Subgroups — #18075403

Order 4864.

462 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	C₂×#11926501	C₂	1	1	yes	maximal, characteristic
2	#14562381	C₂	1	2	yes	maximal, complemented
3	#14561947	C₂	1	2	yes	maximal
4	#14561466	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#1275	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#64515	C₄	1	1	yes	characteristic
7	C₂²×C₈×D₁₉	C₄	1	1	yes	characteristic
8	C₂×#10597318	C₂²	1	1	yes	characteristic
9	#11926597	C₂²	1	2	yes	—
10	#11926501	—	2	2	no	—
11	#11926501	—	2	2	no	—
12	C₂×#10597272	C₄	1	1	yes	characteristic
13	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₈×Dic₁₉	C₄	1	1	yes	characteristic
15	#11926296	—	2	2	no	—
16	#11926294	—	2	2	no	—
17	#11926269	—	2	2	no	—
18	#11926267	—	2	2	no	—
19	C₁₉×#431	C₂²	1	2	yes	—
20	C₂×C₁₉×#97	C₂²	1	1	yes	characteristic
21	C₂²×C₄×D₁₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
25	C₂×(C₂×D₁₉)⋊C₄	D₄	1	1	yes	characteristic
26	C₂×(C₂×D₁₉)⋊C₄	Q₈	1	1	yes	characteristic
27	C₂×Dic₁₉⋊C₄	D₄	1	1	yes	characteristic
28	C₂×Dic₁₉⋊C₄	Q₈	1	1	yes	characteristic
29	C₂×C₄×Dic₁₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×#64515	—	2	2	no	—
33	C₂×#64515	—	2	2	no	—
34	C₂×#64515	C₈	1	4	yes	complemented
35	C₂×C₈×D₁₉	—	2	2	no	—
36	C₂×C₈×D₁₉	—	2	2	no	—
37	C₂×C₈×D₁₉	—	2	1	no	—
38	C₂×C₈×D₁₉	—	2	1	no	—
39	#10597318	—	2	2	no	—
40	#10597318	—	2	2	no	—
41	C₁₉×C₄⋊C₈	—	2	2	no	—
42	C₁₉×C₂²⋊C₈	—	2	2	no	—
43	C₁₉×#97	—	2	2	no	—
44	C₄×C₈×C₁₉	—	2	2	no	—
45	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
46	#10597272	—	2	2	no	—
47	C₈×Dic₁₉	—	2	1	no	—
48	C₈×Dic₁₉	—	2	1	no	—
49	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
50	C₄×C₁₉⋊C₈	—	2	2	no	—
51	C₄×C₁₉⋊C₈	D₄	1	2	yes	—
52	C₄×C₁₉⋊C₈	D₄	1	2	yes	—
53	C₂³×D₁₉	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
57	C₂²×Dic₁₉	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×Dic₁₉	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×Dic₁₉	C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×D₁₉	—	2	1	no	—
61	C₂×C₄×D₁₉	—	2	1	no	—
62	C₂×C₄×D₁₉	—	2	1	no	—
63	C₂×C₄×D₁₉	—	2	1	no	—
64	C₂×C₄×D₁₉	C₂×C₈	1	2	yes	—
65	C₂×C₄×D₁₉	M₁₆	1	2	yes	—
66	C₂×C₈×C₁₉	—	2	1	no	—
67	C₂×C₈×C₁₉	—	2	1	no	—
68	C₂×C₈×C₁₉	—	2	1	no	—
69	C₂×C₈×C₁₉	—	2	1	no	—
70	C₂×C₈×C₁₉	—	4	2	no	—
71	C₂×C₈×C₁₉	C₂×C₈	1	2	yes	—
72	C₂×C₈×C₁₉	M₁₆	1	2	yes	—
73	C₄²×C₁₉	C₂×D₄	1	2	yes	—
74	C₄²×C₁₉	#41	1	2	yes	—
75	(C₂×D₁₉)⋊C₄	—	2	2	no	—
76	(C₂×D₁₉)⋊C₄	—	2	2	no	—
77	Dic₁₉⋊C₄	—	2	2	no	—
78	Dic₁₉⋊C₄	—	2	2	no	—
79	C₄×Dic₁₉	—	2	1	no	—
80	C₄×Dic₁₉	—	2	1	no	—
81	C₂×C₁₉⋊C₈	—	2	1	no	—
82	C₂×C₁₉⋊C₈	—	2	1	no	—
83	C₂×C₁₉⋊C₈	—	4	2	no	—
84	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	2	yes	—
87	C₂×C₁₉⋊C₈	C₂×C₈	1	2	yes	—
88	C₂×C₁₉⋊C₈	M₁₆	1	2	yes	—
89	#64515	—	2	8	no	—
90	C₈×D₁₉	—	2	4	no	—
91	C₈×D₁₉	—	2	4	no	—
92	#8602	—	19	1	no	maximal, Sylow
93	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂²×D₁₉	—	2	1	no	—
95	C₂²×D₁₉	—	2	1	no	—
96	C₂²×D₁₉	—	4	1	no	—
97	C₂²×D₁₉	—	4	1	no	—
98	C₂²×D₁₉	C₄⋊C₈	1	2	yes	—
99	C₂×C₄×C₁₉	—	4	1	no	—
100	C₂×C₄×C₁₉	—	4	1	no	—
101	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
103	C₂×C₄×C₁₉	#121	1	2	yes	—
104	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
105	C₂×C₄×C₁₉	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
107	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
108	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
109	C₂×Dic₁₉	—	2	1	no	—
110	C₂×Dic₁₉	—	2	1	no	—
111	C₂×Dic₁₉	—	2	1	no	—
112	C₂×Dic₁₉	—	2	1	no	—
113	C₂×Dic₁₉	—	4	1	no	—
114	C₂×Dic₁₉	—	4	1	no	—
115	C₂×Dic₁₉	C₄⋊C₈	1	2	yes	—
116	C₄×D₁₉	—	2	2	no	—
117	C₄×D₁₉	—	2	2	no	—
118	C₄×D₁₉	—	2	2	no	—
119	C₄×D₁₉	—	2	2	no	—
120	C₈×C₁₉	—	2	2	no	—
121	C₈×C₁₉	—	2	2	no	—
122	C₈×C₁₉	—	8	2	no	—
123	C₁₉⋊C₈	—	2	1	no	—
124	C₁₉⋊C₈	—	2	1	no	—
125	C₁₉⋊C₈	—	8	2	no	—
126	C₁₉⋊C₈	C₂²⋊C₈	1	4	yes	—
127	C₂×#405	—	19	1	no	—
128	#1564	—	19	2	no	—
129	#1458	—	19	2	no	—
130	#1275	—	19	1	no	—
131	#1275	D₁₉	1	1	yes	characteristic, complemented
132	C₂²×C₁₉	—	2	1	no	—
133	C₂²×C₁₉	—	2	1	no	—
134	C₂²×C₁₉	#380	1	1	yes	characteristic
135	C₂²×C₁₉	C₂×C₄⋊C₈	1	1	yes	characteristic
136	C₂²×C₁₉	#423	1	1	yes	characteristic
137	C₂×D₁₉	—	2	2	no	—
138	C₂×D₁₉	—	4	2	no	—
139	C₂×D₁₉	—	4	1	no	—
140	C₂×D₁₉	—	4	1	no	—
141	C₂×D₁₉	—	4	2	no	—
142	C₄×C₁₉	—	2	2	no	—
143	C₄×C₁₉	—	2	2	no	—
144	C₄×C₁₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
145	C₄×C₁₉	#407	1	1	yes	characteristic
146	C₄×C₁₉	#431	1	1	yes	characteristic
147	C₄×C₁₉	#432	1	1	yes	characteristic
148	Dic₁₉	—	2	2	no	—
149	Dic₁₉	—	4	1	no	—
150	Dic₁₉	—	4	1	no	—
151	C₂²×M₁₆	—	19	1	no	—
152	C₂³×C₈	—	19	1	no	—
153	C₂×#117	—	19	1	no	—
154	#431	—	19	2	no	—
155	#431	C₂×D₁₉	1	2	yes	—
156	#405	—	38	2	no	—
157	#405	—	38	2	no	—
158	C₂×#97	—	19	1	no	—
159	C₂×#97	C₂×D₁₉	1	1	yes	characteristic
160	C₂×C₄×C₈	—	19	1	no	—
161	C₂×C₄×C₈	—	19	1	no	—
162	#335	—	38	2	no	—
163	#334	—	38	2	no	—
164	#324	—	38	2	no	—
165	#323	—	38	2	no	—
166	C₂×C₁₉	—	2	1	no	—
167	C₂×C₁₉	—	2	1	no	—
168	C₂×C₁₉	#1797	1	1	yes	characteristic
169	C₂×C₁₉	#1812	1	1	yes	characteristic
170	C₂×C₁₉	#1813	1	1	yes	characteristic
171	D₁₉	—	4	2	no	—
172	C₂³×C₄	—	19	1	no	—
173	C₂×M₁₆	—	19	4	no	—
174	C₂×M₁₆	—	38	2	no	—
175	C₂×M₁₆	—	38	2	no	—
176	C₂²×C₈	—	19	1	no	—
177	C₂²×C₈	—	19	1	no	—
178	C₂²×C₈	—	38	1	no	—
179	C₂²×C₈	—	38	1	no	—
180	C₂²×C₈	—	38	2	no	—
181	C₂²×C₈	—	38	2	no	—
182	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
183	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
184	#117	—	38	2	no	—
185	#117	—	38	2	no	—
186	C₂×C₄⋊C₄	—	19	1	no	—
187	C₂×C₄⋊C₄	—	19	1	no	—
188	C₂×C₂²⋊C₄	—	19	1	no	—
189	C₂×C₂²⋊C₄	—	19	1	no	—
190	C₂×C₄²	—	19	1	no	—
191	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
192	C₄⋊C₈	—	2	2	no	—
193	C₄⋊C₈	—	38	2	no	—
194	C₂²⋊C₈	—	2	2	no	—
195	C₂²⋊C₈	—	38	2	no	—
196	#97	—	2	2	no	—
197	#97	—	38	2	no	—
198	C₄×C₈	—	2	2	no	—
199	C₄×C₈	—	19	2	no	—
200	C₄×C₈	—	19	2	no	—
201	C₄×C₈	—	38	1	no	—
202	C₄×C₈	—	38	1	no	—
203	C₄×C₈	—	38	2	no	—
204	C₁₉	#8602	1	1	yes	characteristic, complemented, Sylow
205	C₂⁴	—	19	1	no	—
206	C₂²×C₄	—	19	1	no	—
207	C₂²×C₄	—	19	2	no	—
208	C₂²×C₄	—	19	2	no	—
209	C₂²×C₄	—	19	1	no	—
210	C₂²×C₄	—	19	1	no	—
211	C₂²×C₄	—	38	1	no	—
212	C₂²×C₄	—	38	1	no	—
213	C₂²×C₄	—	38	1	no	—
214	C₂²×C₄	—	38	1	no	—
215	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
216	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
217	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
218	M₁₆	—	38	8	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	4	2	no	—
224	C₂×C₈	—	19	2	no	—
225	C₂×C₈	—	19	2	no	—
226	C₂×C₈	—	19	2	no	—
227	C₂×C₈	—	19	1	no	—
228	C₂×C₈	—	19	1	no	—
229	C₂×C₈	—	38	4	no	—
230	C₂×C₈	—	38	4	no	—
231	C₂×C₈	—	38	1	no	—
232	C₂×C₈	—	38	1	no	—
233	C₂×C₈	—	76	2	no	—
234	C₂×C₈	C₈×D₁₉	1	2	yes	—
235	C₂×C₈	#64515	1	2	yes	—
236	C₄⋊C₄	—	38	2	no	—
237	C₄⋊C₄	—	38	2	no	—
238	C₂²⋊C₄	—	38	2	no	—
239	C₂²⋊C₄	—	38	2	no	—
240	C₄²	—	38	1	no	—
241	C₄²	—	38	1	no	—
242	C₄²	#64541	1	2	yes	—
243	C₄²	D₄×D₁₉	1	2	yes	—
244	C₂³	—	19	2	no	—
245	C₂³	—	38	1	no	—
246	C₂³	—	38	1	no	—
247	C₂³	—	76	1	no	—
248	C₂³	—	76	1	no	—
249	C₂³	C₄⋊C₄×D₁₉	1	1	yes	characteristic
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	19	2	no	—
253	C₂×C₄	—	38	1	no	—
254	C₂×C₄	—	38	1	no	—
255	C₂×C₄	—	38	2	no	—
256	C₂×C₄	—	38	2	no	—
257	C₂×C₄	—	38	2	no	—
258	C₂×C₄	—	38	2	no	—
259	C₂×C₄	—	38	1	no	—
260	C₂×C₄	—	38	1	no	—
261	C₂×C₄	—	76	1	no	—
262	C₂×C₄	—	76	1	no	—
263	C₂×C₄	#10597318	1	1	yes	characteristic
264	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
265	C₂×C₄	#10597330	1	2	yes	—
266	C₂×C₄	#10597342	1	2	yes	—
267	C₂×C₄	C₂×C₈×D₁₉	1	1	yes	characteristic
268	C₂×C₄	C₂×#64515	1	1	yes	characteristic
269	C₂×C₄	M₁₆×D₁₉	1	1	yes	characteristic
270	C₂×C₄	M₁₆×D₁₉	1	1	yes	characteristic
271	C₈	—	2	2	no	—
272	C₈	—	2	2	no	—
273	C₈	—	8	2	no	—
274	C₈	—	19	4	no	—
275	C₈	—	38	1	no	—
276	C₈	—	38	1	no	—
277	C₈	—	152	2	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	2	1	no	—
280	C₂²	—	38	2	no	—
281	C₂²	—	76	2	no	—
282	C₂²	—	76	2	no	—
283	C₂²	—	76	1	no	—
284	C₂²	—	76	1	no	—
285	C₂²	#11926403	1	1	yes	characteristic
286	C₂²	C₄⋊C₈×D₁₉	1	1	yes	characteristic
287	C₂²	#11926566	1	1	yes	characteristic
288	C₄	—	2	2	no	—
289	C₄	—	2	2	no	—
290	C₄	—	38	2	no	—
291	C₄	—	76	1	no	—
292	C₄	—	76	1	no	—
293	C₄	C₂²⋊C₈×D₁₉	1	1	yes	characteristic
294	C₄	#11926506	1	1	yes	characteristic
295	C₄	#11926595	1	1	yes	characteristic
296	C₄	#11926599	1	1	yes	characteristic
297	C₂	—	2	1	no	—
298	C₂	—	2	1	no	—
299	C₂	—	76	2	no	—
300	C₂	#14563614	1	1	yes	characteristic
301	C₂	#14563680	1	1	yes	characteristic
302	C₂	#14563684	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10597318
commutator	C₂×C₄×C₁₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1275	C₂
socle	C₂²×C₁₉	#380
prime radical	C₂³×D₁₉	C₄⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1275	D₁₉
hypocenter	C₁₉	#8602
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ #1275

Complements

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

N	H ≅ G/N	type
2. #14562381	C₂	semidirect
4. #14561466	C₂	semidirect
5. C₁₉×#1275	C₂	semidirect
34. C₂×#64515	C₈	semidirect
131. #1275	D₁₉	semidirect
204. C₁₉	#8602	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926501	2
#14562381	2
#14561947	2
#14561466	2
C₁₉×#1275	2
#8602	19