Subgroups — #18075855

Order 4864.

384 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14572610	C₂	1	1	yes	maximal, characteristic
2	C₄×#10597275	C₂	1	1	yes	maximal, characteristic, complemented
3	#14562161	C₂	1	2	yes	maximal
4	#14561490	C₂	1	2	yes	maximal, complemented
5	C₁₉×#1568	C₂	1	1	yes	maximal, characteristic, complemented
6	#11927078	C₂²	1	1	yes	characteristic
7	#11926674	C₄	1	1	yes	characteristic, complemented
8	C₈×Dic₃₈	C₄	1	1	yes	characteristic, complemented
9	#11926623	C₄	1	1	yes	characteristic, complemented
10	#11926615	—	2	1	no	—
11	C₈×D₇₆	C₄	1	1	yes	characteristic, complemented
12	C₈×C₂²⋊D₁₉	—	2	1	no	—
13	#11926603	—	2	1	no	—
14	#11926601	C₂²	1	1	yes	characteristic
15	C₂×#10597275	—	2	1	no	—
16	#11926498	—	2	1	no	—
17	C₄×C₄⋊Dic₁₉	C₂²	1	2	yes	—
18	#11926307	—	2	1	no	—
19	#11926282	—	2	2	no	—
20	#11926279	—	2	2	no	—
21	C₁₉×#432	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
23	C₁₉×#335	—	2	1	no	—
24	C₁₉×#334	—	2	1	no	—
25	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
28	#10597376	D₄	1	1	yes	characteristic
29	#10597375	—	2	1	no	—
30	C₄×C₂²⋊D₁₉	—	2	1	no	—
31	C₂×C₄⋊Dic₁₉	—	2	2	no	—
32	#10597370	D₄	1	1	yes	characteristic
33	C₂×C₄×Dic₁₉	—	2	2	no	—
34	C₂×#64515	—	2	1	no	—
35	C₂×C₈×D₁₉	—	2	1	no	—
36	#10597322	—	2	1	no	—
37	#10597321	Q₈	1	1	yes	characteristic
38	C₄×D₇₆	C₂×C₄	1	1	yes	characteristic
39	#10597318	—	2	1	no	—
40	#10597316	Q₈	1	1	yes	characteristic
41	C₄×Dic₃₈	C₂×C₄	1	1	yes	characteristic
42	C₁₉×C₄⋊C₈	—	2	2	no	—
43	C₁₉×C₂²⋊C₈	—	2	2	no	—
44	C₁₉×#97	—	2	2	no	—
45	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
48	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
49	C₁₉⋊#95	—	2	2	no	—
50	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
51	(C₂×D₁₉)⋊C₈	—	2	1	no	—
52	(C₂×D₁₉)⋊C₈	—	2	1	no	—
53	#10597275	—	2	4	no	—
54	#10597272	—	2	1	no	—
55	Dic₁₉⋊C₈	—	2	1	no	—
56	Dic₁₉⋊C₈	—	2	1	no	—
57	C₈×Dic₁₉	—	2	1	no	—
58	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	1	yes	characteristic
59	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	1	yes	characteristic
60	#10597260	—	2	1	no	—
61	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
64	C₂×C₂²⋊D₁₉	—	2	1	no	—
65	C₂²×Dic₁₉	—	4	2	no	—
66	C₂×D₇₆	C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×D₁₉	—	2	1	no	—
68	C₂×Dic₃₈	C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₁₉	—	2	2	no	—
70	C₂×C₈×C₁₉	—	4	1	no	—
71	C₂×C₈×C₁₉	—	4	2	no	—
72	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
76	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
77	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
79	C₄²×C₁₉	#41	1	1	yes	characteristic
80	C₂²⋊Dic₁₉	—	2	1	no	—
81	(C₂×D₁₉)⋊C₄	—	2	1	no	—
82	(C₂×D₁₉)⋊C₄	—	2	1	no	—
83	(C₂×D₁₉)⋊C₄	—	2	1	no	—
84	(C₂×D₁₉)⋊C₄	—	2	1	no	—
85	C₄⋊Dic₁₉	—	2	4	no	—
86	C₄⋊Dic₁₉	—	4	2	no	—
87	C₄⋊Dic₁₉	C₄⋊C₄	1	1	yes	characteristic
88	C₄⋊Dic₁₉	C₄⋊C₄	1	1	yes	characteristic
89	Dic₁₉⋊C₄	—	2	1	no	—
90	Dic₁₉⋊C₄	—	2	1	no	—
91	Dic₁₉⋊C₄	—	2	1	no	—
92	Dic₁₉⋊C₄	—	2	1	no	—
93	C₄×Dic₁₉	—	2	1	no	—
94	C₄×Dic₁₉	—	8	2	no	—
95	C₂×C₁₉⋊C₈	—	2	1	no	—
96	C₂×C₁₉⋊C₈	—	2	1	no	—
97	#64515	—	4	2	no	—
98	C₈×D₁₉	—	4	2	no	—
99	#8697	—	19	1	no	maximal, Sylow
100	C₂³×C₁₉	#121	1	1	yes	characteristic
101	C₂²×D₁₉	—	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	—	4	1	no	—
107	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
110	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
111	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
112	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
113	C₂²⋊D₁₉	—	4	1	no	—
114	C₂²⋊D₁₉	—	4	1	no	—
115	C₂×Dic₁₉	—	2	1	no	—
116	C₂×Dic₁₉	—	2	1	no	—
117	C₂×Dic₁₉	—	2	1	no	—
118	C₂×Dic₁₉	—	4	4	no	—
119	C₂×Dic₁₉	—	8	2	no	—
120	C₂×Dic₁₉	—	8	2	no	—
121	D₇₆	—	4	1	no	—
122	C₄×D₁₉	—	4	1	no	—
123	C₄×D₁₉	—	4	1	no	—
124	Dic₃₈	—	4	1	no	—
125	C₈×C₁₉	—	2	1	no	—
126	C₈×C₁₉	—	2	1	no	—
127	C₈×C₁₉	—	2	1	no	—
128	C₈×C₁₉	—	8	2	no	—
129	C₈×C₁₉	#104	1	2	yes	—
130	C₁₉⋊C₈	—	4	1	no	—
131	C₁₉⋊C₈	—	4	1	no	—
132	#2966	—	19	1	no	—
133	C₄×#107	—	19	1	no	—
134	#1568	D₁₉	1	1	yes	characteristic, complemented
135	#1509	—	19	2	no	—
136	#1285	—	19	2	no	—
137	C₂²×C₁₉	—	2	1	no	—
138	C₂²×C₁₉	—	2	1	no	—
139	C₂²×C₁₉	#380	1	1	yes	characteristic
140	C₂²×C₁₉	#472	1	1	yes	characteristic
141	C₂²×C₁₉	#473	1	1	yes	characteristic
142	C₂×D₁₉	—	4	1	no	—
143	C₂×D₁₉	—	4	1	no	—
144	C₂×D₁₉	—	4	1	no	—
145	C₄×C₁₉	—	2	1	no	—
146	C₄×C₁₉	—	2	1	no	—
147	C₄×C₁₉	—	2	1	no	—
148	C₄×C₁₉	—	2	1	no	—
149	C₄×C₁₉	—	2	1	no	—
150	C₄×C₁₉	C₂×#104	1	1	yes	characteristic
151	C₄×C₁₉	#421	1	1	yes	characteristic
152	Dic₁₉	—	4	1	no	—
153	Dic₁₉	—	4	1	no	—
154	Dic₁₉	—	4	1	no	—
155	Dic₁₉	—	8	4	no	—
156	#529	—	19	1	no	—
157	#446	—	19	1	no	—
158	C₈×Q₈	—	19	1	no	—
159	#436	—	19	1	no	—
160	#435	—	38	1	no	—
161	C₈×D₄	—	19	1	no	—
162	C₈×D₄	—	38	1	no	—
163	#433	—	38	1	no	—
164	#432	—	19	1	no	—
165	#432	C₂×D₁₉	1	2	yes	—
166	C₂×#107	—	38	1	no	—
167	#405	—	38	1	no	—
168	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
169	C₄×C₄⋊C₄	—	19	2	no	—
170	#340	—	38	1	no	—
171	#335	—	2	1	no	—
172	#334	—	2	1	no	—
173	#328	—	38	2	no	—
174	#327	—	38	2	no	—
175	C₂×C₁₉	—	2	1	no	—
176	C₂×C₁₉	—	2	1	no	—
177	C₂×C₁₉	#1806	1	1	yes	characteristic
178	C₂×C₁₉	#1812	1	1	yes	characteristic
179	C₂×C₁₉	#1816	1	1	yes	characteristic
180	D₁₉	—	8	1	no	—
181	C₂×M₁₆	—	38	1	no	—
182	C₂²×C₈	—	38	1	no	—
183	C₂²×C₈	D₇₆	1	1	yes	characteristic
184	C₂²×C₈	D₇₆	1	1	yes	characteristic
185	#126	—	38	1	no	—
186	#125	—	19	1	no	—
187	#124	—	19	1	no	—
188	#123	—	38	1	no	—
189	C₂²⋊Q₈	—	19	1	no	—
190	#121	—	19	1	no	—
191	C₄×Q₈	—	19	1	no	—
192	C₄×D₄	—	19	1	no	—
193	C₄×D₄	—	38	1	no	—
194	#117	—	38	1	no	—
195	C₂×C₄⋊C₄	—	38	2	no	—
196	C₂×C₄²	—	38	2	no	—
197	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
198	#107	—	38	4	no	—
199	C₄⋊C₈	—	2	2	no	—
200	C₄⋊C₈	—	19	1	no	—
201	C₄⋊C₈	—	19	1	no	—
202	C₄⋊C₈	—	38	1	no	—
203	C₄⋊C₈	—	38	1	no	—
204	C₂²⋊C₈	—	2	2	no	—
205	C₂²⋊C₈	—	38	1	no	—
206	C₂²⋊C₈	—	38	1	no	—
207	C₂²⋊C₈	—	38	1	no	—
208	#97	—	2	2	no	—
209	#97	—	38	1	no	—
210	#97	—	38	1	no	—
211	C₄×C₈	—	38	1	no	—
212	C₄×C₈	C₄×D₁₉	1	1	yes	characteristic
213	C₄×C₈	C₄×D₁₉	1	1	yes	characteristic
214	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
215	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
216	#95	—	38	2	no	—
217	C₁₉	#8697	1	1	yes	characteristic, complemented, Sylow
218	C₂×Q₈	—	19	1	no	—
219	C₂×D₄	—	19	1	no	—
220	C₂×D₄	—	38	1	no	—
221	C₂²×C₄	—	38	1	no	—
222	C₂²×C₄	—	76	2	no	—
223	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
224	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
225	C₂²×C₄	#64545	1	1	yes	characteristic
226	M₁₆	—	76	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	4	1	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	38	1	no	—
231	C₂×C₈	—	38	1	no	—
232	C₂×C₈	—	76	2	no	—
233	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
234	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
235	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
236	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
237	C₄⋊C₄	—	19	1	no	—
238	C₄⋊C₄	—	19	1	no	—
239	C₄⋊C₄	—	38	4	no	—
240	C₄⋊C₄	—	38	1	no	—
241	C₄⋊C₄	—	38	1	no	—
242	C₄⋊C₄	—	38	1	no	—
243	C₄⋊C₄	—	38	1	no	—
244	C₄⋊C₄	—	76	2	no	—
245	C₂²⋊C₄	—	38	1	no	—
246	C₂²⋊C₄	—	38	1	no	—
247	C₂²⋊C₄	—	38	1	no	—
248	C₂²⋊C₄	—	38	1	no	—
249	C₂²⋊C₄	—	38	1	no	—
250	C₄²	—	38	1	no	—
251	C₄²	—	152	2	no	—
252	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
253	C₄²	#64543	1	1	yes	characteristic
254	C₄²	Q₈×D₁₉	1	1	yes	characteristic
255	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
256	C₂³	—	38	1	no	—
257	C₂³	#10597341	1	1	yes	characteristic
258	Q₈	—	76	1	no	—
259	D₄	—	76	1	no	—
260	D₄	—	76	1	no	—
261	D₄	—	76	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	—	4	1	no	—
267	C₂×C₄	—	38	1	no	—
268	C₂×C₄	—	38	1	no	—
269	C₂×C₄	—	38	1	no	—
270	C₂×C₄	—	76	1	no	—
271	C₂×C₄	—	76	1	no	—
272	C₂×C₄	—	76	4	no	—
273	C₂×C₄	—	152	2	no	—
274	C₂×C₄	—	152	2	no	—
275	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
276	C₂×C₄	#10597338	1	1	yes	characteristic
277	C₂×C₄	#10597343	1	1	yes	characteristic
278	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
279	C₂×C₄	#10597384	1	1	yes	characteristic
280	C₂×C₄	#10597392	1	1	yes	characteristic
281	C₈	—	2	1	no	—
282	C₈	—	2	1	no	—
283	C₈	—	2	1	no	—
284	C₈	—	8	2	no	—
285	C₈	—	76	1	no	—
286	C₈	—	76	1	no	—
287	C₈	#10597282	1	2	yes	—
288	C₂²	—	2	1	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	76	1	no	—
291	C₂²	—	76	1	no	—
292	C₂²	—	76	1	no	—
293	C₂²	#11926401	1	1	yes	characteristic
294	C₂²	#11926815	1	1	yes	characteristic
295	C₂²	#11926820	1	1	yes	characteristic
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	76	1	no	—
302	C₄	—	76	1	no	—
303	C₄	—	76	1	no	—
304	C₄	—	152	4	no	—
305	C₄	C₂×#10597282	1	1	yes	characteristic
306	C₄	#11926560	1	1	yes	characteristic
307	C₂	—	2	1	no	—
308	C₂	—	2	1	no	—
309	C₂	—	152	1	no	—
310	C₂	#14563652	1	1	yes	characteristic
311	C₂	#14563678	1	1	yes	characteristic
312	C₂	#14563696	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11926401
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1568	C₂
socle	C₂²×C₁₉	#380
prime radical	#10597376	D₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1568	D₁₉
hypocenter	C₁₉	#8697
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1568

Complements

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

N	H ≅ G/N	type
2. C₄×#10597275	C₂	semidirect
4. #14561490	C₂	semidirect
5. C₁₉×#1568	C₂	semidirect
7. #11926674	C₄	semidirect
8. C₈×Dic₃₈	C₄	semidirect
9. #11926623	C₄	semidirect
11. C₈×D₇₆	C₄	semidirect
134. #1568	D₁₉	semidirect
217. C₁₉	#8697	semidirect

Maximal subgroups

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

subgroup	index
#14572610	2
C₄×#10597275	2
#14562161	2
#14561490	2
C₁₉×#1568	2
#8697	19