Subgroups — #18075680

Order 4864.

450 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926623	C₂	1	1	yes	maximal, characteristic
2	#14562383	C₂	1	1	yes	maximal, characteristic, complemented
3	#14562059	C₂	1	2	yes	maximal
4	#14561530	C₂	1	2	yes	maximal, complemented
5	C₁₉×#1848	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#64515	—	2	1	no	—
7	C₂×C₄×D₇₆	C₂²	1	1	yes	characteristic
8	#11926623	—	2	1	no	—
9	#11926623	—	2	1	no	—
10	#11926623	C₄	1	2	yes	—
11	#11926623	C₄	1	2	yes	—
12	#11926597	C₂²	1	2	yes	—
13	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×(C₂×D₁₉)⋊C₈	—	2	1	no	—
15	#11926297	—	2	1	no	—
16	#11926295	—	2	1	no	—
17	#11926271	—	2	4	no	—
18	C₂×C₁₉×#108	—	2	1	no	—
19	C₂×C₁₉×#108	—	2	1	no	—
20	C₁₉×#423	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
22	C₂²×D₇₆	Q₈	1	1	yes	characteristic
23	C₂²×C₄×D₁₉	—	2	1	no	—
24	C₂×M₁₆×C₁₉	—	2	2	no	—
25	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
28	C₂×(C₂×D₁₉)⋊C₄	—	2	1	no	—
29	C₂×C₄⋊Dic₁₉	Q₈	1	1	yes	characteristic
30	C₂²×C₁₉⋊C₈	—	2	1	no	—
31	C₂×#64515	—	2	2	no	—
32	C₂×#64515	—	2	2	no	—
33	C₂×#64515	—	4	2	no	—
34	C₂×#64515	—	4	2	no	—
35	C₄×D₇₆	—	2	1	no	—
36	C₄×D₇₆	—	2	1	no	—
37	C₄×D₇₆	C₂×C₄	1	2	yes	—
38	C₄×D₇₆	D₄	1	2	yes	—
39	C₁₉×#108	—	4	2	no	—
40	C₁₉×#108	—	4	2	no	—
41	C₁₉×C₄⋊C₈	—	2	2	no	—
42	C₁₉×C₄⋊C₈	—	2	2	no	—
43	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
46	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
47	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
48	(C₂×D₁₉)⋊C₈	—	2	2	no	—
49	(C₂×D₁₉)⋊C₈	—	2	2	no	—
50	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
51	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
52	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	2	yes	—
53	C₄×C₁₉⋊C₈	—	2	2	no	—
54	C₂³×D₁₉	—	2	1	no	—
55	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
58	C₂²×Dic₁₉	—	2	1	no	—
59	C₂×D₇₆	—	2	1	no	—
60	C₂×D₇₆	—	4	2	no	—
61	C₂×D₇₆	C₄⋊C₄	1	2	yes	—
62	C₂×C₄×D₁₉	—	2	2	no	—
63	C₂×C₄×D₁₉	—	2	2	no	—
64	C₂×C₄×D₁₉	—	4	1	no	—
65	C₂×C₄×D₁₉	—	4	1	no	—
66	C₂×C₄×D₁₉	—	4	1	no	—
67	C₂×C₄×D₁₉	—	4	1	no	—
68	M₁₆×C₁₉	—	4	2	no	—
69	M₁₆×C₁₉	—	4	2	no	—
70	C₂×C₈×C₁₉	—	2	1	no	—
71	C₂×C₈×C₁₉	—	2	1	no	—
72	C₂×C₈×C₁₉	—	4	2	no	—
73	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₁₉	D₈	1	2	yes	—
78	C₂×C₈×C₁₉	SD₁₆	1	2	yes	—
79	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₁₉	#41	1	1	yes	characteristic
83	(C₂×D₁₉)⋊C₄	—	2	2	no	—
84	(C₂×D₁₉)⋊C₄	—	2	2	no	—
85	C₄⋊Dic₁₉	—	2	1	no	—
86	C₄⋊Dic₁₉	C₄⋊C₄	1	2	yes	—
87	C₂×C₁₉⋊C₈	—	2	2	no	—
88	C₂×C₁₉⋊C₈	—	4	1	no	—
89	C₂×C₁₉⋊C₈	—	4	1	no	—
90	C₂×C₁₉⋊C₈	—	4	2	no	—
91	#64515	—	4	4	no	—
92	#64515	—	4	4	no	—
93	#8662	—	19	1	no	maximal, Sylow
94	C₂³×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
95	C₂²×D₁₉	—	2	2	no	—
96	C₂²×D₁₉	—	4	2	no	—
97	C₂²×D₁₉	—	4	2	no	—
98	C₂²×D₁₉	—	4	1	no	—
99	C₂²×D₁₉	—	4	1	no	—
100	C₂×C₄×C₁₉	—	2	2	no	—
101	C₂×C₄×C₁₉	—	4	1	no	—
102	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
105	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
106	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
107	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
108	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
109	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
110	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
111	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
112	C₂×Dic₁₉	—	2	2	no	—
113	C₂×Dic₁₉	—	4	1	no	—
114	C₂×Dic₁₉	—	4	1	no	—
115	D₇₆	—	2	4	no	—
116	D₇₆	—	4	2	no	—
117	C₄×D₁₉	—	4	2	no	—
118	C₄×D₁₉	—	4	2	no	—
119	C₄×D₁₉	—	4	2	no	—
120	C₄×D₁₉	—	4	2	no	—
121	C₈×C₁₉	—	2	1	no	—
122	C₈×C₁₉	—	2	1	no	—
123	C₈×C₁₉	—	8	2	no	—
124	C₈×C₁₉	#102	1	2	yes	—
125	C₈×C₁₉	#102	1	2	yes	—
126	C₁₉⋊C₈	—	4	2	no	—
127	C₁₉⋊C₈	—	8	2	no	—
128	C₂×#436	—	19	1	no	—
129	#1848	D₁₉	1	1	yes	characteristic, complemented
130	#1564	—	19	1	no	—
131	#1486	—	19	2	no	—
132	#1301	—	19	2	no	—
133	C₂²×C₁₉	—	2	1	no	—
134	C₂²×C₁₉	—	2	1	no	—
135	C₂²×C₁₉	#380	1	1	yes	characteristic
136	C₂²×C₁₉	#474	1	1	yes	characteristic
137	C₂²×C₁₉	#476	1	1	yes	characteristic
138	C₂×D₁₉	—	4	2	no	—
139	C₂×D₁₉	—	4	2	no	—
140	C₂×D₁₉	—	4	2	no	—
141	C₂×D₁₉	—	8	2	no	—
142	C₂×D₁₉	—	8	2	no	—
143	C₄×C₁₉	—	2	1	no	—
144	C₄×C₁₉	—	2	1	no	—
145	C₄×C₁₉	—	2	1	no	—
146	C₄×C₁₉	C₂×#102	1	1	yes	characteristic
147	C₄×C₁₉	#417	1	1	yes	characteristic
148	C₄×C₁₉	#430	1	2	yes	—
149	C₄×C₁₉	#459	1	1	yes	characteristic
150	C₄×C₁₉	#461	1	1	yes	characteristic
151	Dic₁₉	—	4	2	no	—
152	C₂²×M₁₆	—	38	1	no	—
153	C₂×C₄×D₄	—	19	1	no	—
154	#436	—	19	2	no	—
155	#436	—	19	2	no	—
156	#436	—	38	1	no	—
157	#436	—	38	1	no	—
158	#431	—	19	2	no	—
159	C₂×#108	—	2	1	no	—
160	C₂×#108	—	2	1	no	—
161	#423	C₂×D₁₉	1	2	yes	—
162	C₂×C₄⋊C₈	—	19	1	no	—
163	C₂×C₂²⋊C₈	—	38	1	no	—
164	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
165	#335	—	38	1	no	—
166	#334	—	38	1	no	—
167	D₄⋊C₈	—	38	4	no	—
168	C₂×C₁₉	—	2	1	no	—
169	C₂×C₁₉	—	2	1	no	—
170	C₂×C₁₉	#1802	1	1	yes	characteristic
171	C₂×C₁₉	#1811	1	1	yes	characteristic
172	C₂×C₁₉	#1814	1	1	yes	characteristic
173	D₁₉	—	8	2	no	—
174	C₂²×D₄	—	19	1	no	—
175	C₂³×C₄	—	38	1	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	38	2	no	—
178	C₂×M₁₆	—	38	2	no	—
179	C₂×M₁₆	—	76	2	no	—
180	C₂×M₁₆	—	76	2	no	—
181	C₂²×C₈	—	38	1	no	—
182	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
183	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
184	C₄×D₄	—	19	2	no	—
185	C₄×D₄	—	19	2	no	—
186	C₄×D₄	—	38	1	no	—
187	C₄×D₄	—	38	1	no	—
188	C₂×C₄⋊C₄	—	19	1	no	—
189	C₂×C₂²⋊C₄	—	38	1	no	—
190	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
191	#108	—	4	2	no	—
192	#108	—	4	2	no	—
193	C₄⋊C₈	—	2	2	no	—
194	C₄⋊C₈	—	2	2	no	—
195	C₄⋊C₈	—	19	2	no	—
196	C₄⋊C₈	—	38	1	no	—
197	C₄⋊C₈	—	38	2	no	—
198	C₂²⋊C₈	—	38	2	no	—
199	C₂²⋊C₈	—	38	2	no	—
200	C₂²⋊C₈	—	38	2	no	—
201	C₄×C₈	—	38	2	no	—
202	C₄×C₈	C₄×D₁₉	1	1	yes	characteristic
203	C₄×C₈	C₄×D₁₉	1	1	yes	characteristic
204	C₄×C₈	D₇₆	1	1	yes	characteristic
205	C₄×C₈	D₇₆	1	1	yes	characteristic
206	C₁₉	#8662	1	1	yes	characteristic, complemented, Sylow
207	C₂⁴	—	38	1	no	—
208	C₂×D₄	—	19	2	no	—
209	C₂×D₄	—	38	1	no	—
210	C₂×D₄	—	76	2	no	—
211	C₂²×C₄	—	38	1	no	—
212	C₂²×C₄	—	38	2	no	—
213	C₂²×C₄	—	38	2	no	—
214	C₂²×C₄	—	76	1	no	—
215	C₂²×C₄	—	76	1	no	—
216	C₂²×C₄	—	76	1	no	—
217	C₂²×C₄	—	76	1	no	—
218	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
219	C₂²×C₄	#64545	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
221	M₁₆	—	4	2	no	—
222	M₁₆	—	4	2	no	—
223	M₁₆	—	76	4	no	—
224	M₁₆	—	76	4	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	4	2	no	—
228	C₂×C₈	—	38	2	no	—
229	C₂×C₈	—	76	1	no	—
230	C₂×C₈	—	76	1	no	—
231	C₂×C₈	—	76	2	no	—
232	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
233	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
234	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
235	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
236	C₂×C₈	#64525	1	2	yes	—
237	C₂×C₈	#64527	1	2	yes	—
238	C₄⋊C₄	—	19	2	no	—
239	C₄⋊C₄	—	38	1	no	—
240	C₂²⋊C₄	—	38	2	no	—
241	C₂²⋊C₄	—	38	2	no	—
242	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
243	C₄²	C₂×D₇₆	1	1	yes	characteristic
244	C₄²	D₄×D₁₉	1	1	yes	characteristic
245	C₄²	#64543	1	1	yes	characteristic
246	C₂³	—	38	2	no	—
247	C₂³	—	76	2	no	—
248	C₂³	—	76	2	no	—
249	C₂³	—	76	1	no	—
250	C₂³	—	76	1	no	—
251	C₂³	#10597392	1	1	yes	characteristic
252	D₄	—	38	4	no	—
253	D₄	—	76	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	38	2	no	—
257	C₂×C₄	—	76	1	no	—
258	C₂×C₄	—	76	1	no	—
259	C₂×C₄	—	76	2	no	—
260	C₂×C₄	—	76	2	no	—
261	C₂×C₄	—	76	2	no	—
262	C₂×C₄	—	76	2	no	—
263	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
264	C₂×C₄	#10597338	1	1	yes	characteristic
265	C₂×C₄	#10597341	1	1	yes	characteristic
266	C₂×C₄	#10597343	1	1	yes	characteristic
267	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
268	C₂×C₄	C₂×#64525	1	1	yes	characteristic
269	C₂×C₄	#10597378	1	1	yes	characteristic
270	C₂×C₄	#10597384	1	1	yes	characteristic
271	C₂×C₄	C₂×#64527	1	1	yes	characteristic
272	C₂×C₄	#10597388	1	1	yes	characteristic
273	C₈	—	2	1	no	—
274	C₈	—	2	1	no	—
275	C₈	—	8	2	no	—
276	C₈	—	76	2	no	—
277	C₈	—	152	2	no	—
278	C₈	#10597266	1	2	yes	—
279	C₈	#10597266	1	2	yes	—
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	76	2	no	—
283	C₂²	—	76	2	no	—
284	C₂²	—	76	2	no	—
285	C₂²	—	152	2	no	—
286	C₂²	—	152	2	no	—
287	C₂²	#11926401	1	1	yes	characteristic
288	C₂²	#11926821	1	1	yes	characteristic
289	C₂²	#11926831	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	76	2	no	—
294	C₄	C₂×#10597266	1	1	yes	characteristic
295	C₄	#11926543	1	1	yes	characteristic
296	C₄	#11926591	1	2	yes	—
297	C₄	#11926736	1	1	yes	characteristic
298	C₄	#11926746	1	1	yes	characteristic
299	C₂	—	2	1	no	—
300	C₂	—	2	1	no	—
301	C₂	—	152	2	no	—
302	C₂	#14563632	1	1	yes	characteristic
303	C₂	#14563674	1	1	yes	characteristic
304	C₂	#14563690	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10597384
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1848	C₂
socle	C₂²×C₁₉	#380
prime radical	C₂²×D₇₆	Q₈
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1848	D₁₉
hypocenter	C₁₉	#8662
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

Complements

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

N	H ≅ G/N	type
2. #14562383	C₂	semidirect
4. #14561530	C₂	semidirect
5. C₁₉×#1848	C₂	semidirect
129. #1848	D₁₉	semidirect
206. C₁₉	#8662	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926623	2
#14562383	2
#14562059	2
#14561530	2
C₁₉×#1848	2
#8662	19