Subgroups — #18073683

Order 4864.

462 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊(C₁₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₁₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926266	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×C₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₁₉⋊#1273	C₂	1	1	yes	maximal, characteristic
6	C₁₉⋊#1273	C₂	1	1	yes	maximal, characteristic
7	C₁₉×#1744	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊(C₁₉⋊C₈)	C₄	1	2	yes	—
9	C₂×C₄⋊(C₁₉⋊C₈)	C₄	1	2	yes	—
10	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
14	#11926299	C₄	1	2	yes	—
15	#11926299	C₄	1	2	yes	—
16	#11926266	—	2	2	no	—
17	C₈×C₁₉⋊C₈	—	2	2	no	—
18	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
20	C₄³×C₁₉	C₂²	1	1	yes	characteristic
21	C₁₉×#336	C₄	1	2	yes	—
22	C₁₉×#336	C₄	1	2	yes	—
23	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
32	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
33	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
34	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
35	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₁₉	—	2	1	no	—
37	C₄×C₈×C₁₉	—	2	1	no	—
38	C₄×C₈×C₁₉	D₄	1	2	yes	—
39	C₄×C₈×C₁₉	Q₈	1	2	yes	—
40	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
41	C₄⋊(C₁₉⋊C₈)	C₈	1	8	yes	complemented
42	C₄×C₁₉⋊C₈	—	2	1	no	—
43	C₄×C₁₉⋊C₈	—	2	1	no	—
44	C₄×C₁₉⋊C₈	—	2	1	no	—
45	C₄×C₁₉⋊C₈	—	2	1	no	—
46	C₄×C₁₉⋊C₈	—	2	1	no	—
47	C₄×C₁₉⋊C₈	—	2	1	no	—
48	C₄×C₁₉⋊C₈	—	2	1	no	—
49	C₄×C₁₉⋊C₈	—	2	1	no	—
50	C₂²×C₄×C₁₉	C₄²	1	1	yes	characteristic
51	C₂²×C₄×C₁₉	C₄²	1	1	yes	characteristic
52	C₂²×C₄×C₁₉	C₄²	1	1	yes	characteristic
53	C₂²×C₄×C₁₉	C₄²	1	1	yes	characteristic
54	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₁₉	—	2	2	no	—
58	C₂×C₈×C₁₉	—	2	2	no	—
59	C₂×C₈×C₁₉	—	2	2	no	—
60	C₂×C₈×C₁₉	—	2	2	no	—
61	C₂×C₈×C₁₉	—	2	2	no	—
62	C₂×C₈×C₁₉	C₄⋊C₄	1	4	yes	—
63	C₄²×C₁₉	—	2	2	no	—
64	C₄²×C₁₉	—	2	2	no	—
65	C₄²×C₁₉	—	2	2	no	—
66	C₄²×C₁₉	—	2	2	no	—
67	C₄²×C₁₉	—	2	2	no	—
68	C₄²×C₁₉	C₂×C₈	1	4	yes	—
69	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
71	C₄²×C₁₉	#41	1	1	yes	characteristic
72	C₄²×C₁₉	#41	1	1	yes	characteristic
73	C₂×C₁₉⋊C₈	—	2	2	no	—
74	C₂×C₁₉⋊C₈	—	2	2	no	—
75	C₂×C₁₉⋊C₈	—	2	2	no	—
76	C₂×C₁₉⋊C₈	—	2	2	no	—
77	C₂×C₁₉⋊C₈	—	2	2	no	—
78	C₂×C₁₉⋊C₈	—	2	2	no	—
79	C₂×C₁₉⋊C₈	—	2	2	no	—
80	C₂×C₁₉⋊C₈	—	2	2	no	—
81	C₂×C₁₉⋊C₈	—	2	2	no	—
82	C₂×C₁₉⋊C₈	C₂×C₈	1	4	yes	—
83	C₂×C₁₉⋊C₈	C₂×C₈	1	4	yes	—
84	C₂×C₁₉⋊C₈	M₁₆	1	4	yes	—
85	#8264	—	19	1	no	maximal, Sylow
86	C₂³×C₁₉	C₂×C₄²	1	1	yes	characteristic
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	C₄×C₈	1	2	yes	—
98	C₂×C₄×C₁₉	C₄×C₈	1	2	yes	—
99	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
101	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₉	C₄×Q₈	1	1	yes	characteristic
104	C₂×C₄×C₁₉	C₄×Q₈	1	1	yes	characteristic
105	C₂×C₄×C₁₉	C₂²×C₈	1	2	yes	—
106	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
107	C₂×C₄×C₁₉	#131	1	2	yes	—
108	C₂×C₄×C₁₉	#131	1	2	yes	—
109	C₂×C₄×C₁₉	#131	1	2	yes	—
110	C₂×C₄×C₁₉	#131	1	2	yes	—
111	C₈×C₁₉	—	2	4	no	—
112	C₈×C₁₉	—	4	2	no	—
113	C₁₉⋊C₈	—	2	4	no	—
114	C₁₉⋊C₈	—	2	4	no	—
115	C₁₉⋊C₈	—	2	4	no	—
116	C₁₉⋊C₈	—	4	2	no	—
117	C₄×C₄⋊C₈	—	19	1	no	—
118	C₄×C₄⋊C₈	—	19	1	no	—
119	#1744	D₁₉	1	1	yes	characteristic
120	C₂×#322	—	19	1	no	—
121	C₂×C₈²	—	19	1	no	—
122	#1273	—	19	1	no	—
123	#1273	—	19	1	no	—
124	C₂²×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
125	C₂²×C₁₉	C₂×C₄×C₈	1	1	yes	characteristic
126	C₂²×C₁₉	C₄×M₁₆	1	1	yes	characteristic
127	C₂²×C₁₉	#405	1	1	yes	characteristic
128	C₂²×C₁₉	#405	1	1	yes	characteristic
129	C₂²×C₁₉	#405	1	1	yes	characteristic
130	C₂²×C₁₉	#405	1	1	yes	characteristic
131	C₄×C₁₉	—	2	2	no	—
132	C₄×C₁₉	—	2	2	no	—
133	C₄×C₁₉	—	4	1	no	—
134	C₄×C₁₉	—	4	1	no	—
135	C₄×C₁₉	#424	1	2	yes	—
136	C₄×C₁₉	#433	1	2	yes	—
137	C₄×C₁₉	C₈×D₄	1	2	yes	—
138	C₄×C₁₉	#435	1	2	yes	—
139	C₄×C₁₉	C₈×Q₈	1	2	yes	—
140	C₄×C₁₉	#446	1	2	yes	—
141	C₂×C₄⋊C₈	—	19	2	no	—
142	C₂×C₄⋊C₈	—	19	2	no	—
143	C₂×C₄×C₈	—	19	1	no	—
144	C₂×C₄×C₈	—	19	1	no	—
145	C₂×C₄×C₈	—	19	1	no	—
146	C₂×C₄×C₈	—	19	1	no	—
147	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
148	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
149	C₄³	C₂×D₁₉	1	1	yes	characteristic
150	#336	—	19	2	no	—
151	#336	—	19	2	no	—
152	#336	Dic₁₉	1	2	yes	—
153	#336	Dic₁₉	1	2	yes	—
154	#322	—	38	2	no	—
155	C₈²	—	38	2	no	—
156	C₂×C₁₉	C₈×M₁₆	1	1	yes	characteristic
157	C₂×C₁₉	#1450	1	1	yes	characteristic
158	C₂×C₁₉	#1453	1	1	yes	characteristic
159	C₂×C₁₉	#1454	1	1	yes	characteristic
160	C₂×C₁₉	#1768	1	1	yes	characteristic
161	C₂×C₁₉	C₈×C₄⋊C₄	1	1	yes	characteristic
162	C₂×C₁₉	#1771	1	1	yes	characteristic
163	C₂²×C₈	—	19	2	no	—
164	C₂²×C₈	—	19	2	no	—
165	C₂²×C₈	—	19	2	no	—
166	C₂²×C₈	—	19	2	no	—
167	C₂²×C₈	C₂×Dic₁₉	1	2	yes	—
168	C₂²×C₈	C₂×Dic₁₉	1	2	yes	—
169	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
170	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
171	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
172	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
173	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
174	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
175	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
176	C₄⋊C₈	—	19	8	no	—
177	C₄⋊C₈	—	38	4	no	—
178	C₄×C₈	—	2	1	no	—
179	C₄×C₈	—	2	1	no	—
180	C₄×C₈	—	38	1	no	—
181	C₄×C₈	—	38	1	no	—
182	C₄×C₈	—	38	1	no	—
183	C₄×C₈	—	38	1	no	—
184	C₄×C₈	—	38	1	no	—
185	C₄×C₈	—	38	1	no	—
186	C₄×C₈	—	38	1	no	—
187	C₄×C₈	—	38	1	no	—
188	C₄×C₈	Dic₃₈	1	2	yes	—
189	C₄×C₈	D₇₆	1	2	yes	—
190	C₁₉	#8264	1	1	yes	characteristic, complemented, Sylow
191	C₂²×C₄	C₄×Dic₁₉	1	1	yes	characteristic
192	C₂²×C₄	C₄×Dic₁₉	1	1	yes	characteristic
193	C₂²×C₄	C₄×Dic₁₉	1	1	yes	characteristic
194	C₂²×C₄	C₄×Dic₁₉	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
196	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
197	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	19	4	no	—
204	C₂×C₈	—	19	4	no	—
205	C₂×C₈	—	19	4	no	—
206	C₂×C₈	—	38	2	no	—
207	C₂×C₈	—	38	2	no	—
208	C₂×C₈	—	38	2	no	—
209	C₂×C₈	—	38	2	no	—
210	C₂×C₈	—	38	2	no	—
211	C₂×C₈	—	38	2	no	—
212	C₂×C₈	—	38	2	no	—
213	C₂×C₈	—	38	2	no	—
214	C₂×C₈	—	38	2	no	—
215	C₂×C₈	C₄⋊Dic₁₉	1	4	yes	—
216	C₄²	—	2	2	no	—
217	C₄²	—	2	2	no	—
218	C₄²	—	2	2	no	—
219	C₄²	—	2	2	no	—
220	C₄²	—	2	2	no	—
221	C₄²	C₈×D₁₉	1	4	yes	—
222	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
223	C₄²	C₂×D₇₆	1	1	yes	characteristic
224	C₄²	#64541	1	1	yes	characteristic
225	C₄²	#64541	1	1	yes	characteristic
226	C₂³	C₂×C₄×Dic₁₉	1	1	yes	characteristic
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	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	C₈×Dic₁₉	1	2	yes	—
238	C₂×C₄	C₈×Dic₁₉	1	2	yes	—
239	C₂×C₄	C₄×Dic₃₈	1	1	yes	characteristic
240	C₂×C₄	C₄×Dic₃₈	1	1	yes	characteristic
241	C₂×C₄	C₄×D₇₆	1	1	yes	characteristic
242	C₂×C₄	C₄×D₇₆	1	1	yes	characteristic
243	C₂×C₄	C₂×C₈×D₁₉	1	2	yes	—
244	C₂×C₄	#10597347	1	2	yes	—
245	C₂×C₄	M₁₆×D₁₉	1	2	yes	—
246	C₂×C₄	#10597353	1	2	yes	—
247	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
248	C₂×C₄	#10597372	1	1	yes	characteristic
249	C₂×C₄	#10597394	1	2	yes	—
250	C₂×C₄	#10597394	1	2	yes	—
251	C₈	—	2	4	no	—
252	C₈	—	4	2	no	—
253	C₈	—	38	4	no	—
254	C₈	—	38	4	no	—
255	C₈	—	38	4	no	—
256	C₈	—	76	2	no	—
257	C₂²	C₄×C₄⋊Dic₁₉	1	1	yes	characteristic
258	C₂²	C₂×C₈×Dic₁₉	1	1	yes	characteristic
259	C₂²	M₁₆×Dic₁₉	1	1	yes	characteristic
260	C₂²	#11926499	1	1	yes	characteristic
261	C₂²	#11926500	1	1	yes	characteristic
262	C₂²	#11926500	1	1	yes	characteristic
263	C₂²	#11926502	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	2	2	no	—
266	C₄	—	4	1	no	—
267	C₄	—	4	1	no	—
268	C₄	#11926571	1	2	yes	—
269	C₄	#11926605	1	2	yes	—
270	C₄	C₈×D₇₆	1	2	yes	—
271	C₄	#11926617	1	2	yes	—
272	C₄	C₈×Dic₃₈	1	2	yes	—
273	C₄	#11926672	1	2	yes	—
274	C₂	#14561916	1	1	yes	characteristic
275	C₂	#14561919	1	1	yes	characteristic
276	C₂	#14561930	1	1	yes	characteristic
277	C₂	#14561935	1	1	yes	characteristic
278	C₂	#14563492	1	1	yes	characteristic
279	C₂	C₈×C₄⋊Dic₁₉	1	1	yes	characteristic
280	C₂	#14563507	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₁₉
commutator	C₂²×C₁₉	C₂×C₄×C₈
Fitting	C₁₉×#1744	C₂
socle, prime radical	C₂³×C₁₉	C₂×C₄²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1744	D₁₉
hypocenter	C₁₉	#8264
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄² ◁ #1744

Complements

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

N	H ≅ G/N	type
41. C₄⋊(C₁₉⋊C₈)	C₈	semidirect
190. C₁₉	#8264	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊(C₁₉⋊C₈)	2
C₂×#11926266	2
C₂×C₈×C₁₉⋊C₈	2
C₁₉⋊#1273	2
C₁₉×#1744	2
#8264	19