Subgroups — #18073653

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	C₈×C₄⋊Dic₁₉	C₂	1	1	yes	maximal, characteristic
2	#14562292	C₂	1	1	yes	maximal, characteristic
3	#14562134	C₂	1	1	yes	maximal, characteristic
4	#14561927	C₂	1	1	yes	maximal, characteristic
5	#14561536	C₂	1	1	yes	maximal, characteristic
6	#14561530	C₂	1	1	yes	maximal, characteristic
7	C₁₉×#2960	C₂	1	1	yes	maximal, characteristic
8	#11926597	C₂²	1	1	yes	characteristic
9	#11926597	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₁₉	—	2	1	no	—
12	C₄×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
13	#11926298	—	2	1	no	—
14	#11926293	C₄	1	2	yes	complemented
15	#11926290	C₄	1	2	yes	complemented
16	#11926281	C₄	1	4	yes	—
17	#11926266	—	2	1	no	—
18	#11926266	—	2	1	no	—
19	C₁₉×#533	C₂²	1	1	yes	characteristic
20	C₁₉×#446	—	2	1	no	—
21	C₁₉×#446	C₄	1	2	yes	complemented
22	C₈×Q₈×C₁₉	—	2	1	no	—
23	C₈×Q₈×C₁₉	C₄	1	2	yes	complemented
24	C₁₉×#424	—	2	1	no	—
25	C₁₉×#423	C₂²	1	1	yes	characteristic
26	C₁₉×#405	—	2	1	no	—
27	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
28	C₂×M₁₆×C₁₉	—	2	1	no	—
29	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
31	C₁₉×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
32	C₁₉×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
33	C₁₉×#125	—	2	1	no	—
34	C₁₉×C₂²⋊Q₈	—	2	1	no	—
35	C₁₉×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
36	C₄×Q₈×C₁₉	—	2	1	no	—
37	C₄×Q₈×C₁₉	C₂×C₄	1	2	yes	—
38	C₁₉×#117	—	2	1	no	—
39	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
40	C₂×C₄⋊Dic₁₉	—	2	1	no	—
41	C₂×C₄×Dic₁₉	—	2	1	no	—
42	C₂²×C₁₉⋊C₈	—	2	1	no	—
43	C₁₉×C₄⋊C₈	—	2	2	no	—
44	C₁₉×C₄⋊C₈	—	2	2	no	—
45	C₁₉×C₄⋊C₈	—	2	1	no	—
46	C₁₉×C₄⋊C₈	C₂×C₄	1	2	yes	—
47	C₁₉×#97	—	2	2	no	—
48	C₄×C₈×C₁₉	—	2	2	no	—
49	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
52	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
53	C₁₉⋊#95	C₂×C₄	1	2	yes	—
54	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
55	C₂²⋊(C₁₉⋊C₈)	C₂×C₄	1	2	yes	—
56	C₈×Dic₁₉	—	4	2	no	—
57	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
58	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
59	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	2	yes	—
60	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	2	yes	—
61	C₄×C₁₉⋊C₈	—	2	1	no	—
62	C₄×C₁₉⋊C₈	—	2	1	no	—
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	M₁₆×C₁₉	—	4	1	no	—
70	M₁₆×C₁₉	—	4	1	no	—
71	C₂×C₈×C₁₉	—	2	1	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	2	no	—
76	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
80	C₄⋊C₄×C₁₉	—	2	2	no	—
81	C₄⋊C₄×C₁₉	—	2	1	no	—
82	C₄⋊C₄×C₁₉	—	2	1	no	—
83	C₄⋊C₄×C₁₉	—	2	1	no	—
84	C₄⋊C₄×C₁₉	—	2	2	no	—
85	C₄⋊C₄×C₁₉	C₄²	1	2	yes	—
86	C₂²⋊C₄×C₁₉	—	2	2	no	—
87	C₄²×C₁₉	—	2	2	no	—
88	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₁₉	C₂²×C₄	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	4	no	—
93	C₄×Dic₁₉	—	4	1	no	—
94	C₄×Dic₁₉	—	4	1	no	—
95	C₂×C₁₉⋊C₈	—	2	2	no	—
96	C₂×C₁₉⋊C₈	—	2	2	no	—
97	C₂×C₁₉⋊C₈	—	2	2	no	—
98	C₂×C₁₉⋊C₈	—	4	1	no	—
99	C₂×C₁₉⋊C₈	—	4	1	no	—
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	—	2	1	no	—
122	C₈×C₁₉	—	2	1	no	—
123	C₈×C₁₉	—	2	1	no	—
124	C₈×C₁₉	—	2	1	no	—
125	C₈×C₁₉	—	4	2	no	—
126	C₁₉⋊C₈	—	4	2	no	—
127	C₁₉⋊C₈	—	4	2	no	—
128	C₁₉⋊C₈	—	4	2	no	—
129	#2960	D₁₉	1	1	yes	characteristic
130	C₈×C₄⋊C₄	—	19	1	no	—
131	#1542	—	19	1	no	—
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	—	2	1	no	—
154	#446	Dic₁₉	1	2	yes	complemented
155	C₈×Q₈	—	2	1	no	—
156	C₈×Q₈	Dic₁₉	1	2	yes	complemented
157	#431	—	19	1	no	—
158	#431	—	19	1	no	—
159	#424	—	2	1	no	—
160	#423	C₂×D₁₉	1	1	yes	characteristic
161	C₂×C₄⋊C₈	—	19	1	no	—
162	#405	—	2	1	no	—
163	C₂×C₄×C₈	—	38	1	no	—
164	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
165	C₄×C₄⋊C₄	—	19	1	no	—
166	#336	—	38	1	no	—
167	#333	—	19	2	no	—
168	#332	—	19	2	no	—
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	—	2	1	no	—
178	C₂²×C₈	—	38	1	no	—
179	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
180	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
181	C₄⋊Q₈	C₄×D₁₉	1	1	yes	characteristic
182	C₄⋊Q₈	C₄×D₁₉	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	C₂×Dic₁₉	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	2	no	—
193	C₄⋊C₈	—	2	2	no	—
194	C₄⋊C₈	—	2	1	no	—
195	C₄⋊C₈	—	19	2	no	—
196	C₄⋊C₈	—	19	2	no	—
197	C₄⋊C₈	—	38	1	no	—
198	C₄⋊C₈	—	38	1	no	—
199	C₄⋊C₈	C₂×Dic₁₉	1	2	yes	—
200	C₂²⋊C₈	—	19	2	no	—
201	C₂²⋊C₈	—	38	1	no	—
202	#97	—	2	2	no	—
203	C₄×C₈	—	2	2	no	—
204	C₄×C₈	—	38	1	no	—
205	C₄×C₈	—	38	1	no	—
206	C₄×C₈	—	76	2	no	—
207	C₄×C₈	C₂×Dic₁₉	1	1	yes	characteristic
208	C₄×C₈	C₂×Dic₁₉	1	1	yes	characteristic
209	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
210	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
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	#64541	1	1	yes	characteristic
218	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
219	M₁₆	—	4	1	no	—
220	M₁₆	—	4	1	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	38	2	no	—
227	C₂×C₈	—	38	2	no	—
228	C₂×C₈	—	38	2	no	—
229	C₂×C₈	—	76	1	no	—
230	C₂×C₈	—	76	1	no	—
231	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
232	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
233	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
234	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	2	1	no	—
238	C₄⋊C₄	—	2	1	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	#64541	1	1	yes	characteristic
248	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
249	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
250	C₂³	C₄×C₂²⋊D₁₉	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	C₂×C₄×Dic₁₉	1	1	yes	characteristic
261	C₂×C₄	#10597372	1	1	yes	characteristic
262	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
263	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
264	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
265	C₂×C₄	#10597388	1	1	yes	characteristic
266	C₂×C₄	#10597388	1	1	yes	characteristic
267	C₂×C₄	#10597395	1	1	yes	characteristic
268	C₂×C₄	#10597397	1	1	yes	characteristic
269	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
270	C₈	—	2	1	no	—
271	C₈	—	2	1	no	—
272	C₈	—	2	1	no	—
273	C₈	—	2	1	no	—
274	C₈	—	4	2	no	—
275	C₈	—	76	2	no	—
276	C₈	—	76	2	no	—
277	C₈	—	76	2	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	2	1	no	—
280	C₂²	C₄×C₂²⋊Dic₁₉	1	1	yes	characteristic
281	C₂²	#11926642	1	1	yes	characteristic
282	C₂²	#11926649	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	#11926545	1	1	yes	characteristic
290	C₄	#11926553	1	1	yes	characteristic
291	C₄	#11926642	1	1	yes	characteristic
292	C₄	#11926649	1	1	yes	characteristic
293	C₄	#11926659	1	2	yes	—
294	C₂	—	2	1	no	—
295	C₂	—	2	1	no	—
296	C₂	#14563435	1	1	yes	characteristic
297	C₂	#14563465	1	1	yes	characteristic
298	C₂	#14563470	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₁₉
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#2960	C₂
socle	C₂²×C₁₉	C₄×C₂²⋊C₄
prime radical	C₂³×C₁₉	C₄×D₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#2960	D₁₉
hypocenter	C₁₉	#8258
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2960

Complements

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

N	H ≅ G/N	type
14. #11926293	C₄	semidirect
15. #11926290	C₄	semidirect
21. C₁₉×#446	C₄	semidirect
23. C₈×Q₈×C₁₉	C₄	semidirect
154. #446	Dic₁₉	semidirect
156. C₈×Q₈	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
C₈×C₄⋊Dic₁₉	2
#14562292	2
#14562134	2
#14561927	2
#14561536	2
#14561530	2
C₁₉×#2960	2
#8258	19