Subgroups — #18073658

Order 4864.

406 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14572570	C₂	1	1	yes	maximal, characteristic
2	#14563500	C₂	1	1	yes	maximal, characteristic
3	#14562134	C₂	1	1	yes	maximal, characteristic
4	#14561928	C₂	1	1	yes	maximal, characteristic
5	#14561537	C₂	1	1	yes	maximal, characteristic
6	#14561532	C₂	1	1	yes	maximal, characteristic
7	C₁₉×#1542	C₂	1	1	yes	maximal, characteristic
8	#11926675	—	2	1	no	—
9	#11926675	C₄	1	2	yes	complemented
10	Q₈×C₁₉⋊C₈	—	2	1	no	—
11	Q₈×C₁₉⋊C₈	C₄	1	2	yes	complemented
12	#11926597	C₂²	1	1	yes	characteristic
13	#11926571	—	2	1	no	—
14	#11926568	C₂²	1	1	yes	characteristic
15	#11926500	—	2	1	no	—
16	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₈×Dic₁₉	—	2	1	no	—
18	C₄×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
19	#11926298	—	2	1	no	—
20	#11926281	C₄	1	4	yes	—
21	#11926266	—	2	1	no	—
22	#11926265	—	2	1	no	—
23	C₁₉×#533	C₂²	1	1	yes	characteristic
24	C₁₉×#431	C₂²	1	1	yes	characteristic
25	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
26	C₁₉×#333	C₄	1	2	yes	complemented
27	C₁₉×#332	C₄	1	2	yes	complemented
28	C₂²×C₈×C₁₉	—	2	1	no	—
29	C₁₉×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
30	C₁₉×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
31	C₁₉×#125	—	2	1	no	—
32	C₁₉×C₂²⋊Q₈	—	2	1	no	—
33	C₁₉×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
34	C₄×Q₈×C₁₉	—	2	1	no	—
35	C₄×Q₈×C₁₉	C₂×C₄	1	2	yes	—
36	C₁₉×#117	—	2	1	no	—
37	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
38	C₂×C₄⋊Dic₁₉	—	2	1	no	—
39	C₂×C₄×Dic₁₉	—	2	1	no	—
40	C₂×#64520	—	2	1	no	—
41	C₂²×C₁₉⋊C₈	D₄	1	1	yes	characteristic
42	C₂²×C₁₉⋊C₈	D₄	1	1	yes	characteristic
43	C₁₉×C₄⋊C₈	—	2	1	no	—
44	C₁₉×C₄⋊C₈	C₂×C₄	1	2	yes	—
45	C₁₉×C₄⋊C₈	C₂×C₄	1	2	yes	—
46	C₁₉×C₂²⋊C₈	—	2	1	no	—
47	C₄×C₈×C₁₉	—	2	1	no	—
48	C₁₉⋊#95	C₂×C₄	1	2	yes	—
49	C₂²⋊(C₁₉⋊C₈)	C₂×C₄	1	2	yes	—
50	C₈×Dic₁₉	—	4	2	no	—
51	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
52	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
53	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
54	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
55	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	2	yes	—
56	#10597260	—	2	2	no	—
57	C₄×C₁₉⋊C₈	—	2	1	no	—
58	C₄×C₁₉⋊C₈	—	2	2	no	—
59	C₄×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
60	C₄×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
61	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
62	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
63	C₂×Q₈×C₁₉	—	2	1	no	—
64	C₂×Q₈×C₁₉	C₄²	1	2	yes	—
65	C₂²×C₄×C₁₉	C₂²×C₄	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	2	no	—
69	C₂×C₈×C₁₉	—	2	2	no	—
70	C₂×C₈×C₁₉	—	2	2	no	—
71	C₂×C₈×C₁₉	—	4	1	no	—
72	C₂×C₈×C₁₉	—	4	1	no	—
73	C₄⋊C₄×C₁₉	—	2	2	no	—
74	C₄⋊C₄×C₁₉	—	2	1	no	—
75	C₄⋊C₄×C₁₉	—	2	1	no	—
76	C₄⋊C₄×C₁₉	—	2	1	no	—
77	C₄⋊C₄×C₁₉	—	2	2	no	—
78	C₄⋊C₄×C₁₉	C₄²	1	2	yes	—
79	C₂²⋊C₄×C₁₉	—	2	2	no	—
80	C₄²×C₁₉	—	2	2	no	—
81	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₁₉	#41	1	1	yes	characteristic
85	C₄⋊Dic₁₉	—	2	4	no	—
86	C₄×Dic₁₉	—	4	1	no	—
87	C₄×Dic₁₉	—	4	1	no	—
88	#64520	—	4	1	no	—
89	#64520	—	4	1	no	—
90	C₂×C₁₉⋊C₈	—	2	1	no	—
91	C₂×C₁₉⋊C₈	—	2	1	no	—
92	C₂×C₁₉⋊C₈	—	2	1	no	—
93	C₂×C₁₉⋊C₈	—	2	1	no	—
94	C₂×C₁₉⋊C₈	—	2	2	no	—
95	C₂×C₁₉⋊C₈	—	2	2	no	—
96	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
100	#8258	—	19	1	no	maximal, Sylow
101	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
102	Q₈×C₁₉	—	4	2	no	—
103	Q₈×C₁₉	—	4	2	no	—
104	C₂×C₄×C₁₉	—	2	2	no	—
105	C₂×C₄×C₁₉	—	2	2	no	—
106	C₂×C₄×C₁₉	—	2	2	no	—
107	C₂×C₄×C₁₉	—	4	1	no	—
108	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
109	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
111	C₂×C₄×C₁₉	C₄×D₄	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	AGL(1,ℤ/8)	1	1	yes	characteristic
115	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
116	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
117	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
118	C₂×Dic₁₉	—	2	4	no	—
119	C₂×Dic₁₉	—	4	2	no	—
120	C₂×Dic₁₉	—	4	2	no	—
121	C₈×C₁₉	—	4	2	no	—
122	C₈×C₁₉	—	4	2	no	—
123	C₁₉⋊C₈	—	2	1	no	—
124	C₁₉⋊C₈	—	2	1	no	—
125	C₁₉⋊C₈	—	2	1	no	—
126	C₁₉⋊C₈	—	2	1	no	—
127	C₁₉⋊C₈	—	4	2	no	—
128	C₁₉⋊C₈	—	4	2	no	—
129	#2960	—	19	1	no	—
130	C₈×C₄⋊C₄	—	19	1	no	—
131	#1542	D₁₉	1	1	yes	characteristic
132	#1503	—	19	1	no	—
133	#1452	—	19	1	no	—
134	#1303	—	19	1	no	—
135	#1301	—	19	1	no	—
136	C₂²×C₁₉	—	2	1	no	—
137	C₂²×C₁₉	—	2	1	no	—
138	C₂²×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
139	C₂²×C₁₉	#440	1	1	yes	characteristic
140	C₂²×C₁₉	#441	1	1	yes	characteristic
141	C₄×C₁₉	—	2	1	no	—
142	C₄×C₁₉	—	2	1	no	—
143	C₄×C₁₉	—	2	1	no	—
144	C₄×C₁₉	—	4	2	no	—
145	C₄×C₁₉	—	4	2	no	—
146	C₄×C₁₉	#417	1	1	yes	characteristic
147	C₄×C₁₉	#419	1	1	yes	characteristic
148	C₄×C₁₉	#440	1	1	yes	characteristic
149	C₄×C₁₉	#441	1	1	yes	characteristic
150	C₄×C₁₉	#443	1	2	yes	—
151	Dic₁₉	—	4	4	no	—
152	#533	C₂×D₁₉	1	1	yes	characteristic
153	#446	—	19	2	no	—
154	#446	—	38	1	no	—
155	C₈×Q₈	—	19	2	no	—
156	C₈×Q₈	—	38	1	no	—
157	#431	—	19	1	no	—
158	#431	C₂×D₁₉	1	1	yes	characteristic
159	#424	—	38	1	no	—
160	#423	—	19	1	no	—
161	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
162	#405	—	38	1	no	—
163	C₂×C₄×C₈	—	19	1	no	—
164	C₂×C₄×C₈	—	38	1	no	—
165	C₄×C₄⋊C₄	—	19	1	no	—
166	#336	—	38	1	no	—
167	#333	Dic₁₉	1	2	yes	complemented
168	#332	Dic₁₉	1	2	yes	complemented
169	#328	—	19	4	no	—
170	#322	—	38	1	no	—
171	#322	—	38	1	no	—
172	C₂×C₁₉	—	2	1	no	—
173	C₂×C₁₉	—	2	1	no	—
174	C₂×C₁₉	#1757	1	1	yes	characteristic
175	C₂×C₁₉	#1763	1	1	yes	characteristic
176	C₂×C₁₉	#1764	1	1	yes	characteristic
177	C₂×M₁₆	—	38	1	no	—
178	C₂²×C₈	—	2	1	no	—
179	C₂²×C₈	—	19	1	no	—
180	C₂²×C₈	—	19	1	no	—
181	C₄⋊Q₈	C₂×Dic₁₉	1	1	yes	characteristic
182	C₄⋊Q₈	C₂×Dic₁₉	1	1	yes	characteristic
183	#125	—	2	1	no	—
184	C₂²⋊Q₈	—	2	1	no	—
185	C₂²⋊Q₈	C₄×D₁₉	1	2	yes	—
186	C₄×Q₈	—	2	1	no	—
187	C₄×Q₈	C₄×D₁₉	1	2	yes	—
188	#117	—	2	1	no	—
189	C₂×C₄⋊C₄	—	38	1	no	—
190	C₂×C₄²	—	38	1	no	—
191	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
192	C₄⋊C₈	—	2	1	no	—
193	C₄⋊C₈	—	19	2	no	—
194	C₄⋊C₈	—	38	2	no	—
195	C₄⋊C₈	—	38	2	no	—
196	C₄⋊C₈	—	38	1	no	—
197	C₄⋊C₈	—	38	1	no	—
198	C₄⋊C₈	C₂×Dic₁₉	1	2	yes	—
199	C₄⋊C₈	C₂×Dic₁₉	1	2	yes	—
200	C₂²⋊C₈	—	2	1	no	—
201	C₂²⋊C₈	—	19	2	no	—
202	#97	—	38	2	no	—
203	C₄×C₈	—	2	1	no	—
204	C₄×C₈	—	19	1	no	—
205	C₄×C₈	—	19	1	no	—
206	C₄×C₈	—	19	1	no	—
207	C₄×C₈	—	19	1	no	—
208	C₄×C₈	—	38	2	no	—
209	C₄×C₈	—	38	1	no	—
210	C₄×C₈	—	76	2	no	—
211	#95	—	19	2	no	—
212	C₁₉	#8258	1	1	yes	characteristic, complemented, Sylow
213	C₂×Q₈	—	2	1	no	—
214	C₂×Q₈	C₄×Dic₁₉	1	2	yes	—
215	C₂²×C₄	—	38	2	no	—
216	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
217	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
218	C₂²×C₄	#64543	1	1	yes	characteristic
219	M₁₆	—	76	1	no	—
220	M₁₆	—	76	1	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	—	19	1	no	—
226	C₂×C₈	—	19	1	no	—
227	C₂×C₈	—	19	1	no	—
228	C₂×C₈	—	19	1	no	—
229	C₂×C₈	—	38	1	no	—
230	C₂×C₈	—	38	1	no	—
231	C₂×C₈	—	38	1	no	—
232	C₂×C₈	—	38	1	no	—
233	C₂×C₈	—	38	2	no	—
234	C₂×C₈	—	38	2	no	—
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	2	1	no	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	38	4	no	—
241	C₄⋊C₄	C₄×Dic₁₉	1	2	yes	—
242	C₂²⋊C₄	—	2	2	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	76	1	no	—
245	C₄²	—	76	1	no	—
246	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
247	C₄²	D₄×D₁₉	1	1	yes	characteristic
248	C₄²	#64543	1	1	yes	characteristic
249	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
250	C₂³	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
251	Q₈	—	4	2	no	—
252	Q₈	—	4	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	38	4	no	—
258	C₂×C₄	—	76	2	no	—
259	C₂×C₄	—	76	2	no	—
260	C₂×C₄	#10597323	1	1	yes	characteristic
261	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
262	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
263	C₂×C₄	#10597360	1	1	yes	characteristic
264	C₂×C₄	#10597361	1	1	yes	characteristic
265	C₂×C₄	#10597364	1	1	yes	characteristic
266	C₂×C₄	#10597364	1	1	yes	characteristic
267	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
268	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
269	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
270	C₈	—	4	2	no	—
271	C₈	—	4	2	no	—
272	C₈	—	38	1	no	—
273	C₈	—	38	1	no	—
274	C₈	—	38	1	no	—
275	C₈	—	38	1	no	—
276	C₈	—	76	2	no	—
277	C₈	—	76	2	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	2	1	no	—
280	C₂²	C₂²⋊C₄×Dic₁₉	1	1	yes	characteristic
281	C₂²	#11926647	1	1	yes	characteristic
282	C₂²	#11926648	1	1	yes	characteristic
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	4	2	no	—
287	C₄	—	4	2	no	—
288	C₄	—	76	4	no	—
289	C₄	#11926544	1	1	yes	characteristic
290	C₄	#11926551	1	1	yes	characteristic
291	C₄	#11926645	1	1	yes	characteristic
292	C₄	#11926651	1	1	yes	characteristic
293	C₄	#11926661	1	2	yes	—
294	C₂	—	2	1	no	—
295	C₂	—	2	1	no	—
296	C₂	#14563438	1	1	yes	characteristic
297	C₂	#14563467	1	1	yes	characteristic
298	C₂	#14563475	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×#1542	C₂
socle	C₂²×C₁₉	C₄×C₂²⋊C₄
prime radical	C₂³×C₁₉	C₄×D₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1542	D₁₉
hypocenter	C₁₉	#8258
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1542

Complements

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

N	H ≅ G/N	type
9. #11926675	C₄	semidirect
11. Q₈×C₁₉⋊C₈	C₄	semidirect
26. C₁₉×#333	C₄	semidirect
27. C₁₉×#332	C₄	semidirect
167. #333	Dic₁₉	semidirect
168. #332	Dic₁₉	semidirect
212. C₁₉	#8258	semidirect

Maximal subgroups

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

subgroup	index
#14572570	2
#14563500	2
#14562134	2
#14561928	2
#14561537	2
#14561532	2
C₁₉×#1542	2
#8258	19