Subgroups — #18081136

Order 4864.

388 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14563488	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₁₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926294	C₂	1	2	yes	maximal
4	#14561469	C₂	1	2	yes	maximal
5	C₁₉×#1842	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₁₉⋊C₈)	C₄	1	2	yes	—
7	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
9	C₂×#10597260	—	2	1	no	—
10	C₂×C₄×C₁₉⋊C₈	C₂²	1	2	yes	—
11	#11926299	—	2	1	no	—
12	#11926299	C₄	1	1	yes	characteristic
13	#11926299	C₄	1	1	yes	characteristic
14	#11926294	—	2	4	no	—
15	C₂×C₁₉×C₄⋊C₈	—	2	1	no	—
16	C₂×C₁₉×C₄⋊C₈	—	2	1	no	—
17	C₂×C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
18	C₄³×C₁₉	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₁₉	—	2	2	no	—
20	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
24	C₂×C₄²×C₁₉	Q₈	1	1	yes	characteristic
25	C₂×C₄²×C₁₉	Q₈	1	1	yes	characteristic
26	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
27	C₂²×C₁₉⋊C₈	—	2	1	no	—
28	C₂²×C₁₉⋊C₈	—	2	1	no	—
29	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
30	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
31	C₁₉×C₄⋊C₈	—	2	2	no	—
32	C₁₉×C₄⋊C₈	—	2	2	no	—
33	C₁₉×C₄⋊C₈	—	4	2	no	—
34	C₁₉×C₄⋊C₈	—	4	2	no	—
35	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
36	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
37	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
38	C₄⋊(C₁₉⋊C₈)	D₄	1	2	yes	—
39	C₄⋊(C₁₉⋊C₈)	Q₈	1	2	yes	—
40	#10597260	—	4	2	no	—
41	C₄×C₁₉⋊C₈	—	2	2	no	—
42	C₄×C₁₉⋊C₈	—	2	2	no	—
43	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈×C₁₉	—	4	2	no	—
51	C₂×C₈×C₁₉	—	4	2	no	—
52	C₂×C₈×C₁₉	—	4	2	no	—
53	C₄²×C₁₉	—	2	2	no	—
54	C₄²×C₁₉	—	2	2	no	—
55	C₄²×C₁₉	—	2	2	no	—
56	C₄²×C₁₉	—	2	1	no	—
57	C₄²×C₁₉	—	2	1	no	—
58	C₄²×C₁₉	—	4	1	no	—
59	C₄²×C₁₉	SD₁₆	1	2	yes	—
60	C₄²×C₁₉	SD₁₆	1	2	yes	—
61	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
62	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
63	C₄²×C₁₉	#41	1	1	yes	characteristic
64	C₄²×C₁₉	#41	1	1	yes	characteristic
65	C₂×C₁₉⋊C₈	—	2	2	no	—
66	C₂×C₁₉⋊C₈	—	2	2	no	—
67	C₂×C₁₉⋊C₈	—	2	2	no	—
68	C₂×C₁₉⋊C₈	—	2	2	no	—
69	C₂×C₁₉⋊C₈	—	2	2	no	—
70	C₂×C₁₉⋊C₈	—	2	2	no	—
71	C₂×C₁₉⋊C₈	—	2	2	no	—
72	C₂×C₁₉⋊C₈	—	4	1	no	—
73	C₂×C₁₉⋊C₈	—	4	1	no	—
74	C₂×C₁₉⋊C₈	—	4	1	no	—
75	C₂×C₁₉⋊C₈	—	4	1	no	—
76	C₂×C₁₉⋊C₈	M₁₆	1	4	yes	—
77	#9605	—	19	1	no	maximal, Sylow
78	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₉	—	2	1	no	—
80	C₂×C₄×C₁₉	—	2	1	no	—
81	C₂×C₄×C₁₉	—	2	1	no	—
82	C₂×C₄×C₁₉	—	2	1	no	—
83	C₂×C₄×C₁₉	—	2	1	no	—
84	C₂×C₄×C₁₉	—	2	1	no	—
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	#106	1	2	yes	—
91	C₂×C₄×C₁₉	#106	1	2	yes	—
92	C₂×C₄×C₁₉	#108	1	2	yes	—
93	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
94	C₂×C₄×C₁₉	C₄×Q₈	1	1	yes	characteristic
95	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
96	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
97	C₂×C₄×C₁₉	#125	1	1	yes	characteristic
98	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
99	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
100	C₂×C₄×C₁₉	#131	1	2	yes	—
101	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
102	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
103	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
104	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
105	C₈×C₁₉	—	8	2	no	—
106	C₁₉⋊C₈	—	2	4	no	—
107	C₁₉⋊C₈	—	4	2	no	—
108	C₁₉⋊C₈	—	4	2	no	—
109	C₁₉⋊C₈	—	4	2	no	—
110	#1842	D₁₉	1	1	yes	characteristic
111	#1767	—	19	1	no	—
112	C₄×C₄⋊C₈	—	19	1	no	—
113	C₂×#334	—	19	2	no	—
114	#1277	—	19	2	no	—
115	C₂²×C₁₉	#389	1	1	yes	characteristic
116	C₂²×C₁₉	#423	1	1	yes	characteristic
117	C₂²×C₁₉	#424	1	1	yes	characteristic
118	C₂²×C₁₉	C₂×#106	1	1	yes	characteristic
119	C₂²×C₁₉	#428	1	1	yes	characteristic
120	C₂²×C₁₉	C₂×#108	1	1	yes	characteristic
121	C₂²×C₁₉	#430	1	1	yes	characteristic
122	C₄×C₁₉	—	2	2	no	—
123	C₄×C₁₉	—	2	2	no	—
124	C₄×C₁₉	—	2	2	no	—
125	C₄×C₁₉	—	2	1	no	—
126	C₄×C₁₉	—	2	1	no	—
127	C₄×C₁₉	—	4	1	no	—
128	C₄×C₁₉	#435	1	2	yes	—
129	C₄×C₁₉	#446	1	2	yes	—
130	C₄×C₁₉	#460	1	1	yes	characteristic
131	C₄×C₁₉	#461	1	1	yes	characteristic
132	C₄×C₁₉	#475	1	1	yes	characteristic
133	C₄×C₁₉	#476	1	1	yes	characteristic
134	C₂×C₄⋊C₈	—	2	1	no	—
135	C₂×C₄⋊C₈	—	2	1	no	—
136	C₂×C₄⋊C₈	—	19	1	no	—
137	C₂×C₄⋊C₈	—	19	1	no	—
138	C₂×C₄⋊C₈	—	19	2	no	—
139	C₂×C₄⋊C₈	C₂×D₁₉	1	2	yes	—
140	C₂×#97	—	38	1	no	—
141	C₂×C₄×C₈	—	19	2	no	—
142	C₄³	C₂×D₁₉	1	1	yes	characteristic
143	#336	—	19	1	no	—
144	#336	—	19	1	no	—
145	#336	—	38	1	no	—
146	#334	—	38	4	no	—
147	C₂×C₁₉	#1566	1	1	yes	characteristic
148	C₂×C₁₉	#1569	1	1	yes	characteristic
149	C₂×C₁₉	#1570	1	1	yes	characteristic
150	C₂×C₁₉	#1573	1	1	yes	characteristic
151	C₂×C₁₉	#1940	1	1	yes	characteristic
152	C₂×C₁₉	#1948	1	1	yes	characteristic
153	C₂×C₁₉	#1950	1	1	yes	characteristic
154	C₂²×C₈	—	2	2	no	—
155	C₂²×C₈	—	19	2	no	—
156	C₂²×C₈	—	19	2	no	—
157	C₂²×C₈	—	38	1	no	—
158	C₂²×C₈	—	38	1	no	—
159	C₂×C₄²	Dic₃₈	1	1	yes	characteristic
160	C₂×C₄²	Dic₃₈	1	1	yes	characteristic
161	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
162	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
163	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
164	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
165	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
166	C₄⋊C₈	—	2	2	no	—
167	C₄⋊C₈	—	2	2	no	—
168	C₄⋊C₈	—	4	2	no	—
169	C₄⋊C₈	—	4	2	no	—
170	C₄⋊C₈	—	19	2	no	—
171	C₄⋊C₈	—	19	2	no	—
172	C₄⋊C₈	—	38	4	no	—
173	C₄⋊C₈	—	38	1	no	—
174	C₄⋊C₈	—	38	1	no	—
175	#97	—	76	2	no	—
176	C₄×C₈	—	38	2	no	—
177	C₄×C₈	—	38	2	no	—
178	C₁₉	#9605	1	1	yes	characteristic, complemented, Sylow
179	C₂²×C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
180	C₂²×C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
181	C₂²×C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
182	C₂²×C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
183	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
184	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
185	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
186	C₂×C₈	—	4	2	no	—
187	C₂×C₈	—	4	2	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	—	19	4	no	—
190	C₂×C₈	—	38	2	no	—
191	C₂×C₈	—	38	2	no	—
192	C₂×C₈	—	38	2	no	—
193	C₂×C₈	—	38	2	no	—
194	C₂×C₈	—	38	2	no	—
195	C₂×C₈	—	38	2	no	—
196	C₂×C₈	—	38	2	no	—
197	C₂×C₈	—	76	1	no	—
198	C₂×C₈	—	76	1	no	—
199	C₂×C₈	—	76	1	no	—
200	C₂×C₈	—	76	1	no	—
201	C₄²	—	2	1	no	—
202	C₄²	—	2	1	no	—
203	C₄²	—	2	2	no	—
204	C₄²	—	2	2	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	4	1	no	—
207	C₄²	#64526	1	2	yes	—
208	C₄²	#64527	1	2	yes	—
209	C₄²	D₄×D₁₉	1	1	yes	characteristic
210	C₄²	#64543	1	1	yes	characteristic
211	C₄²	Q₈×D₁₉	1	1	yes	characteristic
212	C₄²	#64545	1	1	yes	characteristic
213	C₂³	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	#10597265	1	2	yes	—
226	C₂×C₄	#10597265	1	2	yes	—
227	C₂×C₄	#10597279	1	2	yes	—
228	C₂×C₄	#10597333	1	1	yes	characteristic
229	C₂×C₄	#10597334	1	1	yes	characteristic
230	C₂×C₄	#10597336	1	1	yes	characteristic
231	C₂×C₄	#10597339	1	1	yes	characteristic
232	C₂×C₄	#10597347	1	2	yes	—
233	C₂×C₄	M₁₆×D₁₉	1	2	yes	—
234	C₂×C₄	#10597378	1	1	yes	characteristic
235	C₂×C₄	C₂×#64526	1	1	yes	characteristic
236	C₂×C₄	#10597384	1	1	yes	characteristic
237	C₂×C₄	C₂×#64527	1	1	yes	characteristic
238	C₂×C₄	#10597388	1	1	yes	characteristic
239	C₂×C₄	#10597392	1	1	yes	characteristic
240	C₈	—	8	2	no	—
241	C₈	—	38	4	no	—
242	C₈	—	76	2	no	—
243	C₈	—	76	2	no	—
244	C₈	—	76	2	no	—
245	C₂²	#11926440	1	1	yes	characteristic
246	C₂²	#11926567	1	1	yes	characteristic
247	C₂²	#11926570	1	1	yes	characteristic
248	C₂²	C₂×#10597265	1	1	yes	characteristic
249	C₂²	#11926584	1	1	yes	characteristic
250	C₂²	C₂×#10597279	1	1	yes	characteristic
251	C₂²	#11926592	1	1	yes	characteristic
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	4	1	no	—
258	C₄	#11926614	1	2	yes	—
259	C₄	#11926673	1	2	yes	—
260	C₄	#11926741	1	1	yes	characteristic
261	C₄	#11926746	1	1	yes	characteristic
262	C₄	#11926826	1	1	yes	characteristic
263	C₄	#11926831	1	1	yes	characteristic
264	C₂	#14562390	1	1	yes	characteristic
265	C₂	#14562400	1	1	yes	characteristic
266	C₂	#14562405	1	1	yes	characteristic
267	C₂	#14562418	1	1	yes	characteristic
268	C₂	#14564356	1	1	yes	characteristic
269	C₂	#14564396	1	1	yes	characteristic
270	C₂	#14564408	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₉
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1842	C₂
socle, prime radical	C₂³×C₁₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1842	D₁₉
hypocenter	C₁₉	#9605
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
178. C₁₉	#9605	semidirect

Maximal subgroups

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

subgroup	index
#14563488	2
C₄×C₄⋊(C₁₉⋊C₈)	2
C₂×#11926294	2
#14561469	2
C₁₉×#1842	2
#9605	19