Subgroups — #18090638

Order 4864.

638 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926582	C₂	1	1	yes	maximal, characteristic
2	C₂×#11926570	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926315	C₂	1	2	yes	maximal
4	#14563346	C₂	1	2	yes	maximal
5	C₂×C₁₉×#341	C₂	1	1	yes	maximal, characteristic
6	C₂²×#64520	C₂²	1	1	yes	characteristic
7	C₂×#10597372	C₂²	1	2	yes	—
8	C₂×#10597372	C₂²	1	1	yes	characteristic
9	#11926582	—	2	4	no	—
10	C₂×#10597275	—	2	1	no	—
11	C₂×#10597274	—	2	1	no	—
12	#11926570	—	2	1	no	—
13	#11926570	—	2	1	no	—
14	#11926570	C₄	1	4	yes	complemented
15	C₂×Dic₁₉⋊C₈	—	2	2	no	—
16	#11926315	—	2	4	no	—
17	#11926305	—	2	4	no	—
18	C₂²×M₁₆×C₁₉	C₂²	1	2	yes	—
19	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
20	C₁₉×#341	—	2	2	no	—
21	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
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	—	2	2	no	—
26	C₂×M₁₆×C₁₉	—	2	2	no	—
27	C₂×M₁₆×C₁₉	Q₈	1	4	yes	—
28	C₂²×C₈×C₁₉	—	2	2	no	—
29	C₂²×C₈×C₁₉	—	2	2	no	—
30	C₂²×C₈×C₁₉	—	2	2	no	—
31	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₉	D₄	1	2	yes	—
35	C₂×C₂²⋊Dic₁₉	—	2	1	no	—
36	C₂×C₂²⋊Dic₁₉	—	2	2	no	—
37	#10597372	—	2	1	no	—
38	#10597372	—	2	1	no	—
39	#10597372	—	2	2	no	—
40	#10597372	—	2	2	no	—
41	#10597372	—	2	2	no	—
42	#10597372	C₂×C₄	1	2	yes	—
43	#10597372	D₄	1	4	yes	—
44	#10597372	D₄	1	2	yes	—
45	C₂×C₄⋊Dic₁₉	—	2	2	no	—
46	C₂×C₄⋊Dic₁₉	Q₈	1	2	yes	—
47	C₂×C₄×Dic₁₉	—	2	1	no	—
48	C₂×C₄×Dic₁₉	—	2	2	no	—
49	C₂×#64520	—	2	1	no	—
50	C₂×#64520	—	2	2	no	—
51	C₂×#64520	—	2	2	no	—
52	C₂×#64520	C₂×C₄	1	2	yes	—
53	C₂²×C₁₉⋊C₈	—	2	1	no	—
54	#10597275	—	2	4	no	—
55	#10597274	—	2	4	no	—
56	Dic₁₉⋊C₈	—	2	4	no	—
57	Dic₁₉⋊C₈	—	2	4	no	—
58	C₂⁴×C₁₉	#41	1	1	yes	characteristic
59	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
65	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	#41	1	2	yes	—
67	C₂²×C₄×C₁₉	#41	1	2	yes	—
68	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
69	C₂²×Dic₁₉	—	2	2	no	—
70	C₂²×Dic₁₉	—	4	2	no	—
71	M₁₆×C₁₉	—	2	4	no	—
72	M₁₆×C₁₉	—	2	4	no	—
73	M₁₆×C₁₉	—	4	2	no	—
74	M₁₆×C₁₉	—	4	2	no	—
75	C₂×C₈×C₁₉	—	2	4	no	—
76	C₂×C₈×C₁₉	—	2	1	no	—
77	C₂×C₈×C₁₉	—	2	1	no	—
78	C₂×C₈×C₁₉	—	2	4	no	—
79	C₂×C₈×C₁₉	—	2	1	no	—
80	C₂×C₈×C₁₉	—	2	1	no	—
81	C₂×C₈×C₁₉	—	2	4	no	—
82	C₂×C₈×C₁₉	—	4	2	no	—
83	C₂×C₈×C₁₉	—	4	2	no	—
84	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
85	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
86	C₂²⋊Dic₁₉	—	2	2	no	—
87	C₂²⋊Dic₁₉	—	2	2	no	—
88	C₂²⋊Dic₁₉	—	2	4	no	—
89	C₂²⋊Dic₁₉	—	4	2	no	—
90	C₄⋊Dic₁₉	—	2	2	no	—
91	C₄⋊Dic₁₉	—	2	4	no	—
92	C₄⋊Dic₁₉	—	4	2	no	—
93	C₄⋊Dic₁₉	C₄⋊C₄	1	4	yes	—
94	C₄×Dic₁₉	—	2	2	no	—
95	C₄×Dic₁₉	—	2	2	no	—
96	C₄×Dic₁₉	—	2	4	no	—
97	C₄×Dic₁₉	—	4	2	no	—
98	#64520	—	2	4	no	—
99	#64520	—	4	1	no	—
100	#64520	—	4	1	no	—
101	C₂×C₁₉⋊C₈	—	2	2	no	—
102	C₂×C₁₉⋊C₈	—	4	2	no	—
103	#11241	—	19	1	no	maximal, Sylow
104	C₂³×C₁₉	—	4	2	no	—
105	C₂³×C₁₉	#117	1	1	yes	characteristic
106	C₂³×C₁₉	#121	1	1	yes	characteristic
107	C₂³×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
108	C₂³×C₁₉	#123	1	2	yes	—
109	C₂³×C₁₉	#125	1	2	yes	—
110	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
114	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
115	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
116	C₂×C₄×C₁₉	#121	1	2	yes	—
117	C₂×C₄×C₁₉	#121	1	2	yes	—
118	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
119	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
120	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
121	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
122	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
123	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
124	C₂×C₄×C₁₉	#125	1	2	yes	—
125	C₂×C₄×C₁₉	#126	1	2	yes	—
126	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
127	C₂×C₄×C₁₉	C₄⋊Q₈	1	2	yes	—
128	C₂×Dic₁₉	—	2	4	no	—
129	C₂×Dic₁₉	—	4	4	no	—
130	C₂×Dic₁₉	—	4	4	no	—
131	C₂×Dic₁₉	—	8	4	no	—
132	C₈×C₁₉	—	2	2	no	—
133	C₈×C₁₉	—	2	2	no	—
134	C₈×C₁₉	—	4	4	no	—
135	C₁₉⋊C₈	—	4	2	no	—
136	C₂×#427	—	19	1	no	—
137	C₂×#424	—	19	1	no	—
138	C₂×#344	—	19	2	no	—
139	C₂×#341	D₁₉	1	1	yes	characteristic
140	#1733	—	19	2	no	—
141	C₂²×C₁₉	—	2	2	no	—
142	C₂²×C₁₉	—	2	2	no	—
143	C₂²×C₁₉	—	2	2	no	—
144	C₂²×C₁₉	—	4	2	no	—
145	C₂²×C₁₉	—	4	2	no	—
146	C₂²×C₁₉	#380	1	1	yes	characteristic
147	C₂²×C₁₉	#387	1	2	yes	—
148	C₂²×C₁₉	#394	1	1	yes	characteristic
149	C₂²×C₁₉	#398	1	1	yes	characteristic
150	C₂²×C₁₉	#398	1	2	yes	—
151	C₄×C₁₉	#386	1	1	yes	characteristic
152	C₄×C₁₉	#389	1	2	yes	—
153	C₄×C₁₉	#391	1	1	yes	characteristic
154	C₄×C₁₉	#395	1	1	yes	characteristic
155	C₄×C₁₉	#396	1	2	yes	—
156	C₄×C₁₉	#400	1	1	yes	characteristic
157	Dic₁₉	—	4	4	no	—
158	Dic₁₉	—	8	4	no	—
159	C₂²×M₁₆	—	19	1	no	—
160	C₂²×M₁₆	C₂×D₁₉	1	2	yes	—
161	C₂³×C₈	C₂×D₁₉	1	1	yes	characteristic
162	C₂×#117	—	19	1	no	—
163	C₂×#117	—	19	2	no	—
164	#427	—	38	4	no	—
165	C₂×#107	—	38	1	no	—
166	C₂×#106	—	38	1	no	—
167	#424	—	19	4	no	—
168	#424	—	38	1	no	—
169	#424	—	38	1	no	—
170	C₂×C₄⋊C₈	—	38	2	no	—
171	#344	—	38	4	no	—
172	#341	—	2	2	no	—
173	#339	—	38	4	no	—
174	C₂×C₁₉	—	2	1	no	—
175	C₂×C₁₉	—	2	2	no	—
176	C₂×C₁₉	—	2	1	no	—
177	C₂×C₁₉	—	2	2	no	—
178	C₂×C₁₉	#1441	1	1	yes	characteristic
179	C₂×C₁₉	#1814	1	2	yes	—
180	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
181	C₂×M₁₆	—	2	2	no	—
182	C₂×M₁₆	—	2	2	no	—
183	C₂×M₁₆	—	2	2	no	—
184	C₂×M₁₆	—	2	2	no	—
185	C₂×M₁₆	—	2	2	no	—
186	C₂×M₁₆	—	19	2	no	—
187	C₂×M₁₆	—	38	1	no	—
188	C₂×M₁₆	—	38	2	no	—
189	C₂×M₁₆	—	38	2	no	—
190	C₂×M₁₆	Dic₃₈	1	4	yes	—
191	C₂²×C₈	—	2	2	no	—
192	C₂²×C₈	—	2	2	no	—
193	C₂²×C₈	—	2	2	no	—
194	C₂²×C₈	—	38	1	no	—
195	C₂²×C₈	C₄×D₁₉	1	2	yes	—
196	C₂²×C₈	D₇₆	1	1	yes	characteristic
197	C₂²×C₈	D₇₆	1	1	yes	characteristic
198	C₂²×C₈	C₂²⋊D₁₉	1	2	yes	—
199	#117	—	19	2	no	—
200	#117	—	19	2	no	—
201	#117	—	19	4	no	—
202	#117	—	38	1	no	—
203	#117	—	38	1	no	—
204	#117	—	38	2	no	—
205	#117	—	38	2	no	—
206	#117	—	38	2	no	—
207	C₂×C₄⋊C₄	—	19	2	no	—
208	C₂×C₄⋊C₄	—	38	2	no	—
209	C₂×C₂²⋊C₄	—	38	1	no	—
210	C₂×C₂²⋊C₄	—	38	2	no	—
211	C₂×C₄²	—	38	1	no	—
212	C₂×C₄²	—	38	2	no	—
213	#107	—	38	4	no	—
214	#106	—	38	4	no	—
215	C₄⋊C₈	—	38	4	no	—
216	C₄⋊C₈	—	38	4	no	—
217	C₁₉	#11241	1	1	yes	characteristic, complemented, Sylow
218	C₂⁴	#64545	1	1	yes	characteristic
219	C₂²×C₄	—	38	2	no	—
220	C₂²×C₄	—	76	2	no	—
221	C₂²×C₄	C₂×Dic₃₈	1	2	yes	—
222	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
224	C₂²×C₄	#64541	1	2	yes	—
225	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
226	C₂²×C₄	D₄×D₁₉	1	2	yes	—
227	C₂²×C₄	#64543	1	1	yes	characteristic
228	C₂²×C₄	#64543	1	2	yes	—
229	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
230	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
231	M₁₆	—	2	4	no	—
232	M₁₆	—	2	4	no	—
233	M₁₆	—	4	2	no	—
234	M₁₆	—	4	2	no	—
235	M₁₆	—	38	4	no	—
236	M₁₆	—	76	1	no	—
237	M₁₆	—	76	1	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	2	1	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	4	2	no	—
246	C₂×C₈	—	4	2	no	—
247	C₂×C₈	—	38	2	no	—
248	C₂×C₈	—	76	2	no	—
249	C₂×C₈	(C₂×D₁₉)⋊C₄	1	2	yes	—
250	C₂×C₈	(C₂×D₁₉)⋊C₄	1	2	yes	—
251	C₄⋊C₄	—	19	4	no	—
252	C₄⋊C₄	—	38	2	no	—
253	C₄⋊C₄	—	38	4	no	—
254	C₄⋊C₄	—	76	2	no	—
255	C₂²⋊C₄	—	38	2	no	—
256	C₂²⋊C₄	—	38	2	no	—
257	C₂²⋊C₄	—	38	4	no	—
258	C₂²⋊C₄	—	76	2	no	—
259	C₄²	—	38	2	no	—
260	C₄²	—	38	2	no	—
261	C₄²	—	38	4	no	—
262	C₄²	—	76	2	no	—
263	C₂³	—	4	2	no	—
264	C₂³	#10597336	1	2	yes	—
265	C₂³	#10597338	1	1	yes	characteristic
266	C₂³	#10597340	1	2	yes	—
267	C₂³	#10597341	1	1	yes	characteristic
268	C₂³	#10597392	1	1	yes	characteristic
269	C₂×C₄	—	38	4	no	—
270	C₂×C₄	—	76	4	no	—
271	C₂×C₄	—	76	4	no	—
272	C₂×C₄	—	152	4	no	—
273	C₂×C₄	C₄×Dic₃₈	1	2	yes	—
274	C₂×C₄	#10597315	1	1	yes	characteristic
275	C₂×C₄	#10597321	1	1	yes	characteristic
276	C₂×C₄	#10597324	1	2	yes	—
277	C₂×C₄	#10597324	1	2	yes	—
278	C₂×C₄	#10597325	1	2	yes	—
279	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	2	yes	—
280	C₂×C₄	#10597328	1	1	yes	characteristic
281	C₂×C₄	#10597330	1	2	yes	—
282	C₂×C₄	#10597332	1	1	yes	characteristic
283	C₂×C₄	#10597334	1	2	yes	—
284	C₂×C₄	#10597335	1	2	yes	—
285	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
286	C₂×C₄	#10597343	1	1	yes	characteristic
287	C₂×C₄	#10597370	1	2	yes	—
288	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
289	C₂×C₄	#10597384	1	1	yes	characteristic
290	C₂×C₄	#10597385	1	2	yes	—
291	C₈	—	2	2	no	—
292	C₈	—	2	2	no	—
293	C₈	—	4	4	no	—
294	C₈	—	76	2	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	4	2	no	—
299	C₂²	—	4	2	no	—
300	C₂²	#11926401	1	1	yes	characteristic
301	C₂²	#11926431	1	2	yes	—
302	C₂²	#11926462	1	1	yes	characteristic
303	C₂²	#11926476	1	1	yes	characteristic
304	C₂²	#11926477	1	2	yes	—
305	C₄	—	76	4	no	—
306	C₄	—	152	4	no	—
307	C₄	#11926421	1	1	yes	characteristic
308	C₄	#11926441	1	2	yes	—
309	C₄	#11926450	1	1	yes	characteristic
310	C₄	#11926465	1	1	yes	characteristic
311	C₄	#11926467	1	2	yes	—
312	C₄	#11926484	1	1	yes	characteristic
313	C₂	—	2	1	no	—
314	C₂	—	2	2	no	—
315	C₂	—	2	1	no	—
316	C₂	—	2	2	no	—
317	C₂	#14561889	1	1	yes	characteristic
318	C₂	#14563690	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#10597384
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₂×C₁₉×#341	C₂
socle	C₂²×C₁₉	#380
prime radical	C₂⁴×C₁₉	#41
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂×#341	D₁₉
hypocenter	C₁₉	#11241
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#341

Complements

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

N	H ≅ G/N	type
14. #11926570	C₄	semidirect
217. C₁₉	#11241	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926582	2
C₂×#11926570	2
C₂×#11926315	2
#14563346	2
C₂×C₁₉×#341	2
#11241	19