Subgroups — #18081898

Order 4864.

394 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14563502	C₂	1	1	yes	maximal, characteristic
2	#14562383	C₂	1	1	yes	maximal, characteristic
3	#14562134	C₂	1	2	yes	maximal
4	#14562038	C₂	1	2	yes	maximal
5	C₁₉×#3160	C₂	1	1	yes	maximal, characteristic
6	#11926597	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
8	C₂×#10597272	—	2	1	no	—
9	C₄×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
10	#11926298	—	2	1	no	—
11	#11926297	—	2	1	no	—
12	#11926295	—	2	1	no	—
13	#11926281	—	2	4	no	—
14	Q₈⋊(C₁₉⋊C₈)	—	2	4	no	—
15	C₁₉×#533	C₂²	1	2	yes	—
16	C₁₉×#500	C₄	1	1	yes	characteristic, complemented
17	C₁₉×#500	C₄	1	1	yes	characteristic, complemented
18	C₁₉×#499	—	2	1	no	—
19	C₁₉×#499	—	2	1	no	—
20	C₁₉×#498	C₄	1	2	yes	complemented
21	C₁₉×#427	—	2	1	no	—
22	C₁₉×#427	—	2	1	no	—
23	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
24	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
26	C₁₉×C₄⋊Q₈	C₂×C₄	1	2	yes	—
27	C₁₉×C₄⋊Q₈	C₂×C₄	1	2	yes	—
28	C₁₉×#125	—	2	2	no	—
29	C₁₉×C₂²⋊Q₈	—	2	2	no	—
30	C₁₉×C₂²⋊Q₈	—	2	2	no	—
31	C₄×Q₈×C₁₉	—	2	2	no	—
32	C₄×Q₈×C₁₉	—	2	2	no	—
33	C₁₉×#117	—	2	2	no	—
34	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
35	C₂×C₄⋊Dic₁₉	—	2	1	no	—
36	C₂×C₄×Dic₁₉	—	2	1	no	—
37	C₂²×C₁₉⋊C₈	—	2	1	no	—
38	C₁₉×#107	—	2	1	no	—
39	C₁₉×#107	—	2	1	no	—
40	C₁₉×#107	—	2	1	no	—
41	C₁₉×#107	—	2	1	no	—
42	C₁₉×#106	—	2	2	no	—
43	C₁₉×#106	—	2	2	no	—
44	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
47	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
48	C₁₉⋊#95	D₄	1	2	yes	—
49	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
50	#10597272	—	4	2	no	—
51	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
52	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
53	C₄⋊(C₁₉⋊C₈)	D₄	1	2	yes	—
54	C₄×C₁₉⋊C₈	—	2	2	no	—
55	C₂×Q₈×C₁₉	—	2	2	no	—
56	C₂×Q₈×C₁₉	—	2	2	no	—
57	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
60	C₂²×Dic₁₉	—	2	2	no	—
61	C₂×C₈×C₁₉	—	2	1	no	—
62	C₂×C₈×C₁₉	—	2	1	no	—
63	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₁₉	SD₁₆	1	2	yes	—
68	C₂×C₈×C₁₉	Q₁₆	1	2	yes	—
69	C₄⋊C₄×C₁₉	—	2	2	no	—
70	C₄⋊C₄×C₁₉	—	2	2	no	—
71	C₄⋊C₄×C₁₉	—	2	2	no	—
72	C₄⋊C₄×C₁₉	—	2	2	no	—
73	C₄⋊C₄×C₁₉	—	4	2	no	—
74	C₄⋊C₄×C₁₉	—	4	2	no	—
75	C₂²⋊C₄×C₁₉	—	4	2	no	—
76	C₄²×C₁₉	—	4	2	no	—
77	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₁₉	#41	1	1	yes	characteristic
81	C₄⋊Dic₁₉	—	4	2	no	—
82	C₄×Dic₁₉	—	4	1	no	—
83	C₄×Dic₁₉	—	4	1	no	—
84	C₂×C₁₉⋊C₈	—	2	2	no	—
85	C₂×C₁₉⋊C₈	—	4	1	no	—
86	C₂×C₁₉⋊C₈	—	4	1	no	—
87	C₂×C₁₉⋊C₈	—	4	2	no	—
88	#9752	—	19	1	no	maximal, Sylow
89	C₂³×C₁₉	#121	1	1	yes	characteristic
90	Q₈×C₁₉	—	4	4	no	—
91	Q₈×C₁₉	—	8	2	no	—
92	C₂×C₄×C₁₉	—	2	2	no	—
93	C₂×C₄×C₁₉	—	4	1	no	—
94	C₂×C₄×C₁₉	—	4	2	no	—
95	C₂×C₄×C₁₉	—	4	2	no	—
96	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
98	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
99	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
100	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
101	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
102	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
103	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
104	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
105	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
106	C₂×Dic₁₉	—	4	2	no	—
107	C₂×Dic₁₉	—	4	2	no	—
108	C₂×Dic₁₉	—	4	2	no	—
109	C₈×C₁₉	—	2	1	no	—
110	C₈×C₁₉	—	2	1	no	—
111	C₈×C₁₉	#103	1	2	yes	—
112	C₈×C₁₉	#103	1	2	yes	—
113	C₁₉⋊C₈	—	4	2	no	—
114	C₁₉⋊C₈	—	8	2	no	—
115	#3160	D₁₉	1	1	yes	characteristic
116	#1770	—	19	1	no	—
117	#1564	—	19	1	no	—
118	#1503	—	19	2	no	—
119	#1481	—	19	2	no	—
120	C₂²×C₁₉	—	2	1	no	—
121	C₂²×C₁₉	—	2	1	no	—
122	C₂²×C₁₉	#390	1	1	yes	characteristic
123	C₂²×C₁₉	#461	1	1	yes	characteristic
124	C₂²×C₁₉	#462	1	1	yes	characteristic
125	C₄×C₁₉	—	2	1	no	—
126	C₄×C₁₉	—	2	1	no	—
127	C₄×C₁₉	—	2	1	no	—
128	C₄×C₁₉	—	8	2	no	—
129	C₄×C₁₉	—	8	2	no	—
130	C₄×C₁₉	C₂×#103	1	1	yes	characteristic
131	C₄×C₁₉	#419	1	1	yes	characteristic
132	C₄×C₁₉	#444	1	2	yes	—
133	C₄×C₁₉	#461	1	1	yes	characteristic
134	C₄×C₁₉	#462	1	1	yes	characteristic
135	Dic₁₉	—	8	2	no	—
136	#533	C₂×D₁₉	1	2	yes	—
137	#500	Dic₁₉	1	1	yes	characteristic, complemented
138	#500	Dic₁₉	1	1	yes	characteristic, complemented
139	#499	—	2	1	no	—
140	#499	—	2	1	no	—
141	#498	Dic₁₉	1	2	yes	complemented
142	#431	—	19	2	no	—
143	#427	—	2	1	no	—
144	#427	—	2	1	no	—
145	C₂×C₄⋊C₈	—	19	1	no	—
146	C₂×#97	—	38	1	no	—
147	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
148	C₄×C₄⋊C₄	—	19	1	no	—
149	#336	—	38	1	no	—
150	#335	—	38	1	no	—
151	#334	—	38	1	no	—
152	#328	—	38	4	no	—
153	Q₈⋊C₈	—	38	4	no	—
154	C₂×C₁₉	—	2	1	no	—
155	C₂×C₁₉	—	2	1	no	—
156	C₂×C₁₉	#1975	1	1	yes	characteristic
157	C₂×C₁₉	#1976	1	1	yes	characteristic
158	C₂×C₁₉	#1981	1	1	yes	characteristic
159	C₂²×C₈	—	38	1	no	—
160	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
161	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
162	C₄⋊Q₈	C₂×Dic₁₉	1	2	yes	—
163	C₄⋊Q₈	C₂×Dic₁₉	1	2	yes	—
164	#125	—	2	2	no	—
165	C₂²⋊Q₈	—	2	2	no	—
166	C₂²⋊Q₈	—	2	2	no	—
167	C₄×Q₈	—	2	2	no	—
168	C₄×Q₈	—	2	2	no	—
169	#117	—	2	2	no	—
170	C₂×C₄⋊C₄	—	38	1	no	—
171	C₂×C₄²	—	38	1	no	—
172	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
173	#107	—	2	1	no	—
174	#107	—	2	1	no	—
175	#107	—	2	1	no	—
176	#107	—	2	1	no	—
177	#106	—	2	2	no	—
178	#106	—	2	2	no	—
179	C₄⋊C₈	—	19	2	no	—
180	C₄⋊C₈	—	38	1	no	—
181	C₄⋊C₈	—	38	2	no	—
182	C₂²⋊C₈	—	38	2	no	—
183	#97	—	76	2	no	—
184	C₄×C₈	—	38	2	no	—
185	C₄×C₈	C₂×Dic₁₉	1	1	yes	characteristic
186	C₄×C₈	C₂×Dic₁₉	1	1	yes	characteristic
187	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
188	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
189	#95	—	19	2	no	—
190	C₁₉	#9752	1	1	yes	characteristic, complemented, Sylow
191	C₂×Q₈	—	2	2	no	—
192	C₂×Q₈	—	2	2	no	—
193	C₂²×C₄	—	38	2	no	—
194	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
195	C₂²×C₄	#64543	1	1	yes	characteristic
196	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	—	38	2	no	—
200	C₂×C₈	—	76	1	no	—
201	C₂×C₈	—	76	1	no	—
202	C₂×C₈	—	76	2	no	—
203	C₂×C₈	#64527	1	2	yes	—
204	C₂×C₈	#64528	1	2	yes	—
205	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
206	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
207	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
208	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	4	2	no	—
214	C₄⋊C₄	—	4	2	no	—
215	C₄⋊C₄	—	76	2	no	—
216	C₂²⋊C₄	—	4	2	no	—
217	C₄²	—	4	2	no	—
218	C₄²	—	76	1	no	—
219	C₄²	—	76	1	no	—
220	C₄²	D₄×D₁₉	1	1	yes	characteristic
221	C₄²	#64543	1	1	yes	characteristic
222	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
223	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
224	C₂³	#10597384	1	1	yes	characteristic
225	Q₈	—	4	4	no	—
226	Q₈	—	8	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	4	1	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	76	2	no	—
232	C₂×C₄	—	76	2	no	—
233	C₂×C₄	—	76	2	no	—
234	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
235	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
236	C₂×C₄	#10597382	1	1	yes	characteristic
237	C₂×C₄	#10597384	1	1	yes	characteristic
238	C₂×C₄	#10597386	1	1	yes	characteristic
239	C₂×C₄	C₂×#64527	1	1	yes	characteristic
240	C₂×C₄	#10597388	1	1	yes	characteristic
241	C₂×C₄	#10597388	1	1	yes	characteristic
242	C₂×C₄	C₂×#64528	1	1	yes	characteristic
243	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	76	2	no	—
247	C₈	—	152	2	no	—
248	C₈	#10597292	1	2	yes	—
249	C₈	#10597292	1	2	yes	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	#11926447	1	1	yes	characteristic
253	C₂²	#11926746	1	1	yes	characteristic
254	C₂²	#11926751	1	1	yes	characteristic
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	8	2	no	—
259	C₄	—	8	2	no	—
260	C₄	—	152	2	no	—
261	C₄	C₂×#10597292	1	1	yes	characteristic
262	C₄	#11926553	1	1	yes	characteristic
263	C₄	#11926666	1	2	yes	—
264	C₄	#11926746	1	1	yes	characteristic
265	C₄	#11926751	1	1	yes	characteristic
266	C₂	—	2	1	no	—
267	C₂	—	2	1	no	—
268	C₂	#14564529	1	1	yes	characteristic
269	C₂	#14564534	1	1	yes	characteristic
270	C₂	#14564559	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10597384
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#3160	C₂
socle	C₂²×C₁₉	#390
prime radical	C₂³×C₁₉	#121
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#3160	D₁₉
hypocenter	C₁₉	#9752
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3160

Complements

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

N	H ≅ G/N	type
16. C₁₉×#500	C₄	semidirect
17. C₁₉×#500	C₄	semidirect
20. C₁₉×#498	C₄	semidirect
137. #500	Dic₁₉	semidirect
138. #500	Dic₁₉	semidirect
141. #498	Dic₁₉	semidirect
190. C₁₉	#9752	semidirect

Maximal subgroups

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

subgroup	index
#14563502	2
#14562383	2
#14562134	2
#14562038	2
C₁₉×#3160	2
#9752	19