Subgroups — #18073647

Order 4864.

474 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14572346	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₄×C₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#14562282	C₂	1	1	yes	maximal, characteristic
4	#14562129	C₂	1	1	yes	maximal, characteristic
5	#14561928	C₂	1	1	yes	maximal, characteristic, complemented
6	#14561530	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1303	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10597353	—	2	1	no	—
9	#11927094	C₂²	1	1	yes	characteristic
10	#11926621	—	2	1	no	—
11	#11926621	C₄	1	2	yes	complemented
12	#11926609	—	2	1	no	—
13	#11926609	C₄	1	2	yes	complemented
14	#11926597	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
16	#11926512	—	2	1	no	—
17	C₂×C₄×C₁₉⋊C₈	—	2	1	no	—
18	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
19	#11926299	—	2	1	no	—
20	#11926289	C₄	1	2	yes	complemented
21	#11926287	C₄	1	2	yes	complemented
22	#11926277	C₄	1	4	yes	—
23	#11926266	—	2	1	no	—
24	#11926265	—	2	1	no	—
25	C₁₉×#431	C₂²	1	1	yes	characteristic
26	C₁₉×#423	C₂²	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
28	C₂×#64541	—	2	1	no	—
29	C₂×M₁₆×C₁₉	—	2	1	no	—
30	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
31	C₂×C₄²×C₁₉	—	2	1	no	—
32	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
33	#10597376	—	2	1	no	—
34	#10597376	C₂×C₄	1	2	yes	—
35	C₂²×C₁₉⋊C₈	—	2	1	no	—
36	C₂²×C₁₉⋊C₈	D₄	1	1	yes	characteristic
37	C₂²×C₁₉⋊C₈	D₄	1	1	yes	characteristic
38	#10597353	—	4	2	no	—
39	#10597353	—	4	2	no	—
40	C₂×#64515	—	2	2	no	—
41	C₂×C₈×D₁₉	—	2	2	no	—
42	#10597321	—	2	1	no	—
43	#10597320	C₂×C₄	1	1	yes	characteristic
44	C₄×D₇₆	—	2	1	no	—
45	C₄×D₇₆	C₂×C₄	1	2	yes	—
46	#10597315	C₂×C₄	1	1	yes	characteristic
47	C₁₉×C₄⋊C₈	—	2	1	no	—
48	C₁₉×C₄⋊C₈	C₂×C₄	1	2	yes	—
49	C₁₉×C₄⋊C₈	C₂×C₄	1	2	yes	—
50	C₁₉×C₂²⋊C₈	—	2	1	no	—
51	C₄×C₈×C₁₉	—	2	1	no	—
52	C₁₉×#95	C₂×C₄	1	2	yes	—
53	C₂²⋊(C₁₉⋊C₈)	C₂×C₄	1	2	yes	—
54	(C₂×D₁₉)⋊C₈	—	2	2	no	—
55	(C₂×D₁₉)⋊C₈	—	2	2	no	—
56	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
57	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
58	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	2	yes	—
59	C₄×C₁₉⋊C₈	—	2	1	no	—
60	C₄×C₁₉⋊C₈	—	4	2	no	—
61	C₄×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
62	C₄×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
63	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
64	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	—	2	2	no	—
66	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
69	C₂×C₂²⋊D₁₉	—	2	2	no	—
70	#64541	—	4	1	no	—
71	#64541	—	4	1	no	—
72	#64541	—	4	1	no	—
73	#64541	—	4	1	no	—
74	C₂×D₇₆	—	2	1	no	—
75	C₂×D₇₆	—	2	1	no	—
76	C₂×D₇₆	C₄²	1	2	yes	—
77	C₂×C₄×D₁₉	—	2	2	no	—
78	C₂×Dic₃₈	—	2	1	no	—
79	M₁₆×C₁₉	—	4	1	no	—
80	M₁₆×C₁₉	—	4	1	no	—
81	C₂×C₈×C₁₉	—	2	2	no	—
82	C₂×C₈×C₁₉	—	2	2	no	—
83	C₄⋊C₄×C₁₉	—	2	4	no	—
84	C₄²×C₁₉	—	4	1	no	—
85	C₄²×C₁₉	—	4	1	no	—
86	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₁₉	#41	1	1	yes	characteristic
90	(C₂×D₁₉)⋊C₄	—	2	2	no	—
91	(C₂×D₁₉)⋊C₄	—	2	2	no	—
92	C₄⋊Dic₁₉	—	2	1	no	—
93	C₄⋊Dic₁₉	C₄²	1	2	yes	—
94	C₂×C₁₉⋊C₈	—	2	1	no	—
95	C₂×C₁₉⋊C₈	—	2	1	no	—
96	C₂×C₁₉⋊C₈	—	2	1	no	—
97	C₂×C₁₉⋊C₈	—	2	1	no	—
98	C₂×C₁₉⋊C₈	—	2	2	no	—
99	C₂×C₁₉⋊C₈	—	2	2	no	—
100	C₂×C₁₉⋊C₈	—	4	1	no	—
101	C₂×C₁₉⋊C₈	—	4	1	no	—
102	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
106	#64515	—	4	4	no	—
107	C₈×D₁₉	—	4	4	no	—
108	#8257	—	19	1	no	maximal, Sylow
109	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
110	C₂²×D₁₉	—	2	2	no	—
111	C₂×C₄×C₁₉	—	2	2	no	—
112	C₂×C₄×C₁₉	—	2	4	no	—
113	C₂×C₄×C₁₉	—	4	1	no	—
114	C₂×C₄×C₁₉	—	4	2	no	—
115	C₂×C₄×C₁₉	—	4	2	no	—
116	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
117	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
119	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
123	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
124	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
126	C₂²⋊D₁₉	—	4	2	no	—
127	C₂²⋊D₁₉	—	4	2	no	—
128	C₂×Dic₁₉	—	2	2	no	—
129	D₇₆	—	4	2	no	—
130	D₇₆	—	4	1	no	—
131	D₇₆	—	4	1	no	—
132	D₇₆	—	4	2	no	—
133	C₄×D₁₉	—	4	2	no	—
134	C₄×D₁₉	—	4	2	no	—
135	Dic₃₈	—	4	1	no	—
136	Dic₃₈	—	4	1	no	—
137	C₈×C₁₉	—	4	2	no	—
138	C₈×C₁₉	—	4	2	no	—
139	C₁₉⋊C₈	—	2	1	no	—
140	C₁₉⋊C₈	—	2	1	no	—
141	C₁₉⋊C₈	—	2	1	no	—
142	C₁₉⋊C₈	—	2	1	no	—
143	C₁₉⋊C₈	—	4	2	no	—
144	C₁₉⋊C₈	—	4	2	no	—
145	#2931	—	19	1	no	—
146	C₈×C₄⋊C₄	—	19	1	no	—
147	#1540	—	19	1	no	—
148	#1502	—	19	1	no	—
149	#1452	—	19	1	no	—
150	#1303	D₁₉	1	1	yes	characteristic, complemented
151	#1301	—	19	1	no	—
152	C₂²×C₁₉	—	2	1	no	—
153	C₂²×C₁₉	—	2	1	no	—
154	C₂²×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
155	C₂²×C₁₉	#440	1	1	yes	characteristic
156	C₂²×C₁₉	#442	1	1	yes	characteristic
157	C₂×D₁₉	—	4	2	no	—
158	C₂×D₁₉	—	4	2	no	—
159	C₂×D₁₉	—	4	2	no	—
160	C₄×C₁₉	—	2	1	no	—
161	C₄×C₁₉	—	2	1	no	—
162	C₄×C₁₉	—	2	1	no	—
163	C₄×C₁₉	—	4	4	no	—
164	C₄×C₁₉	#417	1	1	yes	characteristic
165	C₄×C₁₉	#418	1	1	yes	characteristic
166	C₄×C₁₉	#440	1	1	yes	characteristic
167	C₄×C₁₉	#442	1	1	yes	characteristic
168	C₄×C₁₉	#443	1	2	yes	—
169	Dic₁₉	—	4	2	no	—
170	C₂×#131	—	38	1	no	—
171	#532	—	19	1	no	—
172	#436	—	19	2	no	—
173	#436	—	38	1	no	—
174	C₈×D₄	—	19	2	no	—
175	C₈×D₄	—	38	1	no	—
176	#431	—	19	1	no	—
177	#431	C₂×D₁₉	1	1	yes	characteristic
178	#423	C₂×D₁₉	1	1	yes	characteristic
179	C₂×C₄⋊C₈	—	19	1	no	—
180	#408	—	38	1	no	—
181	C₂×C₄×C₈	—	19	1	no	—
182	C₂×C₄×C₈	—	38	1	no	—
183	C₄×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
184	#336	—	38	1	no	—
185	#332	—	19	2	no	—
186	#331	—	19	2	no	—
187	#327	—	19	4	no	—
188	#322	—	38	1	no	—
189	#322	—	38	1	no	—
190	C₂×C₁₉	—	2	1	no	—
191	C₂×C₁₉	—	2	1	no	—
192	C₂×C₁₉	#1757	1	1	yes	characteristic
193	C₂×C₁₉	#1762	1	1	yes	characteristic
194	C₂×C₁₉	#1764	1	1	yes	characteristic
195	D₁₉	—	8	2	no	—
196	C₂×#41	—	38	1	no	—
197	#131	—	76	2	no	—
198	#131	—	76	2	no	—
199	C₂×M₁₆	—	2	1	no	—
200	C₂×M₁₆	—	38	2	no	—
201	C₂²×C₈	—	19	1	no	—
202	C₂²×C₈	—	19	1	no	—
203	C₂²×C₈	—	38	2	no	—
204	C₂²×C₈	—	38	1	no	—
205	C₄⋊Q₈	—	19	1	no	—
206	#127	—	19	1	no	—
207	#124	—	38	1	no	—
208	#121	—	19	2	no	—
209	#121	—	38	1	no	—
210	C₄×D₄	—	19	2	no	—
211	C₄×D₄	—	38	1	no	—
212	C₂×C₄⋊C₄	—	2	1	no	—
213	C₂×C₄²	—	2	1	no	—
214	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
215	C₄⋊C₈	—	2	1	no	—
216	C₄⋊C₈	—	19	2	no	—
217	C₄⋊C₈	—	38	1	no	—
218	C₄⋊C₈	—	38	1	no	—
219	C₄⋊C₈	C₄×D₁₉	1	2	yes	—
220	C₄⋊C₈	C₄×D₁₉	1	2	yes	—
221	C₂²⋊C₈	—	2	1	no	—
222	C₂²⋊C₈	—	19	2	no	—
223	C₂²⋊C₈	—	38	2	no	—
224	C₂²⋊C₈	—	38	2	no	—
225	C₄×C₈	—	2	1	no	—
226	C₄×C₈	—	19	1	no	—
227	C₄×C₈	—	19	1	no	—
228	C₄×C₈	—	19	1	no	—
229	C₄×C₈	—	19	1	no	—
230	C₄×C₈	—	38	1	no	—
231	C₄×C₈	—	76	2	no	—
232	#95	C₄×D₁₉	1	2	yes	—
233	C₁₉	#8257	1	1	yes	characteristic, complemented, Sylow
234	#41	—	76	1	no	—
235	#41	—	76	1	no	—
236	#41	—	76	1	no	—
237	#41	—	76	1	no	—
238	C₂×Q₈	—	38	1	no	—
239	C₂×D₄	—	19	2	no	—
240	C₂×D₄	—	38	2	no	—
241	C₂×D₄	—	38	1	no	—
242	C₂×D₄	—	38	1	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	38	2	no	—
245	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
246	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
247	C₂²×C₄	#64543	1	1	yes	characteristic
248	M₁₆	—	4	1	no	—
249	M₁₆	—	4	1	no	—
250	M₁₆	—	76	4	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	19	1	no	—
254	C₂×C₈	—	19	1	no	—
255	C₂×C₈	—	19	1	no	—
256	C₂×C₈	—	19	1	no	—
257	C₂×C₈	—	38	1	no	—
258	C₂×C₈	—	38	1	no	—
259	C₂×C₈	—	38	1	no	—
260	C₂×C₈	—	38	1	no	—
261	C₂×C₈	—	38	2	no	—
262	C₂×C₈	—	38	2	no	—
263	C₂×C₈	—	76	4	no	—
264	C₂×C₈	—	76	1	no	—
265	C₂×C₈	—	76	1	no	—
266	C₄⋊C₄	—	2	4	no	—
267	C₄⋊C₄	—	19	2	no	—
268	C₄⋊C₄	—	38	1	no	—
269	C₂²⋊C₄	—	38	2	no	—
270	C₂²⋊C₄	—	38	2	no	—
271	C₄²	—	4	1	no	—
272	C₄²	—	4	1	no	—
273	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
274	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
275	C₄²	D₄×D₁₉	1	1	yes	characteristic
276	C₄²	#64545	1	1	yes	characteristic
277	C₂³	—	38	2	no	—
278	C₂³	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
279	Q₈	—	76	1	no	—
280	Q₈	—	76	1	no	—
281	D₄	—	76	2	no	—
282	D₄	—	76	2	no	—
283	D₄	—	76	2	no	—
284	D₄	—	76	1	no	—
285	D₄	—	76	1	no	—
286	D₄	—	76	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	38	2	no	—
293	C₂×C₄	—	76	2	no	—
294	C₂×C₄	—	76	2	no	—
295	C₂×C₄	C₄²×D₁₉	1	1	yes	characteristic
296	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
297	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
298	C₂×C₄	#10597338	1	1	yes	characteristic
299	C₂×C₄	#10597339	1	1	yes	characteristic
300	C₂×C₄	#10597339	1	1	yes	characteristic
301	C₂×C₄	#10597357	1	1	yes	characteristic
302	C₂×C₄	#10597360	1	1	yes	characteristic
303	C₂×C₄	#10597360	1	1	yes	characteristic
304	C₂×C₄	#10597364	1	1	yes	characteristic
305	C₈	—	4	2	no	—
306	C₈	—	4	2	no	—
307	C₈	—	38	1	no	—
308	C₈	—	38	1	no	—
309	C₈	—	38	1	no	—
310	C₈	—	38	1	no	—
311	C₈	—	76	2	no	—
312	C₈	—	76	2	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	76	2	no	—
316	C₂²	—	76	2	no	—
317	C₂²	—	76	2	no	—
318	C₂²	#11926391	1	1	yes	characteristic
319	C₂²	#11926647	1	1	yes	characteristic
320	C₂²	#11926653	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	4	4	no	—
325	C₄	—	76	2	no	—
326	C₄	#11926544	1	1	yes	characteristic
327	C₄	#11926547	1	1	yes	characteristic
328	C₄	#11926643	1	1	yes	characteristic
329	C₄	#11926655	1	1	yes	characteristic
330	C₄	#11926658	1	2	yes	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	152	2	no	—
334	C₂	#14563434	1	1	yes	characteristic
335	C₂	#14563459	1	1	yes	characteristic
336	C₂	#14563469	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₁₉
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1303	C₂
socle	C₂²×C₁₉	C₄×C₂²⋊C₄
prime radical	#11927094	C₂²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1303	D₁₉
hypocenter	C₁₉	#8257
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1303

Complements

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

N	H ≅ G/N	type
2. C₄⋊C₄×C₁₉⋊C₈	C₂	semidirect
5. #14561928	C₂	semidirect
6. #14561530	C₂	semidirect
7. C₁₉×#1303	C₂	semidirect
11. #11926621	C₄	semidirect
13. #11926609	C₄	semidirect
20. #11926289	C₄	semidirect
21. #11926287	C₄	semidirect
150. #1303	D₁₉	semidirect
233. C₁₉	#8257	semidirect

Maximal subgroups

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

subgroup	index
#14572346	2
C₄⋊C₄×C₁₉⋊C₈	2
#14562282	2
#14562129	2
#14561928	2
#14561530	2
C₁₉×#1303	2
#8257	19