Subgroups — #18082521

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×C₄⋊(C₁₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₁₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#14561476	C₂	1	4	yes	maximal
4	C₄×C₁₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
6	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄×C₁₉⋊C₈	—	2	1	no	—
8	C₂×C₄×C₁₉⋊C₈	C₂²	1	2	yes	—
9	#11926299	—	2	1	no	—
10	#11926299	—	2	1	no	—
11	C₂×C₁₉×C₄⋊Q₈	C₄	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₁₉	—	2	1	no	—
13	C₁₉×#391	—	2	1	no	—
14	C₁₉×#389	—	2	1	no	—
15	C₁₉×#389	C₄	1	2	yes	—
16	C₁₉×#384	C₄	1	1	yes	characteristic
17	C₄×C₄⋊C₄×C₁₉	—	2	1	no	—
18	C₄×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
19	C₄³×C₁₉	C₂²	1	1	yes	characteristic
20	C₂²×Q₈×C₁₉	—	2	1	no	—
21	C₁₉×C₄⋊Q₈	—	2	2	no	—
22	C₁₉×C₄⋊Q₈	C₈	1	4	yes	complemented
23	C₄×Q₈×C₁₉	—	4	4	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	2	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	—	2	2	no	—
31	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₉	C₂×C₄	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	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
38	C₂²×C₁₉⋊C₈	—	2	2	no	—
39	C₂²×C₁₉⋊C₈	—	2	2	no	—
40	C₁₉×#95	—	2	2	no	—
41	C₁₉×#95	—	2	2	no	—
42	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
43	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
44	C₄⋊(C₁₉⋊C₈)	Q₈	1	4	yes	complemented
45	C₄×C₁₉⋊C₈	—	2	4	no	—
46	C₄×C₁₉⋊C₈	—	4	2	no	—
47	C₂×Q₈×C₁₉	—	2	4	no	—
48	C₂×Q₈×C₁₉	—	4	2	no	—
49	C₂×Q₈×C₁₉	—	4	2	no	—
50	C₂²×C₄×C₁₉	—	2	2	no	—
51	C₂²×C₄×C₁₉	—	2	2	no	—
52	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₉	C₂²⋊C₄	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	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
59	C₄⋊C₄×C₁₉	—	2	4	no	—
60	C₄⋊C₄×C₁₉	—	2	2	no	—
61	C₄⋊C₄×C₁₉	—	2	4	no	—
62	C₄⋊C₄×C₁₉	—	4	2	no	—
63	C₄⋊C₄×C₁₉	—	4	4	no	—
64	C₄⋊C₄×C₁₉	—	4	2	no	—
65	C₄⋊C₄×C₁₉	C₂×C₈	1	4	yes	—
66	C₄²×C₁₉	—	2	2	no	—
67	C₄²×C₁₉	—	2	2	no	—
68	C₄²×C₁₉	—	2	2	no	—
69	C₄²×C₁₉	—	4	1	no	—
70	C₄²×C₁₉	—	4	4	no	—
71	C₄²×C₁₉	C₂×C₈	1	2	yes	—
72	C₄²×C₁₉	M₁₆	1	2	yes	—
73	C₄²×C₁₉	SD₁₆	1	2	yes	—
74	C₄²×C₁₉	Q₁₆	1	2	yes	—
75	C₄²×C₁₉	C₂×Q₈	1	2	yes	—
76	C₄²×C₁₉	#41	1	2	yes	—
77	C₂×C₁₉⋊C₈	—	2	4	no	—
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	#9864	—	19	1	no	maximal, Sylow
84	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	Q₈×C₁₉	—	4	8	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	4	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	—	4	2	no	—
97	C₂×C₄×C₁₉	—	4	2	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	C₂²⋊C₈	1	2	yes	—
102	C₂×C₄×C₁₉	C₂²⋊C₈	1	2	yes	—
103	C₂×C₄×C₁₉	#103	1	2	yes	—
104	C₂×C₄×C₁₉	#103	1	2	yes	—
105	C₂×C₄×C₁₉	#104	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	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
112	C₂×C₄×C₁₉	#131	1	2	yes	—
113	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
114	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
115	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
116	C₁₉⋊C₈	—	4	4	no	—
117	C₁₉⋊C₈	—	4	4	no	—
118	C₄×C₄⋊Q₈	D₁₉	1	1	yes	characteristic
119	C₄×C₄⋊C₈	—	19	1	no	—
120	C₄×C₄⋊C₈	—	19	1	no	—
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	4	no	—
134	C₄×C₁₉	—	4	1	no	—
135	C₄×C₁₉	—	8	2	no	—
136	C₄×C₁₉	Q₈⋊C₈	1	4	yes	—
137	C₄×C₁₉	C₈×Q₈	1	2	yes	—
138	C₄×C₁₉	#446	1	2	yes	—
139	C₄×C₁₉	#476	1	2	yes	—
140	C₄×C₁₉	#477	1	2	yes	—
141	C₂×C₄⋊Q₈	Dic₁₉	1	1	yes	characteristic
142	C₂×C₄×Q₈	—	2	1	no	—
143	C₂×C₄⋊C₈	—	19	2	no	—
144	C₂×C₄⋊C₈	—	38	1	no	—
145	C₂×C₄×C₈	—	19	2	no	—
146	C₂×C₄×C₈	—	38	1	no	—
147	#391	—	2	1	no	—
148	#389	—	2	1	no	—
149	#389	Dic₁₉	1	2	yes	—
150	#384	Dic₁₉	1	1	yes	characteristic
151	C₄×C₄⋊C₄	—	2	1	no	—
152	C₄×C₄⋊C₄	C₂×D₁₉	1	2	yes	—
153	C₄³	C₂×D₁₉	1	1	yes	characteristic
154	#336	—	38	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	—	2	1	no	—
164	C₂²×C₈	—	38	2	no	—
165	C₂²×C₈	—	38	2	no	—
166	C₄⋊Q₈	—	2	2	no	—
167	C₄⋊Q₈	C₁₉⋊C₈	1	4	yes	complemented
168	C₄×Q₈	—	4	4	no	—
169	C₂×C₄⋊C₄	—	2	1	no	—
170	C₂×C₄⋊C₄	—	2	1	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	—	2	1	no	—
173	C₂×C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
174	C₂×C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
175	C₂×C₄²	—	2	2	no	—
176	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
177	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
178	C₂×C₄²	C₂²⋊D₁₉	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	C₄⋊C₈	—	19	4	no	—
184	C₄⋊C₈	—	38	4	no	—
185	C₄⋊C₈	—	38	2	no	—
186	C₄×C₈	—	38	4	no	—
187	C₄×C₈	—	76	2	no	—
188	#95	—	2	2	no	—
189	#95	—	2	2	no	—
190	C₁₉	#9864	1	1	yes	characteristic, complemented, Sylow
191	C₂×Q₈	—	2	4	no	—
192	C₂×Q₈	—	4	2	no	—
193	C₂×Q₈	—	4	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	2	no	—
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	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	4	no	—
204	C₂×C₈	—	38	4	no	—
205	C₂×C₈	—	76	2	no	—
206	C₂×C₈	—	76	2	no	—
207	C₂×C₈	—	76	2	no	—
208	C₂×C₈	—	76	2	no	—
209	C₄⋊C₄	—	2	4	no	—
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	4	2	no	—
213	C₄⋊C₄	—	4	4	no	—
214	C₄⋊C₄	—	4	2	no	—
215	C₄⋊C₄	C₂×C₁₉⋊C₈	1	4	yes	—
216	C₄²	—	2	2	no	—
217	C₄²	—	2	2	no	—
218	C₄²	—	2	2	no	—
219	C₄²	—	4	1	no	—
220	C₄²	—	4	4	no	—
221	C₄²	C₂×C₁₉⋊C₈	1	2	yes	—
222	C₄²	#64520	1	2	yes	—
223	C₄²	#64527	1	2	yes	—
224	C₄²	#64528	1	2	yes	—
225	C₄²	Q₈×D₁₉	1	2	yes	—
226	C₄²	#64545	1	2	yes	—
227	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
228	Q₈	—	4	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	4	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	C₂²⋊(C₁₉⋊C₈)	1	2	yes	—
245	C₂×C₄	C₂²⋊(C₁₉⋊C₈)	1	2	yes	—
246	C₂×C₄	#10597292	1	2	yes	—
247	C₂×C₄	#10597292	1	2	yes	—
248	C₂×C₄	#10597294	1	2	yes	—
249	C₂×C₄	C₂²×C₁₉⋊C₈	1	1	yes	characteristic
250	C₂×C₄	C₂×#64520	1	1	yes	characteristic
251	C₂×C₄	#10597378	1	2	yes	—
252	C₂×C₄	C₂×#64527	1	1	yes	characteristic
253	C₂×C₄	C₂×#64528	1	1	yes	characteristic
254	C₂×C₄	#10597390	1	1	yes	characteristic
255	C₂×C₄	Q₈×Dic₁₉	1	2	yes	—
256	C₂×C₄	#10597392	1	2	yes	—
257	C₂×C₄	#10597393	1	1	yes	characteristic
258	C₂×C₄	#10597394	1	2	yes	—
259	C₈	—	76	4	no	—
260	C₈	—	76	4	no	—
261	C₂²	#11926451	1	1	yes	characteristic
262	C₂²	C₂×C₂²⋊(C₁₉⋊C₈)	1	1	yes	characteristic
263	C₂²	#11926511	1	1	yes	characteristic
264	C₂²	C₂×#10597292	1	1	yes	characteristic
265	C₂²	#11926549	1	1	yes	characteristic
266	C₂²	C₂×#10597294	1	1	yes	characteristic
267	C₂²	#11926561	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	—	4	4	no	—
274	C₄	—	8	2	no	—
275	C₄	Q₈⋊(C₁₉⋊C₈)	1	4	yes	—
276	C₄	Q₈×C₁₉⋊C₈	1	2	yes	—
277	C₄	#11926675	1	2	yes	—
278	C₄	#11926831	1	2	yes	—
279	C₄	#11926836	1	2	yes	—
280	C₂	C₂×Q₈⋊(C₁₉⋊C₈)	1	1	yes	characteristic
281	C₂	#14562026	1	1	yes	characteristic
282	C₂	#14562038	1	1	yes	characteristic
283	C₂	#14562052	1	1	yes	characteristic
284	C₂	#14564572	1	1	yes	characteristic
285	C₂	#14564580	1	1	yes	characteristic
286	C₂	#14564584	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₉
commutator	C₂×C₄×C₁₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₄×C₁₉×C₄⋊Q₈	C₂
socle, prime radical	C₂³×C₁₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₄×C₄⋊Q₈	D₁₉
hypocenter	C₁₉	#9864
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
22. C₁₉×C₄⋊Q₈	C₈	semidirect
44. C₄⋊(C₁₉⋊C₈)	Q₈	semidirect
167. C₄⋊Q₈	C₁₉⋊C₈	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×C₄⋊(C₁₉⋊C₈)	2
#14561476	2
C₄×C₁₉×C₄⋊Q₈	2
#9864	19