Subgroups — #18088351

Order 4864.

1110 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14566042	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄²×Dic₁₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926305	C₂	1	4	yes	maximal
4	C₂×C₁₉×#423	C₂	1	1	yes	maximal, characteristic
5	C₂×#10597372	C₂²	1	2	yes	—
6	C₂²×C₄×Dic₁₉	—	2	2	no	—
7	C₂²×C₄×Dic₁₉	C₂²	1	2	yes	—
8	#11926414	C₄	1	2	yes	complemented
9	#11926410	C₄	1	2	yes	complemented
10	#11926404	—	2	2	no	—
11	C₄×C₄⋊Dic₁₉	—	2	2	no	—
12	C₄²×Dic₁₉	—	2	2	no	—
13	C₄²×Dic₁₉	C₄	1	4	yes	complemented
14	#11926305	—	2	8	no	—
15	C₂²×M₁₆×C₁₉	C₂²	1	2	yes	—
16	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
17	C₁₉×#423	—	2	2	no	—
18	C₁₉×#423	C₄	1	4	yes	complemented
19	C₂×C₁₉×C₄⋊C₈	—	2	2	no	—
20	C₂³×C₄×C₁₉	D₄	1	2	yes	—
21	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
22	C₂³×Dic₁₉	—	2	2	no	—
23	C₂×M₁₆×C₁₉	—	2	2	no	—
24	C₂×M₁₆×C₁₉	—	2	4	no	—
25	C₂×M₁₆×C₁₉	—	2	4	no	—
26	C₂×M₁₆×C₁₉	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₁₉	—	2	2	no	—
28	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₉	D₄	1	2	yes	—
33	C₂×C₄²×C₁₉	D₄	1	2	yes	—
34	C₂×C₄²×C₁₉	Q₈	1	2	yes	—
35	C₂×C₂²⋊Dic₁₉	—	2	2	no	—
36	#10597372	—	2	4	no	—
37	#10597372	—	2	4	no	—
38	C₂×C₄⋊Dic₁₉	—	2	4	no	—
39	C₂×C₄⋊Dic₁₉	C₂×C₄	1	4	yes	—
40	C₂×C₄×Dic₁₉	—	2	16	no	—
41	C₂×C₄×Dic₁₉	—	2	8	no	—
42	C₂×C₄×Dic₁₉	—	2	2	no	—
43	C₂×C₄×Dic₁₉	—	2	2	no	—
44	C₂×C₄×Dic₁₉	—	2	2	no	—
45	C₂×C₄×Dic₁₉	—	2	2	no	—
46	C₂×C₄×Dic₁₉	C₂×C₄	1	4	yes	—
47	C₂×C₄×Dic₁₉	D₄	1	4	yes	—
48	C₂×C₄×Dic₁₉	Q₈	1	4	yes	—
49	C₁₉×C₄⋊C₈	—	2	8	no	—
50	C₁₉⋊#95	—	2	4	no	—
51	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₉	—	2	8	no	—
53	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
55	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₁₉	C₄⋊C₄	1	4	yes	—
57	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
65	C₂²×C₄×C₁₉	#41	1	2	yes	—
66	C₂²×C₄×C₁₉	#41	1	2	yes	—
67	C₂²×Dic₁₉	—	2	16	no	—
68	C₂²×Dic₁₉	—	2	4	no	—
69	C₂²×Dic₁₉	—	2	4	no	—
70	C₂²×Dic₁₉	—	2	4	no	—
71	C₂²×Dic₁₉	—	2	4	no	—
72	M₁₆×C₁₉	—	2	8	no	—
73	M₁₆×C₁₉	—	4	4	no	—
74	C₂×C₈×C₁₉	—	2	4	no	—
75	C₂×C₈×C₁₉	—	4	2	no	—
76	C₂×C₈×C₁₉	—	4	2	no	—
77	C₄²×C₁₉	C₄²	1	2	yes	—
78	C₄²×C₁₉	C₄²	1	2	yes	—
79	C₄²×C₁₉	C₂²⋊C₄	1	4	yes	—
80	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
81	C₄²×C₁₉	C₄⋊C₄	1	4	yes	—
82	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
83	C₂²⋊Dic₁₉	—	4	4	no	—
84	C₄⋊Dic₁₉	—	2	8	no	—
85	C₄⋊Dic₁₉	—	4	8	no	—
86	C₄×Dic₁₉	—	2	32	no	—
87	C₄×Dic₁₉	—	2	4	no	—
88	C₄×Dic₁₉	—	2	4	no	—
89	C₄×Dic₁₉	—	2	4	no	—
90	C₄×Dic₁₉	—	4	2	no	—
91	C₄×Dic₁₉	—	4	2	no	—
92	C₄×Dic₁₉	C₄⋊C₄	1	8	yes	complemented
93	#10839	—	19	1	no	maximal, Sylow
94	C₂³×C₁₉	—	2	4	no	—
95	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂³×C₁₉	#121	1	2	yes	—
99	C₂³×C₁₉	C₂²⋊Q₈	1	2	yes	—
100	C₂×C₄×C₁₉	—	2	4	no	—
101	C₂×C₄×C₁₉	—	2	8	no	—
102	C₂×C₄×C₁₉	—	2	8	no	—
103	C₂×C₄×C₁₉	#95	1	4	yes	—
104	C₂×C₄×C₁₉	#95	1	4	yes	—
105	C₂×C₄×C₁₉	#104	1	8	yes	—
106	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
107	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
109	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₉	#117	1	2	yes	—
112	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
115	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
116	C₂×C₄×C₁₉	#121	1	2	yes	—
117	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
118	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
119	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
120	C₂×C₄×C₁₉	#125	1	2	yes	—
121	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
122	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
123	C₂×C₄×C₁₉	C₄⋊Q₈	1	2	yes	—
124	C₂×Dic₁₉	—	2	16	no	—
125	C₂×Dic₁₉	—	2	16	no	—
126	C₂×Dic₁₉	—	2	16	no	—
127	C₂×Dic₁₉	—	2	8	no	—
128	C₂×Dic₁₉	—	4	4	no	—
129	C₂×Dic₁₉	—	4	8	no	—
130	C₈×C₁₉	—	4	4	no	—
131	C₂×#423	D₁₉	1	1	yes	characteristic
132	#2285	—	19	1	no	—
133	C₂×C₄³	—	19	1	no	—
134	C₂×#339	—	19	4	no	—
135	C₂²×C₁₉	—	2	4	no	—
136	C₂²×C₁₉	—	2	2	no	—
137	C₂²×C₁₉	—	2	2	no	—
138	C₂²×C₁₉	—	2	4	no	—
139	C₂²×C₁₉	C₂×#95	1	1	yes	characteristic
140	C₂²×C₁₉	#380	1	2	yes	—
141	C₂²×C₁₉	#389	1	2	yes	—
142	C₂²×C₁₉	#390	1	1	yes	characteristic
143	C₂²×C₁₉	#391	1	1	yes	characteristic
144	C₂²×C₁₉	C₂×#104	1	4	yes	—
145	C₄×C₁₉	—	2	4	no	—
146	C₄×C₁₉	#339	1	8	yes	complemented
147	C₄×C₁₉	C₄×C₄⋊C₄	1	2	yes	—
148	C₄×C₁₉	#382	1	1	yes	characteristic
149	C₄×C₁₉	#384	1	1	yes	characteristic
150	C₄×C₁₉	#389	1	2	yes	—
151	C₄×C₁₉	#390	1	1	yes	characteristic
152	C₄×C₁₉	#391	1	1	yes	characteristic
153	Dic₁₉	—	2	16	no	—
154	Dic₁₉	—	8	4	no	—
155	C₂²×M₁₆	C₂×D₁₉	1	2	yes	—
156	C₂×#117	—	19	2	no	—
157	C₂²×C₄²	—	19	2	no	—
158	C₂²×C₄²	—	38	2	no	—
159	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
160	#423	—	2	2	no	—
161	#423	Dic₁₉	1	4	yes	complemented
162	C₂×C₄⋊C₈	—	2	2	no	—
163	#384	—	19	2	no	—
164	#382	—	19	2	no	—
165	#380	—	38	2	no	—
166	C₄×C₄⋊C₄	—	38	2	no	—
167	C₄³	—	19	4	no	—
168	C₄³	—	38	2	no	—
169	#339	—	38	8	no	—
170	C₂×C₁₉	—	2	2	no	—
171	C₂×C₁₉	—	2	2	no	—
172	C₂×C₁₉	C₄⋊#95	1	1	yes	characteristic
173	C₂×C₁₉	C₂×#339	1	2	yes	—
174	C₂×C₁₉	#1962	1	2	yes	—
175	C₂×C₁₉	#1994	1	2	yes	—
176	C₂³×C₄	—	38	2	no	—
177	C₂³×C₄	D₇₆	1	2	yes	—
178	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	2	4	no	—
181	C₂×M₁₆	—	2	4	no	—
182	C₂×M₁₆	C₂×Dic₁₉	1	4	yes	—
183	C₂²×C₈	—	2	2	no	—
184	#117	—	38	4	no	—
185	#117	—	38	4	no	—
186	C₂×C₄⋊C₄	—	19	4	no	—
187	C₂×C₄⋊C₄	—	38	4	no	—
188	C₂×C₂²⋊C₄	—	38	2	no	—
189	C₂×C₄²	—	19	4	no	—
190	C₂×C₄²	—	19	4	no	—
191	C₂×C₄²	—	19	4	no	—
192	C₂×C₄²	—	38	16	no	—
193	C₂×C₄²	—	38	8	no	—
194	C₂×C₄²	—	38	2	no	—
195	C₂×C₄²	—	38	2	no	—
196	C₂×C₄²	—	38	2	no	—
197	C₂×C₄²	—	38	2	no	—
198	C₂×C₄²	Dic₃₈	1	2	yes	—
199	C₂×C₄²	C₄×D₁₉	1	2	yes	—
200	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
201	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
202	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
203	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
204	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
205	C₄⋊C₈	—	2	8	no	—
206	#95	—	38	4	no	—
207	C₁₉	#10839	1	1	yes	characteristic, complemented, Sylow
208	C₂⁴	C₂×D₇₆	1	1	yes	characteristic
209	C₂²×C₄	—	2	8	no	—
210	C₂²×C₄	—	38	16	no	—
211	C₂²×C₄	—	38	4	no	—
212	C₂²×C₄	—	38	4	no	—
213	C₂²×C₄	—	38	4	no	—
214	C₂²×C₄	—	38	4	no	—
215	C₂²×C₄	C₄⋊Dic₁₉	1	4	yes	—
216	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
217	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	4	yes	—
218	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
219	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
222	C₂²×C₄	D₄×D₁₉	1	2	yes	—
223	C₂²×C₄	#64543	1	2	yes	—
224	C₂²×C₄	Q₈×D₁₉	1	2	yes	—
225	C₂²×C₄	#64545	1	2	yes	—
226	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
229	M₁₆	—	2	8	no	—
230	M₁₆	—	4	4	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	—	4	2	no	—
234	C₄⋊C₄	—	38	8	no	—
235	C₄⋊C₄	—	76	8	no	—
236	C₂²⋊C₄	—	76	4	no	—
237	C₄²	—	19	8	no	—
238	C₄²	—	38	32	no	—
239	C₄²	—	38	4	no	—
240	C₄²	—	38	4	no	—
241	C₄²	—	38	4	no	—
242	C₄²	—	76	2	no	—
243	C₄²	—	76	2	no	—
244	C₄²	C₄×Dic₁₉	1	2	yes	—
245	C₄²	C₄×Dic₁₉	1	2	yes	—
246	C₄²	Dic₁₉⋊C₄	1	2	yes	—
247	C₄²	Dic₁₉⋊C₄	1	4	yes	—
248	C₄²	Dic₁₉⋊C₄	1	2	yes	—
249	C₄²	C₂²⋊Dic₁₉	1	4	yes	—
250	C₂³	—	2	4	no	—
251	C₂³	#10597341	1	2	yes	—
252	C₂³	#10597343	1	2	yes	—
253	C₂³	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
254	C₂³	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
255	C₂³	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	8	no	—
258	C₂×C₄	—	2	8	no	—
259	C₂×C₄	—	38	16	no	—
260	C₂×C₄	—	38	16	no	—
261	C₂×C₄	—	38	16	no	—
262	C₂×C₄	—	38	8	no	—
263	C₂×C₄	—	76	8	no	—
264	C₂×C₄	—	76	4	no	—
265	C₂×C₄	#10597282	1	8	yes	—
266	C₂×C₄	C₁₉⋊#95	1	4	yes	—
267	C₂×C₄	C₁₉⋊#95	1	4	yes	—
268	C₂×C₄	#10597333	1	2	yes	—
269	C₂×C₄	#10597334	1	2	yes	—
270	C₂×C₄	#10597336	1	2	yes	—
271	C₂×C₄	C₄⋊C₄×D₁₉	1	2	yes	—
272	C₂×C₄	#10597338	1	2	yes	—
273	C₂×C₄	#10597339	1	2	yes	—
274	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
275	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
276	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
277	C₂×C₄	D₄×Dic₁₉	1	2	yes	—
278	C₂×C₄	#10597382	1	1	yes	characteristic
279	C₂×C₄	#10597384	1	2	yes	—
280	C₂×C₄	#10597386	1	1	yes	characteristic
281	C₂×C₄	#10597390	1	1	yes	characteristic
282	C₂×C₄	Q₈×Dic₁₉	1	2	yes	—
283	C₂×C₄	#10597392	1	2	yes	—
284	C₂×C₄	#10597393	1	1	yes	characteristic
285	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
286	C₈	—	4	4	no	—
287	C₂²	—	2	4	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	4	no	—
291	C₂²	C₂×C₁₉⋊#95	1	1	yes	characteristic
292	C₂²	#11926401	1	2	yes	—
293	C₂²	#11926443	1	2	yes	—
294	C₂²	#11926445	1	1	yes	characteristic
295	C₂²	#11926449	1	1	yes	characteristic
296	C₂²	C₂×#10597282	1	4	yes	—
297	C₄	—	2	4	no	—
298	C₄	—	38	16	no	—
299	C₄	—	152	4	no	—
300	C₄	#11926305	1	8	yes	complemented
301	C₄	C₄⋊C₄×Dic₁₉	1	2	yes	—
302	C₄	#11926409	1	1	yes	characteristic
303	C₄	#11926413	1	1	yes	characteristic
304	C₄	#11926440	1	2	yes	—
305	C₄	#11926447	1	1	yes	characteristic
306	C₄	#11926451	1	1	yes	characteristic
307	C₂	—	2	2	no	—
308	C₂	—	2	2	no	—
309	C₂	#14561864	1	1	yes	characteristic
310	C₂	C₂×#11926305	1	2	yes	—
311	C₂	#14564462	1	2	yes	—
312	C₂	#14564611	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₉
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₂×C₁₉×#423	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₉	C₂×D₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂×#423	D₁₉
hypocenter	C₁₉	#10839
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
8. #11926414	C₄	semidirect
9. #11926410	C₄	semidirect
13. C₄²×Dic₁₉	C₄	semidirect
18. C₁₉×#423	C₄	semidirect
92. C₄×Dic₁₉	C₄⋊C₄	semidirect
146. C₄×C₁₉	#339	semidirect
161. #423	Dic₁₉	semidirect
207. C₁₉	#10839	semidirect
300. C₄	#11926305	semidirect

Maximal subgroups

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

subgroup	index
#14566042	2
C₂×C₄²×Dic₁₉	2
C₂×#11926305	2
C₂×C₁₉×#423	2
#10839	19