Subgroups — #18088267

Order 4864.

894 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926307	C₂	1	2	yes	maximal
2	C₂×#11926307	C₂	1	2	yes	maximal
3	#14561865	C₂	1	2	yes	maximal
4	C₂²×C₄×C₈×C₁₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₁₉	C₂²	1	4	yes	—
6	C₂×C₁₉⋊#95	—	2	2	no	—
7	C₂×C₁₉⋊#95	—	2	2	no	—
8	#11926307	—	2	4	no	—
9	#11926307	C₄	1	8	yes	complemented
10	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₉	—	2	4	no	—
14	C₂×C₄×C₈×C₁₉	C₄	1	4	yes	complemented
15	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
16	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
17	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
18	C₂³×Dic₁₉	—	2	4	no	—
19	C₂²×C₈×C₁₉	—	2	2	no	—
20	C₂²×C₈×C₁₉	—	2	2	no	—
21	C₂²×C₈×C₁₉	—	2	2	no	—
22	C₂²×C₈×C₁₉	—	2	2	no	—
23	C₂²×C₈×C₁₉	C₂×C₄	1	4	yes	—
24	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₁₉	D₄	1	4	yes	—
27	C₂×C₄²×C₁₉	—	2	2	no	—
28	C₂×C₄²×C₁₉	—	2	2	no	—
29	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₁₉	Q₈	1	2	yes	—
31	C₂×C₄⋊Dic₁₉	—	2	4	no	—
32	C₂×C₄⋊Dic₁₉	—	2	4	no	—
33	C₂×C₄⋊Dic₁₉	—	2	4	no	—
34	C₂×C₄⋊Dic₁₉	—	2	4	no	—
35	C₂×C₄⋊Dic₁₉	—	2	4	no	—
36	C₂×C₄⋊Dic₁₉	C₂×C₄	1	8	yes	—
37	C₄×C₈×C₁₉	—	2	8	no	—
38	C₁₉⋊#95	—	4	4	no	—
39	C₁₉⋊#95	—	4	4	no	—
40	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₉	—	2	2	no	—
42	C₂²×C₄×C₁₉	—	2	2	no	—
43	C₂²×C₄×C₁₉	—	2	2	no	—
44	C₂²×C₄×C₁₉	—	2	2	no	—
45	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
46	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₁₉	D₈	1	2	yes	—
48	C₂²×C₄×C₁₉	SD₁₆	1	4	yes	—
49	C₂²×C₄×C₁₉	Q₁₆	1	2	yes	—
50	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
51	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
53	C₂²×C₄×C₁₉	C₂×Q₈	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	2	yes	—
58	C₂²×Dic₁₉	—	2	8	no	—
59	C₂²×Dic₁₉	—	4	4	no	—
60	C₂²×Dic₁₉	—	4	4	no	—
61	C₂²×Dic₁₉	—	4	4	no	—
62	C₂²×Dic₁₉	—	4	4	no	—
63	C₂²×Dic₁₉	—	4	4	no	—
64	C₂²×Dic₁₉	—	4	4	no	—
65	C₂×C₈×C₁₉	—	2	2	no	—
66	C₂×C₈×C₁₉	—	2	2	no	—
67	C₂×C₈×C₁₉	—	2	2	no	—
68	C₂×C₈×C₁₉	—	2	2	no	—
69	C₂×C₈×C₁₉	—	2	8	no	—
70	C₂×C₈×C₁₉	—	2	8	no	—
71	C₂×C₈×C₁₉	C₄²	1	4	yes	—
72	C₂×C₈×C₁₉	C₂²⋊C₄	1	4	yes	—
73	C₄²×C₁₉	—	2	2	no	—
74	C₄²×C₁₉	—	2	2	no	—
75	C₄²×C₁₉	—	2	4	no	—
76	C₄⋊Dic₁₉	—	2	8	no	—
77	C₄⋊Dic₁₉	—	2	8	no	—
78	C₄⋊Dic₁₉	—	4	4	no	—
79	C₄⋊Dic₁₉	—	4	4	no	—
80	#10820	—	19	1	no	maximal, Sylow
81	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂³×C₁₉	C₄×D₄	1	2	yes	—
83	C₂³×C₁₉	#121	1	2	yes	—
84	C₂³×C₁₉	#124	1	2	yes	—
85	C₂³×C₁₉	C₂×D₈	1	1	yes	characteristic
86	C₂³×C₁₉	C₂×SD₁₆	1	2	yes	—
87	C₂³×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
88	C₂³×C₁₉	#135	1	2	yes	—
89	C₂³×C₁₉	#135	1	2	yes	—
90	C₂×C₄×C₁₉	—	2	2	no	—
91	C₂×C₄×C₁₉	—	2	2	no	—
92	C₂×C₄×C₁₉	—	2	2	no	—
93	C₂×C₄×C₁₉	—	2	2	no	—
94	C₂×C₄×C₁₉	—	2	8	no	—
95	C₂×C₄×C₁₉	—	2	4	no	—
96	C₂×C₄×C₁₉	—	2	4	no	—
97	C₂×C₄×C₁₉	#106	1	4	yes	—
98	C₂×C₄×C₁₉	#107	1	4	yes	—
99	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
100	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₁₉	#117	1	4	yes	—
102	C₂×C₄×C₁₉	#117	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	1	yes	characteristic
106	C₂×C₄×C₁₉	C₄×D₄	1	4	yes	—
107	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
108	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₄×C₁₉	#123	1	4	yes	—
110	C₂×C₄×C₁₉	#125	1	2	yes	—
111	C₂×C₄×C₁₉	#126	1	4	yes	—
112	C₂×Dic₁₉	—	4	8	no	—
113	C₂×Dic₁₉	—	4	8	no	—
114	C₂×Dic₁₉	—	4	8	no	—
115	C₂×Dic₁₉	—	8	4	no	—
116	C₂×Dic₁₉	—	8	4	no	—
117	C₂×Dic₁₉	—	8	4	no	—
118	C₂×Dic₁₉	—	8	4	no	—
119	C₈×C₁₉	—	2	4	no	—
120	C₈×C₁₉	—	2	4	no	—
121	C₂²×C₄×C₈	D₁₉	1	1	yes	characteristic
122	C₂×#340	—	19	2	no	—
123	C₂×#340	—	19	2	no	—
124	C₄⋊#95	—	19	2	no	—
125	C₂²×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
126	C₂²×C₁₉	#380	1	1	yes	characteristic
127	C₂²×C₁₉	#382	1	1	yes	characteristic
128	C₂²×C₁₉	#388	1	4	yes	—
129	C₂²×C₁₉	C₂×#106	1	1	yes	characteristic
130	C₂²×C₁₉	C₂×#107	1	1	yes	characteristic
131	C₂²×C₁₉	#427	1	1	yes	characteristic
132	C₂²×C₁₉	#427	1	1	yes	characteristic
133	C₂²×C₁₉	C₄×D₈	1	2	yes	—
134	C₂²×C₁₉	C₄×SD₁₆	1	4	yes	—
135	C₂²×C₁₉	C₄×Q₁₆	1	2	yes	—
136	C₂²×C₁₉	#465	1	4	yes	—
137	C₂²×C₁₉	#466	1	2	yes	—
138	C₂²×C₁₉	#467	1	2	yes	—
139	C₂²×C₁₉	#486	1	2	yes	—
140	C₂²×C₁₉	#487	1	2	yes	—
141	C₂²×C₁₉	#488	1	4	yes	—
142	C₄×C₁₉	—	2	4	no	—
143	C₄×C₁₉	—	2	4	no	—
144	C₄×C₁₉	#376	1	1	yes	characteristic
145	C₄×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
146	C₄×C₁₉	#381	1	1	yes	characteristic
147	C₄×C₁₉	#383	1	1	yes	characteristic
148	C₄×C₁₉	#387	1	4	yes	—
149	Dic₁₉	—	8	8	no	—
150	C₂³×C₈	C₂×D₁₉	1	1	yes	characteristic
151	C₂³×C₈	C₂×D₁₉	1	1	yes	characteristic
152	C₂²×C₄⋊C₄	—	19	4	no	—
153	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
154	C₂×C₄×C₈	—	2	4	no	—
155	C₂×C₄×C₈	Dic₁₉	1	4	yes	complemented
156	C₂×#95	—	38	2	no	—
157	C₂×#95	—	38	2	no	—
158	#340	—	19	8	no	—
159	#340	—	38	4	no	—
160	C₂×C₁₉	#1434	1	1	yes	characteristic
161	C₂×C₁₉	C₄×#106	1	1	yes	characteristic
162	C₂×C₁₉	C₄×#107	1	1	yes	characteristic
163	C₂×C₁₉	#1807	1	1	yes	characteristic
164	C₂×C₁₉	#1808	1	1	yes	characteristic
165	C₂×C₁₉	#1834	1	1	yes	characteristic
166	C₂×C₁₉	#1835	1	1	yes	characteristic
167	C₂×C₁₉	#1933	1	4	yes	—
168	C₂×C₁₉	#1934	1	4	yes	—
169	C₂³×C₄	—	38	4	no	—
170	C₂³×C₄	D₇₆	1	1	yes	characteristic
171	C₂³×C₄	D₇₆	1	1	yes	characteristic
172	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
173	C₂²×C₈	—	2	2	no	—
174	C₂²×C₈	—	2	2	no	—
175	C₂²×C₈	—	2	2	no	—
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	C₄×D₁₉	1	4	yes	—
178	C₂²×C₈	C₂×Dic₁₉	1	2	yes	—
179	C₂²×C₈	C₂×Dic₁₉	1	2	yes	—
180	C₂²×C₈	C₂²⋊D₁₉	1	4	yes	—
181	C₂×C₄⋊C₄	—	19	8	no	—
182	C₂×C₄⋊C₄	—	38	4	no	—
183	C₂×C₄⋊C₄	—	38	4	no	—
184	C₂×C₄⋊C₄	—	38	4	no	—
185	C₂×C₄⋊C₄	—	38	4	no	—
186	C₂×C₄⋊C₄	—	38	4	no	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	—	2	2	no	—
189	C₂×C₄²	Dic₃₈	1	2	yes	—
190	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
191	C₄×C₈	—	2	8	no	—
192	#95	—	76	4	no	—
193	#95	—	76	4	no	—
194	C₁₉	#10820	1	1	yes	characteristic, complemented, Sylow
195	C₂⁴	C₂×D₇₆	1	1	yes	characteristic
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	38	8	no	—
201	C₂²×C₄	—	76	4	no	—
202	C₂²×C₄	—	76	4	no	—
203	C₂²×C₄	—	76	4	no	—
204	C₂²×C₄	—	76	4	no	—
205	C₂²×C₄	—	76	4	no	—
206	C₂²×C₄	—	76	4	no	—
207	C₂²×C₄	#64516	1	4	yes	—
208	C₂²×C₄	D₁₅₂	1	2	yes	—
209	C₂²×C₄	Dic₇₆	1	2	yes	—
210	C₂²×C₄	C₄⋊Dic₁₉	1	2	yes	—
211	C₂²×C₄	C₄⋊Dic₁₉	1	2	yes	—
212	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₄×D₁₉	1	2	yes	—
214	C₂²×C₄	#64541	1	2	yes	—
215	C₂²×C₄	#64541	1	2	yes	—
216	C₂²×C₄	#64541	1	2	yes	—
217	C₂²×C₄	#64541	1	2	yes	—
218	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	8	no	—
225	C₂×C₈	—	2	8	no	—
226	C₂×C₈	C₄×Dic₁₉	1	4	yes	—
227	C₂×C₈	C₂²⋊Dic₁₉	1	4	yes	—
228	C₄⋊C₄	—	38	8	no	—
229	C₄⋊C₄	—	38	8	no	—
230	C₄⋊C₄	—	76	4	no	—
231	C₄⋊C₄	—	76	4	no	—
232	C₄²	—	2	2	no	—
233	C₄²	—	2	4	no	—
234	C₄²	—	2	2	no	—
235	C₂³	C₄×D₇₆	1	2	yes	—
236	C₂³	#10597321	1	2	yes	—
237	C₂³	C₂×#64516	1	2	yes	—
238	C₂³	C₂×D₁₅₂	1	1	yes	characteristic
239	C₂³	#10597350	1	2	yes	—
240	C₂³	#10597350	1	2	yes	—
241	C₂³	C₂×Dic₇₆	1	1	yes	characteristic
242	C₂³	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
243	C₂³	#10597376	1	2	yes	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	8	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	76	8	no	—
252	C₂×C₄	—	76	8	no	—
253	C₂×C₄	—	76	8	no	—
254	C₂×C₄	—	152	4	no	—
255	C₂×C₄	—	152	4	no	—
256	C₂×C₄	—	152	4	no	—
257	C₂×C₄	—	152	4	no	—
258	C₂×C₄	#10597274	1	4	yes	—
259	C₂×C₄	#10597275	1	4	yes	—
260	C₂×C₄	C₄×Dic₃₈	1	2	yes	—
261	C₂×C₄	#10597316	1	2	yes	—
262	C₂×C₄	#10597318	1	4	yes	—
263	C₂×C₄	#10597322	1	4	yes	—
264	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
265	C₂×C₄	#10597370	1	2	yes	—
266	C₂×C₄	#10597372	1	1	yes	characteristic
267	C₂×C₄	#10597372	1	1	yes	characteristic
268	C₂×C₄	#10597372	1	1	yes	characteristic
269	C₂×C₄	#10597372	1	1	yes	characteristic
270	C₂×C₄	C₄×C₂²⋊D₁₉	1	4	yes	—
271	C₂×C₄	#10597375	1	4	yes	—
272	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
273	C₈	—	2	4	no	—
274	C₈	—	2	4	no	—
275	C₂²	C₄×C₄⋊Dic₁₉	1	1	yes	characteristic
276	C₂²	#11926404	1	1	yes	characteristic
277	C₂²	#11926410	1	1	yes	characteristic
278	C₂²	#11926435	1	4	yes	—
279	C₂²	C₂×#10597274	1	1	yes	characteristic
280	C₂²	C₂×#10597275	1	1	yes	characteristic
281	C₂²	#11926582	1	1	yes	characteristic
282	C₂²	#11926582	1	1	yes	characteristic
283	C₂²	C₄×D₁₅₂	1	2	yes	—
284	C₂²	C₄×#64516	1	4	yes	—
285	C₂²	C₄×Dic₇₆	1	2	yes	—
286	C₂²	#11926773	1	4	yes	—
287	C₂²	#11926780	1	2	yes	—
288	C₂²	#11926785	1	2	yes	—
289	C₂²	#11926891	1	2	yes	—
290	C₂²	#11926895	1	2	yes	—
291	C₂²	#11926899	1	4	yes	—
292	C₄	—	2	4	no	—
293	C₄	—	2	4	no	—
294	C₄	—	152	8	no	—
295	C₄	#11926390	1	1	yes	characteristic
296	C₄	C₄×C₂²⋊Dic₁₉	1	1	yes	characteristic
297	C₄	#11926407	1	1	yes	characteristic
298	C₄	#11926412	1	1	yes	characteristic
299	C₄	#11926428	1	4	yes	—
300	C₂	#14561862	1	1	yes	characteristic
301	C₂	C₄×#10597274	1	1	yes	characteristic
302	C₂	C₄×#10597275	1	1	yes	characteristic
303	C₂	#14563660	1	1	yes	characteristic
304	C₂	#14563664	1	1	yes	characteristic
305	C₂	#14563763	1	1	yes	characteristic
306	C₂	#14563766	1	1	yes	characteristic
307	C₂	#14564311	1	4	yes	—
308	C₂	#14564318	1	4	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₇₆
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₂²×C₄×C₈×C₁₉	C₂
socle, prime radical	C₂⁴×C₁₉	C₂×D₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂²×C₄×C₈	D₁₉
hypocenter	C₁₉	#10820
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
9. #11926307	C₄	semidirect
14. C₂×C₄×C₈×C₁₉	C₄	semidirect
155. C₂×C₄×C₈	Dic₁₉	semidirect
194. C₁₉	#10820	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926307	2
#14561865	2
C₂²×C₄×C₈×C₁₉	2
#10820	19