Subgroups — #18088273

Order 4864.

718 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×Dic₁₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#11926309	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926308	C₂	1	2	yes	maximal
4	#14563328	C₂	1	2	yes	maximal
5	C₂×C₁₉×#341	C₂	1	1	yes	maximal, characteristic
6	C₂²×#64520	C₂²	1	2	yes	—
7	C₂³×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₄×Dic₁₉	C₂²	1	1	yes	characteristic
9	C₂×C₈×Dic₁₉	—	2	4	no	—
10	C₂×C₈×Dic₁₉	C₄	1	4	yes	—
11	#11926309	C₄	1	4	yes	complemented
12	#11926308	—	2	4	no	—
13	#11926298	—	2	4	no	—
14	C₂²×M₁₆×C₁₉	C₂²	1	2	yes	—
15	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
16	C₁₉×#341	C₄	1	4	yes	complemented
17	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
18	C₂³×Dic₁₉	Q₈	1	2	yes	—
19	C₂×M₁₆×C₁₉	—	2	2	no	—
20	C₂×M₁₆×C₁₉	—	2	2	no	—
21	C₂×M₁₆×C₁₉	—	2	2	no	—
22	C₂×M₁₆×C₁₉	—	2	2	no	—
23	C₂×M₁₆×C₁₉	—	2	2	no	—
24	C₂×M₁₆×C₁₉	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₁₉	—	2	2	no	—
26	C₂²×C₈×C₁₉	—	2	2	no	—
27	C₂²×C₈×C₁₉	—	2	2	no	—
28	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
31	C₂×C₄×Dic₁₉	—	2	2	no	—
32	C₂×C₄×Dic₁₉	—	2	2	no	—
33	C₂×C₄×Dic₁₉	C₂×C₄	1	2	yes	—
34	C₂×C₄×Dic₁₉	D₄	1	2	yes	—
35	C₂×#64520	—	2	2	no	—
36	C₂×#64520	—	2	2	no	—
37	C₂×#64520	—	2	2	no	—
38	C₂×#64520	—	2	2	no	—
39	C₂×#64520	—	2	2	no	—
40	C₂×#64520	C₂×C₄	1	4	yes	—
41	C₂²×C₁₉⋊C₈	—	2	2	no	—
42	C₂²×C₁₉⋊C₈	—	2	1	no	—
43	C₂²×C₁₉⋊C₈	—	2	1	no	—
44	C₂²×C₁₉⋊C₈	—	2	2	no	—
45	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
46	C₂²×C₁₉⋊C₈	D₄	1	4	yes	—
47	C₈×Dic₁₉	—	2	8	no	—
48	C₂⁴×C₁₉	C₂×Q₈	1	1	yes	characteristic
49	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
53	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₉	#41	1	2	yes	—
55	C₂²×C₄×C₁₉	#41	1	2	yes	—
56	C₂²×C₄×C₁₉	#41	1	2	yes	—
57	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
59	C₂²×Dic₁₉	—	2	4	no	—
60	C₂²×Dic₁₉	—	2	8	no	—
61	C₂²×Dic₁₉	C₄⋊C₄	1	4	yes	—
62	M₁₆×C₁₉	—	2	4	no	—
63	M₁₆×C₁₉	—	2	4	no	—
64	M₁₆×C₁₉	—	4	2	no	—
65	M₁₆×C₁₉	—	4	2	no	—
66	C₂×C₈×C₁₉	—	2	2	no	—
67	C₂×C₈×C₁₉	—	2	2	no	—
68	C₂×C₈×C₁₉	—	2	4	no	—
69	C₂×C₈×C₁₉	—	2	4	no	—
70	C₂×C₈×C₁₉	—	2	4	no	—
71	C₂×C₈×C₁₉	—	4	2	no	—
72	C₂×C₈×C₁₉	—	4	2	no	—
73	C₂×C₈×C₁₉	C₄²	1	4	yes	—
74	C₄×Dic₁₉	—	2	2	no	—
75	C₄×Dic₁₉	—	2	2	no	—
76	C₄×Dic₁₉	—	2	4	no	—
77	#64520	—	2	4	no	—
78	#64520	—	2	4	no	—
79	#64520	—	4	2	no	—
80	#64520	—	4	2	no	—
81	C₂×C₁₉⋊C₈	—	2	2	no	—
82	C₂×C₁₉⋊C₈	—	2	2	no	—
83	C₂×C₁₉⋊C₈	—	2	4	no	—
84	C₂×C₁₉⋊C₈	—	2	4	no	—
85	C₂×C₁₉⋊C₈	—	2	4	no	—
86	C₂×C₁₉⋊C₈	—	4	2	no	—
87	C₂×C₁₉⋊C₈	—	4	2	no	—
88	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	4	yes	—
89	#10822	—	19	1	no	maximal, Sylow
90	C₂³×C₁₉	—	2	4	no	—
91	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂³×C₁₉	C₄×Q₈	1	1	yes	characteristic
93	C₂³×C₁₉	C₄×Q₈	1	1	yes	characteristic
94	C₂³×C₁₉	C₂²⋊Q₈	1	2	yes	—
95	C₂³×C₁₉	#125	1	2	yes	—
96	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
97	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
99	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
100	C₂×C₄×C₁₉	#117	1	2	yes	—
101	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
102	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
103	C₂×C₄×C₁₉	#117	1	2	yes	—
104	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
105	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
107	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₉	#121	1	2	yes	—
109	C₂×C₄×C₁₉	#123	1	2	yes	—
110	C₂×C₄×C₁₉	#123	1	2	yes	—
111	C₂×C₄×C₁₉	#124	1	2	yes	—
112	C₂×C₄×C₁₉	#126	1	2	yes	—
113	C₂×C₄×C₁₉	#126	1	2	yes	—
114	C₂×Dic₁₉	—	2	16	no	—
115	C₂×Dic₁₉	—	2	4	no	—
116	C₂×Dic₁₉	—	2	4	no	—
117	C₂×Dic₁₉	#108	1	8	yes	—
118	C₈×C₁₉	—	2	4	no	—
119	C₈×C₁₉	—	4	4	no	—
120	C₁₉⋊C₈	—	2	4	no	—
121	C₁₉⋊C₈	—	4	4	no	—
122	C₂²×C₄×C₈	—	19	1	no	—
123	C₂×#341	—	19	2	no	—
124	C₂×#341	—	19	1	no	—
125	C₂×#341	D₁₉	1	1	yes	characteristic
126	#1728	—	19	2	no	—
127	C₂²×C₁₉	—	2	4	no	—
128	C₂²×C₁₉	—	2	4	no	—
129	C₂²×C₁₉	—	2	2	no	—
130	C₂²×C₁₉	—	2	2	no	—
131	C₂²×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
132	C₂²×C₁₉	#380	1	1	yes	characteristic
133	C₂²×C₁₉	#382	1	1	yes	characteristic
134	C₂²×C₁₉	#387	1	2	yes	—
135	C₂²×C₁₉	#387	1	2	yes	—
136	C₂²×C₁₉	C₂×#108	1	4	yes	—
137	C₄×C₁₉	#376	1	1	yes	characteristic
138	C₄×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
139	C₄×C₁₉	#381	1	1	yes	characteristic
140	C₄×C₁₉	#383	1	1	yes	characteristic
141	C₄×C₁₉	#388	1	2	yes	—
142	C₄×C₁₉	#388	1	2	yes	—
143	Dic₁₉	—	2	8	no	—
144	C₂²×M₁₆	—	19	2	no	—
145	C₂²×M₁₆	C₂×D₁₉	1	2	yes	—
146	C₂³×C₈	—	19	1	no	—
147	C₂³×C₈	C₂×D₁₉	1	1	yes	characteristic
148	C₂²×C₄²	—	19	1	no	—
149	C₂×C₄×C₈	—	19	4	no	—
150	C₂×C₄×C₈	—	38	4	no	—
151	#341	—	19	4	no	—
152	#341	—	38	4	no	—
153	#341	Dic₁₉	1	4	yes	complemented
154	#336	—	38	4	no	—
155	C₂×C₁₉	—	2	2	no	—
156	C₂×C₁₉	—	2	2	no	—
157	C₂×C₁₉	#1434	1	1	yes	characteristic
158	C₂×C₁₉	C₄×#108	1	2	yes	—
159	C₂×C₁₉	#1810	1	2	yes	—
160	C₂×C₁₉	#1837	1	2	yes	—
161	C₂³×C₄	—	19	2	no	—
162	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
163	C₂×M₁₆	—	2	2	no	—
164	C₂×M₁₆	—	2	2	no	—
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	19	4	no	—
169	C₂×M₁₆	—	38	2	no	—
170	C₂×M₁₆	—	38	2	no	—
171	C₂×M₁₆	—	38	2	no	—
172	C₂×M₁₆	—	38	2	no	—
173	C₂×M₁₆	—	38	2	no	—
174	C₂×M₁₆	C₂×Dic₁₉	1	4	yes	—
175	C₂²×C₈	—	2	2	no	—
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	—	2	2	no	—
178	C₂²×C₈	—	19	2	no	—
179	C₂²×C₈	—	19	4	no	—
180	C₂²×C₈	—	38	2	no	—
181	C₂²×C₈	—	38	1	no	—
182	C₂²×C₈	—	38	1	no	—
183	C₂²×C₈	—	38	2	no	—
184	C₂²×C₈	C₄×D₁₉	1	2	yes	—
185	C₂²×C₈	C₄×D₁₉	1	2	yes	—
186	C₂²×C₈	C₂×Dic₁₉	1	2	yes	—
187	C₂×C₄²	—	19	2	no	—
188	C₂×C₄²	—	19	2	no	—
189	C₂×C₄²	—	38	2	no	—
190	C₂×C₄²	—	38	2	no	—
191	C₄×C₈	—	38	8	no	—
192	C₁₉	#10822	1	1	yes	characteristic, complemented, Sylow
193	C₂⁴	Q₈×D₁₉	1	1	yes	characteristic
194	C₂²×C₄	—	19	4	no	—
195	C₂²×C₄	—	38	8	no	—
196	C₂²×C₄	—	38	4	no	—
197	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
198	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
199	C₂²×C₄	#64541	1	2	yes	—
200	C₂²×C₄	#64541	1	2	yes	—
201	C₂²×C₄	D₄×D₁₉	1	2	yes	—
202	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
203	C₂²×C₄	#64543	1	2	yes	—
204	C₂²×C₄	#64543	1	1	yes	characteristic
205	C₂²×C₄	#64545	1	1	yes	characteristic
206	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
207	M₁₆	—	2	4	no	—
208	M₁₆	—	2	4	no	—
209	M₁₆	—	4	2	no	—
210	M₁₆	—	4	2	no	—
211	M₁₆	—	38	4	no	—
212	M₁₆	—	38	4	no	—
213	M₁₆	—	76	2	no	—
214	M₁₆	—	76	2	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	4	2	no	—
221	C₂×C₈	—	4	2	no	—
222	C₂×C₈	—	19	4	no	—
223	C₂×C₈	—	38	2	no	—
224	C₂×C₈	—	38	4	no	—
225	C₂×C₈	—	38	2	no	—
226	C₂×C₈	—	38	4	no	—
227	C₂×C₈	—	38	4	no	—
228	C₂×C₈	—	76	2	no	—
229	C₂×C₈	—	76	2	no	—
230	C₂×C₈	C₄×Dic₁₉	1	4	yes	—
231	C₄²	—	38	2	no	—
232	C₄²	—	38	2	no	—
233	C₄²	—	38	4	no	—
234	C₂³	—	2	4	no	—
235	C₂³	#10597333	1	1	yes	characteristic
236	C₂³	#10597335	1	2	yes	—
237	C₂³	C₄⋊C₄×D₁₉	1	1	yes	characteristic
238	C₂³	#10597342	1	2	yes	—
239	C₂³	Q₈×Dic₁₉	1	1	yes	characteristic
240	C₂×C₄	—	19	8	no	—
241	C₂×C₄	—	38	16	no	—
242	C₂×C₄	—	38	4	no	—
243	C₂×C₄	—	38	4	no	—
244	C₂×C₄	#10597318	1	1	yes	characteristic
245	C₂×C₄	#10597318	1	1	yes	characteristic
246	C₂×C₄	#10597318	1	2	yes	—
247	C₂×C₄	#10597323	1	1	yes	characteristic
248	C₂×C₄	#10597325	1	2	yes	—
249	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
250	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	2	yes	—
251	C₂×C₄	#10597329	1	2	yes	—
252	C₂×C₄	#10597330	1	2	yes	—
253	C₂×C₄	#10597331	1	2	yes	—
254	C₂×C₄	#10597338	1	1	yes	characteristic
255	C₂×C₄	#10597339	1	1	yes	characteristic
256	C₂×C₄	#10597340	1	2	yes	—
257	C₂×C₄	#10597344	1	2	yes	—
258	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
259	C₂×C₄	#10597372	1	2	yes	—
260	C₂×C₄	D₄×Dic₁₉	1	2	yes	—
261	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
262	C₈	—	2	4	no	—
263	C₈	—	4	4	no	—
264	C₈	—	38	4	no	—
265	C₈	—	76	4	no	—
266	C₂²	—	2	4	no	—
267	C₂²	—	2	4	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	2	no	—
270	C₂²	C₄⋊C₄×Dic₁₉	1	1	yes	characteristic
271	C₂²	#11926403	1	1	yes	characteristic
272	C₂²	#11926408	1	1	yes	characteristic
273	C₂²	#11926426	1	2	yes	—
274	C₂²	#11926430	1	2	yes	—
275	C₂²	#108×D₁₉	1	4	yes	—
276	C₄	—	38	8	no	—
277	C₄	#11926390	1	1	yes	characteristic
278	C₄	C₂²⋊C₄×Dic₁₉	1	1	yes	characteristic
279	C₄	#11926406	1	1	yes	characteristic
280	C₄	#11926411	1	1	yes	characteristic
281	C₄	#11926433	1	2	yes	—
282	C₄	#11926437	1	2	yes	—
283	C₂	—	2	2	no	—
284	C₂	—	2	2	no	—
285	C₂	#14561860	1	1	yes	characteristic
286	C₂	#108×Dic₁₉	1	2	yes	—
287	C₂	#14563672	1	2	yes	—
288	C₂	#14563771	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₉
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₂×C₁₉×#341	C₂
socle	C₂³×C₁₉	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₉	C₂×Q₈
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂×#341	D₁₉
hypocenter	C₁₉	#10822
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#341

Complements

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

N	H ≅ G/N	type
11. #11926309	C₄	semidirect
16. C₁₉×#341	C₄	semidirect
153. #341	Dic₁₉	semidirect
192. C₁₉	#10822	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₈×Dic₁₉	2
C₂×#11926309	2
C₂×#11926308	2
#14563328	2
C₂×C₁₉×#341	2
#10822	19