Subgroups — #18082700

Order 4864.

438 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14563787	C₂	1	1	yes	maximal, characteristic
2	#14563488	C₂	1	1	yes	maximal, characteristic
3	#14561518	C₂	1	2	yes	maximal
4	#14561518	C₂	1	2	yes	maximal
5	C₁₉×#2859	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
7	C₂×C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
8	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
9	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
10	C₂×#10597260	—	2	1	no	—
11	#11926299	—	2	1	no	—
12	C₂×C₁₉×#125	C₄	1	2	yes	—
13	C₁₉×#389	—	2	1	no	—
14	C₁₉×#389	—	2	1	no	—
15	C₁₉×#389	—	2	1	no	—
16	C₁₉×#389	—	2	1	no	—
17	C₁₉×#384	C₂²	1	1	yes	characteristic
18	C₁₉×#384	C₂²	1	1	yes	characteristic
19	C₁₉×#382	C₄	1	2	yes	—
20	C₄³×C₁₉	C₂²	1	1	yes	characteristic
21	C₁₉×#125	—	4	4	no	—
22	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
23	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
24	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
25	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
26	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
27	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
28	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
29	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
37	C₂²×C₁₉⋊C₈	—	2	2	no	—
38	C₂²×C₁₉⋊C₈	—	2	2	no	—
39	C₁₉×#95	—	2	2	no	—
40	C₁₉×#95	—	2	2	no	—
41	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
42	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
43	C₄⋊(C₁₉⋊C₈)	—	4	2	no	—
44	C₄⋊(C₁₉⋊C₈)	—	4	2	no	—
45	C₄⋊(C₁₉⋊C₈)	Q₈	1	4	yes	—
46	#10597260	—	4	2	no	—
47	C₂²×C₄×C₁₉	—	2	2	no	—
48	C₂²×C₄×C₁₉	—	2	2	no	—
49	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
56	C₄⋊C₄×C₁₉	—	4	2	no	—
57	C₄⋊C₄×C₁₉	—	4	2	no	—
58	C₄⋊C₄×C₁₉	—	4	4	no	—
59	C₄⋊C₄×C₁₉	—	4	4	no	—
60	C₄⋊C₄×C₁₉	—	4	2	no	—
61	C₄⋊C₄×C₁₉	—	4	1	no	—
62	C₄⋊C₄×C₁₉	—	4	1	no	—
63	C₄⋊C₄×C₁₉	—	4	1	no	—
64	C₄⋊C₄×C₁₉	—	4	1	no	—
65	C₄⋊C₄×C₁₉	—	4	1	no	—
66	C₄⋊C₄×C₁₉	—	4	1	no	—
67	C₄²×C₁₉	—	2	2	no	—
68	C₄²×C₁₉	—	2	2	no	—
69	C₄²×C₁₉	—	2	2	no	—
70	C₄²×C₁₉	—	2	1	no	—
71	C₄²×C₁₉	—	2	1	no	—
72	C₄²×C₁₉	—	2	1	no	—
73	C₄²×C₁₉	—	2	1	no	—
74	C₄²×C₁₉	—	2	1	no	—
75	C₄²×C₁₉	—	2	1	no	—
76	C₄²×C₁₉	C₂×Q₈	1	2	yes	—
77	C₄²×C₁₉	#41	1	2	yes	—
78	C₂×C₁₉⋊C₈	—	2	4	no	—
79	C₂×C₁₉⋊C₈	—	4	2	no	—
80	C₂×C₁₉⋊C₈	—	4	2	no	—
81	C₂×C₁₉⋊C₈	—	4	2	no	—
82	C₂×C₁₉⋊C₈	—	4	2	no	—
83	C₂×C₁₉⋊C₈	—	4	2	no	—
84	#9906	—	19	1	no	maximal, Sylow
85	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₉	—	2	1	no	—
87	C₂×C₄×C₁₉	—	2	1	no	—
88	C₂×C₄×C₁₉	—	2	1	no	—
89	C₂×C₄×C₁₉	—	2	1	no	—
90	C₂×C₄×C₁₉	—	2	1	no	—
91	C₂×C₄×C₁₉	—	2	1	no	—
92	C₂×C₄×C₁₉	—	2	1	no	—
93	C₂×C₄×C₁₉	—	2	1	no	—
94	C₂×C₄×C₁₉	—	2	1	no	—
95	C₂×C₄×C₁₉	—	2	1	no	—
96	C₂×C₄×C₁₉	—	2	1	no	—
97	C₂×C₄×C₁₉	—	2	1	no	—
98	C₂×C₄×C₁₉	—	4	2	no	—
99	C₂×C₄×C₁₉	—	4	2	no	—
100	C₂×C₄×C₁₉	—	4	2	no	—
101	C₂×C₄×C₁₉	—	4	2	no	—
102	C₂×C₄×C₁₉	—	4	2	no	—
103	C₂×C₄×C₁₉	—	4	2	no	—
104	C₂×C₄×C₁₉	#100	1	2	yes	—
105	C₂×C₄×C₁₉	#101	1	2	yes	—
106	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
107	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
108	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
109	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
110	C₂×C₄×C₁₉	#135	1	2	yes	—
111	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
112	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
113	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
114	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
115	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
116	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
117	C₁₉⋊C₈	—	4	4	no	—
118	C₁₉⋊C₈	—	8	2	no	—
119	#2859	D₁₉	1	1	yes	characteristic
120	#1842	—	19	1	no	—
121	#1767	—	19	1	no	—
122	#1296	—	19	2	no	—
123	#1296	—	19	2	no	—
124	C₂²×C₁₉	#391	1	1	yes	characteristic
125	C₂²×C₁₉	C₂×#100	1	1	yes	characteristic
126	C₂²×C₁₉	C₂×#101	1	1	yes	characteristic
127	C₂²×C₁₉	#416	1	1	yes	characteristic
128	C₂²×C₁₉	#417	1	1	yes	characteristic
129	C₂²×C₁₉	#417	1	1	yes	characteristic
130	C₂²×C₁₉	#417	1	1	yes	characteristic
131	C₄×C₁₉	—	2	2	no	—
132	C₄×C₁₉	—	2	2	no	—
133	C₄×C₁₉	—	2	2	no	—
134	C₄×C₁₉	—	4	1	no	—
135	C₄×C₁₉	—	4	1	no	—
136	C₄×C₁₉	—	8	2	no	—
137	C₄×C₁₉	—	8	2	no	—
138	C₄×C₁₉	#478	1	2	yes	—
139	C₄×C₁₉	#479	1	2	yes	—
140	C₄×C₁₉	#490	1	1	yes	characteristic
141	C₄×C₁₉	#491	1	1	yes	characteristic
142	C₄×C₁₉	#501	1	1	yes	characteristic
143	C₄×C₁₉	#501	1	1	yes	characteristic
144	C₂×#125	Dic₁₉	1	2	yes	—
145	C₂×C₄⋊C₈	—	19	2	no	—
146	C₂×C₄⋊C₈	—	19	2	no	—
147	C₂×C₄⋊C₈	—	38	1	no	—
148	C₂×C₄⋊C₈	—	38	1	no	—
149	C₂×#97	—	38	1	no	—
150	#389	—	2	1	no	—
151	#389	—	2	1	no	—
152	#389	—	2	1	no	—
153	#389	—	2	1	no	—
154	#384	C₂×D₁₉	1	1	yes	characteristic
155	#384	C₂×D₁₉	1	1	yes	characteristic
156	#382	Dic₁₉	1	2	yes	—
157	C₄³	C₂×D₁₉	1	1	yes	characteristic
158	#336	—	38	1	no	—
159	C₂×C₁₉	#1551	1	1	yes	characteristic
160	C₂×C₁₉	#1552	1	1	yes	characteristic
161	C₂×C₁₉	#1560	1	1	yes	characteristic
162	C₂×C₁₉	#1561	1	1	yes	characteristic
163	C₂×C₁₉	#1989	1	1	yes	characteristic
164	C₂×C₁₉	#1992	1	1	yes	characteristic
165	C₂×C₁₉	#1997	1	1	yes	characteristic
166	C₂²×C₈	—	38	2	no	—
167	C₂²×C₈	—	38	2	no	—
168	#125	—	4	4	no	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	—	2	1	no	—
172	C₂×C₄⋊C₄	—	2	1	no	—
173	C₂×C₄⋊C₄	—	2	1	no	—
174	C₂×C₄⋊C₄	—	2	1	no	—
175	C₂×C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
176	C₂×C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
177	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
178	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
179	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
180	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
181	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
182	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
183	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
184	C₄⋊C₈	—	19	4	no	—
185	C₄⋊C₈	—	38	2	no	—
186	C₄⋊C₈	—	38	4	no	—
187	C₄⋊C₈	—	76	2	no	—
188	C₄⋊C₈	—	76	2	no	—
189	#97	—	76	2	no	—
190	#95	—	2	2	no	—
191	#95	—	2	2	no	—
192	C₁₉	#9906	1	1	yes	characteristic, complemented, Sylow
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
196	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
197	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
198	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
199	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
202	C₂×C₈	—	38	4	no	—
203	C₂×C₈	—	76	2	no	—
204	C₂×C₈	—	76	2	no	—
205	C₂×C₈	—	76	2	no	—
206	C₂×C₈	—	76	2	no	—
207	C₂×C₈	—	76	2	no	—
208	C₄⋊C₄	—	4	2	no	—
209	C₄⋊C₄	—	4	2	no	—
210	C₄⋊C₄	—	4	4	no	—
211	C₄⋊C₄	—	4	1	no	—
212	C₄⋊C₄	—	4	1	no	—
213	C₄⋊C₄	—	4	1	no	—
214	C₄⋊C₄	—	4	2	no	—
215	C₄⋊C₄	—	4	1	no	—
216	C₄⋊C₄	—	4	1	no	—
217	C₄⋊C₄	—	4	1	no	—
218	C₄⋊C₄	—	4	4	no	—
219	C₄²	—	2	1	no	—
220	C₄²	—	2	1	no	—
221	C₄²	—	2	1	no	—
222	C₄²	—	2	2	no	—
223	C₄²	—	2	1	no	—
224	C₄²	—	2	1	no	—
225	C₄²	—	2	1	no	—
226	C₄²	—	2	2	no	—
227	C₄²	—	2	2	no	—
228	C₄²	Q₈×D₁₉	1	2	yes	—
229	C₄²	#64545	1	2	yes	—
230	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	#10597290	1	2	yes	—
250	C₂×C₄	#10597293	1	2	yes	—
251	C₂×C₄	#10597390	1	1	yes	characteristic
252	C₂×C₄	Q₈×Dic₁₉	1	2	yes	—
253	C₂×C₄	#10597392	1	2	yes	—
254	C₂×C₄	#10597393	1	1	yes	characteristic
255	C₂×C₄	#10597395	1	1	yes	characteristic
256	C₂×C₄	#10597395	1	1	yes	characteristic
257	C₂×C₄	#10597395	1	1	yes	characteristic
258	C₂×C₄	#10597396	1	2	yes	—
259	C₂×C₄	#10597397	1	1	yes	characteristic
260	C₂×C₄	#10597397	1	1	yes	characteristic
261	C₂×C₄	#10597397	1	1	yes	characteristic
262	C₈	—	76	4	no	—
263	C₈	—	152	2	no	—
264	C₂²	#11926451	1	1	yes	characteristic
265	C₂²	C₂×#10597290	1	1	yes	characteristic
266	C₂²	C₂×#10597293	1	1	yes	characteristic
267	C₂²	#11926540	1	1	yes	characteristic
268	C₂²	#11926545	1	1	yes	characteristic
269	C₂²	#11926545	1	1	yes	characteristic
270	C₂²	#11926545	1	1	yes	characteristic
271	C₄	—	2	2	no	—
272	C₄	—	2	2	no	—
273	C₄	—	2	2	no	—
274	C₄	—	4	1	no	—
275	C₄	—	4	1	no	—
276	C₄	—	8	2	no	—
277	C₄	—	8	2	no	—
278	C₄	#11926841	1	2	yes	—
279	C₄	#11926848	1	2	yes	—
280	C₄	#11926910	1	1	yes	characteristic
281	C₄	#11926914	1	1	yes	characteristic
282	C₄	#11926954	1	1	yes	characteristic
283	C₄	#11926954	1	1	yes	characteristic
284	C₂	#14562330	1	1	yes	characteristic
285	C₂	#14562335	1	1	yes	characteristic
286	C₂	#14562370	1	1	yes	characteristic
287	C₂	#14562373	1	1	yes	characteristic
288	C₂	#14564593	1	1	yes	characteristic
289	C₂	#14564605	1	1	yes	characteristic
290	C₂	#14564625	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×#2859	C₂
socle, prime radical	C₂³×C₁₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#2859	D₁₉
hypocenter	C₁₉	#9906
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2859

Complements

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

N	H ≅ G/N	type
192. C₁₉	#9906	semidirect

Maximal subgroups

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

subgroup	index
#14563787	2
#14563488	2
#14561518	2
C₁₉×#2859	2
#9906	19