Subgroups — #18081069

Order 4864.

356 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14564554	C₂	1	1	yes	maximal, characteristic
2	#14564410	C₂	1	1	yes	maximal, characteristic, complemented
3	#14562282	C₂	1	1	yes	maximal, characteristic
4	#14562087	C₂	1	1	yes	maximal, characteristic
5	#14561537	C₂	1	1	yes	maximal, characteristic, complemented
6	#14561475	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1451	C₂	1	1	yes	maximal, characteristic, complemented
8	#11927094	C₂²	1	1	yes	characteristic
9	C₂×#10597279	—	2	1	no	—
10	#11926568	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×#10597282	—	2	1	no	—
13	C₂×#10597282	—	2	1	no	—
14	C₄×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
15	#11926308	—	2	1	no	—
16	#11926289	C₄	1	1	yes	characteristic, complemented
17	#11926289	C₄	1	1	yes	characteristic, complemented
18	#11926287	C₄	1	1	yes	characteristic, complemented
19	#11926287	C₄	1	1	yes	characteristic, complemented
20	#11926273	—	2	1	no	—
21	#11926273	—	2	1	no	—
22	C₁₉×#431	C₂²	1	1	yes	characteristic
23	C₄×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
25	C₁₉×#322	—	2	1	no	—
26	C₁₉×#322	—	2	1	no	—
27	C₂×#64541	—	2	1	no	—
28	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
29	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₉	—	2	1	no	—
31	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
32	#10597376	—	2	1	no	—
33	#10597376	—	2	1	no	—
34	C₂×C₄⋊Dic₁₉	—	2	1	no	—
35	C₂×C₄×Dic₁₉	—	2	1	no	—
36	C₂×#64520	—	2	1	no	—
37	C₂²×C₁₉⋊C₈	—	2	1	no	—
38	#10597321	—	2	1	no	—
39	#10597320	C₂×C₄	1	1	yes	characteristic
40	C₄×D₇₆	—	2	1	no	—
41	C₄×D₇₆	—	2	1	no	—
42	#10597315	C₂×C₄	1	1	yes	characteristic
43	C₁₉×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
44	C₁₉×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
45	C₁₉×C₂²⋊C₈	—	2	1	no	—
46	C₁₉×#97	—	2	1	no	—
47	C₄×C₈×C₁₉	—	2	1	no	—
48	C₄×C₈×C₁₉	—	2	1	no	—
49	C₄×C₈×C₁₉	—	2	1	no	—
50	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
51	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
52	C₁₉⋊#95	—	2	1	no	—
53	#10597282	—	4	2	no	—
54	#10597282	—	4	2	no	—
55	#10597279	—	4	2	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)	C₂×C₄	1	1	yes	characteristic
60	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
64	C₂×C₂²⋊D₁₉	—	4	1	no	—
65	C₂²×Dic₁₉	—	4	1	no	—
66	#64541	—	4	1	no	—
67	#64541	—	4	1	no	—
68	#64541	—	4	1	no	—
69	#64541	—	4	1	no	—
70	C₂×D₇₆	—	2	1	no	—
71	C₂×D₇₆	—	2	1	no	—
72	C₂×D₇₆	—	2	1	no	—
73	C₂×C₄×D₁₉	—	4	1	no	—
74	C₂×Dic₃₈	—	2	1	no	—
75	M₁₆×C₁₉	—	2	1	no	—
76	M₁₆×C₁₉	—	2	1	no	—
77	M₁₆×C₁₉	—	2	1	no	—
78	M₁₆×C₁₉	—	2	1	no	—
79	C₂×C₈×C₁₉	—	2	1	no	—
80	C₂×C₈×C₁₉	—	2	1	no	—
81	C₂×C₈×C₁₉	—	2	1	no	—
82	C₂×C₈×C₁₉	—	2	1	no	—
83	C₂×C₈×C₁₉	—	2	1	no	—
84	C₂×C₈×C₁₉	—	2	1	no	—
85	C₂×C₈×C₁₉	—	4	2	no	—
86	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₁₉	#41	1	1	yes	characteristic
89	C₄²×C₁₉	#41	1	1	yes	characteristic
90	(C₂×D₁₉)⋊C₄	—	4	1	no	—
91	(C₂×D₁₉)⋊C₄	—	4	1	no	—
92	C₄⋊Dic₁₉	—	2	2	no	—
93	C₄⋊Dic₁₉	—	2	1	no	—
94	C₄⋊Dic₁₉	—	2	1	no	—
95	C₄⋊Dic₁₉	—	4	1	no	—
96	C₄×Dic₁₉	—	4	1	no	—
97	C₄×Dic₁₉	—	4	1	no	—
98	#64520	—	4	1	no	—
99	#64520	—	4	1	no	—
100	C₂×C₁₉⋊C₈	—	2	1	no	—
101	C₂×C₁₉⋊C₈	—	2	1	no	—
102	C₂×C₁₉⋊C₈	—	4	1	no	—
103	C₂×C₁₉⋊C₈	—	4	1	no	—
104	C₂×C₁₉⋊C₈	—	4	1	no	—
105	#9592	—	19	1	no	maximal, Sylow
106	C₂³×C₁₉	#124	1	1	yes	characteristic
107	C₂²×D₁₉	—	4	1	no	—
108	C₂×C₄×C₁₉	—	2	1	no	—
109	C₂×C₄×C₁₉	—	2	1	no	—
110	C₂×C₄×C₁₉	—	4	1	no	—
111	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
112	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
115	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
116	C₂×C₄×C₁₉	#126	1	1	yes	characteristic
117	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
118	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
119	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
120	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
121	C₂²⋊D₁₉	—	8	1	no	—
122	C₂²⋊D₁₉	—	8	1	no	—
123	C₂×Dic₁₉	—	4	2	no	—
124	C₂×Dic₁₉	—	4	1	no	—
125	C₂×Dic₁₉	—	8	1	no	—
126	C₂×Dic₁₉	—	8	1	no	—
127	D₇₆	—	4	1	no	—
128	D₇₆	—	4	1	no	—
129	D₇₆	—	4	1	no	—
130	D₇₆	—	4	1	no	—
131	D₇₆	—	8	1	no	—
132	C₄×D₁₉	—	8	1	no	—
133	C₄×D₁₉	—	8	1	no	—
134	Dic₃₈	—	4	1	no	—
135	Dic₃₈	—	4	1	no	—
136	C₈×C₁₉	—	2	2	no	—
137	C₈×C₁₉	—	4	1	no	—
138	C₈×C₁₉	—	4	1	no	—
139	C₈×C₁₉	—	4	1	no	—
140	C₈×C₁₉	—	4	1	no	—
141	C₈×C₁₉	—	4	1	no	—
142	C₁₉⋊C₈	—	4	1	no	—
143	C₁₉⋊C₈	—	4	1	no	—
144	C₁₉⋊C₈	—	8	1	no	—
145	#1980	—	19	1	no	—
146	#1950	—	19	1	no	—
147	#1540	—	19	1	no	—
148	#1492	—	19	1	no	—
149	#1451	D₁₉	1	1	yes	characteristic, complemented
150	#1303	—	19	1	no	—
151	#1279	—	19	1	no	—
152	C₂²×C₁₉	—	2	1	no	—
153	C₂²×C₁₉	—	2	1	no	—
154	C₂²×C₁₉	#388	1	1	yes	characteristic
155	C₂²×C₁₉	#489	1	1	yes	characteristic
156	C₂²×C₁₉	#490	1	1	yes	characteristic
157	C₂×D₁₉	—	8	1	no	—
158	C₂×D₁₉	—	8	1	no	—
159	C₂×D₁₉	—	8	1	no	—
160	C₄×C₁₉	—	2	1	no	—
161	C₄×C₁₉	—	2	1	no	—
162	C₄×C₁₉	—	2	1	no	—
163	C₄×C₁₉	#440	1	1	yes	characteristic
164	C₄×C₁₉	#442	1	1	yes	characteristic
165	C₄×C₁₉	#443	1	1	yes	characteristic
166	C₄×C₁₉	#444	1	1	yes	characteristic
167	C₄×C₁₉	#468	1	1	yes	characteristic
168	C₄×C₁₉	#469	1	1	yes	characteristic
169	Dic₁₉	—	8	2	no	—
170	Dic₁₉	—	8	1	no	—
171	#532	—	19	1	no	—
172	#431	C₂×D₁₉	1	1	yes	characteristic
173	C₂×#108	—	38	1	no	—
174	#423	—	19	1	no	—
175	C₂×C₄⋊C₈	—	19	1	no	—
176	C₂×#104	—	38	1	no	—
177	C₂×#104	—	38	1	no	—
178	C₄×M₁₆	C₂×D₁₉	1	1	yes	characteristic
179	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
180	C₄×C₄⋊C₄	—	19	1	no	—
181	#341	—	38	1	no	—
182	#332	—	19	1	no	—
183	#332	—	19	1	no	—
184	#331	—	19	1	no	—
185	#331	—	19	1	no	—
186	D₄⋊C₈	—	38	1	no	—
187	D₄⋊C₈	—	38	1	no	—
188	#322	—	2	1	no	—
189	#322	—	2	1	no	—
190	C₂×C₁₉	—	2	1	no	—
191	C₂×C₁₉	—	2	1	no	—
192	C₂×C₁₉	#1929	1	1	yes	characteristic
193	C₂×C₁₉	#1932	1	1	yes	characteristic
194	C₂×C₁₉	#1936	1	1	yes	characteristic
195	D₁₉	—	16	1	no	—
196	C₂×#41	—	38	1	no	—
197	C₂×M₁₆	—	38	1	no	—
198	C₂×M₁₆	D₇₆	1	1	yes	characteristic
199	C₂×M₁₆	D₇₆	1	1	yes	characteristic
200	C₂²×C₈	—	2	1	no	—
201	C₂²×C₈	—	38	1	no	—
202	C₄⋊Q₈	—	19	1	no	—
203	#127	—	19	1	no	—
204	#124	—	38	1	no	—
205	#121	—	38	1	no	—
206	#121	—	38	1	no	—
207	C₄×D₄	—	38	1	no	—
208	C₄×D₄	—	38	1	no	—
209	C₂×C₄⋊C₄	—	38	1	no	—
210	C₂×C₄²	—	38	1	no	—
211	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
212	#108	—	76	2	no	—
213	C₄⋊C₈	—	19	1	no	—
214	C₄⋊C₈	—	19	1	no	—
215	C₄⋊C₈	—	38	1	no	—
216	C₄⋊C₈	—	38	1	no	—
217	C₄⋊C₈	—	38	1	no	—
218	C₄⋊C₈	C₄×D₁₉	1	1	yes	characteristic
219	C₄⋊C₈	C₄×D₁₉	1	1	yes	characteristic
220	#104	—	76	2	no	—
221	#104	—	76	2	no	—
222	C₂²⋊C₈	—	2	1	no	—
223	#97	—	2	1	no	—
224	C₄×C₈	—	2	1	no	—
225	C₄×C₈	—	2	1	no	—
226	C₄×C₈	—	2	1	no	—
227	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
228	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
229	#95	—	38	1	no	—
230	C₁₉	#9592	1	1	yes	characteristic, complemented, Sylow
231	#41	—	76	1	no	—
232	#41	—	76	1	no	—
233	#41	—	76	1	no	—
234	#41	—	76	1	no	—
235	C₂×Q₈	—	38	1	no	—
236	C₂×D₄	—	38	1	no	—
237	C₂×D₄	—	38	1	no	—
238	C₂×D₄	—	38	1	no	—
239	C₂×D₄	—	76	1	no	—
240	C₂²×C₄	—	76	1	no	—
241	C₂²×C₄	—	76	1	no	—
242	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
243	C₂²×C₄	#64541	1	1	yes	characteristic
244	C₂²×C₄	#64541	1	1	yes	characteristic
245	M₁₆	—	2	1	no	—
246	M₁₆	—	2	1	no	—
247	M₁₆	—	2	1	no	—
248	M₁₆	—	2	1	no	—
249	M₁₆	—	76	1	no	—
250	M₁₆	—	76	1	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	4	2	no	—
258	C₂×C₈	—	38	1	no	—
259	C₂×C₈	—	38	1	no	—
260	C₂×C₈	—	76	1	no	—
261	C₂×C₈	—	76	1	no	—
262	C₂×C₈	—	76	1	no	—
263	C₄⋊C₄	—	38	2	no	—
264	C₄⋊C₄	—	38	1	no	—
265	C₄⋊C₄	—	38	1	no	—
266	C₄⋊C₄	—	76	1	no	—
267	C₂²⋊C₄	—	76	1	no	—
268	C₂²⋊C₄	—	76	1	no	—
269	C₄²	—	76	1	no	—
270	C₄²	—	76	1	no	—
271	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
272	C₄²	#64541	1	1	yes	characteristic
273	C₄²	#64541	1	1	yes	characteristic
274	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
275	C₂³	—	76	1	no	—
276	C₂³	#10597321	1	1	yes	characteristic
277	Q₈	—	76	1	no	—
278	Q₈	—	76	1	no	—
279	D₄	—	76	1	no	—
280	D₄	—	76	1	no	—
281	D₄	—	76	1	no	—
282	D₄	—	76	1	no	—
283	D₄	—	152	1	no	—
284	D₄	—	152	1	no	—
285	D₄	—	152	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	76	2	no	—
290	C₂×C₄	—	76	1	no	—
291	C₂×C₄	—	152	1	no	—
292	C₂×C₄	—	152	1	no	—
293	C₂×C₄	—	152	1	no	—
294	C₂×C₄	—	152	1	no	—
295	C₂×C₄	#10597318	1	1	yes	characteristic
296	C₂×C₄	C₄×D₇₆	1	1	yes	characteristic
297	C₂×C₄	#10597322	1	1	yes	characteristic
298	C₂×C₄	#10597354	1	1	yes	characteristic
299	C₂×C₄	#10597354	1	1	yes	characteristic
300	C₂×C₄	#10597354	1	1	yes	characteristic
301	C₂×C₄	#10597355	1	1	yes	characteristic
302	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
303	C₂×C₄	#10597375	1	1	yes	characteristic
304	C₂×C₄	#10597376	1	1	yes	characteristic
305	C₈	—	2	2	no	—
306	C₈	—	4	1	no	—
307	C₈	—	4	1	no	—
308	C₈	—	4	1	no	—
309	C₈	—	4	1	no	—
310	C₈	—	4	1	no	—
311	C₈	—	76	1	no	—
312	C₈	—	76	1	no	—
313	C₈	—	152	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	152	1	no	—
317	C₂²	—	152	1	no	—
318	C₂²	—	152	1	no	—
319	C₂²	#11926435	1	1	yes	characteristic
320	C₂²	#11926904	1	1	yes	characteristic
321	C₂²	#11926909	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	152	2	no	—
326	C₄	—	152	1	no	—
327	C₄	#11926646	1	1	yes	characteristic
328	C₄	#11926657	1	1	yes	characteristic
329	C₄	#11926659	1	1	yes	characteristic
330	C₄	#11926664	1	1	yes	characteristic
331	C₄	#11926790	1	1	yes	characteristic
332	C₄	#11926797	1	1	yes	characteristic
333	C₂	—	2	1	no	—
334	C₂	—	2	1	no	—
335	C₂	—	304	1	no	—
336	C₂	#14564283	1	1	yes	characteristic
337	C₂	#14564304	1	1	yes	characteristic
338	C₂	#14564332	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10597376
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1451	C₂
socle	C₂²×C₁₉	#388
prime radical	#11927094	C₂²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1451	D₁₉
hypocenter	C₁₉	#9592
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1451

Complements

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

N	H ≅ G/N	type
2. #14564410	C₂	semidirect
5. #14561537	C₂	semidirect
6. #14561475	C₂	semidirect
7. C₁₉×#1451	C₂	semidirect
16. #11926289	C₄	semidirect
17. #11926289	C₄	semidirect
18. #11926287	C₄	semidirect
19. #11926287	C₄	semidirect
149. #1451	D₁₉	semidirect
230. C₁₉	#9592	semidirect

Maximal subgroups

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

subgroup	index
#14564554	2
#14564410	2
#14562282	2
#14562087	2
#14561537	2
#14561475	2
C₁₉×#1451	2
#9592	19