Subgroups — #18069122

Order 4864.

476 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14563329	C₂	1	1	yes	maximal, characteristic
2	#14561862	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1817	C₂	1	1	yes	maximal, characteristic
4	C₂²×#64520	—	2	1	no	—
5	C₂×C₁₉⋊#95	—	2	1	no	—
6	C₂×C₁₉⋊#95	—	2	1	no	—
7	C₂×C₁₉⋊#95	C₄	1	2	yes	—
8	#11926299	—	2	1	no	—
9	#11926299	C₄	1	2	yes	complemented
10	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
11	C₂×C₁₉×C₂²⋊C₈	—	2	1	no	—
12	C₁₉×#336	—	2	1	no	—
13	C₁₉×#336	C₄	1	2	yes	complemented
14	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
15	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
16	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
17	C₂³×Dic₁₉	—	2	2	no	—
18	C₂²×C₈×C₁₉	—	2	2	no	—
19	C₂×C₄²×C₁₉	—	4	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	Q₈	1	1	yes	characteristic
24	C₂×#64520	—	2	4	no	—
25	C₂×#64520	—	4	2	no	—
26	C₂×#64520	—	4	1	no	—
27	C₂×#64520	—	4	1	no	—
28	C₂²×C₁₉⋊C₈	—	2	2	no	—
29	C₁₉×C₂²⋊C₈	—	4	2	no	—
30	C₁₉⋊#95	—	2	4	no	—
31	C₁₉⋊#95	—	4	1	no	—
32	C₁₉⋊#95	—	4	1	no	—
33	C₁₉⋊#95	—	4	2	no	—
34	C₂⁴×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₉	—	2	2	no	—
36	C₂²×C₄×C₁₉	—	2	2	no	—
37	C₂²×C₄×C₁₉	—	2	2	no	—
38	C₂²×C₄×C₁₉	—	2	2	no	—
39	C₂²×C₄×C₁₉	—	2	2	no	—
40	C₂²×C₄×C₁₉	—	4	1	no	—
41	C₂²×C₄×C₁₉	—	4	1	no	—
42	C₂²×C₄×C₁₉	—	4	1	no	—
43	C₂²×C₄×C₁₉	C₄²	1	1	yes	characteristic
44	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₉	SD₁₆	1	1	yes	characteristic
51	C₂²×C₄×C₁₉	SD₁₆	1	1	yes	characteristic
52	C₂²×C₄×C₁₉	Q₁₆	1	1	yes	characteristic
53	C₂²×Dic₁₉	—	2	4	no	—
54	C₂²×Dic₁₉	—	4	2	no	—
55	C₂²×Dic₁₉	—	4	2	no	—
56	C₂²×Dic₁₉	—	4	2	no	—
57	C₂²×Dic₁₉	—	4	2	no	—
58	C₂²×Dic₁₉	—	4	2	no	—
59	C₂²×Dic₁₉	—	4	2	no	—
60	C₂×C₈×C₁₉	—	4	2	no	—
61	C₂×C₈×C₁₉	—	4	2	no	—
62	C₂×C₈×C₁₉	—	4	2	no	—
63	C₄²×C₁₉	—	4	1	no	—
64	C₄²×C₁₉	—	4	1	no	—
65	C₄²×C₁₉	—	4	1	no	—
66	C₄²×C₁₉	—	4	1	no	—
67	#64520	—	4	4	no	—
68	#64520	—	4	4	no	—
69	C₂×C₁₉⋊C₈	—	2	4	no	—
70	C₂×C₁₉⋊C₈	—	4	2	no	—
71	C₂×C₁₉⋊C₈	—	4	2	no	—
72	#7037	—	19	1	no	maximal, Sylow
73	C₂³×C₁₉	—	2	1	no	—
74	C₂³×C₁₉	—	2	1	no	—
75	C₂³×C₁₉	—	2	1	no	—
76	C₂³×C₁₉	—	2	1	no	—
77	C₂³×C₁₉	#95	1	1	yes	characteristic
78	C₂³×C₁₉	#99	1	2	yes	—
79	C₂³×C₁₉	#102	1	1	yes	characteristic
80	C₂³×C₁₉	#103	1	1	yes	characteristic
81	C₂³×C₁₉	#104	1	2	yes	—
82	C₂×C₄×C₁₉	—	2	2	no	—
83	C₂×C₄×C₁₉	—	2	2	no	—
84	C₂×C₄×C₁₉	—	2	2	no	—
85	C₂×C₄×C₁₉	—	2	2	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	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	—	4	4	no	—
102	C₂×C₄×C₁₉	—	4	4	no	—
103	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
104	C₂×C₄×C₁₉	#103	1	1	yes	characteristic
105	C₂×C₄×C₁₉	#104	1	2	yes	—
106	C₂×C₄×C₁₉	#106	1	1	yes	characteristic
107	C₂×C₄×C₁₉	#107	1	1	yes	characteristic
108	C₂×Dic₁₉	—	4	4	no	—
109	C₂×Dic₁₉	—	4	4	no	—
110	C₂×Dic₁₉	—	4	4	no	—
111	C₂×Dic₁₉	—	8	2	no	—
112	C₂×Dic₁₉	—	8	2	no	—
113	C₂×Dic₁₉	—	8	2	no	—
114	C₂×Dic₁₉	—	8	2	no	—
115	C₈×C₁₉	—	8	2	no	—
116	C₁₉⋊C₈	—	4	4	no	—
117	#1817	D₁₉	1	1	yes	characteristic
118	#1728	—	19	1	no	—
119	#1434	—	19	1	no	—
120	C₂²×C₁₉	—	2	2	no	—
121	C₂²×C₁₉	—	2	1	no	—
122	C₂²×C₁₉	—	2	1	no	—
123	C₂²×C₁₉	—	2	2	no	—
124	C₂²×C₁₉	—	2	2	no	—
125	C₂²×C₁₉	—	2	1	no	—
126	C₂²×C₁₉	—	2	1	no	—
127	C₂²×C₁₉	—	2	2	no	—
128	C₂²×C₁₉	#327	1	2	yes	—
129	C₂²×C₁₉	#328	1	2	yes	—
130	C₂²×C₁₉	H(3,ℤ/4)	1	1	yes	characteristic
131	C₂²×C₁₉	#339	1	1	yes	characteristic
132	C₂²×C₁₉	#339	1	1	yes	characteristic
133	C₂²×C₁₉	#340	1	1	yes	characteristic
134	C₂²×C₁₉	#342	1	1	yes	characteristic
135	C₂²×C₁₉	#344	1	1	yes	characteristic
136	C₂²×C₁₉	#344	1	1	yes	characteristic
137	C₄×C₁₉	—	2	2	no	—
138	C₄×C₁₉	—	2	1	no	—
139	C₄×C₁₉	—	2	1	no	—
140	C₄×C₁₉	—	2	1	no	—
141	C₄×C₁₉	—	2	1	no	—
142	C₄×C₁₉	—	4	1	no	—
143	C₄×C₁₉	—	4	1	no	—
144	C₄×C₁₉	—	4	1	no	—
145	Dic₁₉	—	8	4	no	—
146	C₂²×M₁₆	—	38	1	no	—
147	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
148	C₂×C₂²⋊C₈	—	2	1	no	—
149	C₂×#95	—	19	2	no	—
150	C₂×#95	—	38	1	no	—
151	C₂×#95	—	38	1	no	—
152	#336	—	2	1	no	—
153	#336	—	19	2	no	—
154	#336	—	38	1	no	—
155	#336	Dic₁₉	1	2	yes	complemented
156	C₂×C₁₉	—	2	2	no	—
157	C₂×C₁₉	—	2	2	no	—
158	C₂×C₁₉	#1284	1	1	yes	characteristic
159	C₂×C₁₉	#1286	1	1	yes	characteristic
160	C₂×C₁₉	#1288	1	1	yes	characteristic
161	C₂×C₁₉	#1294	1	1	yes	characteristic
162	C₂×C₁₉	#1300	1	1	yes	characteristic
163	C₂×C₁₉	#1303	1	1	yes	characteristic
164	C₂×C₁₉	#1306	1	1	yes	characteristic
165	C₂³×C₄	—	38	2	no	—
166	C₂³×C₄	C₄×D₁₉	1	1	yes	characteristic
167	C₂³×C₄	D₇₆	1	1	yes	characteristic
168	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
169	C₂×M₁₆	—	38	4	no	—
170	C₂×M₁₆	—	76	2	no	—
171	C₂×M₁₆	—	76	1	no	—
172	C₂×M₁₆	—	76	1	no	—
173	C₂²×C₈	—	2	2	no	—
174	C₂²×C₈	—	38	2	no	—
175	C₂×C₄²	—	4	2	no	—
176	C₂×C₄²	Dic₃₈	1	1	yes	characteristic
177	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
178	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
179	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
180	C₂²⋊C₈	—	4	2	no	—
181	#95	—	38	4	no	—
182	#95	—	76	2	no	—
183	#95	—	76	1	no	—
184	#95	—	76	1	no	—
185	C₁₉	#7037	1	1	yes	characteristic, complemented, Sylow
186	C₂⁴	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	4	1	no	—
193	C₂²×C₄	—	4	1	no	—
194	C₂²×C₄	—	4	1	no	—
195	C₂²×C₄	—	38	4	no	—
196	C₂²×C₄	—	76	2	no	—
197	C₂²×C₄	—	76	2	no	—
198	C₂²×C₄	—	76	2	no	—
199	C₂²×C₄	—	76	2	no	—
200	C₂²×C₄	—	76	2	no	—
201	C₂²×C₄	—	76	2	no	—
202	C₂²×C₄	#64516	1	1	yes	characteristic
203	C₂²×C₄	#64516	1	1	yes	characteristic
204	C₂²×C₄	D₁₅₂	1	1	yes	characteristic
205	C₂²×C₄	Dic₇₆	1	1	yes	characteristic
206	C₂²×C₄	C₄×Dic₁₉	1	1	yes	characteristic
207	C₂²×C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
208	C₂²×C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
209	C₂²×C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
210	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
211	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
212	M₁₆	—	76	4	no	—
213	M₁₆	—	76	4	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	—	4	2	no	—
217	C₂×C₈	—	38	4	no	—
218	C₂×C₈	—	76	2	no	—
219	C₂×C₈	—	76	2	no	—
220	C₄²	—	4	1	no	—
221	C₄²	—	4	1	no	—
222	C₄²	—	4	1	no	—
223	C₄²	—	4	1	no	—
224	C₂³	—	2	1	no	—
225	C₂³	—	2	1	no	—
226	C₂³	—	2	1	no	—
227	C₂³	—	2	1	no	—
228	C₂³	#10597263	1	2	yes	—
229	C₂³	#10597273	1	1	yes	characteristic
230	C₂³	#10597278	1	1	yes	characteristic
231	C₂³	#10597282	1	2	yes	—
232	C₂³	C₁₉⋊#95	1	1	yes	characteristic
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
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	—	4	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	76	4	no	—
255	C₂×C₄	—	76	4	no	—
256	C₂×C₄	—	76	4	no	—
257	C₂×C₄	—	152	2	no	—
258	C₂×C₄	—	152	2	no	—
259	C₂×C₄	—	152	2	no	—
260	C₂×C₄	—	152	2	no	—
261	C₂×C₄	#10597273	1	1	yes	characteristic
262	C₂×C₄	#10597274	1	1	yes	characteristic
263	C₂×C₄	#10597275	1	1	yes	characteristic
264	C₂×C₄	#10597278	1	1	yes	characteristic
265	C₂×C₄	#10597294	1	2	yes	—
266	C₈	—	8	2	no	—
267	C₈	—	76	4	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	#11926279	1	2	yes	—
277	C₂²	#11926282	1	2	yes	—
278	C₂²	#11926300	1	1	yes	characteristic
279	C₂²	#11926303	1	1	yes	characteristic
280	C₂²	#11926305	1	1	yes	characteristic
281	C₂²	#11926307	1	1	yes	characteristic
282	C₂²	#11926310	1	1	yes	characteristic
283	C₂²	#11926315	1	1	yes	characteristic
284	C₂²	#11926316	1	1	yes	characteristic
285	C₄	—	2	2	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	4	1	no	—
291	C₄	—	4	1	no	—
292	C₄	—	4	1	no	—
293	C₄	—	152	4	no	—
294	C₂	—	2	2	no	—
295	C₂	—	2	2	no	—
296	C₂	#14561487	1	1	yes	characteristic
297	C₂	#14561494	1	1	yes	characteristic
298	C₂	#14561501	1	1	yes	characteristic
299	C₂	#14561515	1	1	yes	characteristic
300	C₂	#14561527	1	1	yes	characteristic
301	C₂	#14561537	1	1	yes	characteristic
302	C₂	#14561544	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₁₉⋊#95
commutator	C₂²×C₄×C₁₉	C₄²
Frattini	C₂²×C₄²	C₂×D₁₉
Fitting	C₁₉×#1817	C₂
socle	C₂³×C₁₉	#95
prime radical	C₂⁴×C₁₉	C₂²⋊C₄
cosocle	C₂²×C₄²×C₁₉	C₂²
hypercenter	#1817	D₁₉
hypocenter	C₁₉	#7037
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #1817

Complements

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

N	H ≅ G/N	type
9. #11926299	C₄	semidirect
13. C₁₉×#336	C₄	semidirect
155. #336	Dic₁₉	semidirect
185. C₁₉	#7037	semidirect

Maximal subgroups

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

subgroup	index
#14563329	2
#14561862	2
C₁₉×#1817	2
#7037	19