Subgroups — #18081830

Order 4864.

458 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	#14564584	C₂	1	1	yes	maximal, characteristic
2	#14564559	C₂	1	1	yes	maximal, characteristic
3	#14563502	C₂	1	1	yes	maximal, characteristic
4	C₂×#11926297	C₂	1	1	yes	maximal, characteristic
5	C₂×#11926293	C₂	1	1	yes	maximal, characteristic
6	C₂×#11926290	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₉×#498	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
11	C₂×#10597292	—	2	2	no	—
12	C₂×#10597272	—	2	1	no	—
13	C₄×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
14	#11926306	—	2	2	no	—
15	#11926298	—	2	1	no	—
16	#11926297	—	2	2	no	—
17	#11926293	—	2	2	no	—
18	#11926290	—	2	2	no	—
19	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
20	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
21	C₁₉×#498	—	2	2	no	—
22	C₁₉×#498	C₄	1	4	yes	complemented
23	C₂×C₁₉×#106	—	2	1	no	—
24	C₂×C₁₉×#106	—	2	1	no	—
25	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
26	C₂²×Q₈×C₁₉	—	2	1	no	—
27	C₂²×Q₈×C₁₉	—	2	1	no	—
28	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
30	C₁₉×C₄⋊Q₈	—	2	1	no	—
31	C₁₉×C₄⋊Q₈	—	2	1	no	—
32	C₁₉×C₄⋊Q₈	—	2	1	no	—
33	C₁₉×C₄⋊Q₈	—	2	1	no	—
34	C₁₉×C₄⋊Q₈	—	2	1	no	—
35	C₁₉×C₄⋊Q₈	—	2	1	no	—
36	C₁₉×C₄⋊Q₈	C₂×C₄	1	2	yes	—
37	C₁₉×C₄⋊Q₈	C₂×C₄	1	2	yes	—
38	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
39	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
40	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
41	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
42	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
43	C₂×C₄⋊Dic₁₉	D₄	1	2	yes	—
44	C₂×C₄×Dic₁₉	—	2	1	no	—
45	C₂²×C₁₉⋊C₈	—	2	1	no	—
46	C₂²×C₁₉⋊C₈	—	2	1	no	—
47	C₂²×C₁₉⋊C₈	—	2	1	no	—
48	C₁₉×#106	—	2	4	no	—
49	C₁₉×#106	—	2	4	no	—
50	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
51	C₄×C₈×C₁₉	D₄	1	2	yes	—
52	#10597292	—	4	4	no	—
53	C₁₉⋊#95	—	2	1	no	—
54	#10597272	—	4	2	no	—
55	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
56	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
57	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
58	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
59	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
60	C₄⋊(C₁₉⋊C₈)	D₄	1	2	yes	—
61	C₂×Q₈×C₁₉	—	2	2	no	—
62	C₂×Q₈×C₁₉	—	2	2	no	—
63	C₂×Q₈×C₁₉	—	4	1	no	—
64	C₂×Q₈×C₁₉	—	4	1	no	—
65	C₂×Q₈×C₁₉	—	4	2	no	—
66	C₂×Q₈×C₁₉	—	4	2	no	—
67	C₂×Q₈×C₁₉	—	8	2	no	—
68	C₂²×C₄×C₁₉	—	4	1	no	—
69	C₂²×C₄×C₁₉	—	4	1	no	—
70	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
73	C₂²×Dic₁₉	—	2	2	no	—
74	C₂×C₈×C₁₉	—	2	1	no	—
75	C₂×C₈×C₁₉	—	2	1	no	—
76	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
77	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
78	C₂×C₈×C₁₉	SD₁₆	1	2	yes	—
79	C₂×C₈×C₁₉	Q₁₆	1	2	yes	—
80	C₄⋊C₄×C₁₉	—	2	2	no	—
81	C₄⋊C₄×C₁₉	—	2	2	no	—
82	C₄⋊C₄×C₁₉	—	2	2	no	—
83	C₄⋊C₄×C₁₉	—	2	2	no	—
84	C₄⋊C₄×C₁₉	—	4	1	no	—
85	C₄⋊C₄×C₁₉	—	4	1	no	—
86	C₄⋊C₄×C₁₉	—	4	1	no	—
87	C₄⋊C₄×C₁₉	—	4	1	no	—
88	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₁₉	#41	1	1	yes	characteristic
92	C₄⋊Dic₁₉	—	2	2	no	—
93	C₄⋊Dic₁₉	—	2	2	no	—
94	C₄×Dic₁₉	—	4	1	no	—
95	C₄×Dic₁₉	—	4	1	no	—
96	C₂×C₁₉⋊C₈	—	2	2	no	—
97	C₂×C₁₉⋊C₈	—	4	1	no	—
98	C₂×C₁₉⋊C₈	—	4	1	no	—
99	C₂×C₁₉⋊C₈	—	4	1	no	—
100	C₂×C₁₉⋊C₈	—	4	1	no	—
101	C₂×C₁₉⋊C₈	—	4	2	no	—
102	C₂×C₁₉⋊C₈	—	4	2	no	—
103	#9741	—	19	1	no	maximal, Sylow
104	C₂³×C₁₉	#121	1	1	yes	characteristic
105	Q₈×C₁₉	—	4	4	no	—
106	Q₈×C₁₉	—	8	2	no	—
107	Q₈×C₁₉	—	8	2	no	—
108	Q₈×C₁₉	—	8	1	no	—
109	Q₈×C₁₉	—	8	1	no	—
110	C₂×C₄×C₁₉	—	2	2	no	—
111	C₂×C₄×C₁₉	—	4	2	no	—
112	C₂×C₄×C₁₉	—	4	2	no	—
113	C₂×C₄×C₁₉	—	8	2	no	—
114	C₂×C₄×C₁₉	—	8	2	no	—
115	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
119	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
120	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
121	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
122	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
123	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
124	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
125	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
126	C₂×C₄×C₁₉	#137	1	2	yes	—
127	C₂×Dic₁₉	—	4	2	no	—
128	C₂×Dic₁₉	—	4	2	no	—
129	C₂×Dic₁₉	—	4	2	no	—
130	C₈×C₁₉	—	2	2	no	—
131	C₈×C₁₉	#103	1	4	yes	complemented
132	C₁₉⋊C₈	—	4	2	no	—
133	C₁₉⋊C₈	—	8	1	no	—
134	C₁₉⋊C₈	—	8	1	no	—
135	C₂×#498	D₁₉	1	1	yes	characteristic
136	#1987	—	19	1	no	—
137	#1981	—	19	1	no	—
138	#1770	—	19	1	no	—
139	C₂×#335	—	19	1	no	—
140	C₂×#333	—	19	1	no	—
141	C₂×#332	—	19	1	no	—
142	C₂²×C₁₉	#390	1	1	yes	characteristic
143	C₂²×C₁₉	#461	1	1	yes	characteristic
144	C₂²×C₁₉	#462	1	1	yes	characteristic
145	C₂²×C₁₉	#468	1	2	yes	—
146	C₂²×C₁₉	#471	1	2	yes	—
147	C₄×C₁₉	—	2	1	no	—
148	C₄×C₁₉	—	2	1	no	—
149	C₄×C₁₉	—	8	2	no	—
150	C₄×C₁₉	—	8	2	no	—
151	C₄×C₁₉	C₂×#103	1	1	yes	characteristic
152	C₄×C₁₉	#419	1	1	yes	characteristic
153	C₄×C₁₉	#440	1	2	yes	—
154	C₄×C₁₉	#461	1	1	yes	characteristic
155	C₄×C₁₉	#462	1	1	yes	characteristic
156	C₄×C₁₉	#496	1	1	yes	characteristic
157	C₄×C₁₉	#497	1	1	yes	characteristic
158	Dic₁₉	—	8	2	no	—
159	C₂×C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
160	C₂×C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
161	#498	—	2	2	no	—
162	#498	Dic₁₉	1	4	yes	complemented
163	C₂×#106	—	2	1	no	—
164	C₂×#106	—	2	1	no	—
165	C₂×C₄⋊C₈	—	19	1	no	—
166	C₂×C₄⋊C₈	—	19	1	no	—
167	C₂×C₄⋊C₈	—	19	1	no	—
168	C₂×#103	—	38	2	no	—
169	C₂×#97	—	38	1	no	—
170	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
171	C₄×C₄⋊C₄	—	19	1	no	—
172	#340	—	38	2	no	—
173	#336	—	38	1	no	—
174	#335	—	38	2	no	—
175	#333	—	38	2	no	—
176	#332	—	38	2	no	—
177	C₂×C₁₉	#1691	1	2	yes	—
178	C₂×C₁₉	#1692	1	2	yes	—
179	C₂×C₁₉	#1973	1	1	yes	characteristic
180	C₂×C₁₉	#1977	1	1	yes	characteristic
181	C₂×C₁₉	#1981	1	1	yes	characteristic
182	C₂²×Q₈	—	2	1	no	—
183	C₂²×Q₈	—	2	1	no	—
184	C₂²×C₈	—	38	1	no	—
185	C₂²×C₈	—	38	1	no	—
186	C₂²×C₈	—	38	1	no	—
187	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
188	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
189	C₄⋊Q₈	—	2	1	no	—
190	C₄⋊Q₈	—	2	1	no	—
191	C₄⋊Q₈	—	2	1	no	—
192	C₄⋊Q₈	—	2	1	no	—
193	C₄⋊Q₈	—	2	1	no	—
194	C₄⋊Q₈	—	2	1	no	—
195	C₄⋊Q₈	C₂×Dic₁₉	1	2	yes	—
196	C₄⋊Q₈	C₂×Dic₁₉	1	2	yes	—
197	C₂×C₄⋊C₄	—	2	1	no	—
198	C₂×C₄⋊C₄	—	2	1	no	—
199	C₂×C₄⋊C₄	—	2	1	no	—
200	C₂×C₄⋊C₄	—	2	1	no	—
201	C₂×C₄⋊C₄	—	19	2	no	—
202	C₂×C₄²	—	38	1	no	—
203	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
204	#106	—	2	4	no	—
205	#106	—	2	4	no	—
206	C₄⋊C₈	—	19	2	no	—
207	C₄⋊C₈	—	38	1	no	—
208	C₄⋊C₈	—	38	1	no	—
209	C₄⋊C₈	—	38	1	no	—
210	C₄⋊C₈	—	38	1	no	—
211	C₄⋊C₈	—	38	1	no	—
212	#103	—	76	4	no	—
213	#97	—	76	2	no	—
214	C₄×C₈	C₂×Dic₁₉	1	2	yes	—
215	C₄×C₈	C₂²⋊D₁₉	1	2	yes	—
216	#95	—	38	1	no	—
217	C₁₉	#9741	1	1	yes	characteristic, complemented, Sylow
218	C₂×Q₈	—	2	2	no	—
219	C₂×Q₈	—	2	2	no	—
220	C₂×Q₈	—	4	1	no	—
221	C₂×Q₈	—	4	1	no	—
222	C₂×Q₈	—	4	2	no	—
223	C₂×Q₈	—	4	2	no	—
224	C₂×Q₈	—	8	2	no	—
225	C₂²×C₄	—	4	1	no	—
226	C₂²×C₄	—	4	1	no	—
227	C₂²×C₄	—	38	2	no	—
228	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
229	C₂²×C₄	#64543	1	1	yes	characteristic
230	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	2	1	no	—
233	C₂×C₈	—	38	2	no	—
234	C₂×C₈	—	76	1	no	—
235	C₂×C₈	—	76	1	no	—
236	C₂×C₈	—	76	1	no	—
237	C₂×C₈	—	76	1	no	—
238	C₂×C₈	—	76	2	no	—
239	C₂×C₈	—	76	2	no	—
240	C₂×C₈	#64527	1	2	yes	—
241	C₂×C₈	#64528	1	2	yes	—
242	C₂×C₈	C₂²⋊Dic₁₉	1	2	yes	—
243	C₂×C₈	C₂²⋊Dic₁₉	1	2	yes	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	4	1	no	—
249	C₄⋊C₄	—	4	1	no	—
250	C₄⋊C₄	—	4	1	no	—
251	C₄⋊C₄	—	4	1	no	—
252	C₄⋊C₄	—	38	2	no	—
253	C₄⋊C₄	—	38	2	no	—
254	C₄²	—	76	1	no	—
255	C₄²	—	76	1	no	—
256	C₄²	D₄×D₁₉	1	1	yes	characteristic
257	C₄²	#64543	1	1	yes	characteristic
258	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
259	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
260	C₂³	#10597384	1	1	yes	characteristic
261	Q₈	—	4	4	no	—
262	Q₈	—	8	2	no	—
263	Q₈	—	8	2	no	—
264	Q₈	—	8	1	no	—
265	Q₈	—	8	1	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	8	2	no	—
270	C₂×C₄	—	8	2	no	—
271	C₂×C₄	—	76	2	no	—
272	C₂×C₄	—	76	2	no	—
273	C₂×C₄	—	76	2	no	—
274	C₂×C₄	#10597360	1	2	yes	—
275	C₂×C₄	#10597361	1	2	yes	—
276	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
277	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
278	C₂×C₄	#10597382	1	1	yes	characteristic
279	C₂×C₄	#10597384	1	1	yes	characteristic
280	C₂×C₄	#10597386	1	1	yes	characteristic
281	C₂×C₄	C₂×#64527	1	1	yes	characteristic
282	C₂×C₄	#10597388	1	1	yes	characteristic
283	C₂×C₄	#10597388	1	1	yes	characteristic
284	C₂×C₄	C₂×#64528	1	1	yes	characteristic
285	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
286	C₈	—	2	2	no	—
287	C₈	—	76	2	no	—
288	C₈	—	152	1	no	—
289	C₈	—	152	1	no	—
290	C₈	#10597292	1	4	yes	complemented
291	C₂²	#11926447	1	1	yes	characteristic
292	C₂²	#11926746	1	1	yes	characteristic
293	C₂²	#11926751	1	1	yes	characteristic
294	C₂²	#11926787	1	2	yes	—
295	C₂²	#11926804	1	2	yes	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	8	2	no	—
299	C₄	—	8	2	no	—
300	C₄	—	152	2	no	—
301	C₄	C₂×#10597292	1	1	yes	characteristic
302	C₄	#11926553	1	1	yes	characteristic
303	C₄	#11926645	1	2	yes	—
304	C₄	#11926746	1	1	yes	characteristic
305	C₄	#11926751	1	1	yes	characteristic
306	C₄	#11926930	1	1	yes	characteristic
307	C₄	#11926935	1	1	yes	characteristic
308	C₂	#14563131	1	2	yes	—
309	C₂	#14563138	1	2	yes	—
310	C₂	#14564518	1	1	yes	characteristic
311	C₂	#14564540	1	1	yes	characteristic
312	C₂	#14564559	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10597384
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₁₉×#498	C₂
socle, prime radical	C₂³×C₁₉	#121
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×#498	D₁₉
hypocenter	C₁₉	#9741
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#498

Complements

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

N	H ≅ G/N	type
22. C₁₉×#498	C₄	semidirect
131. C₈×C₁₉	#103	semidirect
162. #498	Dic₁₉	semidirect
217. C₁₉	#9741	semidirect
290. C₈	#10597292	semidirect

Maximal subgroups

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

subgroup	index
#14564584	2
#14564559	2
#14563502	2
C₂×#11926297	2
C₂×#11926293	2
C₂×#11926290	2
C₂×C₁₉×#498	2
#9741	19