Subgroups — #18073196

Order 4864.

606 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, complemented
2	C₄×#10597260	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₄⋊(C₁₉⋊C₈)	C₂	1	2	yes	maximal
4	#14561476	C₂	1	2	yes	maximal, complemented
5	C₁₉×#2293	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
7	C₂×#10597260	—	2	2	no	—
8	C₂×C₄×C₁₉⋊C₈	—	2	1	no	—
9	C₂×C₄×C₁₉⋊C₈	C₂²	1	2	yes	—
10	#11926299	—	2	1	no	—
11	D₄⋊(C₁₉⋊C₈)	C₄	1	8	yes	complemented
12	C₂×C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
13	C₂×C₁₉×#117	—	2	1	no	—
14	C₁₉×#390	C₄	1	2	yes	—
15	C₁₉×#389	C₄	1	2	yes	—
16	C₁₉×#388	—	2	2	no	—
17	C₄×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
18	C₄×C₂²⋊C₄×C₁₉	—	2	1	no	—
19	C₄³×C₁₉	C₂²	1	1	yes	characteristic
20	C₂²×D₄×C₁₉	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₉	—	2	1	no	—
22	C₄×D₄×C₁₉	—	2	2	no	—
23	C₄×D₄×C₁₉	C₂×C₄	1	4	yes	—
24	C₁₉×#117	—	4	4	no	—
25	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
26	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
27	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
29	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
30	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
31	C₂×C₄²×C₁₉	—	2	1	no	—
32	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₉	D₄	1	2	yes	—
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	1	no	—
41	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
42	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	4	yes	—
43	#10597260	—	2	8	no	—
44	C₄×C₁₉⋊C₈	—	2	2	no	—
45	C₄×C₁₉⋊C₈	—	4	2	no	—
46	C₄×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
47	C₂⁴×C₁₉	—	2	1	no	—
48	C₂×D₄×C₁₉	—	2	1	no	—
49	C₂×D₄×C₁₉	—	2	4	no	—
50	C₂×D₄×C₁₉	C₄²	1	2	yes	—
51	C₂²×C₄×C₁₉	—	2	1	no	—
52	C₂²×C₄×C₁₉	—	2	4	no	—
53	C₂²×C₄×C₁₉	—	2	2	no	—
54	C₂²×C₄×C₁₉	—	4	2	no	—
55	C₂²×C₄×C₁₉	—	4	1	no	—
56	C₂²×C₄×C₁₉	—	4	1	no	—
57	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
62	C₄⋊C₄×C₁₉	—	2	1	no	—
63	C₄⋊C₄×C₁₉	—	2	4	no	—
64	C₄⋊C₄×C₁₉	—	4	4	no	—
65	C₄⋊C₄×C₁₉	C₄²	1	2	yes	—
66	C₂²⋊C₄×C₁₉	—	2	4	no	—
67	C₂²⋊C₄×C₁₉	—	4	4	no	—
68	C₂²⋊C₄×C₁₉	—	4	4	no	—
69	C₄²×C₁₉	—	2	4	no	—
70	C₄²×C₁₉	—	2	4	no	—
71	C₄²×C₁₉	—	2	2	no	—
72	C₄²×C₁₉	—	2	1	no	—
73	C₄²×C₁₉	—	2	1	no	—
74	C₄²×C₁₉	—	4	2	no	—
75	C₄²×C₁₉	C₂²×C₄	1	2	yes	—
76	C₄²×C₁₉	#41	1	2	yes	—
77	C₂×C₁₉⋊C₈	—	2	8	no	—
78	C₂×C₁₉⋊C₈	—	2	4	no	—
79	C₂×C₁₉⋊C₈	—	2	4	no	—
80	C₂×C₁₉⋊C₈	—	4	2	no	—
81	C₂×C₁₉⋊C₈	—	4	2	no	—
82	#8181	—	19	1	no	maximal, Sylow
83	C₂³×C₁₉	—	2	2	no	—
84	C₂³×C₁₉	—	4	2	no	—
85	C₂³×C₁₉	—	4	1	no	—
86	C₂³×C₁₉	—	4	1	no	—
87	C₂³×C₁₉	—	4	2	no	—
88	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	D₄×C₁₉	—	2	4	no	—
90	D₄×C₁₉	—	4	2	no	—
91	C₂×C₄×C₁₉	—	2	2	no	—
92	C₂×C₄×C₁₉	—	2	2	no	—
93	C₂×C₄×C₁₉	—	2	2	no	—
94	C₂×C₄×C₁₉	—	2	2	no	—
95	C₂×C₄×C₁₉	—	2	2	no	—
96	C₂×C₄×C₁₉	—	2	2	no	—
97	C₂×C₄×C₁₉	—	2	2	no	—
98	C₂×C₄×C₁₉	—	4	1	no	—
99	C₂×C₄×C₁₉	—	4	1	no	—
100	C₂×C₄×C₁₉	—	4	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	—	4	2	no	—
105	C₂×C₄×C₁₉	#104	1	4	yes	—
106	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
107	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
108	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₁₉	#131	1	4	yes	—
111	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
112	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
113	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
114	C₁₉⋊C₈	—	2	8	no	—
115	C₁₉⋊C₈	—	4	4	no	—
116	#2293	D₁₉	1	1	yes	characteristic
117	C₄×C₄⋊C₈	—	19	1	no	—
118	C₄×#97	—	19	1	no	—
119	C₂×D₄⋊C₈	—	19	2	no	—
120	#1279	—	19	2	no	—
121	C₂²×C₁₉	—	4	2	no	—
122	C₂²×C₁₉	—	4	2	no	—
123	C₂²×C₁₉	—	4	2	no	—
124	C₂²×C₁₉	—	8	2	no	—
125	C₂²×C₁₉	—	8	2	no	—
126	C₂²×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
127	C₂²×C₁₉	#408	1	2	yes	—
128	C₂²×C₁₉	#417	1	1	yes	characteristic
129	C₂²×C₁₉	#418	1	1	yes	characteristic
130	C₂²×C₁₉	C₂×#104	1	2	yes	—
131	C₄×C₁₉	—	2	4	no	—
132	C₄×C₁₉	—	2	2	no	—
133	C₄×C₁₉	—	4	2	no	—
134	C₄×C₁₉	—	4	2	no	—
135	C₄×C₁₉	—	8	2	no	—
136	C₄×C₁₉	#405	1	2	yes	—
137	C₄×C₁₉	#433	1	2	yes	—
138	C₄×C₁₉	#440	1	2	yes	—
139	C₄×C₁₉	#442	1	2	yes	—
140	C₂×C₄×D₄	C₂×D₁₉	1	1	yes	characteristic
141	C₂×#117	—	2	1	no	—
142	C₂×C₄⋊C₈	—	19	2	no	—
143	C₂×#97	—	38	2	no	—
144	C₂×C₄×C₈	—	19	2	no	—
145	C₂×C₄×C₈	—	38	1	no	—
146	#390	Dic₁₉	1	2	yes	—
147	#389	Dic₁₉	1	2	yes	—
148	#388	—	2	2	no	—
149	C₄×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
150	C₄×C₂²⋊C₄	—	2	1	no	—
151	C₄³	C₂×D₁₉	1	1	yes	characteristic
152	#336	—	38	1	no	—
153	D₄⋊C₈	—	19	8	no	—
154	C₂×C₁₉	—	8	2	no	—
155	C₂×C₁₉	#1483	1	2	yes	—
156	C₂×C₁₉	#1484	1	2	yes	—
157	C₂×C₁₉	#1750	1	1	yes	characteristic
158	C₂×C₁₉	C₄×#104	1	1	yes	characteristic
159	C₂×C₁₉	#1762	1	1	yes	characteristic
160	C₂²×D₄	C₂×Dic₁₉	1	1	yes	characteristic
161	C₂³×C₄	—	2	1	no	—
162	C₂²×C₈	—	38	2	no	—
163	C₂²×C₈	—	38	2	no	—
164	C₄×D₄	—	2	2	no	—
165	C₄×D₄	C₄×D₁₉	1	4	yes	—
166	#117	—	4	4	no	—
167	C₂×C₄⋊C₄	—	2	2	no	—
168	C₂×C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
169	C₂×C₄⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
170	C₂×C₂²⋊C₄	—	2	1	no	—
171	C₂×C₂²⋊C₄	—	2	2	no	—
172	C₂×C₂²⋊C₄	—	2	2	no	—
173	C₂×C₄²	—	2	1	no	—
174	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
175	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
176	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
177	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
178	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
179	C₄⋊C₈	—	19	4	no	—
180	C₄⋊C₈	—	38	2	no	—
181	#97	—	38	8	no	—
182	C₄×C₈	—	19	4	no	—
183	C₄×C₈	—	38	2	no	—
184	C₄×C₈	—	76	2	no	—
185	#95	—	2	2	no	—
186	#95	—	2	1	no	—
187	C₁₉	#8181	1	1	yes	characteristic, complemented, Sylow
188	C₂⁴	—	2	1	no	—
189	C₂×D₄	—	2	4	no	—
190	C₂×D₄	—	2	1	no	—
191	C₂×D₄	C₄×Dic₁₉	1	2	yes	—
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	—	4	1	no	—
197	C₂²×C₄	—	4	1	no	—
198	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
199	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
200	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
202	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
203	C₂×C₈	—	38	8	no	—
204	C₂×C₈	—	38	4	no	—
205	C₂×C₈	—	38	4	no	—
206	C₂×C₈	—	76	2	no	—
207	C₂×C₈	—	76	2	no	—
208	C₄⋊C₄	—	2	1	no	—
209	C₄⋊C₄	—	2	4	no	—
210	C₄⋊C₄	—	4	4	no	—
211	C₄⋊C₄	C₄×Dic₁₉	1	2	yes	—
212	C₂²⋊C₄	—	2	4	no	—
213	C₂²⋊C₄	—	4	4	no	—
214	C₂²⋊C₄	—	4	4	no	—
215	C₄²	—	2	2	no	—
216	C₄²	—	2	4	no	—
217	C₄²	—	2	1	no	—
218	C₄²	—	2	1	no	—
219	C₄²	—	2	4	no	—
220	C₄²	—	4	2	no	—
221	C₄²	C₂×C₄×D₁₉	1	2	yes	—
222	C₄²	#64541	1	2	yes	—
223	C₂³	—	2	2	no	—
224	C₂³	—	4	1	no	—
225	C₂³	—	4	1	no	—
226	C₂³	—	4	2	no	—
227	C₂³	—	4	2	no	—
228	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
229	D₄	—	2	4	no	—
230	D₄	—	4	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	4	1	no	—
239	C₂×C₄	—	4	1	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	#10597294	1	4	yes	—
246	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
247	C₂×C₄	#10597372	1	1	yes	characteristic
248	C₂×C₄	C₄×C₂²⋊D₁₉	1	2	yes	—
249	C₂×C₄	C₄×C₂²⋊D₁₉	1	2	yes	—
250	C₂×C₄	#10597378	1	2	yes	—
251	C₂×C₄	#10597394	1	4	yes	—
252	C₂×C₄	#10597395	1	1	yes	characteristic
253	C₂×C₄	#10597397	1	1	yes	characteristic
254	C₈	—	38	8	no	—
255	C₈	—	76	4	no	—
256	C₂²	—	4	2	no	—
257	C₂²	—	4	2	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	8	2	no	—
260	C₂²	—	8	2	no	—
261	C₂²	C₄×C₂²⋊Dic₁₉	1	1	yes	characteristic
262	C₂²	#11926511	1	2	yes	—
263	C₂²	#11926545	1	1	yes	characteristic
264	C₂²	#11926549	1	1	yes	characteristic
265	C₂²	C₂×#10597294	1	2	yes	—
266	C₄	—	2	4	no	—
267	C₄	—	2	2	no	—
268	C₄	—	4	2	no	—
269	C₄	—	4	2	no	—
270	C₄	—	8	2	no	—
271	C₄	#11926500	1	2	yes	—
272	C₄	#11926605	1	2	yes	—
273	C₄	#11926644	1	2	yes	—
274	C₄	#11926654	1	2	yes	—
275	C₂	—	8	2	no	—
276	C₂	#14562048	1	2	yes	—
277	C₂	#14562052	1	2	yes	—
278	C₂	#14563401	1	1	yes	characteristic
279	C₂	C₄×#10597294	1	1	yes	characteristic
280	C₂	#14563460	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₉
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#2293	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂²×D₄×C₁₉	C₂×C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#2293	D₁₉
hypocenter	C₁₉	#8181
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #2293

Complements

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

N	H ≅ G/N	type
1. C₄×C₄⋊(C₁₉⋊C₈)	C₂	semidirect
2. C₄×#10597260	C₂	semidirect
4. #14561476	C₂	semidirect
11. D₄⋊(C₁₉⋊C₈)	C₄	semidirect
187. C₁₉	#8181	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
C₄×#10597260	2
C₂×D₄⋊(C₁₉⋊C₈)	2
#14561476	2
C₁₉×#2293	2
#8181	19