11926593	C₂	1	1	yes	maximal, characteristic
2	C₂×#11926592	C₂	1	2	yes	maximal
3	C₂×C₁₉⋊#338	C₂	1	1	yes	maximal, characteristic
4	#14563329	C₂	1	2	yes	maximal
5	C₁₉×#1728	C₂	1	1	yes	maximal, characteristic
6	C₂²×#64520	C₂²	1	2	yes	—
7	C₂²×#64520	C₂²	1	2	yes	—
8	#11926593	—	2	4	no	—
9	#11926592	—	2	4	no	—
10	#11926592	—	2	4	no	—
11	C₂×#10597276	—	2	2	no	—
12	C₂×#10597279	—	2	4	no	—
13	C₁₉⋊#338	C₄	1	4	yes	complemented
14	#11926299	—	2	4	no	—
15	C₂²×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
16	C₂²×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
17	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
18	C₁₉×#336	—	2	2	no	—
19	C₂³×C₄×C₁₉	Q₈	1	2	yes	—
20	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
21	C₂×M₁₆×C₁₉	—	2	2	no	—
22	C₂×M₁₆×C₁₉	—	2	2	no	—
23	C₂×M₁₆×C₁₉	—	2	1	no	—
24	C₂×M₁₆×C₁₉	—	2	2	no	—
25	C₂×M₁₆×C₁₉	—	2	2	no	—
26	C₂×M₁₆×C₁₉	C₂×C₄	1	2	yes	—
27	C₂×M₁₆×C₁₉	D₄	1	2	yes	—
28	C₂×M₁₆×C₁₉	Q₈	1	2	yes	—
29	C₂²×C₈×C₁₉	—	2	1	no	—
30	C₂²×C₈×C₁₉	D₄	1	2	yes	—
31	C₂×C₄²×C₁₉	—	2	2	no	—
32	C₂×C₄²×C₁₉	—	2	1	no	—
33	C₂×C₄²×C₁₉	—	2	1	no	—
34	C₂×C₄²×C₁₉	—	2	2	no	—
35	C₂×#64520	—	2	2	no	—
36	C₂×#64520	—	2	2	no	—
37	C₂×#64520	—	2	2	no	—
38	C₂×#64520	—	2	2	no	—
39	C₂×#64520	—	2	2	no	—
40	C₂×#64520	—	2	2	no	—
41	C₂×#64520	—	2	2	no	—
42	C₂×#64520	—	2	2	no	—
43	C₂×#64520	—	2	2	no	—
44	C₂×#64520	C₂×C₄	1	4	yes	—
45	C₂×#64520	D₄	1	4	yes	—
46	C₂×#64520	Q₈	1	4	yes	—
47	C₂²×C₁₉⋊C₈	—	2	2	no	—
48	C₂²×C₁₉⋊C₈	D₄	1	4	yes	—
49	#10597279	—	2	16	no	—
50	#10597276	—	2	8	no	—
51	C₂⁴×C₁₉	C₂×Q₈	1	1	yes	characteristic
52	C₂²×C₄×C₁₉	—	2	4	no	—
53	C₂²×C₄×C₁₉	—	2	2	no	—
54	C₂²×C₄×C₁₉	—	4	4	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	2	yes	—
62	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	#41	1	2	yes	—
65	M₁₆×C₁₉	—	2	4	no	—
66	M₁₆×C₁₉	—	2	4	no	—
67	M₁₆×C₁₉	—	2	4	no	—
68	M₁₆×C₁₉	—	4	2	no	—
69	C₂×C₈×C₁₉	—	2	2	no	—
70	C₂×C₈×C₁₉	—	2	2	no	—
71	C₂×C₈×C₁₉	—	2	2	no	—
72	C₂×C₈×C₁₉	—	4	1	no	—
73	C₂×C₈×C₁₉	—	4	1	no	—
74	C₂×C₈×C₁₉	C₄⋊C₄	1	4	yes	—
75	C₄²×C₁₉	—	4	1	no	—
76	C₄²×C₁₉	—	4	2	no	—
77	C₄²×C₁₉	—	4	1	no	—
78	#64520	—	2	4	no	—
79	#64520	—	2	4	no	—
80	#64520	—	2	4	no	—
81	#64520	—	2	4	no	—
82	#64520	—	2	4	no	—
83	#64520	—	2	4	no	—
84	#64520	—	4	2	no	—
85	#64520	—	4	2	no	—
86	C₂×C₁₉⋊C₈	—	2	4	no	—
87	C₂×C₁₉⋊C₈	—	2	4	no	—
88	C₂×C₁₉⋊C₈	—	2	4	no	—
89	C₂×C₁₉⋊C₈	—	4	2	no	—
90	C₂×C₁₉⋊C₈	—	4	2	no	—
91	C₂×C₁₉⋊C₈	C₄⋊C₄	1	8	yes	—
92	#11338	—	19	1	no	maximal, Sylow
93	C₂³×C₁₉	—	4	2	no	—
94	C₂³×C₁₉	C₄×Q₈	1	1	yes	characteristic
95	C₂³×C₁₉	C₂²⋊Q₈	1	2	yes	—
96	C₂³×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
97	C₂³×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
98	C₂³×C₁₉	C₄⋊Q₈	1	2	yes	—
99	C₂×C₄×C₁₉	—	2	8	no	—
100	C₂×C₄×C₁₉	—	2	4	no	—
101	C₂×C₄×C₁₉	—	4	4	no	—
102	C₂×C₄×C₁₉	—	4	2	no	—
103	C₂×C₄×C₁₉	—	4	2	no	—
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	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
109	C₂×C₄×C₁₉	#120	1	2	yes	—
110	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
111	C₂×C₄×C₁₉	#121	1	2	yes	—
112	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
113	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
114	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
115	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
116	C₂×C₄×C₁₉	#123	1	2	yes	—
117	C₂×C₄×C₁₉	#125	1	1	yes	characteristic
118	C₂×C₄×C₁₉	#125	1	2	yes	—
119	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
120	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
121	C₂×C₄×C₁₉	C₄⋊Q₈	1	2	yes	—
122	C₈×C₁₉	—	2	4	no	—
123	C₈×C₁₉	—	4	2	no	—
124	C₁₉⋊C₈	—	2	8	no	—
125	C₁₉⋊C₈	—	4	4	no	—
126	C₂×#430	—	19	2	no	—
127	C₂×#430	—	19	1	no	—
128	C₂×#338	—	19	1	no	—
129	#1728	—	19	2	no	—
130	#1728	D₁₉	1	1	yes	characteristic
131	C₂²×C₁₉	—	2	4	no	—
132	C₂²×C₁₉	—	2	2	no	—
133	C₂²×C₁₉	—	4	1	no	—
134	C₂²×C₁₉	—	4	1	no	—
135	C₂²×C₁₉	—	4	2	no	—
136	C₂²×C₁₉	#389	1	2	yes	—
137	C₂²×C₁₉	#389	1	1	yes	characteristic
138	C₂²×C₁₉	#395	1	1	yes	characteristic
139	C₂²×C₁₉	#395	1	2	yes	—
140	C₂²×C₁₉	#399	1	1	yes	characteristic
141	C₄×C₁₉	—	4	4	no	—
142	C₄×C₁₉	#384	1	1	yes	characteristic
143	C₄×C₁₉	#385	1	1	yes	characteristic
144	C₄×C₁₉	#385	1	2	yes	—
145	C₄×C₁₉	#392	1	1	yes	characteristic
146	C₄×C₁₉	#398	1	1	yes	characteristic
147	C₄×C₁₉	#398	1	2	yes	—
148	C₂²×M₁₆	—	19	2	no	—
149	C₂²×M₁₆	—	19	2	no	—
150	C₂²×M₁₆	C₂×D₁₉	1	1	yes	characteristic
151	C₂²×M₁₆	C₂×D₁₉	1	1	yes	characteristic
152	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
153	#430	—	38	4	no	—
154	#430	—	38	4	no	—
155	#430	—	38	4	no	—
156	C₂×#108	—	38	4	no	—
157	C₂×#108	—	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	#2091	1	2	yes	—
167	C₂³×C₄	Dic₃₈	1	2	yes	—
168	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	1	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	19	4	no	—
175	C₂×M₁₆	—	19	4	no	—
176	C₂×M₁₆	—	19	4	no	—
177	C₂×M₁₆	—	38	2	no	—
178	C₂×M₁₆	—	38	2	no	—
179	C₂×M₁₆	—	38	2	no	—
180	C₂×M₁₆	—	38	2	no	—
181	C₂×M₁₆	—	38	2	no	—
182	C₂×M₁₆	—	38	2	no	—
183	C₂×M₁₆	—	38	2	no	—
184	C₂×M₁₆	—	38	2	no	—
185	C₂×M₁₆	—	38	2	no	—
186	C₂×M₁₆	Dic₃₈	1	2	yes	—
187	C₂×M₁₆	C₄×D₁₉	1	2	yes	—
188	C₂×M₁₆	D₇₆	1	2	yes	—
189	C₂²×C₈	—	2	1	no	—
190	C₂²×C₈	—	19	4	no	—
191	C₂²×C₈	—	38	2	no	—
192	C₂²×C₈	C₂²⋊D₁₉	1	2	yes	—
193	C₂×C₄²	—	2	1	no	—
194	C₂×C₄²	—	2	2	no	—
195	C₂×C₄²	—	2	2	no	—
196	C₂×C₄²	—	2	1	no	—
197	#108	—	38	8	no	—
198	#108	—	38	16	no	—
199	C₁₉	#11338	1	1	yes	characteristic, complemented, Sylow
200	C₂⁴	C₂×Dic₃₈	1	1	yes	characteristic
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	4	4	no	—
204	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
206	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
207	C₂²×C₄	#64541	1	1	yes	characteristic
208	C₂²×C₄	#64541	1	1	yes	characteristic
209	C₂²×C₄	D₄×D₁₉	1	2	yes	—
210	C₂²×C₄	D₄×D₁₉	1	2	yes	—
211	C₂²×C₄	#64543	1	2	yes	—
212	C₂²×C₄	Q₈×D₁₉	1	2	yes	—
213	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
214	M₁₆	—	2	4	no	—
215	M₁₆	—	2	4	no	—
216	M₁₆	—	2	4	no	—
217	M₁₆	—	4	2	no	—
218	M₁₆	—	38	4	no	—
219	M₁₆	—	38	4	no	—
220	M₁₆	—	38	4	no	—
221	M₁₆	—	38	4	no	—
222	M₁₆	—	38	4	no	—
223	M₁₆	—	38	4	no	—
224	M₁₆	—	76	2	no	—
225	M₁₆	—	76	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	—	4	1	no	—
231	C₂×C₈	—	19	8	no	—
232	C₂×C₈	—	38	4	no	—
233	C₂×C₈	—	38	4	no	—
234	C₂×C₈	—	38	4	no	—
235	C₂×C₈	—	76	2	no	—
236	C₂×C₈	—	76	2	no	—
237	C₂×C₈	Dic₁₉⋊C₄	1	4	yes	—
238	C₄²	—	4	1	no	—
239	C₄²	—	4	1	no	—
240	C₄²	—	4	2	no	—
241	C₂³	—	4	2	no	—
242	C₂³	C₄×Dic₃₈	1	1	yes	characteristic
243	C₂³	#10597315	1	1	yes	characteristic
244	C₂³	#10597324	1	2	yes	—
245	C₂³	#10597334	1	2	yes	—
246	C₂³	#10597370	1	1	yes	characteristic
247	C₂×C₄	—	2	8	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	#10597316	1	1	yes	characteristic
253	C₂×C₄	C₄×D₇₆	1	1	yes	characteristic
254	C₂×C₄	#10597324	1	2	yes	—
255	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	2	yes	—
256	C₂×C₄	#10597328	1	2	yes	—
257	C₂×C₄	#10597329	1	2	yes	—
258	C₂×C₄	#10597330	1	2	yes	—
259	C₂×C₄	#10597334	1	2	yes	—
260	C₂×C₄	#10597335	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	#10597376	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	C₈	—	76	4	no	—
274	C₂²	—	2	4	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	4	1	no	—
277	C₂²	—	4	1	no	—
278	C₂²	—	4	2	no	—
279	C₂²	#11926441	1	2	yes	—
280	C₂²	#11926442	1	1	yes	characteristic
281	C₂²	#11926464	1	1	yes	characteristic
282	C₂²	#11926465	1	2	yes	—
283	C₂²	#11926480	1	1	yes	characteristic
284	C₄	—	4	4	no	—
285	C₄	#11926413	1	1	yes	characteristic
286	C₄	#11926416	1	1	yes	characteristic
287	C₄	#11926418	1	2	yes	—
288	C₄	#11926453	1	1	yes	characteristic
289	C₄	#11926474	1	1	yes	characteristic
290	C₄	#11926478	1	2	yes	—
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	#14565151	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#10597386
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₁₉×#1728	C₂
socle	C₂²×C₁₉	#399
prime radical	C₂⁴×C₁₉	C₂×Q₈
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	#1728	D₁₉
hypocenter	C₁₉	#11338
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1728

Complements

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

N	H ≅ G/N	type
13. C₁₉⋊#338	C₄	semidirect
199. C₁₉	#11338	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926593	2
C₂×#11926592	2
C₂×C₁₉⋊#338	2
#14563329	2
C₁₉×#1728	2
#11338	19