Subgroups — #18070630

Order 4864.

406 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14566112	C₂	1	1	yes	maximal, characteristic
2	#14563778	C₂	1	1	yes	maximal, characteristic
3	#14561460	C₂	1	2	yes	maximal
4	#14561459	C₂	1	2	yes	maximal
5	C₁₉×#1831	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₃₈	C₄	1	1	yes	characteristic
7	#11926450	C₄	1	1	yes	characteristic
8	#11926450	C₂²	1	1	yes	characteristic
9	#11926442	C₄	1	1	yes	characteristic
10	#11926442	C₂²	1	1	yes	characteristic
11	#11926428	—	2	1	no	—
12	#11926428	—	2	1	no	—
13	C₄×Dic₁₉⋊C₄	—	2	1	no	—
14	C₄×C₄⋊Dic₁₉	C₄	1	1	yes	characteristic
15	#11926390	—	2	1	no	—
16	#11926299	—	2	1	no	—
17	#11926299	—	2	1	no	—
18	#11926299	C₂²	1	2	yes	—
19	C₂×C₁₉×C₄⋊C₈	—	2	1	no	—
20	C₄³×C₁₉	C₂²	1	1	yes	characteristic
21	C₁₉×#336	—	2	1	no	—
22	C₁₉×#336	C₂²	1	2	yes	—
23	C₂²×Dic₃₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₉	—	2	2	no	—
25	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
32	C₂×C₄⋊Dic₁₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄⋊Dic₁₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄⋊Dic₁₉	C₂×C₄	1	1	yes	characteristic
35	C₂×Dic₁₉⋊C₄	—	2	1	no	—
36	C₂×Dic₁₉⋊C₄	—	2	1	no	—
37	C₂×Dic₁₉⋊C₄	—	2	1	no	—
38	C₂×C₄×Dic₁₉	—	2	1	no	—
39	C₂×C₄×Dic₁₉	—	2	1	no	—
40	C₂²×C₁₉⋊C₈	—	2	2	no	—
41	C₄×Dic₃₈	—	2	2	no	—
42	C₄×Dic₃₈	C₈	1	4	yes	complemented
43	C₁₉×C₄⋊C₈	—	4	2	no	—
44	C₁₉⋊#95	—	2	1	no	—
45	C₁₉⋊#95	—	2	1	no	—
46	C₁₉⋊#95	—	2	1	no	—
47	C₁₉⋊#95	—	2	1	no	—
48	C₁₉⋊#95	—	2	1	no	—
49	C₁₉⋊#95	—	2	1	no	—
50	C₁₉⋊#95	—	2	1	no	—
51	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
58	C₂²×Dic₁₉	—	2	1	no	—
59	C₂²×Dic₁₉	—	2	1	no	—
60	C₂²×Dic₁₉	—	2	1	no	—
61	C₂²×Dic₁₉	—	2	1	no	—
62	C₂×Dic₃₈	—	2	1	no	—
63	C₂×Dic₃₈	—	4	1	no	—
64	C₂×Dic₃₈	—	4	1	no	—
65	C₂×Dic₃₈	C₂×C₈	1	2	yes	—
66	C₂×C₈×C₁₉	—	4	2	no	—
67	C₂×C₈×C₁₉	—	4	2	no	—
68	C₂×C₈×C₁₉	—	4	2	no	—
69	C₄²×C₁₉	—	2	1	no	—
70	C₄²×C₁₉	—	2	1	no	—
71	C₄²×C₁₉	—	2	2	no	—
72	C₄²×C₁₉	—	4	1	no	—
73	C₄²×C₁₉	—	4	1	no	—
74	C₄²×C₁₉	—	4	1	no	—
75	C₄²×C₁₉	—	4	1	no	—
76	C₄²×C₁₉	C₂×C₈	1	2	yes	—
77	C₄²×C₁₉	M₁₆	1	2	yes	—
78	C₄⋊Dic₁₉	—	2	1	no	—
79	C₄⋊Dic₁₉	—	4	1	no	—
80	C₄⋊Dic₁₉	—	4	1	no	—
81	C₄⋊Dic₁₉	C₂×C₈	1	2	yes	—
82	Dic₁₉⋊C₄	—	2	4	no	—
83	Dic₁₉⋊C₄	—	4	2	no	—
84	Dic₁₉⋊C₄	—	4	1	no	—
85	Dic₁₉⋊C₄	—	4	1	no	—
86	C₄×Dic₁₉	—	2	4	no	—
87	C₄×Dic₁₉	—	4	2	no	—
88	C₂×C₁₉⋊C₈	—	4	2	no	—
89	C₂×C₁₉⋊C₈	—	4	2	no	—
90	C₂×C₁₉⋊C₈	—	4	2	no	—
91	#7596	—	19	1	no	maximal, Sylow
92	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	—	2	1	no	—
98	C₂×C₄×C₁₉	—	2	1	no	—
99	C₂×C₄×C₁₉	—	2	1	no	—
100	C₂×C₄×C₁₉	—	2	1	no	—
101	C₂×C₄×C₁₉	—	2	1	no	—
102	C₂×C₄×C₁₉	—	2	1	no	—
103	C₂×C₄×C₁₉	—	2	1	no	—
104	C₂×C₄×C₁₉	—	2	1	no	—
105	C₂×C₄×C₁₉	—	2	1	no	—
106	C₂×C₄×C₁₉	—	2	1	no	—
107	C₂×C₄×C₁₉	—	2	1	no	—
108	C₂×C₄×C₁₉	—	2	1	no	—
109	C₂×C₄×C₁₉	C₂²⋊C₈	1	2	yes	—
110	C₂×C₄×C₁₉	C₂²⋊C₈	1	2	yes	—
111	C₂×C₄×C₁₉	C₂²×C₈	1	1	yes	characteristic
112	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
113	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
114	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
115	C₂×C₄×C₁₉	#137	1	2	yes	—
116	C₂×Dic₁₉	—	2	2	no	—
117	C₂×Dic₁₉	—	2	2	no	—
118	C₂×Dic₁₉	—	4	1	no	—
119	C₂×Dic₁₉	—	4	1	no	—
120	C₂×Dic₁₉	—	4	1	no	—
121	C₂×Dic₁₉	—	4	1	no	—
122	C₂×Dic₁₉	—	4	1	no	—
123	C₂×Dic₁₉	—	4	1	no	—
124	C₂×Dic₁₉	—	4	1	no	—
125	C₂×Dic₁₉	—	4	1	no	—
126	C₂×Dic₁₉	—	4	1	no	—
127	C₂×Dic₁₉	—	4	1	no	—
128	Dic₃₈	—	4	2	no	—
129	Dic₃₈	—	4	2	no	—
130	C₈×C₁₉	—	8	2	no	—
131	C₁₉⋊C₈	—	8	2	no	—
132	#2295	—	19	1	no	—
133	#1839	—	19	1	no	—
134	#1831	D₁₉	1	1	yes	characteristic
135	#1272	—	19	2	no	—
136	#1271	—	19	2	no	—
137	C₂²×C₁₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
138	C₂²×C₁₉	#410	1	1	yes	characteristic
139	C₂²×C₁₉	#413	1	1	yes	characteristic
140	C₂²×C₁₉	#417	1	1	yes	characteristic
141	C₂²×C₁₉	#419	1	1	yes	characteristic
142	C₂²×C₁₉	#421	1	1	yes	characteristic
143	C₂²×C₁₉	#421	1	1	yes	characteristic
144	C₄×C₁₉	—	2	1	no	—
145	C₄×C₁₉	—	2	1	no	—
146	C₄×C₁₉	—	2	2	no	—
147	C₄×C₁₉	—	2	2	no	—
148	C₄×C₁₉	—	2	1	no	—
149	C₄×C₁₉	—	2	1	no	—
150	C₄×C₁₉	—	2	2	no	—
151	C₄×C₁₉	—	4	1	no	—
152	C₄×C₁₉	—	4	1	no	—
153	Dic₁₉	—	4	2	no	—
154	Dic₁₉	—	4	2	no	—
155	Dic₁₉	—	8	1	no	—
156	Dic₁₉	—	8	1	no	—
157	C₂×C₄×Q₈	—	19	1	no	—
158	C₂×C₄⋊C₈	—	2	1	no	—
159	#391	—	19	1	no	—
160	#391	—	19	1	no	—
161	#389	—	19	1	no	—
162	#389	—	19	1	no	—
163	#387	—	38	1	no	—
164	#387	—	38	1	no	—
165	C₄×C₄⋊C₄	—	19	1	no	—
166	C₄×C₄⋊C₄	—	38	1	no	—
167	#376	—	38	1	no	—
168	C₄³	C₂×D₁₉	1	1	yes	characteristic
169	#336	—	2	1	no	—
170	#336	—	19	2	no	—
171	#336	—	38	1	no	—
172	#336	—	38	1	no	—
173	#336	C₂×D₁₉	1	2	yes	—
174	C₂×C₁₉	#1458	1	1	yes	characteristic
175	C₂×C₁₉	#1477	1	1	yes	characteristic
176	C₂×C₁₉	#1479	1	1	yes	characteristic
177	C₂×C₁₉	#1512	1	1	yes	characteristic
178	C₂×C₁₉	#1515	1	1	yes	characteristic
179	C₂×C₁₉	#1535	1	1	yes	characteristic
180	C₂×C₁₉	#1559	1	1	yes	characteristic
181	C₂²×Q₈	—	19	1	no	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	38	2	no	—
184	C₄×Q₈	—	19	4	no	—
185	C₄×Q₈	—	38	2	no	—
186	C₂×C₄⋊C₄	—	19	1	no	—
187	C₂×C₄⋊C₄	—	19	1	no	—
188	C₂×C₄⋊C₄	—	19	1	no	—
189	C₂×C₄⋊C₄	—	38	1	no	—
190	C₂×C₄⋊C₄	—	38	1	no	—
191	C₂×C₄⋊C₄	—	38	1	no	—
192	C₂×C₄²	—	38	1	no	—
193	C₂×C₄²	—	38	1	no	—
194	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
195	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
196	C₂×C₄²	D₇₆	1	1	yes	characteristic
197	C₂×C₄²	D₇₆	1	1	yes	characteristic
198	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
199	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
200	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
201	C₄⋊C₈	—	4	2	no	—
202	#95	—	38	1	no	—
203	#95	—	38	1	no	—
204	#95	—	38	1	no	—
205	#95	—	38	1	no	—
206	#95	—	38	1	no	—
207	#95	—	38	1	no	—
208	#95	—	38	1	no	—
209	C₁₉	#7596	1	1	yes	characteristic, complemented, Sylow
210	C₂×Q₈	—	19	2	no	—
211	C₂×Q₈	—	38	1	no	—
212	C₂×Q₈	—	76	1	no	—
213	C₂×Q₈	—	76	1	no	—
214	C₂²×C₄	—	38	1	no	—
215	C₂²×C₄	—	38	1	no	—
216	C₂²×C₄	—	38	1	no	—
217	C₂²×C₄	—	38	1	no	—
218	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
219	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
220	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
221	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	—	4	2	no	—
228	C₂×C₈	—	76	2	no	—
229	C₂×C₈	—	76	2	no	—
230	C₂×C₈	—	76	2	no	—
231	C₄⋊C₄	—	19	2	no	—
232	C₄⋊C₄	—	38	4	no	—
233	C₄⋊C₄	—	38	1	no	—
234	C₄⋊C₄	—	76	1	no	—
235	C₄⋊C₄	—	76	1	no	—
236	C₄⋊C₄	—	76	2	no	—
237	C₄⋊C₄	—	76	1	no	—
238	C₄⋊C₄	—	76	1	no	—
239	C₄²	—	2	1	no	—
240	C₄²	—	2	1	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	4	1	no	—
243	C₄²	—	4	1	no	—
244	C₄²	—	4	1	no	—
245	C₄²	—	4	1	no	—
246	C₄²	—	38	4	no	—
247	C₄²	—	76	2	no	—
248	C₄²	C₈×D₁₉	1	2	yes	—
249	C₄²	#64515	1	2	yes	—
250	C₂³	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
251	Q₈	—	76	2	no	—
252	Q₈	—	76	2	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	38	2	no	—
270	C₂×C₄	—	38	2	no	—
271	C₂×C₄	—	76	1	no	—
272	C₂×C₄	—	76	1	no	—
273	C₂×C₄	—	76	1	no	—
274	C₂×C₄	—	76	1	no	—
275	C₂×C₄	—	76	1	no	—
276	C₂×C₄	—	76	1	no	—
277	C₂×C₄	—	76	1	no	—
278	C₂×C₄	—	76	1	no	—
279	C₂×C₄	—	76	1	no	—
280	C₂×C₄	—	76	1	no	—
281	C₂×C₄	(C₂×D₁₉)⋊C₈	1	2	yes	—
282	C₂×C₄	(C₂×D₁₉)⋊C₈	1	2	yes	—
283	C₂×C₄	C₂×C₈×D₁₉	1	1	yes	characteristic
284	C₂×C₄	C₂×#64515	1	1	yes	characteristic
285	C₂×C₄	#10597355	1	2	yes	—
286	C₂×C₄	#10597378	1	1	yes	characteristic
287	C₂×C₄	#10597388	1	1	yes	characteristic
288	C₈	—	8	2	no	—
289	C₈	—	152	2	no	—
290	C₂²	C₂×(C₂×D₁₉)⋊C₈	1	1	yes	characteristic
291	C₂²	#11926517	1	1	yes	characteristic
292	C₂²	#11926526	1	1	yes	characteristic
293	C₂²	#11926543	1	1	yes	characteristic
294	C₂²	#11926554	1	1	yes	characteristic
295	C₂²	#11926560	1	1	yes	characteristic
296	C₂²	#11926562	1	1	yes	characteristic
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	2	2	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	2	no	—
303	C₄	—	2	2	no	—
304	C₄	—	4	1	no	—
305	C₄	—	4	1	no	—
306	C₄	—	76	2	no	—
307	C₄	—	76	2	no	—
308	C₄	—	152	1	no	—
309	C₄	—	152	1	no	—
310	C₂	#14561947	1	1	yes	characteristic
311	C₂	#14562020	1	1	yes	characteristic
312	C₂	#14562031	1	1	yes	characteristic
313	C₂	#14562171	1	1	yes	characteristic
314	C₂	#14562187	1	1	yes	characteristic
315	C₂	#14562264	1	1	yes	characteristic
316	C₂	#14562363	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₁₉)⋊C₄
commutator	C₂×C₄×C₁₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1831	C₂
socle, prime radical	C₂³×C₁₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1831	D₁₉
hypocenter	C₁₉	#7596
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

Complements

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

N	H ≅ G/N	type
42. C₄×Dic₃₈	C₈	semidirect
209. C₁₉	#7596	semidirect

Maximal subgroups

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

subgroup	index
#14566112	2
#14563778	2
#14561460	2
#14561459	2
C₁₉×#1831	2
#7596	19