Subgroups — #18086742

Order 4864.

622 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14564563	C₂	1	2	yes	maximal, complemented
2	#14564353	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11926288	C₂	1	2	yes	maximal, complemented
4	#14561470	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₁₉×#493	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
8	C₂×Dic₁₉⋊C₈	—	2	1	no	—
9	C₂×#10597289	—	2	2	no	—
10	C₂×#10597289	—	2	2	no	—
11	C₄×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
12	#11926298	—	2	1	no	—
13	#11926288	—	2	4	no	—
14	C₂²×D₈×C₁₉	—	2	1	no	—
15	C₂²×D₈×C₁₉	—	2	1	no	—
16	C₂×C₁₉×#127	C₂²	1	2	yes	—
17	C₁₉×#493	—	2	4	no	—
18	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
19	C₂²×D₄×C₁₉	—	2	2	no	—
20	C₂²×D₄×C₁₉	—	2	2	no	—
21	C₂²×D₄×C₁₉	—	2	2	no	—
22	C₂×D₈×C₁₉	—	2	4	no	—
23	C₂×D₈×C₁₉	—	2	4	no	—
24	C₂×D₈×C₁₉	—	4	2	no	—
25	C₂×D₈×C₁₉	—	4	2	no	—
26	C₂×D₈×C₁₉	—	4	2	no	—
27	C₂×D₈×C₁₉	—	4	2	no	—
28	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
30	C₁₉×#127	—	2	2	no	—
31	C₁₉×#127	—	2	2	no	—
32	C₁₉×#127	—	2	2	no	—
33	C₁₉×#127	D₄	1	4	yes	—
34	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
35	C₂×C₄⋊Dic₁₉	D₄	1	1	yes	characteristic
36	C₂×C₄⋊Dic₁₉	D₄	1	1	yes	characteristic
37	C₂×C₄×Dic₁₉	—	2	1	no	—
38	C₂²×C₁₉⋊C₈	—	2	2	no	—
39	C₂²×C₁₉⋊C₈	—	2	1	no	—
40	C₄×C₈×C₁₉	—	2	2	no	—
41	#10597289	—	4	4	no	—
42	#10597289	—	4	4	no	—
43	C₁₉⋊#95	—	2	1	no	—
44	Dic₁₉⋊C₈	—	4	2	no	—
45	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
46	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
47	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
48	C₂⁴×C₁₉	—	4	2	no	—
49	C₂×D₄×C₁₉	—	2	4	no	—
50	C₂×D₄×C₁₉	—	2	4	no	—
51	C₂×D₄×C₁₉	—	2	4	no	—
52	C₂×D₄×C₁₉	—	4	2	no	—
53	C₂×D₄×C₁₉	—	4	2	no	—
54	C₂×D₄×C₁₉	—	4	2	no	—
55	C₂×D₄×C₁₉	—	4	2	no	—
56	C₂×D₄×C₁₉	—	4	2	no	—
57	C₂×D₄×C₁₉	—	4	2	no	—
58	C₂×D₄×C₁₉	—	4	2	no	—
59	C₂×D₄×C₁₉	—	4	2	no	—
60	C₂×D₄×C₁₉	—	4	2	no	—
61	C₂×D₄×C₁₉	—	4	2	no	—
62	C₂×D₄×C₁₉	—	4	2	no	—
63	C₂×D₄×C₁₉	—	8	2	no	—
64	C₂×D₄×C₁₉	—	8	2	no	—
65	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
68	C₂²×Dic₁₉	—	2	1	no	—
69	C₂²×Dic₁₉	—	2	1	no	—
70	D₈×C₁₉	—	4	8	no	—
71	D₈×C₁₉	—	4	8	no	—
72	C₂×C₈×C₁₉	—	2	1	no	—
73	C₂×C₈×C₁₉	—	2	1	no	—
74	C₂×C₈×C₁₉	—	2	1	no	—
75	C₂×C₈×C₁₉	—	2	1	no	—
76	C₂×C₈×C₁₉	D₈	1	2	yes	—
77	C₂×C₈×C₁₉	SD₁₆	1	2	yes	—
78	C₄²×C₁₉	C₂×D₄	1	2	yes	—
79	C₄²×C₁₉	#41	1	2	yes	—
80	C₄⋊Dic₁₉	—	2	1	no	—
81	C₄⋊Dic₁₉	—	2	2	no	—
82	C₄⋊Dic₁₉	D₈	1	2	yes	complemented
83	C₄×Dic₁₉	—	4	1	no	—
84	C₄×Dic₁₉	—	4	1	no	—
85	C₂×C₁₉⋊C₈	—	4	1	no	—
86	C₂×C₁₉⋊C₈	—	4	2	no	—
87	C₂×C₁₉⋊C₈	—	4	1	no	—
88	C₂×C₁₉⋊C₈	—	4	2	no	—
89	C₂×C₁₉⋊C₈	—	4	2	no	—
90	C₂×C₁₉⋊C₈	—	4	1	no	—
91	#10536	—	19	1	no	maximal, Sylow
92	C₂³×C₁₉	—	4	4	no	—
93	C₂³×C₁₉	—	8	2	no	—
94	C₂³×C₁₉	—	8	2	no	—
95	C₂³×C₁₉	—	8	2	no	—
96	C₂³×C₁₉	—	8	2	no	—
97	C₂³×C₁₉	—	8	2	no	—
98	C₂³×C₁₉	—	8	2	no	—
99	C₂³×C₁₉	#121	1	1	yes	characteristic
100	D₄×C₁₉	—	4	4	no	—
101	D₄×C₁₉	—	4	4	no	—
102	D₄×C₁₉	—	4	4	no	—
103	D₄×C₁₉	—	4	4	no	—
104	D₄×C₁₉	—	8	4	no	—
105	D₄×C₁₉	—	8	4	no	—
106	D₄×C₁₉	—	8	2	no	—
107	D₄×C₁₉	—	8	2	no	—
108	D₄×C₁₉	—	8	2	no	—
109	D₄×C₁₉	—	8	2	no	—
110	C₂×C₄×C₁₉	—	2	1	no	—
111	C₂×C₄×C₁₉	—	2	1	no	—
112	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
113	C₂×C₄×C₁₉	#121	1	2	yes	—
114	C₂×C₄×C₁₉	#123	1	2	yes	—
115	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
116	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
117	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
119	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
120	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
121	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
122	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
123	C₂×Dic₁₉	—	2	2	no	—
124	C₂×Dic₁₉	—	4	1	no	—
125	C₂×Dic₁₉	—	4	1	no	—
126	C₂×Dic₁₉	—	4	1	no	—
127	C₂×Dic₁₉	—	4	1	no	—
128	C₂×Dic₁₉	—	4	1	no	—
129	C₈×C₁₉	—	2	2	no	—
130	C₈×C₁₉	—	2	2	no	—
131	C₁₉⋊C₈	—	8	1	no	—
132	C₁₉⋊C₈	—	8	2	no	—
133	C₂×#493	D₁₉	1	1	yes	characteristic
134	#1982	—	19	2	no	—
135	#1939	—	19	1	no	—
136	C₂×#331	—	19	2	no	—
137	#1277	—	19	1	no	—
138	C₂²×C₁₉	—	8	4	no	—
139	C₂²×C₁₉	—	8	4	no	—
140	C₂²×C₁₉	—	8	4	no	—
141	C₂²×C₁₉	—	16	2	no	—
142	C₂²×C₁₉	—	16	2	no	—
143	C₂²×C₁₉	—	16	2	no	—
144	C₂²×C₁₉	—	16	2	no	—
145	C₂²×C₁₉	#394	1	1	yes	characteristic
146	C₂²×C₁₉	#459	1	1	yes	characteristic
147	C₂²×C₁₉	#460	1	1	yes	characteristic
148	C₂²×C₁₉	#466	1	1	yes	characteristic
149	C₂²×C₁₉	#469	1	1	yes	characteristic
150	C₂²×C₁₉	#472	1	1	yes	characteristic
151	C₂²×C₁₉	#472	1	1	yes	characteristic
152	C₄×C₁₉	—	2	2	no	—
153	C₄×C₁₉	#447	1	1	yes	characteristic
154	C₄×C₁₉	#450	1	1	yes	characteristic
155	C₄×C₁₉	#459	1	2	yes	—
156	C₄×C₁₉	#463	1	2	yes	—
157	C₄×C₁₉	#486	1	1	yes	characteristic
158	C₄×C₁₉	#490	1	1	yes	characteristic
159	Dic₁₉	—	4	2	no	—
160	Dic₁₉	—	8	1	no	—
161	C₂²×D₈	—	2	1	no	—
162	C₂²×D₈	—	2	1	no	—
163	C₂×#127	C₂×D₁₉	1	2	yes	—
164	#493	—	2	4	no	—
165	C₂×C₄⋊C₈	—	19	1	no	—
166	C₂×C₄⋊C₈	—	19	2	no	—
167	C₂×C₄⋊C₈	—	38	1	no	—
168	C₂×#102	—	38	2	no	—
169	C₂×#102	—	38	2	no	—
170	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
171	C₄×C₄⋊C₄	—	19	1	no	—
172	#336	—	38	1	no	—
173	#331	—	38	4	no	—
174	C₂×C₁₉	—	16	4	no	—
175	C₂×C₁₉	#1667	1	1	yes	characteristic
176	C₂×C₁₉	#1668	1	1	yes	characteristic
177	C₂×C₁₉	#1687	1	1	yes	characteristic
178	C₂×C₁₉	#1688	1	1	yes	characteristic
179	C₂×C₁₉	#2041	1	1	yes	characteristic
180	C₂×C₁₉	#2045	1	1	yes	characteristic
181	C₂×C₁₉	#2054	1	1	yes	characteristic
182	C₂²×D₄	—	2	2	no	—
183	C₂²×D₄	—	2	2	no	—
184	C₂²×D₄	—	2	2	no	—
185	C₂×D₈	—	2	4	no	—
186	C₂×D₈	—	2	4	no	—
187	C₂×D₈	—	4	2	no	—
188	C₂×D₈	—	4	2	no	—
189	C₂×D₈	—	4	2	no	—
190	C₂×D₈	—	4	2	no	—
191	C₂²×C₈	—	38	2	no	—
192	C₂²×C₈	—	38	1	no	—
193	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
194	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
195	#127	—	2	2	no	—
196	#127	—	2	2	no	—
197	#127	—	2	2	no	—
198	#127	C₂²⋊D₁₉	1	4	yes	—
199	C₂×C₄⋊C₄	—	19	1	no	—
200	C₂×C₄⋊C₄	—	19	1	no	—
201	C₂×C₄²	—	38	1	no	—
202	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
203	C₄⋊C₈	—	38	2	no	—
204	C₄⋊C₈	—	38	2	no	—
205	C₄⋊C₈	—	38	2	no	—
206	C₄⋊C₈	—	76	2	no	—
207	#102	—	76	4	no	—
208	#102	—	76	4	no	—
209	C₄×C₈	—	2	2	no	—
210	#95	—	38	1	no	—
211	C₁₉	#10536	1	1	yes	characteristic, complemented, Sylow
212	C₂⁴	—	4	2	no	—
213	C₂×D₄	—	2	4	no	—
214	C₂×D₄	—	2	4	no	—
215	C₂×D₄	—	2	4	no	—
216	C₂×D₄	—	4	2	no	—
217	C₂×D₄	—	4	2	no	—
218	C₂×D₄	—	4	2	no	—
219	C₂×D₄	—	4	2	no	—
220	C₂×D₄	—	4	2	no	—
221	C₂×D₄	—	4	2	no	—
222	C₂×D₄	—	4	2	no	—
223	C₂×D₄	—	4	2	no	—
224	C₂×D₄	—	4	2	no	—
225	C₂×D₄	—	4	2	no	—
226	C₂×D₄	—	4	2	no	—
227	C₂×D₄	—	8	2	no	—
228	C₂×D₄	—	8	2	no	—
229	C₂²×C₄	—	38	1	no	—
230	C₂²×C₄	—	38	1	no	—
231	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
232	C₂²×C₄	#64543	1	1	yes	characteristic
233	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
234	D₈	—	4	8	no	—
235	D₈	—	4	8	no	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	76	1	no	—
241	C₂×C₈	—	76	1	no	—
242	C₂×C₈	—	76	2	no	—
243	C₂×C₈	—	76	1	no	—
244	C₂×C₈	—	76	2	no	—
245	C₂×C₈	—	76	2	no	—
246	C₂×C₈	#64525	1	2	yes	—
247	C₂×C₈	#64526	1	2	yes	—
248	C₄⋊C₄	—	19	2	no	—
249	C₄⋊C₄	—	38	1	no	—
250	C₄⋊C₄	—	38	2	no	—
251	C₄²	—	76	1	no	—
252	C₄²	—	76	1	no	—
253	C₄²	#64543	1	2	yes	—
254	C₄²	C₂×C₂²⋊D₁₉	1	2	yes	—
255	C₂³	—	4	4	no	—
256	C₂³	—	8	2	no	—
257	C₂³	—	8	2	no	—
258	C₂³	—	8	2	no	—
259	C₂³	—	8	2	no	—
260	C₂³	—	8	2	no	—
261	C₂³	—	8	2	no	—
262	C₂³	#10597384	1	1	yes	characteristic
263	D₄	—	4	4	no	—
264	D₄	—	4	4	no	—
265	D₄	—	4	4	no	—
266	D₄	—	4	4	no	—
267	D₄	—	8	4	no	—
268	D₄	—	8	4	no	—
269	D₄	—	8	2	no	—
270	D₄	—	8	2	no	—
271	D₄	—	8	2	no	—
272	D₄	—	8	2	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	38	2	no	—
276	C₂×C₄	—	76	1	no	—
277	C₂×C₄	—	76	1	no	—
278	C₂×C₄	—	76	1	no	—
279	C₂×C₄	—	76	1	no	—
280	C₂×C₄	—	76	1	no	—
281	C₂×C₄	D₈×D₁₉	1	1	yes	characteristic
282	C₂×C₄	#10597357	1	2	yes	—
283	C₂×C₄	#10597358	1	1	yes	characteristic
284	C₂×C₄	C₂×#64525	1	1	yes	characteristic
285	C₂×C₄	#10597378	1	1	yes	characteristic
286	C₂×C₄	#10597378	1	1	yes	characteristic
287	C₂×C₄	C₂×#64526	1	1	yes	characteristic
288	C₂×C₄	#10597381	1	2	yes	—
289	C₂×C₄	#10597382	1	1	yes	characteristic
290	C₂×C₄	#10597385	1	2	yes	—
291	C₂×C₄	#10597399	1	1	yes	characteristic
292	C₈	—	2	2	no	—
293	C₈	—	2	2	no	—
294	C₈	—	152	1	no	—
295	C₈	—	152	2	no	—
296	C₂²	—	8	4	no	—
297	C₂²	—	8	4	no	—
298	C₂²	—	8	4	no	—
299	C₂²	—	16	2	no	—
300	C₂²	—	16	2	no	—
301	C₂²	—	16	2	no	—
302	C₂²	—	16	2	no	—
303	C₂²	#11926458	1	1	yes	characteristic
304	C₂²	#11926736	1	1	yes	characteristic
305	C₂²	#11926741	1	1	yes	characteristic
306	C₂²	#11926777	1	1	yes	characteristic
307	C₂²	#11926794	1	1	yes	characteristic
308	C₂²	#11926811	1	1	yes	characteristic
309	C₂²	#11926811	1	1	yes	characteristic
310	C₄	—	2	2	no	—
311	C₄	—	76	2	no	—
312	C₄	—	152	1	no	—
313	C₄	#11926678	1	1	yes	characteristic
314	C₄	#11926695	1	1	yes	characteristic
315	C₄	#11926737	1	2	yes	—
316	C₄	#11926757	1	2	yes	—
317	C₄	#11926890	1	1	yes	characteristic
318	C₄	#11926908	1	1	yes	characteristic
319	C₂	—	16	4	no	—
320	C₂	#14562987	1	1	yes	characteristic
321	C₂	#14562992	1	1	yes	characteristic
322	C₂	#14563111	1	1	yes	characteristic
323	C₂	#14563116	1	1	yes	characteristic
324	C₂	#14564866	1	1	yes	characteristic
325	C₂	#14564888	1	1	yes	characteristic
326	C₂	#14564943	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10597384
commutator, cosocle	C₂×C₄²×C₁₉	C₂³
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting, prime radical	C₂×C₁₉×#493	C₂
socle	C₂³×C₁₉	#121
hypercenter	C₂×#493	D₁₉
hypocenter	C₁₉	#10536
LCS	G ▷ C₂×C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#493

Complements

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

N	H ≅ G/N	type
1. #14564563	C₂	semidirect
2. #14564353	C₂	semidirect
3. C₂×#11926288	C₂	semidirect
4. #14561470	C₂	semidirect
82. C₄⋊Dic₁₉	D₈	semidirect
211. C₁₉	#10536	semidirect

Maximal subgroups

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

subgroup	index
#14564563	2
#14564353	2
C₂×#11926288	2
#14561470	2
C₂×C₁₉×#493	2
#10536	19