Subgroups — #18082519

Order 4864.

528 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#10597315	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₁₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#14561475	C₂	1	4	yes	maximal
4	C₄×C₁₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂×#10597315	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₃₈	—	2	1	no	—
7	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄×C₁₉⋊C₈	—	2	1	no	—
9	#11926450	—	2	1	no	—
10	#11926442	—	2	1	no	—
11	#11926442	C₄	1	2	yes	—
12	#11926414	C₄	1	1	yes	characteristic
13	C₄×C₄⋊Dic₁₉	—	2	1	no	—
14	C₄×C₄⋊Dic₁₉	C₂²	1	2	yes	—
15	#11926299	—	2	1	no	—
16	C₂×C₁₉×C₄⋊C₈	—	2	1	no	—
17	C₂×C₄×C₈×C₁₉	C₂²	1	2	yes	—
18	C₄³×C₁₉	C₂²	1	1	yes	characteristic
19	C₁₉×#336	—	2	1	no	—
20	C₂²×Dic₃₈	—	2	1	no	—
21	C₂²×C₈×C₁₉	—	2	2	no	—
22	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊Dic₁₉	—	2	1	no	—
30	C₂×C₄⋊Dic₁₉	—	2	1	no	—
31	C₂×C₄⋊Dic₁₉	—	2	1	no	—
32	C₂×C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
34	C₂×Dic₁₉⋊C₄	—	2	2	no	—
35	C₂×C₄×Dic₁₉	—	2	2	no	—
36	C₂²×C₁₉⋊C₈	—	2	2	no	—
37	#10597315	—	2	2	no	—
38	#10597315	C₈	1	4	yes	complemented
39	C₄×Dic₃₈	—	4	4	no	—
40	C₁₉×C₄⋊C₈	—	2	4	no	—
41	C₄×C₈×C₁₉	—	2	4	no	—
42	C₁₉⋊#95	—	2	2	no	—
43	C₁₉⋊#95	—	2	2	no	—
44	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
45	C₄⋊(C₁₉⋊C₈)	Q₈	1	4	yes	complemented
46	C₄×C₁₉⋊C₈	—	4	2	no	—
47	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
54	C₂²×Dic₁₉	—	2	2	no	—
55	C₂²×Dic₁₉	—	2	2	no	—
56	C₂×Dic₃₈	—	2	4	no	—
57	C₂×Dic₃₈	—	4	2	no	—
58	C₂×Dic₃₈	—	4	2	no	—
59	C₂×C₈×C₁₉	—	2	4	no	—
60	C₂×C₈×C₁₉	—	4	2	no	—
61	C₂×C₈×C₁₉	—	4	2	no	—
62	C₄²×C₁₉	—	2	2	no	—
63	C₄²×C₁₉	—	2	2	no	—
64	C₄²×C₁₉	—	2	2	no	—
65	C₄²×C₁₉	—	4	1	no	—
66	C₄²×C₁₉	C₂×C₈	1	2	yes	—
67	C₄²×C₁₉	M₁₆	1	2	yes	—
68	C₄²×C₁₉	SD₁₆	1	2	yes	—
69	C₄²×C₁₉	Q₁₆	1	2	yes	—
70	C₄²×C₁₉	C₂×Q₈	1	2	yes	—
71	C₄²×C₁₉	#41	1	2	yes	—
72	C₄⋊Dic₁₉	—	2	4	no	—
73	C₄⋊Dic₁₉	—	2	2	no	—
74	C₄⋊Dic₁₉	—	2	4	no	—
75	C₄⋊Dic₁₉	—	4	2	no	—
76	C₄⋊Dic₁₉	—	4	2	no	—
77	C₄⋊Dic₁₉	C₂×C₈	1	4	yes	—
78	Dic₁₉⋊C₄	—	4	4	no	—
79	C₄×Dic₁₉	—	4	4	no	—
80	C₂×C₁₉⋊C₈	—	2	4	no	—
81	C₂×C₁₉⋊C₈	—	4	2	no	—
82	C₂×C₁₉⋊C₈	—	4	2	no	—
83	#9864	—	19	1	no	maximal, Sylow
84	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₉	—	2	1	no	—
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	C₂²⋊C₈	1	2	yes	—
95	C₂×C₄×C₁₉	C₂²⋊C₈	1	2	yes	—
96	C₂×C₄×C₁₉	#103	1	2	yes	—
97	C₂×C₄×C₁₉	#103	1	2	yes	—
98	C₂×C₄×C₁₉	#104	1	2	yes	—
99	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
100	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
101	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
102	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
103	C₂×C₄×C₁₉	C₂²×C₈	1	1	yes	characteristic
104	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
105	C₂×C₄×C₁₉	#131	1	2	yes	—
106	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
107	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
108	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
109	C₂×Dic₁₉	—	2	4	no	—
110	C₂×Dic₁₉	—	4	2	no	—
111	C₂×Dic₁₉	—	4	2	no	—
112	C₂×Dic₁₉	—	4	2	no	—
113	C₂×Dic₁₉	—	4	2	no	—
114	C₂×Dic₁₉	—	4	2	no	—
115	Dic₃₈	—	4	8	no	—
116	C₈×C₁₉	—	4	4	no	—
117	C₁₉⋊C₈	—	4	4	no	—
118	C₄×C₄⋊Q₈	—	19	1	no	—
119	C₄×C₄⋊C₈	—	19	1	no	—
120	C₄×C₄⋊C₈	D₁₉	1	1	yes	characteristic
121	#1279	—	19	4	no	—
122	C₂²×C₁₉	#391	1	1	yes	characteristic
123	C₂²×C₁₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
124	C₂²×C₁₉	#408	1	1	yes	characteristic
125	C₂²×C₁₉	C₂×#103	1	1	yes	characteristic
126	C₂²×C₁₉	#418	1	1	yes	characteristic
127	C₂²×C₁₉	C₂×#104	1	1	yes	characteristic
128	C₂²×C₁₉	#421	1	1	yes	characteristic
129	C₄×C₁₉	—	2	1	no	—
130	C₄×C₁₉	—	2	1	no	—
131	C₄×C₁₉	—	2	2	no	—
132	C₄×C₁₉	—	2	2	no	—
133	C₄×C₁₉	—	4	1	no	—
134	C₄×C₁₉	Q₈⋊C₈	1	4	yes	—
135	C₄×C₁₉	C₈×Q₈	1	2	yes	—
136	C₄×C₁₉	#446	1	2	yes	—
137	C₄×C₁₉	#476	1	2	yes	—
138	C₄×C₁₉	#477	1	2	yes	—
139	Dic₁₉	—	4	4	no	—
140	Dic₁₉	—	8	2	no	—
141	C₂×C₄⋊Q₈	—	19	1	no	—
142	C₂×C₄×Q₈	—	38	1	no	—
143	C₂×C₄⋊C₈	—	2	1	no	—
144	C₂×C₄⋊C₈	—	19	2	no	—
145	C₂×C₄×C₈	—	38	1	no	—
146	C₂×C₄×C₈	C₂×D₁₉	1	2	yes	—
147	#391	—	38	1	no	—
148	#389	—	19	2	no	—
149	#389	—	38	1	no	—
150	#384	—	19	1	no	—
151	C₄×C₄⋊C₄	—	19	2	no	—
152	C₄×C₄⋊C₄	—	38	1	no	—
153	C₄³	C₂×D₁₉	1	1	yes	characteristic
154	#336	—	2	1	no	—
155	#336	—	38	1	no	—
156	C₂×C₁₉	C₂×Q₈⋊C₈	1	1	yes	characteristic
157	C₂×C₁₉	#1478	1	1	yes	characteristic
158	C₂×C₁₉	#1481	1	1	yes	characteristic
159	C₂×C₁₉	#1484	1	1	yes	characteristic
160	C₂×C₁₉	#1984	1	1	yes	characteristic
161	C₂×C₁₉	#1986	1	1	yes	characteristic
162	C₂×C₁₉	#1987	1	1	yes	characteristic
163	C₂²×Q₈	—	38	1	no	—
164	C₂²×C₈	—	2	2	no	—
165	C₂²×C₈	—	38	2	no	—
166	C₄⋊Q₈	—	19	4	no	—
167	C₄⋊Q₈	—	38	2	no	—
168	C₄×Q₈	—	76	4	no	—
169	C₂×C₄⋊C₄	—	19	2	no	—
170	C₂×C₄⋊C₄	—	19	2	no	—
171	C₂×C₄⋊C₄	—	38	1	no	—
172	C₂×C₄⋊C₄	—	38	1	no	—
173	C₂×C₄⋊C₄	—	38	2	no	—
174	C₂×C₄⋊C₄	—	38	1	no	—
175	C₂×C₄²	—	38	2	no	—
176	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
177	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
178	C₂×C₄²	D₇₆	1	1	yes	characteristic
179	C₂×C₄²	D₇₆	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	C₄⋊C₈	—	2	4	no	—
184	C₄⋊C₈	—	19	4	no	—
185	C₄⋊C₈	—	38	2	no	—
186	C₄×C₈	—	2	4	no	—
187	C₄×C₈	—	76	2	no	—
188	#95	—	38	2	no	—
189	#95	—	38	2	no	—
190	C₁₉	#9864	1	1	yes	characteristic, complemented, Sylow
191	C₂×Q₈	—	38	4	no	—
192	C₂×Q₈	—	76	2	no	—
193	C₂×Q₈	—	76	2	no	—
194	C₂²×C₄	—	38	2	no	—
195	C₂²×C₄	—	38	2	no	—
196	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
197	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
198	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
199	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
202	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	—	4	2	no	—
205	C₂×C₈	—	4	2	no	—
206	C₂×C₈	—	38	4	no	—
207	C₂×C₈	—	76	2	no	—
208	C₂×C₈	—	76	2	no	—
209	C₄⋊C₄	—	19	4	no	—
210	C₄⋊C₄	—	38	4	no	—
211	C₄⋊C₄	—	38	2	no	—
212	C₄⋊C₄	—	38	4	no	—
213	C₄⋊C₄	—	76	4	no	—
214	C₄⋊C₄	—	76	2	no	—
215	C₄⋊C₄	—	76	2	no	—
216	C₄²	—	2	2	no	—
217	C₄²	—	2	2	no	—
218	C₄²	—	2	2	no	—
219	C₄²	—	4	1	no	—
220	C₄²	—	76	4	no	—
221	C₄²	C₈×D₁₉	1	2	yes	—
222	C₄²	#64515	1	2	yes	—
223	C₄²	#64516	1	2	yes	—
224	C₄²	Dic₇₆	1	2	yes	—
225	C₄²	#64543	1	2	yes	—
226	C₄²	Q₈×D₁₉	1	2	yes	—
227	C₂³	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
228	Q₈	—	76	8	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
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	—	38	4	no	—
239	C₂×C₄	—	76	2	no	—
240	C₂×C₄	—	76	2	no	—
241	C₂×C₄	—	76	2	no	—
242	C₂×C₄	—	76	2	no	—
243	C₂×C₄	—	76	2	no	—
244	C₂×C₄	#10597262	1	2	yes	—
245	C₂×C₄	#10597273	1	2	yes	—
246	C₂×C₄	#10597273	1	2	yes	—
247	C₂×C₄	(C₂×D₁₉)⋊C₈	1	2	yes	—
248	C₂×C₄	(C₂×D₁₉)⋊C₈	1	2	yes	—
249	C₂×C₄	#10597333	1	2	yes	—
250	C₂×C₄	#10597343	1	2	yes	—
251	C₂×C₄	C₂×C₈×D₁₉	1	1	yes	characteristic
252	C₂×C₄	C₂×#64515	1	1	yes	characteristic
253	C₂×C₄	C₂×#64516	1	1	yes	characteristic
254	C₂×C₄	C₂×Dic₇₆	1	1	yes	characteristic
255	C₂×C₄	#10597353	1	2	yes	—
256	C₂×C₄	#10597354	1	2	yes	—
257	C₂×C₄	#10597382	1	1	yes	characteristic
258	C₂×C₄	#10597390	1	1	yes	characteristic
259	C₈	—	4	4	no	—
260	C₈	—	76	4	no	—
261	C₂²	#11926449	1	1	yes	characteristic
262	C₂²	C₂×(C₂×D₁₉)⋊C₈	1	1	yes	characteristic
263	C₂²	#11926512	1	1	yes	characteristic
264	C₂²	C₂×#10597273	1	1	yes	characteristic
265	C₂²	#11926550	1	1	yes	characteristic
266	C₂²	C₂×#10597262	1	1	yes	characteristic
267	C₂²	#11926562	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	2	no	—
271	C₄	—	2	2	no	—
272	C₄	—	4	1	no	—
273	C₄	—	76	4	no	—
274	C₄	—	152	2	no	—
275	C₄	#11926276	1	4	yes	—
276	C₄	#11926668	1	2	yes	—
277	C₄	#11926671	1	2	yes	—
278	C₄	#11926835	1	2	yes	—
279	C₄	#11926840	1	2	yes	—
280	C₂	C₂×#11926276	1	1	yes	characteristic
281	C₂	#14562027	1	1	yes	characteristic
282	C₂	#14562039	1	1	yes	characteristic
283	C₂	#14562053	1	1	yes	characteristic
284	C₂	#14564570	1	1	yes	characteristic
285	C₂	#14564578	1	1	yes	characteristic
286	C₂	#14564582	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₇₆
commutator	C₂×C₄×C₁₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₄×C₁₉×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₁₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₄×C₄⋊C₈	D₁₉
hypocenter	C₁₉	#9864
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
38. #10597315	C₈	semidirect
45. C₄⋊(C₁₉⋊C₈)	Q₈	semidirect
190. C₁₉	#9864	semidirect

Maximal subgroups

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

subgroup	index
C₄×#10597315	2
C₄×C₄⋊(C₁₉⋊C₈)	2
#14561475	2
C₄×C₁₉×C₄⋊C₈	2
#9864	19