Subgroups — #18087727

Order 4864.

870 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×Dic₁₉	C₂	1	2	yes	maximal
2	C₂²×#10597272	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926309	C₂	1	1	yes	maximal, characteristic
4	C₂×#11926308	C₂	1	2	yes	maximal
5	C₂×C₁₉×#341	C₂	1	1	yes	maximal, characteristic
6	C₂²×#64520	C₂²	1	2	yes	—
7	C₂³×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₄×Dic₁₉	C₂²	1	1	yes	characteristic
9	M₁₆×Dic₁₉	—	2	4	no	—
10	M₁₆×Dic₁₉	C₄	1	8	yes	—
11	C₂×#10597272	—	2	4	no	—
12	C₂×#10597272	—	2	2	no	—
13	C₂×#10597272	C₄	1	4	yes	—
14	C₂×C₈×Dic₁₉	—	2	2	no	—
15	#11926309	C₄	1	4	yes	complemented
16	#11926308	C₄	1	8	yes	complemented
17	C₂²×M₁₆×C₁₉	C₂²	1	2	yes	—
18	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
19	C₁₉×#341	C₄	1	4	yes	complemented
20	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
21	C₂³×Dic₁₉	Q₈	1	2	yes	—
22	C₂×M₁₆×C₁₉	—	2	2	no	—
23	C₂×M₁₆×C₁₉	—	2	2	no	—
24	C₂×M₁₆×C₁₉	—	2	2	no	—
25	C₂×M₁₆×C₁₉	C₂×C₄	1	4	yes	—
26	C₂×M₁₆×C₁₉	C₂×C₄	1	4	yes	—
27	C₂×M₁₆×C₁₉	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₁₉	—	2	2	no	—
29	C₂²×C₈×C₁₉	—	2	2	no	—
30	C₂²×C₈×C₁₉	—	2	2	no	—
31	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
34	C₂×C₄×Dic₁₉	C₂×C₄	1	2	yes	—
35	C₂×C₄×Dic₁₉	C₂×C₄	1	4	yes	—
36	C₂×C₄×Dic₁₉	D₄	1	2	yes	—
37	C₂×C₄×Dic₁₉	D₄	1	4	yes	—
38	C₂×#64520	—	2	2	no	—
39	C₂×#64520	—	2	2	no	—
40	C₂×#64520	—	2	2	no	—
41	C₂×#64520	C₂×C₄	1	4	yes	—
42	C₂×#64520	C₂×C₄	1	4	yes	—
43	C₂×#64520	C₂×C₄	1	4	yes	—
44	C₂²×C₁₉⋊C₈	—	2	2	no	—
45	C₂²×C₁₉⋊C₈	—	2	2	no	—
46	C₂²×C₁₉⋊C₈	—	2	1	no	—
47	C₂²×C₁₉⋊C₈	—	2	1	no	—
48	C₂²×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
49	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
50	#10597272	—	2	8	no	—
51	#10597272	—	2	8	no	—
52	C₈×Dic₁₉	—	2	8	no	—
53	C₂⁴×C₁₉	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	#41	1	2	yes	—
62	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
64	C₂²×Dic₁₉	—	4	4	no	—
65	C₂²×Dic₁₉	C₄⋊C₄	1	4	yes	—
66	C₂²×Dic₁₉	C₄⋊C₄	1	8	yes	—
67	M₁₆×C₁₉	—	2	4	no	—
68	M₁₆×C₁₉	—	2	4	no	—
69	M₁₆×C₁₉	—	2	4	no	—
70	M₁₆×C₁₉	C₄²	1	8	yes	—
71	C₂×C₈×C₁₉	—	2	2	no	—
72	C₂×C₈×C₁₉	—	2	4	no	—
73	C₂×C₈×C₁₉	—	2	2	no	—
74	C₂×C₈×C₁₉	—	2	4	no	—
75	C₂×C₈×C₁₉	—	2	4	no	—
76	C₂×C₈×C₁₉	—	2	4	no	—
77	C₂×C₈×C₁₉	—	2	4	no	—
78	C₂×C₈×C₁₉	C₄²	1	4	yes	—
79	C₄×Dic₁₉	C₄²	1	4	yes	—
80	C₄×Dic₁₉	C₂²⋊C₄	1	4	yes	—
81	C₄×Dic₁₉	C₂²⋊C₄	1	4	yes	—
82	C₄×Dic₁₉	C₂²⋊C₄	1	4	yes	—
83	#64520	—	2	4	no	—
84	#64520	—	2	4	no	—
85	#64520	—	2	4	no	—
86	#64520	C₄²	1	8	yes	—
87	C₂×C₁₉⋊C₈	—	2	4	no	—
88	C₂×C₁₉⋊C₈	—	2	2	no	—
89	C₂×C₁₉⋊C₈	—	2	2	no	—
90	C₂×C₁₉⋊C₈	—	2	4	no	—
91	C₂×C₁₉⋊C₈	—	2	4	no	—
92	C₂×C₁₉⋊C₈	—	2	4	no	—
93	C₂×C₁₉⋊C₈	—	2	4	no	—
94	C₂×C₁₉⋊C₈	C₄²	1	4	yes	—
95	#10705	—	19	1	no	maximal, Sylow
96	C₂³×C₁₉	—	2	4	no	—
97	C₂³×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
98	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂³×C₁₉	C₄×Q₈	1	1	yes	characteristic
100	C₂³×C₁₉	C₄×Q₈	1	1	yes	characteristic
101	C₂³×C₁₉	C₄×Q₈	1	2	yes	—
102	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
103	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
104	C₂×C₄×C₁₉	C₂×C₄²	1	2	yes	—
105	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
106	C₂×C₄×C₁₉	C₂×C₄²	1	2	yes	—
107	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
109	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
110	C₂×C₄×C₁₉	#117	1	2	yes	—
111	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
112	C₂×C₄×C₁₉	#117	1	2	yes	—
113	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
120	C₂×Dic₁₉	—	2	8	no	—
121	C₂×Dic₁₉	—	4	8	no	—
122	C₂×Dic₁₉	#95	1	8	yes	—
123	C₈×C₁₉	—	2	4	no	—
124	C₈×C₁₉	—	2	8	no	—
125	C₁₉⋊C₈	—	2	4	no	—
126	C₁₉⋊C₈	—	2	8	no	—
127	C₂×C₄×M₁₆	—	19	2	no	—
128	C₂²×#97	—	19	1	no	—
129	C₂×#341	—	19	2	no	—
130	C₂×#341	—	19	1	no	—
131	C₂×#341	D₁₉	1	1	yes	characteristic
132	C₂²×C₁₉	—	2	4	no	—
133	C₂²×C₁₉	—	2	4	no	—
134	C₂²×C₁₉	—	2	2	no	—
135	C₂²×C₁₉	—	2	2	no	—
136	C₂²×C₁₉	C₂×#95	1	1	yes	characteristic
137	C₂²×C₁₉	C₄×C₄⋊C₄	1	2	yes	—
138	C₂²×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
139	C₂²×C₁₉	C₄×C₄⋊C₄	1	2	yes	—
140	C₂²×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
141	C₂²×C₁₉	#430	1	4	yes	—
142	C₄×C₁₉	C₄³	1	1	yes	characteristic
143	C₄×C₁₉	#376	1	1	yes	characteristic
144	C₄×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
145	C₄×C₁₉	C₄×C₂²⋊C₄	1	2	yes	—
146	C₄×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
147	C₄×C₁₉	C₄×C₂²⋊C₄	1	2	yes	—
148	Dic₁₉	—	2	8	no	—
149	C₂²×M₁₆	—	19	2	no	—
150	C₂²×M₁₆	C₂×D₁₉	1	2	yes	—
151	C₂³×C₈	—	19	1	no	—
152	C₂³×C₈	C₂×D₁₉	1	1	yes	characteristic
153	C₂²×C₄²	—	19	1	no	—
154	C₄×M₁₆	—	19	8	no	—
155	C₄×M₁₆	—	38	4	no	—
156	C₂×#97	—	19	4	no	—
157	C₂×#97	—	38	4	no	—
158	C₂×#97	—	38	2	no	—
159	C₂×C₄×C₈	—	38	2	no	—
160	#341	—	19	8	no	—
161	#341	—	19	4	no	—
162	#341	Dic₁₉	1	4	yes	complemented
163	C₂×C₁₉	—	2	2	no	—
164	C₂×C₁₉	—	2	2	no	—
165	C₂×C₁₉	C₄×#95	1	1	yes	characteristic
166	C₂×C₁₉	#1738	1	2	yes	—
167	C₂×C₁₉	#1778	1	2	yes	—
168	C₂×C₁₉	#1778	1	2	yes	—
169	C₂³×C₄	—	19	2	no	—
170	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	19	4	no	—
175	C₂×M₁₆	—	19	4	no	—
176	C₂×M₁₆	—	19	4	no	—
177	C₂×M₁₆	—	38	2	no	—
178	C₂×M₁₆	—	38	2	no	—
179	C₂×M₁₆	—	38	2	no	—
180	C₂×M₁₆	C₄×D₁₉	1	4	yes	—
181	C₂×M₁₆	C₄×D₁₉	1	4	yes	—
182	C₂×M₁₆	C₂×Dic₁₉	1	4	yes	—
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	—	2	2	no	—
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	—	19	4	no	—
187	C₂²×C₈	—	19	2	no	—
188	C₂²×C₈	—	38	2	no	—
189	C₂²×C₈	—	38	2	no	—
190	C₂²×C₈	—	38	1	no	—
191	C₂²×C₈	—	38	1	no	—
192	C₂²×C₈	C₄×D₁₉	1	2	yes	—
193	C₂²×C₈	C₄×D₁₉	1	2	yes	—
194	C₂²×C₈	C₂×Dic₁₉	1	2	yes	—
195	C₂×C₄²	—	19	2	no	—
196	C₂×C₄²	—	19	4	no	—
197	C₂×C₄²	—	19	4	no	—
198	C₂×C₄²	—	19	2	no	—
199	#97	—	38	8	no	—
200	#97	—	38	8	no	—
201	C₄×C₈	—	38	8	no	—
202	C₁₉	#10705	1	1	yes	characteristic, complemented, Sylow
203	C₂⁴	Q₈×D₁₉	1	1	yes	characteristic
204	C₂²×C₄	—	19	4	no	—
205	C₂²×C₄	—	19	8	no	—
206	C₂²×C₄	—	76	4	no	—
207	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
208	C₂²×C₄	C₂×C₄×D₁₉	1	2	yes	—
209	C₂²×C₄	C₂×C₄×D₁₉	1	2	yes	—
210	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
211	C₂²×C₄	D₄×D₁₉	1	2	yes	—
212	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
213	C₂²×C₄	#64543	1	2	yes	—
214	C₂²×C₄	#64543	1	1	yes	characteristic
215	C₂²×C₄	#64545	1	1	yes	characteristic
216	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
217	M₁₆	—	2	4	no	—
218	M₁₆	—	2	4	no	—
219	M₁₆	—	2	4	no	—
220	M₁₆	—	19	8	no	—
221	M₁₆	—	38	4	no	—
222	M₁₆	—	38	4	no	—
223	M₁₆	—	38	4	no	—
224	M₁₆	C₄×Dic₁₉	1	8	yes	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	19	4	no	—
233	C₂×C₈	—	38	4	no	—
234	C₂×C₈	—	38	2	no	—
235	C₂×C₈	—	38	4	no	—
236	C₂×C₈	—	38	4	no	—
237	C₂×C₈	—	38	2	no	—
238	C₂×C₈	—	38	4	no	—
239	C₂×C₈	—	38	4	no	—
240	C₂×C₈	C₄×Dic₁₉	1	4	yes	—
241	C₄²	—	19	4	no	—
242	C₄²	—	19	4	no	—
243	C₄²	—	19	4	no	—
244	C₄²	—	19	4	no	—
245	C₂³	—	2	4	no	—
246	C₂³	#10597333	1	2	yes	—
247	C₂³	#10597333	1	1	yes	characteristic
248	C₂³	C₄⋊C₄×D₁₉	1	2	yes	—
249	C₂³	C₄⋊C₄×D₁₉	1	1	yes	characteristic
250	C₂³	Q₈×Dic₁₉	1	1	yes	characteristic
251	C₂×C₄	—	19	8	no	—
252	C₂×C₄	—	38	8	no	—
253	C₂×C₄	—	76	8	no	—
254	C₂×C₄	C₄²×D₁₉	1	1	yes	characteristic
255	C₂×C₄	C₄²×D₁₉	1	1	yes	characteristic
256	C₂×C₄	C₄²×D₁₉	1	2	yes	—
257	C₂×C₄	#10597323	1	1	yes	characteristic
258	C₂×C₄	#10597323	1	2	yes	—
259	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
260	C₂×C₄	C₂²⋊C₄×D₁₉	1	2	yes	—
261	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	2	yes	—
262	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	2	yes	—
263	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	2	yes	—
264	C₂×C₄	#10597338	1	1	yes	characteristic
265	C₂×C₄	#10597338	1	2	yes	—
266	C₂×C₄	#10597339	1	1	yes	characteristic
267	C₂×C₄	#10597339	1	2	yes	—
268	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
269	C₂×C₄	C₂×C₄×Dic₁₉	1	2	yes	—
270	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
271	C₂×C₄	D₄×Dic₁₉	1	2	yes	—
272	C₈	—	2	4	no	—
273	C₈	—	2	8	no	—
274	C₈	—	38	4	no	—
275	C₈	—	38	8	no	—
276	C₂²	—	2	4	no	—
277	C₂²	—	2	4	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	#95×D₁₉	1	1	yes	characteristic
281	C₂²	#11926395	1	2	yes	—
282	C₂²	#11926395	1	1	yes	characteristic
283	C₂²	C₄⋊C₄×Dic₁₉	1	2	yes	—
284	C₂²	C₄⋊C₄×Dic₁₉	1	1	yes	characteristic
285	C₂²	#11926591	1	4	yes	—
286	C₄	—	38	8	no	—
287	C₄	C₄²×Dic₁₉	1	1	yes	characteristic
288	C₄	#11926389	1	1	yes	characteristic
289	C₄	#11926391	1	1	yes	characteristic
290	C₄	#11926391	1	2	yes	—
291	C₄	C₂²⋊C₄×Dic₁₉	1	1	yes	characteristic
292	C₄	C₂²⋊C₄×Dic₁₉	1	2	yes	—
293	C₂	—	2	2	no	—
294	C₂	—	2	2	no	—
295	C₂	#95×Dic₁₉	1	1	yes	characteristic
296	C₂	#14563360	1	2	yes	—
297	C₂	#14563540	1	2	yes	—
298	C₂	#14563543	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₉
commutator	C₄×C₁₉	C₄³
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₂×C₁₉×#341	C₂
socle	C₂³×C₁₉	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₉	C₂×Q₈
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂×#341	D₁₉
hypocenter	C₁₉	#10705
LCS	G ▷ C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#341

Complements

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

N	H ≅ G/N	type
15. #11926309	C₄	semidirect
16. #11926308	C₄	semidirect
19. C₁₉×#341	C₄	semidirect
162. #341	Dic₁₉	semidirect
202. C₁₉	#10705	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×Dic₁₉	2
C₂²×#10597272	2
C₂×#11926309	2
C₂×#11926308	2
C₂×C₁₉×#341	2
#10705	19