Subgroups — #18088343

Order 4864.

842 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#64520	C₂	1	1	yes	maximal, characteristic
2	#14566042	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926300	C₂	1	4	yes	maximal
4	C₁₉×#2285	C₂	1	1	yes	maximal, characteristic
5	C₂²×#64520	C₂²	1	2	yes	—
6	C₂×#10597372	C₂²	1	2	yes	—
7	C₄×#64520	—	2	1	no	—
8	C₄×#64520	—	2	1	no	—
9	C₄×#64520	—	2	1	no	—
10	C₄×#64520	—	2	1	no	—
11	C₂×#10597260	C₄	1	2	yes	complemented
12	C₂×C₄×C₁₉⋊C₈	C₄	1	2	yes	complemented
13	#11926414	—	2	1	no	—
14	#11926410	—	2	1	no	—
15	#11926404	—	2	2	no	—
16	C₄×C₄⋊Dic₁₉	C₄	1	4	yes	complemented
17	#11926300	—	2	8	no	—
18	C₂×C₁₉×#117	C₂²	1	2	yes	—
19	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
20	C₁₉×#384	—	2	1	no	—
21	C₁₉×#382	—	2	1	no	—
22	C₁₉×#380	—	2	2	no	—
23	C₄×C₄⋊C₄×C₁₉	C₄	1	4	yes	complemented
24	C₂³×C₄×C₁₉	D₄	1	2	yes	—
25	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
26	C₁₉×#117	—	2	4	no	—
27	C₁₉×#117	—	2	4	no	—
28	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
29	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
30	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
31	C₂×C₄²×C₁₉	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₉	Q₈	1	2	yes	—
41	C₂×C₂²⋊Dic₁₉	—	2	2	no	—
42	#10597372	—	2	4	no	—
43	#10597372	—	2	4	no	—
44	C₂×C₄⋊Dic₁₉	—	2	4	no	—
45	C₂×C₄⋊Dic₁₉	—	2	2	no	—
46	C₂×C₄×Dic₁₉	C₂×C₄	1	4	yes	—
47	C₂×#64520	—	2	2	no	—
48	C₂×#64520	—	2	2	no	—
49	C₂×#64520	—	2	2	no	—
50	C₂×#64520	—	2	2	no	—
51	C₂×#64520	D₄	1	4	yes	—
52	C₂×#64520	Q₈	1	4	yes	—
53	C₂²×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
54	C₁₉×#95	—	2	4	no	—
55	C₁₉⋊#95	—	2	4	no	—
56	#10597260	—	2	2	no	—
57	#10597260	—	2	2	no	—
58	C₄×C₁₉⋊C₈	—	2	2	no	—
59	C₄×C₁₉⋊C₈	—	2	2	no	—
60	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	—	2	4	no	—
62	C₂²×C₄×C₁₉	—	4	4	no	—
63	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
67	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
69	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
76	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
77	C₂²×C₄×C₁₉	#41	1	2	yes	—
78	C₂²×C₄×C₁₉	#41	1	2	yes	—
79	C₂²×Dic₁₉	—	2	4	no	—
80	C₄⋊C₄×C₁₉	—	4	8	no	—
81	C₄⋊C₄×C₁₉	—	4	4	no	—
82	C₂²⋊C₄×C₁₉	—	4	2	no	—
83	C₂²⋊C₄×C₁₉	—	4	2	no	—
84	C₄²×C₁₉	—	2	8	no	—
85	C₄²×C₁₉	C₄²	1	2	yes	—
86	C₄²×C₁₉	C₄²	1	2	yes	—
87	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
88	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
89	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
90	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
91	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
92	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
93	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
94	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
95	C₂²⋊Dic₁₉	—	4	2	no	—
96	C₂²⋊Dic₁₉	—	4	2	no	—
97	C₄⋊Dic₁₉	—	4	8	no	—
98	C₄⋊Dic₁₉	—	4	4	no	—
99	C₄×Dic₁₉	—	2	8	no	—
100	#64520	—	2	4	no	—
101	#64520	—	2	4	no	—
102	#64520	—	2	4	no	—
103	#64520	—	2	4	no	—
104	C₂×C₁₉⋊C₈	—	2	4	no	—
105	C₂×C₁₉⋊C₈	—	2	4	no	—
106	C₂×C₁₉⋊C₈	C₄⋊C₄	1	8	yes	complemented
107	#10837	—	19	1	no	maximal, Sylow
108	C₂³×C₁₉	—	4	2	no	—
109	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂³×C₁₉	#124	1	1	yes	characteristic
113	C₂³×C₁₉	#124	1	1	yes	characteristic
114	C₂³×C₁₉	#127	1	1	yes	characteristic
115	C₂³×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×C₁₉	—	2	4	no	—
117	C₂×C₄×C₁₉	—	2	4	no	—
118	C₂×C₄×C₁₉	—	2	4	no	—
119	C₂×C₄×C₁₉	—	4	2	no	—
120	C₂×C₄×C₁₉	—	4	2	no	—
121	C₂×C₄×C₁₉	—	4	2	no	—
122	C₂×C₄×C₁₉	—	4	8	no	—
123	C₂×C₄×C₁₉	—	4	4	no	—
124	C₂×C₄×C₁₉	#95	1	2	yes	—
125	C₂×C₄×C₁₉	#95	1	2	yes	—
126	C₂×C₄×C₁₉	#95	1	2	yes	—
127	C₂×C₄×C₁₉	#95	1	2	yes	—
128	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
129	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
131	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₁₉	#117	1	2	yes	—
134	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
135	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
136	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
137	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
138	C₂×C₄×C₁₉	#121	1	2	yes	—
139	C₂×C₄×C₁₉	#121	1	2	yes	—
140	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
141	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
142	C₂×C₄×C₁₉	#125	1	2	yes	—
143	C₂×C₄×C₁₉	C₄⋊Q₈	1	2	yes	—
144	C₂×Dic₁₉	—	4	8	no	—
145	C₂×Dic₁₉	—	4	4	no	—
146	C₁₉⋊C₈	—	2	8	no	—
147	C₂×C₄×M₁₆	—	19	1	no	—
148	#2285	—	19	1	no	—
149	#2285	D₁₉	1	1	yes	characteristic
150	C₂×H(3,ℤ/4)	—	19	4	no	—
151	C₂²×C₁₉	—	2	2	no	—
152	C₂²×C₁₉	—	2	2	no	—
153	C₂²×C₁₉	—	2	2	no	—
154	C₂²×C₁₉	—	4	1	no	—
155	C₂²×C₁₉	—	4	1	no	—
156	C₂²×C₁₉	—	4	1	no	—
157	C₂²×C₁₉	—	4	1	no	—
158	C₂²×C₁₉	C₂×#95	1	1	yes	characteristic
159	C₂²×C₁₉	#382	1	1	yes	characteristic
160	C₂²×C₁₉	#384	1	1	yes	characteristic
161	C₂²×C₁₉	#390	1	1	yes	characteristic
162	C₂²×C₁₉	#390	1	1	yes	characteristic
163	C₂²×C₁₉	#391	1	1	yes	characteristic
164	C₂²×C₁₉	#391	1	1	yes	characteristic
165	C₄×C₁₉	—	2	2	no	—
166	C₄×C₁₉	—	2	2	no	—
167	C₄×C₁₉	—	2	2	no	—
168	C₄×C₁₉	—	8	4	no	—
169	C₄×C₁₉	H(3,ℤ/4)	1	4	yes	complemented
170	C₄×C₁₉	C₄×C₄⋊C₄	1	2	yes	—
171	C₄×C₁₉	#380	1	2	yes	—
172	C₄×C₁₉	#389	1	2	yes	—
173	C₄×C₁₉	#389	1	2	yes	—
174	Dic₁₉	—	8	4	no	—
175	C₂²×M₁₆	—	19	2	no	—
176	C₂×#117	—	19	2	no	—
177	C₂×#117	C₂×D₁₉	1	2	yes	—
178	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
179	C₄×M₁₆	—	38	1	no	—
180	C₄×M₁₆	—	38	1	no	—
181	C₄×M₁₆	—	38	1	no	—
182	C₄×M₁₆	—	38	1	no	—
183	C₂×#97	—	19	2	no	—
184	C₂×C₄×C₈	—	19	2	no	—
185	#384	—	2	1	no	—
186	#384	—	38	1	no	—
187	#382	—	2	1	no	—
188	#382	—	38	1	no	—
189	#380	—	2	2	no	—
190	#380	—	38	2	no	—
191	C₄×C₄⋊C₄	—	19	4	no	—
192	C₄×C₄⋊C₄	Dic₁₉	1	4	yes	complemented
193	H(3,ℤ/4)	—	38	8	no	—
194	C₂×C₁₉	—	2	1	no	—
195	C₂×C₁₉	—	2	1	no	—
196	C₂×C₁₉	—	2	1	no	—
197	C₂×C₁₉	—	2	1	no	—
198	C₂×C₁₉	—	2	1	no	—
199	C₂×C₁₉	—	2	1	no	—
200	C₂×C₁₉	C₄⋊#95	1	1	yes	characteristic
201	C₂×C₁₉	C₂×H(3,ℤ/4)	1	2	yes	—
202	C₂³×C₄	C₂²⋊D₁₉	1	2	yes	—
203	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
204	C₂×M₁₆	—	19	4	no	—
205	C₂×M₁₆	—	19	4	no	—
206	C₂×M₁₆	—	38	2	no	—
207	C₂×M₁₆	—	38	2	no	—
208	C₂×M₁₆	—	38	2	no	—
209	C₂×M₁₆	—	38	2	no	—
210	C₂²×C₈	—	19	4	no	—
211	#117	—	2	4	no	—
212	#117	—	2	4	no	—
213	#117	—	38	4	no	—
214	#117	—	38	4	no	—
215	C₂×C₄⋊C₄	—	2	4	no	—
216	C₂×C₄⋊C₄	—	2	2	no	—
217	C₂×C₄⋊C₄	—	38	4	no	—
218	C₂×C₄⋊C₄	—	38	2	no	—
219	C₂×C₂²⋊C₄	—	2	2	no	—
220	C₂×C₂²⋊C₄	—	38	2	no	—
221	C₂×C₄²	—	19	4	no	—
222	C₂×C₄²	Dic₃₈	1	2	yes	—
223	C₂×C₄²	C₄×D₁₉	1	2	yes	—
224	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
225	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
226	C₂×C₄²	D₇₆	1	1	yes	characteristic
227	C₂×C₄²	D₇₆	1	1	yes	characteristic
228	C₂×C₄²	C₂×Dic₁₉	1	4	yes	—
229	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
230	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
231	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
232	#97	—	38	2	no	—
233	#97	—	38	2	no	—
234	C₄×C₈	—	38	2	no	—
235	C₄×C₈	—	38	2	no	—
236	#95	—	2	4	no	—
237	#95	—	38	4	no	—
238	C₁₉	#10837	1	1	yes	characteristic, complemented, Sylow
239	C₂⁴	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
240	C₂²×C₄	—	2	4	no	—
241	C₂²×C₄	—	4	4	no	—
242	C₂²×C₄	—	38	4	no	—
243	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
244	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
245	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
246	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
247	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
248	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
249	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
250	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
251	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
252	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
253	C₂²×C₄	D₄×D₁₉	1	2	yes	—
254	C₂²×C₄	#64543	1	2	yes	—
255	C₂²×C₄	Q₈×D₁₉	1	2	yes	—
256	C₂²×C₄	#64545	1	2	yes	—
257	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
258	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
259	M₁₆	—	38	4	no	—
260	M₁₆	—	38	4	no	—
261	M₁₆	—	38	4	no	—
262	M₁₆	—	38	4	no	—
263	C₂×C₈	—	19	8	no	—
264	C₂×C₈	—	38	4	no	—
265	C₂×C₈	—	38	4	no	—
266	C₄⋊C₄	—	4	8	no	—
267	C₄⋊C₄	—	4	4	no	—
268	C₄⋊C₄	—	76	8	no	—
269	C₄⋊C₄	—	76	4	no	—
270	C₂²⋊C₄	—	4	2	no	—
271	C₂²⋊C₄	—	4	2	no	—
272	C₂²⋊C₄	—	76	2	no	—
273	C₂²⋊C₄	—	76	2	no	—
274	C₄²	—	2	8	no	—
275	C₄²	—	38	8	no	—
276	C₄²	C₄×Dic₁₉	1	2	yes	—
277	C₄²	C₄×Dic₁₉	1	2	yes	—
278	C₄²	Dic₁₉⋊C₄	1	2	yes	—
279	C₄²	Dic₁₉⋊C₄	1	2	yes	—
280	C₄²	C₄⋊Dic₁₉	1	2	yes	—
281	C₄²	C₄⋊Dic₁₉	1	2	yes	—
282	C₄²	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
283	C₄²	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
284	C₄²	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
285	C₄²	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
286	C₂³	—	4	2	no	—
287	C₂³	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
288	C₂³	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
289	C₂³	#10597382	1	1	yes	characteristic
290	C₂³	#10597386	1	1	yes	characteristic
291	C₂³	#10597390	1	1	yes	characteristic
292	C₂³	#10597393	1	1	yes	characteristic
293	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	8	no	—
301	C₂×C₄	—	4	4	no	—
302	C₂×C₄	—	76	8	no	—
303	C₂×C₄	—	76	4	no	—
304	C₂×C₄	C₁₉⋊#95	1	2	yes	—
305	C₂×C₄	C₁₉⋊#95	1	2	yes	—
306	C₂×C₄	C₁₉⋊#95	1	2	yes	—
307	C₂×C₄	C₁₉⋊#95	1	2	yes	—
308	C₂×C₄	#10597333	1	2	yes	—
309	C₂×C₄	#10597334	1	2	yes	—
310	C₂×C₄	#10597336	1	2	yes	—
311	C₂×C₄	C₄⋊C₄×D₁₉	1	2	yes	—
312	C₂×C₄	#10597338	1	2	yes	—
313	C₂×C₄	#10597339	1	2	yes	—
314	C₂×C₄	#10597341	1	2	yes	—
315	C₂×C₄	#10597343	1	2	yes	—
316	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
317	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
318	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
319	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
320	C₂×C₄	D₄×Dic₁₉	1	2	yes	—
321	C₂×C₄	#10597384	1	2	yes	—
322	C₂×C₄	Q₈×Dic₁₉	1	2	yes	—
323	C₂×C₄	#10597392	1	2	yes	—
324	C₈	—	38	8	no	—
325	C₂²	—	2	2	no	—
326	C₂²	—	2	2	no	—
327	C₂²	—	2	2	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	—	4	1	no	—
331	C₂²	—	4	1	no	—
332	C₂²	C₂×C₁₉⋊#95	1	1	yes	characteristic
333	C₂²	#11926409	1	1	yes	characteristic
334	C₂²	#11926413	1	1	yes	characteristic
335	C₂²	#11926445	1	1	yes	characteristic
336	C₂²	#11926447	1	1	yes	characteristic
337	C₂²	#11926449	1	1	yes	characteristic
338	C₂²	#11926451	1	1	yes	characteristic
339	C₄	—	2	2	no	—
340	C₄	—	2	2	no	—
341	C₄	—	2	2	no	—
342	C₄	—	8	4	no	—
343	C₄	—	152	4	no	—
344	C₄	#11926300	1	4	yes	complemented
345	C₄	C₄⋊C₄×Dic₁₉	1	2	yes	—
346	C₄	#11926401	1	2	yes	—
347	C₄	#11926440	1	2	yes	—
348	C₄	#11926443	1	2	yes	—
349	C₂	—	2	1	no	—
350	C₂	—	2	1	no	—
351	C₂	—	2	1	no	—
352	C₂	—	2	1	no	—
353	C₂	—	2	1	no	—
354	C₂	—	2	1	no	—
355	C₂	#14561864	1	1	yes	characteristic
356	C₂	C₂×#11926300	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₁₉
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₁₉×#2285	C₂
socle	C₂²×C₁₉	C₂×#95
prime radical	C₂⁴×C₁₉	C₂×D₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	#2285	D₁₉
hypocenter	C₁₉	#10837
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ #2285

Complements

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

N	H ≅ G/N	type
11. C₂×#10597260	C₄	semidirect
12. C₂×C₄×C₁₉⋊C₈	C₄	semidirect
16. C₄×C₄⋊Dic₁₉	C₄	semidirect
23. C₄×C₄⋊C₄×C₁₉	C₄	semidirect
106. C₂×C₁₉⋊C₈	C₄⋊C₄	semidirect
169. C₄×C₁₉	H(3,ℤ/4)	semidirect
192. C₄×C₄⋊C₄	Dic₁₉	semidirect
238. C₁₉	#10837	semidirect
344. C₄	#11926300	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#64520	2
#14566042	2
C₂×#11926300	2
C₁₉×#2285	2
#10837	19