Subgroups — #18088272

Order 4864.

718 conjugacy classes of subgroups.

The subgroup lattice has 302 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	2	no	—
10	C₂×C₈×Dic₁₉	—	2	2	no	—
11	C₂×C₈×Dic₁₉	C₄	1	4	yes	—
12	#11926309	—	2	2	no	—
13	#11926308	C₄	1	8	yes	complemented
14	#11926298	—	2	4	no	—
15	C₂²×M₁₆×C₁₉	C₂²	1	2	yes	—
16	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
17	C₁₉×#341	—	2	2	no	—
18	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
19	C₂³×Dic₁₉	Q₈	1	2	yes	—
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	—	2	2	no	—
25	C₂×M₁₆×C₁₉	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₁₉	—	2	2	no	—
27	C₂²×C₈×C₁₉	—	2	2	no	—
28	C₂²×C₈×C₁₉	—	2	2	no	—
29	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₁₉	D₄	1	2	yes	—
31	C₂²×C₈×C₁₉	D₄	1	2	yes	—
32	C₂×C₄×Dic₁₉	—	2	1	no	—
33	C₂×C₄×Dic₁₉	—	2	1	no	—
34	C₂×C₄×Dic₁₉	—	2	1	no	—
35	C₂×C₄×Dic₁₉	—	2	1	no	—
36	C₂×C₄×Dic₁₉	C₂×C₄	1	2	yes	—
37	C₂×C₄×Dic₁₉	D₄	1	2	yes	—
38	C₂×#64520	—	2	2	no	—
39	C₂×#64520	—	2	2	no	—
40	C₂×#64520	—	2	2	no	—
41	C₂×#64520	—	2	2	no	—
42	C₂×#64520	—	2	2	no	—
43	C₂×#64520	C₂×C₄	1	4	yes	—
44	C₂²×C₁₉⋊C₈	—	2	1	no	—
45	C₂²×C₁₉⋊C₈	—	2	1	no	—
46	C₂²×C₁₉⋊C₈	—	2	2	no	—
47	C₂²×C₁₉⋊C₈	—	2	2	no	—
48	C₂²×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
49	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
50	C₈×Dic₁₉	—	2	8	no	—
51	C₂⁴×C₁₉	C₂×Q₈	1	1	yes	characteristic
52	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	#41	1	2	yes	—
58	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
59	C₂²×C₄×C₁₉	#41	1	2	yes	—
60	C₂²×C₄×C₁₉	#41	1	2	yes	—
61	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
62	C₂²×Dic₁₉	—	2	2	no	—
63	C₂²×Dic₁₉	—	2	2	no	—
64	C₂²×Dic₁₉	—	2	8	no	—
65	C₂²×Dic₁₉	C₄⋊C₄	1	4	yes	—
66	M₁₆×C₁₉	—	2	4	no	—
67	M₁₆×C₁₉	—	2	4	no	—
68	M₁₆×C₁₉	—	4	2	no	—
69	M₁₆×C₁₉	—	4	2	no	—
70	C₂×C₈×C₁₉	—	2	2	no	—
71	C₂×C₈×C₁₉	—	2	2	no	—
72	C₂×C₈×C₁₉	—	2	4	no	—
73	C₂×C₈×C₁₉	—	2	4	no	—
74	C₂×C₈×C₁₉	—	2	4	no	—
75	C₂×C₈×C₁₉	—	4	2	no	—
76	C₂×C₈×C₁₉	—	4	2	no	—
77	C₂×C₈×C₁₉	C₂²⋊C₄	1	4	yes	—
78	C₄×Dic₁₉	—	2	2	no	—
79	C₄×Dic₁₉	—	2	2	no	—
80	C₄×Dic₁₉	—	2	2	no	—
81	C₄×Dic₁₉	—	2	2	no	—
82	#64520	—	2	4	no	—
83	#64520	—	2	4	no	—
84	#64520	—	4	2	no	—
85	#64520	—	4	2	no	—
86	C₂×C₁₉⋊C₈	—	2	2	no	—
87	C₂×C₁₉⋊C₈	—	2	2	no	—
88	C₂×C₁₉⋊C₈	—	2	4	no	—
89	C₂×C₁₉⋊C₈	—	2	4	no	—
90	C₂×C₁₉⋊C₈	—	2	4	no	—
91	C₂×C₁₉⋊C₈	—	4	2	no	—
92	C₂×C₁₉⋊C₈	—	4	2	no	—
93	C₂×C₁₉⋊C₈	C₄²	1	4	yes	—
94	#10822	—	19	1	no	maximal, Sylow
95	C₂³×C₁₉	—	2	4	no	—
96	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂³×C₁₉	C₄×Q₈	1	2	yes	—
98	C₂³×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂³×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
100	C₂³×C₁₉	#125	1	2	yes	—
101	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
102	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
104	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
105	C₂×C₄×C₁₉	#117	1	2	yes	—
106	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
107	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
108	C₂×C₄×C₁₉	#117	1	2	yes	—
109	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
113	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
114	C₂×C₄×C₁₉	#123	1	2	yes	—
115	C₂×C₄×C₁₉	#123	1	2	yes	—
116	C₂×C₄×C₁₉	#124	1	2	yes	—
117	C₂×C₄×C₁₉	#126	1	2	yes	—
118	C₂×C₄×C₁₉	#126	1	2	yes	—
119	C₂×Dic₁₉	—	2	8	no	—
120	C₂×Dic₁₉	—	2	8	no	—
121	C₂×Dic₁₉	—	2	4	no	—
122	C₂×Dic₁₉	—	2	4	no	—
123	C₂×Dic₁₉	#108	1	8	yes	—
124	C₈×C₁₉	—	2	4	no	—
125	C₈×C₁₉	—	4	4	no	—
126	C₁₉⋊C₈	—	2	4	no	—
127	C₁₉⋊C₈	—	4	4	no	—
128	C₂²×C₄×C₈	—	19	1	no	—
129	C₂×#341	—	19	2	no	—
130	C₂×#341	—	19	1	no	—
131	C₂×#341	D₁₉	1	1	yes	characteristic
132	#1728	—	19	2	no	—
133	C₂²×C₁₉	—	2	2	no	—
134	C₂²×C₁₉	—	2	2	no	—
135	C₂²×C₁₉	—	2	4	no	—
136	C₂²×C₁₉	—	2	4	no	—
137	C₂²×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
138	C₂²×C₁₉	#380	1	1	yes	characteristic
139	C₂²×C₁₉	#382	1	1	yes	characteristic
140	C₂²×C₁₉	#387	1	2	yes	—
141	C₂²×C₁₉	#387	1	2	yes	—
142	C₂²×C₁₉	C₂×#108	1	4	yes	—
143	C₄×C₁₉	#376	1	1	yes	characteristic
144	C₄×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
145	C₄×C₁₉	#381	1	1	yes	characteristic
146	C₄×C₁₉	#383	1	1	yes	characteristic
147	C₄×C₁₉	#388	1	2	yes	—
148	C₄×C₁₉	#388	1	2	yes	—
149	Dic₁₉	—	2	8	no	—
150	C₂²×M₁₆	—	19	2	no	—
151	C₂²×M₁₆	C₂×D₁₉	1	2	yes	—
152	C₂³×C₈	—	19	1	no	—
153	C₂³×C₈	C₂×D₁₉	1	1	yes	characteristic
154	C₂²×C₄²	—	19	1	no	—
155	C₂×C₄×C₈	—	19	4	no	—
156	C₂×C₄×C₈	—	38	2	no	—
157	C₂×C₄×C₈	—	38	2	no	—
158	#341	—	2	2	no	—
159	#341	—	19	8	no	—
160	#341	—	38	2	no	—
161	#336	—	38	4	no	—
162	C₂×C₁₉	—	2	2	no	—
163	C₂×C₁₉	—	2	2	no	—
164	C₂×C₁₉	#1434	1	1	yes	characteristic
165	C₂×C₁₉	C₄×#108	1	2	yes	—
166	C₂×C₁₉	#1810	1	2	yes	—
167	C₂×C₁₉	#1837	1	2	yes	—
168	C₂³×C₄	—	19	2	no	—
169	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	19	4	no	—
176	C₂×M₁₆	—	38	2	no	—
177	C₂×M₁₆	—	38	2	no	—
178	C₂×M₁₆	—	38	2	no	—
179	C₂×M₁₆	—	38	2	no	—
180	C₂×M₁₆	—	38	2	no	—
181	C₂×M₁₆	C₄×D₁₉	1	4	yes	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	—	2	2	no	—
185	C₂²×C₈	—	19	4	no	—
186	C₂²×C₈	—	19	2	no	—
187	C₂²×C₈	—	38	1	no	—
188	C₂²×C₈	—	38	2	no	—
189	C₂²×C₈	—	38	1	no	—
190	C₂²×C₈	—	38	2	no	—
191	C₂²×C₈	C₄×D₁₉	1	2	yes	—
192	C₂²×C₈	D₇₆	1	2	yes	—
193	C₂²×C₈	C₂²⋊D₁₉	1	2	yes	—
194	C₂×C₄²	—	19	2	no	—
195	C₂×C₄²	—	19	2	no	—
196	C₂×C₄²	—	38	1	no	—
197	C₂×C₄²	—	38	1	no	—
198	C₂×C₄²	—	38	1	no	—
199	C₂×C₄²	—	38	1	no	—
200	C₄×C₈	—	38	8	no	—
201	C₁₉	#10822	1	1	yes	characteristic, complemented, Sylow
202	C₂⁴	Q₈×D₁₉	1	1	yes	characteristic
203	C₂²×C₄	—	19	4	no	—
204	C₂²×C₄	—	38	8	no	—
205	C₂²×C₄	—	38	2	no	—
206	C₂²×C₄	—	38	2	no	—
207	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
208	C₂²×C₄	C₂×C₄×D₁₉	1	2	yes	—
209	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
210	C₂²×C₄	#64541	1	2	yes	—
211	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
212	C₂²×C₄	#64543	1	2	yes	—
213	C₂²×C₄	#64543	1	2	yes	—
214	C₂²×C₄	#64543	1	1	yes	characteristic
215	C₂²×C₄	#64545	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
217	M₁₆	—	2	4	no	—
218	M₁₆	—	2	4	no	—
219	M₁₆	—	4	2	no	—
220	M₁₆	—	4	2	no	—
221	M₁₆	—	38	4	no	—
222	M₁₆	—	38	4	no	—
223	M₁₆	—	76	2	no	—
224	M₁₆	—	76	2	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	4	2	no	—
231	C₂×C₈	—	4	2	no	—
232	C₂×C₈	—	19	4	no	—
233	C₂×C₈	—	38	2	no	—
234	C₂×C₈	—	38	4	no	—
235	C₂×C₈	—	38	2	no	—
236	C₂×C₈	—	38	4	no	—
237	C₂×C₈	—	38	4	no	—
238	C₂×C₈	—	76	2	no	—
239	C₂×C₈	—	76	2	no	—
240	C₂×C₈	(C₂×D₁₉)⋊C₄	1	4	yes	—
241	C₄²	—	38	2	no	—
242	C₄²	—	38	2	no	—
243	C₄²	—	38	2	no	—
244	C₄²	—	38	2	no	—
245	C₂³	—	2	4	no	—
246	C₂³	#10597333	1	2	yes	—
247	C₂³	#10597335	1	2	yes	—
248	C₂³	C₄⋊C₄×D₁₉	1	1	yes	characteristic
249	C₂³	#10597343	1	1	yes	characteristic
250	C₂³	#10597392	1	1	yes	characteristic
251	C₂×C₄	—	19	8	no	—
252	C₂×C₄	—	38	8	no	—
253	C₂×C₄	—	38	8	no	—
254	C₂×C₄	—	38	4	no	—
255	C₂×C₄	—	38	4	no	—
256	C₂×C₄	C₄²×D₁₉	1	1	yes	characteristic
257	C₂×C₄	#10597318	1	1	yes	characteristic
258	C₂×C₄	C₄×D₇₆	1	2	yes	—
259	C₂×C₄	#10597323	1	1	yes	characteristic
260	C₂×C₄	#10597323	1	2	yes	—
261	C₂×C₄	#10597323	1	1	yes	characteristic
262	C₂×C₄	#10597325	1	2	yes	—
263	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	2	yes	—
264	C₂×C₄	#10597331	1	2	yes	—
265	C₂×C₄	#10597332	1	2	yes	—
266	C₂×C₄	#10597338	1	1	yes	characteristic
267	C₂×C₄	#10597338	1	2	yes	—
268	C₂×C₄	#10597341	1	1	yes	characteristic
269	C₂×C₄	#10597344	1	2	yes	—
270	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
271	C₂×C₄	C₄×C₂²⋊D₁₉	1	2	yes	—
272	C₂×C₄	#10597381	1	2	yes	—
273	C₂×C₄	#10597384	1	1	yes	characteristic
274	C₈	—	2	4	no	—
275	C₈	—	4	4	no	—
276	C₈	—	38	4	no	—
277	C₈	—	76	4	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	2	4	no	—
281	C₂²	—	2	4	no	—
282	C₂²	#11926395	1	1	yes	characteristic
283	C₂²	#11926401	1	1	yes	characteristic
284	C₂²	#11926408	1	1	yes	characteristic
285	C₂²	#11926425	1	2	yes	—
286	C₂²	#11926429	1	2	yes	—
287	C₂²	#108×D₁₉	1	4	yes	—
288	C₄	—	38	8	no	—
289	C₄	#11926389	1	1	yes	characteristic
290	C₄	C₄×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
291	C₄	#11926405	1	1	yes	characteristic
292	C₄	#11926411	1	1	yes	characteristic
293	C₄	#11926432	1	2	yes	—
294	C₄	#11926436	1	2	yes	—
295	C₂	—	2	2	no	—
296	C₂	—	2	2	no	—
297	C₂	#14561859	1	1	yes	characteristic
298	C₂	#14563536	1	2	yes	—
299	C₂	#14563670	1	2	yes	—
300	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
13. #11926308	C₄	semidirect
201. 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