Subgroups — #18070510

Order 4864.

444 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	C₄×#10597316	C₂	1	1	yes	maximal, characteristic
2	#14563381	C₂	1	1	yes	maximal, characteristic
3	#14561459	C₂	1	2	yes	maximal
4	#14561457	C₂	1	2	yes	maximal
5	C₁₉×#1744	C₂	1	1	yes	maximal, characteristic
6	C₂×#10597316	C₄	1	1	yes	characteristic
7	C₂×C₄×C₁₉⋊C₈	—	2	1	no	—
8	#11926442	C₂²	1	1	yes	characteristic
9	#11926442	C₂²	1	1	yes	characteristic
10	#11926428	—	2	1	no	—
11	#11926428	—	2	1	no	—
12	#11926410	C₄	1	1	yes	characteristic
13	C₄×Dic₁₉⋊C₄	—	2	1	no	—
14	C₄×Dic₁₉⋊C₄	—	2	1	no	—
15	C₄×C₄⋊Dic₁₉	C₄	1	1	yes	characteristic
16	C₄×C₄⋊Dic₁₉	C₄	1	1	yes	characteristic
17	#11926299	—	2	1	no	—
18	#11926299	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₁₉	—	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×C₈×C₁₉	—	2	2	no	—
24	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₉	D₄	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	C₂³	1	1	yes	characteristic
31	C₂×C₄⋊Dic₁₉	C₂×C₄	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×Dic₁₉⋊C₄	—	2	1	no	—
39	C₂×C₄×Dic₁₉	—	2	1	no	—
40	C₂×C₄×Dic₁₉	—	2	1	no	—
41	C₂²×C₁₉⋊C₈	—	2	2	no	—
42	#10597316	—	2	2	no	—
43	#10597316	C₈	1	4	yes	complemented
44	C₄×C₈×C₁₉	—	2	4	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	C₄×C₁₉⋊C₈	—	2	4	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	C₂²×Dic₁₉	—	2	1	no	—
60	C₂²×Dic₁₉	—	2	1	no	—
61	C₂²×Dic₁₉	—	2	1	no	—
62	C₂²×Dic₁₉	—	2	1	no	—
63	C₂×C₈×C₁₉	—	2	4	no	—
64	C₂×C₈×C₁₉	—	4	2	no	—
65	C₂×C₈×C₁₉	—	4	2	no	—
66	C₄²×C₁₉	—	2	2	no	—
67	C₄²×C₁₉	—	2	2	no	—
68	C₄²×C₁₉	—	2	2	no	—
69	C₄²×C₁₉	—	2	2	no	—
70	C₄²×C₁₉	—	2	2	no	—
71	C₄²×C₁₉	—	4	1	no	—
72	C₄²×C₁₉	C₂×C₈	1	2	yes	—
73	C₄²×C₁₉	M₁₆	1	2	yes	—
74	C₄⋊Dic₁₉	—	2	1	no	—
75	C₄⋊Dic₁₉	—	2	1	no	—
76	C₄⋊Dic₁₉	—	4	1	no	—
77	C₄⋊Dic₁₉	—	4	1	no	—
78	C₄⋊Dic₁₉	C₂×C₈	1	2	yes	—
79	C₄⋊Dic₁₉	C₂×C₈	1	2	yes	—
80	Dic₁₉⋊C₄	—	2	4	no	—
81	Dic₁₉⋊C₄	—	2	4	no	—
82	Dic₁₉⋊C₄	—	4	2	no	—
83	Dic₁₉⋊C₄	—	4	2	no	—
84	C₄×Dic₁₉	—	4	2	no	—
85	C₄×Dic₁₉	—	4	2	no	—
86	C₂×C₁₉⋊C₈	—	2	4	no	—
87	C₂×C₁₉⋊C₈	—	4	2	no	—
88	C₂×C₁₉⋊C₈	—	4	2	no	—
89	#7568	—	19	1	no	maximal, Sylow
90	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₉	—	2	1	no	—
92	C₂×C₄×C₁₉	—	2	1	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	—	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	C₂²⋊C₈	1	2	yes	—
105	C₂×C₄×C₁₉	C₂²⋊C₈	1	2	yes	—
106	C₂×C₄×C₁₉	#101	1	2	yes	—
107	C₂×C₄×C₁₉	#104	1	2	yes	—
108	C₂×C₄×C₁₉	#104	1	2	yes	—
109	C₂×C₄×C₁₉	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
111	C₂×C₄×C₁₉	#135	1	2	yes	—
112	C₂×C₄×C₁₉	#135	1	2	yes	—
113	C₂×Dic₁₉	—	2	2	no	—
114	C₂×Dic₁₉	—	2	2	no	—
115	C₂×Dic₁₉	—	4	1	no	—
116	C₂×Dic₁₉	—	4	1	no	—
117	C₂×Dic₁₉	—	4	1	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	C₈×C₁₉	—	4	4	no	—
126	C₁₉⋊C₈	—	4	4	no	—
127	C₄×#125	—	19	1	no	—
128	#1744	—	19	1	no	—
129	#1744	D₁₉	1	1	yes	characteristic
130	#1271	—	19	2	no	—
131	#1271	—	19	2	no	—
132	C₂²×C₁₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
133	C₂²×C₁₉	#410	1	1	yes	characteristic
134	C₂²×C₁₉	C₂×#101	1	1	yes	characteristic
135	C₂²×C₁₉	#416	1	1	yes	characteristic
136	C₂²×C₁₉	#416	1	1	yes	characteristic
137	C₂²×C₁₉	C₂×#104	1	1	yes	characteristic
138	C₂²×C₁₉	C₂×#104	1	1	yes	characteristic
139	C₄×C₁₉	—	2	1	no	—
140	C₄×C₁₉	—	2	1	no	—
141	C₄×C₁₉	—	2	2	no	—
142	C₄×C₁₉	—	2	2	no	—
143	C₄×C₁₉	—	2	2	no	—
144	C₄×C₁₉	—	2	2	no	—
145	C₄×C₁₉	—	4	1	no	—
146	C₄×C₁₉	#330	1	4	yes	—
147	Dic₁₉	—	4	2	no	—
148	Dic₁₉	—	4	2	no	—
149	Dic₁₉	—	8	1	no	—
150	Dic₁₉	—	8	1	no	—
151	C₂×#125	—	19	1	no	—
152	C₂×C₄×C₈	—	2	1	no	—
153	C₂×C₄×C₈	—	38	1	no	—
154	#389	—	19	1	no	—
155	#389	—	19	1	no	—
156	#387	—	38	1	no	—
157	#387	—	38	1	no	—
158	#382	—	19	1	no	—
159	C₄×C₄⋊C₄	—	19	1	no	—
160	C₄×C₄⋊C₄	—	19	1	no	—
161	C₄×C₄⋊C₄	—	38	1	no	—
162	C₄×C₄⋊C₄	—	38	1	no	—
163	C₄³	C₂×D₁₉	1	1	yes	characteristic
164	#336	—	2	1	no	—
165	#336	—	19	2	no	—
166	#336	—	38	1	no	—
167	#336	C₂×D₁₉	1	2	yes	—
168	C₂×C₁₉	#1461	1	1	yes	characteristic
169	C₂×C₁₉	#1476	1	1	yes	characteristic
170	C₂×C₁₉	#1476	1	1	yes	characteristic
171	C₂×C₁₉	#1500	1	1	yes	characteristic
172	C₂×C₁₉	#1500	1	1	yes	characteristic
173	C₂×C₁₉	C₂×#330	1	1	yes	characteristic
174	C₂×C₁₉	#1544	1	1	yes	characteristic
175	C₂²×C₈	—	2	2	no	—
176	C₂²×C₈	—	38	2	no	—
177	#125	—	19	4	no	—
178	#125	—	38	2	no	—
179	C₂×C₄⋊C₄	—	19	1	no	—
180	C₂×C₄⋊C₄	—	19	1	no	—
181	C₂×C₄⋊C₄	—	19	1	no	—
182	C₂×C₄⋊C₄	—	19	1	no	—
183	C₂×C₄⋊C₄	—	38	1	no	—
184	C₂×C₄⋊C₄	—	38	1	no	—
185	C₂×C₄⋊C₄	—	38	1	no	—
186	C₂×C₄⋊C₄	—	38	1	no	—
187	C₂×C₄²	—	38	1	no	—
188	C₂×C₄²	—	38	1	no	—
189	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
190	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
191	C₂×C₄²	D₇₆	1	1	yes	characteristic
192	C₂×C₄²	D₇₆	1	1	yes	characteristic
193	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
194	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
195	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
196	C₄×C₈	—	2	4	no	—
197	C₄×C₈	—	38	4	no	—
198	#95	—	38	1	no	—
199	#95	—	38	1	no	—
200	#95	—	38	1	no	—
201	#95	—	38	1	no	—
202	#95	—	38	1	no	—
203	#95	—	38	1	no	—
204	C₁₉	#7568	1	1	yes	characteristic, complemented, Sylow
205	C₂²×C₄	—	38	1	no	—
206	C₂²×C₄	—	38	1	no	—
207	C₂²×C₄	—	38	1	no	—
208	C₂²×C₄	—	38	1	no	—
209	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
210	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
211	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
212	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
214	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	4	2	no	—
218	C₂×C₈	—	4	2	no	—
219	C₂×C₈	—	38	4	no	—
220	C₂×C₈	—	76	2	no	—
221	C₂×C₈	—	76	2	no	—
222	C₄⋊C₄	—	19	2	no	—
223	C₄⋊C₄	—	19	2	no	—
224	C₄⋊C₄	—	38	4	no	—
225	C₄⋊C₄	—	38	4	no	—
226	C₄⋊C₄	—	38	1	no	—
227	C₄⋊C₄	—	38	1	no	—
228	C₄⋊C₄	—	76	1	no	—
229	C₄⋊C₄	—	76	1	no	—
230	C₄⋊C₄	—	76	2	no	—
231	C₄⋊C₄	—	76	2	no	—
232	C₄²	—	2	2	no	—
233	C₄²	—	2	2	no	—
234	C₄²	—	2	2	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	2	2	no	—
237	C₄²	—	4	1	no	—
238	C₄²	—	76	2	no	—
239	C₄²	—	76	2	no	—
240	C₄²	C₈×D₁₉	1	2	yes	—
241	C₄²	#64515	1	2	yes	—
242	C₂³	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	38	2	no	—
257	C₂×C₄	—	38	2	no	—
258	C₂×C₄	—	76	1	no	—
259	C₂×C₄	—	76	1	no	—
260	C₂×C₄	—	76	1	no	—
261	C₂×C₄	—	76	1	no	—
262	C₂×C₄	—	76	1	no	—
263	C₂×C₄	—	76	1	no	—
264	C₂×C₄	—	76	1	no	—
265	C₂×C₄	—	76	1	no	—
266	C₂×C₄	—	76	1	no	—
267	C₂×C₄	—	76	1	no	—
268	C₂×C₄	#10597262	1	2	yes	—
269	C₂×C₄	(C₂×D₁₉)⋊C₈	1	2	yes	—
270	C₂×C₄	(C₂×D₁₉)⋊C₈	1	2	yes	—
271	C₂×C₄	#10597281	1	2	yes	—
272	C₂×C₄	#10597282	1	2	yes	—
273	C₂×C₄	C₂×C₈×D₁₉	1	1	yes	characteristic
274	C₂×C₄	C₂×#64515	1	1	yes	characteristic
275	C₂×C₄	#10597350	1	2	yes	—
276	C₂×C₄	#10597396	1	2	yes	—
277	C₈	—	4	4	no	—
278	C₈	—	76	4	no	—
279	C₂²	C₂×(C₂×D₁₉)⋊C₈	1	1	yes	characteristic
280	C₂²	#11926517	1	1	yes	characteristic
281	C₂²	C₂×#10597281	1	1	yes	characteristic
282	C₂²	#11926538	1	1	yes	characteristic
283	C₂²	#11926541	1	1	yes	characteristic
284	C₂²	C₂×#10597282	1	1	yes	characteristic
285	C₂²	C₂×#10597262	1	1	yes	characteristic
286	C₄	—	2	2	no	—
287	C₄	—	2	2	no	—
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	2	2	no	—
291	C₄	—	2	2	no	—
292	C₄	—	4	1	no	—
293	C₄	—	76	2	no	—
294	C₄	—	76	2	no	—
295	C₄	—	152	1	no	—
296	C₄	—	152	1	no	—
297	C₄	#11926285	1	4	yes	—
298	C₂	#14561958	1	1	yes	characteristic
299	C₂	#14562015	1	1	yes	characteristic
300	C₂	#14562018	1	1	yes	characteristic
301	C₂	#14562119	1	1	yes	characteristic
302	C₂	#14562122	1	1	yes	characteristic
303	C₂	C₂×#11926285	1	1	yes	characteristic
304	C₂	#14562299	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	(C₂×D₁₉)⋊C₄
commutator	C₂×C₄×C₁₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1744	C₂
socle, prime radical	C₂³×C₁₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1744	D₁₉
hypocenter	C₁₉	#7568
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

Complements

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

N	H ≅ G/N	type
43. #10597316	C₈	semidirect
204. C₁₉	#7568	semidirect

Maximal subgroups

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

subgroup	index
C₄×#10597316	2
#14563381	2
#14561459	2
#14561457	2
C₁₉×#1744	2
#7568	19