Subgroups — #18090464

Order 4864.

850 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14566253	C₂	1	1	yes	maximal, characteristic
2	C₂×#11926313	C₂	1	1	yes	maximal, characteristic
3	#14563354	C₂	1	2	yes	maximal
4	C₂×#11926304	C₂	1	2	yes	maximal
5	C₂×C₁₉×#384	C₂	1	1	yes	maximal, characteristic
6	C₂²×#64520	C₂²	1	2	yes	—
7	C₂×#10597372	C₂²	1	2	yes	—
8	#11926443	—	2	4	no	—
9	#11926404	—	2	2	no	—
10	#11926313	—	2	2	no	—
11	#11926306	—	2	4	no	—
12	#11926304	—	2	4	no	—
13	C₂²×C₄⋊C₄×C₁₉	—	2	2	no	—
14	C₂²×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
15	C₂²×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
17	C₁₉×#384	—	2	2	no	—
18	C₁₉×#384	C₄	1	4	yes	complemented
19	C₂³×C₄×C₁₉	—	2	1	no	—
20	C₂³×C₄×C₁₉	—	2	1	no	—
21	C₂³×C₄×C₁₉	D₄	1	2	yes	—
22	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
24	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
25	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
26	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
27	C₂×C₄⋊C₄×C₁₉	—	2	8	no	—
28	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
29	C₂×C₄⋊C₄×C₁₉	—	4	8	no	—
30	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
31	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
32	C₂×C₄⋊C₄×C₁₉	D₄	1	2	yes	—
33	C₂×C₄²×C₁₉	—	2	1	no	—
34	C₂×C₄²×C₁₉	—	2	1	no	—
35	C₂×C₄²×C₁₉	—	2	1	no	—
36	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₁₉	D₄	1	2	yes	—
38	C₂×C₄²×C₁₉	Q₈	1	2	yes	—
39	C₂×C₂²⋊Dic₁₉	—	2	2	no	—
40	#10597372	—	2	2	no	—
41	#10597372	—	2	2	no	—
42	#10597372	—	2	2	no	—
43	#10597372	—	2	2	no	—
44	C₂×C₄⋊Dic₁₉	—	2	4	no	—
45	C₂×C₄⋊Dic₁₉	D₄	1	4	yes	—
46	C₂×C₄×Dic₁₉	—	2	2	no	—
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	—	2	2	no	—
52	C₂×#64520	Q₈	1	4	yes	—
53	C₂²×C₁₉⋊C₈	—	2	2	no	—
54	C₁₉⋊#95	—	2	4	no	—
55	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₉	—	2	4	no	—
57	C₂²×C₄×C₁₉	—	2	4	no	—
58	C₂²×C₄×C₁₉	—	2	2	no	—
59	C₂²×C₄×C₁₉	—	2	2	no	—
60	C₂²×C₄×C₁₉	—	2	2	no	—
61	C₂²×C₄×C₁₉	—	2	2	no	—
62	C₂²×C₄×C₁₉	—	2	2	no	—
63	C₂²×C₄×C₁₉	—	2	2	no	—
64	C₂²×C₄×C₁₉	—	4	4	no	—
65	C₂²×C₄×C₁₉	—	4	4	no	—
66	C₂²×C₄×C₁₉	—	4	1	no	—
67	C₂²×C₄×C₁₉	—	4	1	no	—
68	C₂²×C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
69	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
74	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
75	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
76	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
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	—	2	4	no	—
81	C₄⋊C₄×C₁₉	—	2	4	no	—
82	C₄⋊C₄×C₁₉	—	2	4	no	—
83	C₄⋊C₄×C₁₉	—	4	16	no	—
84	C₄⋊C₄×C₁₉	—	4	2	no	—
85	C₄²×C₁₉	—	2	2	no	—
86	C₄²×C₁₉	—	2	2	no	—
87	C₄²×C₁₉	—	2	2	no	—
88	C₄²×C₁₉	C₄⋊C₄	1	4	yes	complemented
89	C₂²⋊Dic₁₉	—	4	2	no	—
90	C₂²⋊Dic₁₉	—	4	2	no	—
91	C₄⋊Dic₁₉	—	2	4	no	—
92	C₄⋊Dic₁₉	—	2	4	no	—
93	C₄⋊Dic₁₉	—	8	4	no	—
94	C₄×Dic₁₉	—	4	2	no	—
95	C₄×Dic₁₉	—	4	2	no	—
96	#64520	—	2	4	no	—
97	#64520	—	2	4	no	—
98	#64520	—	4	2	no	—
99	#64520	—	4	2	no	—
100	C₂×C₁₉⋊C₈	—	2	4	no	—
101	C₂×C₁₉⋊C₈	—	4	2	no	—
102	C₂×C₁₉⋊C₈	—	4	2	no	—
103	#11210	—	19	1	no	maximal, Sylow
104	C₂³×C₁₉	—	2	2	no	—
105	C₂³×C₁₉	—	2	2	no	—
106	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂³×C₁₉	#120	1	1	yes	characteristic
108	C₂³×C₁₉	#121	1	1	yes	characteristic
109	C₂³×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
110	C₂³×C₁₉	#123	1	1	yes	characteristic
111	C₂³×C₁₉	#124	1	1	yes	characteristic
112	C₂³×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
113	C₂×C₄×C₁₉	—	2	4	no	—
114	C₂×C₄×C₁₉	—	2	8	no	—
115	C₂×C₄×C₁₉	—	2	4	no	—
116	C₂×C₄×C₁₉	—	2	8	no	—
117	C₂×C₄×C₁₉	—	2	4	no	—
118	C₂×C₄×C₁₉	—	2	4	no	—
119	C₂×C₄×C₁₉	—	4	4	no	—
120	C₂×C₄×C₁₉	—	4	4	no	—
121	C₂×C₄×C₁₉	—	4	4	no	—
122	C₂×C₄×C₁₉	—	4	2	no	—
123	C₂×C₄×C₁₉	—	8	2	no	—
124	C₂×C₄×C₁₉	—	8	2	no	—
125	C₂×C₄×C₁₉	—	8	2	no	—
126	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
128	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
129	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
130	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
131	C₂×C₄×C₁₉	#121	1	2	yes	—
132	C₂×C₄×C₁₉	#121	1	2	yes	—
133	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
134	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
135	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
136	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
137	C₂×C₄×C₁₉	#123	1	2	yes	—
138	C₂×C₄×C₁₉	#125	1	2	yes	—
139	C₂×C₄×C₁₉	#125	1	2	yes	—
140	C₂×C₄×C₁₉	#126	1	2	yes	—
141	C₂×C₄×C₁₉	C₄⋊Q₈	1	2	yes	—
142	C₂×Dic₁₉	—	4	4	no	—
143	C₂×Dic₁₉	—	4	4	no	—
144	C₂×Dic₁₉	—	4	4	no	—
145	C₁₉⋊C₈	—	4	4	no	—
146	#2316	—	19	1	no	—
147	C₂×#384	D₁₉	1	1	yes	characteristic
148	C₂×#343	—	19	1	no	—
149	#1736	—	19	2	no	—
150	C₂×#339	—	19	2	no	—
151	C₂²×C₁₉	—	2	2	no	—
152	C₂²×C₁₉	—	2	2	no	—
153	C₂²×C₁₉	—	2	1	no	—
154	C₂²×C₁₉	—	2	1	no	—
155	C₂²×C₁₉	—	2	1	no	—
156	C₂²×C₁₉	—	2	1	no	—
157	C₂²×C₁₉	—	2	2	no	—
158	C₂²×C₁₉	—	2	2	no	—
159	C₂²×C₁₉	#380	1	1	yes	characteristic
160	C₂²×C₁₉	#386	1	1	yes	characteristic
161	C₂²×C₁₉	#391	1	1	yes	characteristic
162	C₂²×C₁₉	#394	1	1	yes	characteristic
163	C₂²×C₁₉	#395	1	1	yes	characteristic
164	C₂²×C₁₉	#398	1	1	yes	characteristic
165	C₂²×C₁₉	#400	1	1	yes	characteristic
166	C₂²×C₁₉	#453	1	2	yes	—
167	C₂²×C₁₉	#456	1	2	yes	—
168	C₄×C₁₉	—	2	8	no	—
169	C₄×C₁₉	—	4	4	no	—
170	C₄×C₁₉	—	8	2	no	—
171	C₄×C₁₉	#387	1	2	yes	—
172	C₄×C₁₉	#389	1	2	yes	—
173	C₄×C₁₉	#396	1	2	yes	—
174	C₄×C₁₉	#398	1	2	yes	—
175	Dic₁₉	—	8	4	no	—
176	C₂²×M₁₆	—	19	2	no	—
177	C₂×#117	—	19	2	no	—
178	C₂²×C₄⋊C₄	—	2	2	no	—
179	C₂²×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
180	C₂²×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
181	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
182	#389	—	38	4	no	—
183	#384	—	2	2	no	—
184	#384	Dic₁₉	1	4	yes	complemented
185	#380	—	38	2	no	—
186	#343	—	38	2	no	—
187	#340	—	38	4	no	—
188	#339	—	38	4	no	—
189	C₂×C₁₉	—	2	1	no	—
190	C₂×C₁₉	—	2	1	no	—
191	C₂×C₁₉	—	2	1	no	—
192	C₂×C₁₉	—	2	1	no	—
193	C₂×C₁₉	#1441	1	1	yes	characteristic
194	C₂×C₁₉	#1886	1	1	yes	characteristic
195	C₂×C₁₉	#1887	1	1	yes	characteristic
196	C₂×C₁₉	#2039	1	2	yes	—
197	C₂×C₁₉	#2060	1	2	yes	—
198	C₂³×C₄	—	2	1	no	—
199	C₂³×C₄	—	2	1	no	—
200	C₂³×C₄	C₂²⋊D₁₉	1	2	yes	—
201	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
202	C₂×M₁₆	—	19	4	no	—
203	C₂×M₁₆	—	38	2	no	—
204	C₂×M₁₆	—	38	2	no	—
205	C₂×M₁₆	—	38	2	no	—
206	C₂×M₁₆	—	38	2	no	—
207	C₂×M₁₆	—	38	2	no	—
208	C₂²×C₈	—	38	2	no	—
209	#117	—	38	2	no	—
210	#117	—	38	2	no	—
211	#117	—	38	2	no	—
212	#117	—	38	2	no	—
213	C₂×C₄⋊C₄	—	2	2	no	—
214	C₂×C₄⋊C₄	—	2	2	no	—
215	C₂×C₄⋊C₄	—	2	2	no	—
216	C₂×C₄⋊C₄	—	2	2	no	—
217	C₂×C₄⋊C₄	—	2	8	no	—
218	C₂×C₄⋊C₄	—	2	1	no	—
219	C₂×C₄⋊C₄	—	4	8	no	—
220	C₂×C₄⋊C₄	—	19	4	no	—
221	C₂×C₄⋊C₄	—	38	4	no	—
222	C₂×C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
223	C₂×C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
224	C₂×C₄⋊C₄	C₂²⋊D₁₉	1	2	yes	—
225	C₂×C₂²⋊C₄	—	38	2	no	—
226	C₂×C₄²	—	2	1	no	—
227	C₂×C₄²	—	2	1	no	—
228	C₂×C₄²	—	2	1	no	—
229	C₂×C₄²	—	38	2	no	—
230	C₂×C₄²	Dic₃₈	1	2	yes	—
231	C₂×C₄²	D₇₆	1	2	yes	—
232	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
233	#95	—	38	4	no	—
234	C₁₉	#11210	1	1	yes	characteristic, complemented, Sylow
235	C₂⁴	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	—	2	4	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	4	4	no	—
245	C₂²×C₄	—	4	4	no	—
246	C₂²×C₄	—	4	1	no	—
247	C₂²×C₄	—	4	1	no	—
248	C₂²×C₄	—	38	4	no	—
249	C₂²×C₄	C₂²⋊Dic₁₉	1	4	yes	—
250	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
251	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
252	C₂²×C₄	#64541	1	1	yes	characteristic
253	C₂²×C₄	#64541	1	1	yes	characteristic
254	C₂²×C₄	D₄×D₁₉	1	2	yes	—
255	C₂²×C₄	#64543	1	2	yes	—
256	C₂²×C₄	Q₈×D₁₉	1	2	yes	—
257	C₂²×C₄	#64545	1	2	yes	—
258	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
259	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
260	M₁₆	—	38	4	no	—
261	M₁₆	—	38	4	no	—
262	M₁₆	—	76	2	no	—
263	M₁₆	—	76	2	no	—
264	C₂×C₈	—	38	4	no	—
265	C₂×C₈	—	76	2	no	—
266	C₂×C₈	—	76	2	no	—
267	C₄⋊C₄	—	2	4	no	—
268	C₄⋊C₄	—	2	4	no	—
269	C₄⋊C₄	—	2	4	no	—
270	C₄⋊C₄	—	4	16	no	—
271	C₄⋊C₄	—	4	2	no	—
272	C₄⋊C₄	—	38	4	no	—
273	C₄⋊C₄	—	38	4	no	—
274	C₄⋊C₄	—	152	4	no	—
275	C₂²⋊C₄	—	76	2	no	—
276	C₂²⋊C₄	—	76	2	no	—
277	C₄²	—	2	2	no	—
278	C₄²	—	2	2	no	—
279	C₄²	—	2	2	no	—
280	C₄²	—	76	2	no	—
281	C₄²	—	76	2	no	—
282	C₄²	C₄⋊Dic₁₉	1	4	yes	complemented
283	C₂³	—	2	2	no	—
284	C₂³	—	2	2	no	—
285	C₂³	#10597370	1	1	yes	characteristic
286	C₂³	#10597376	1	1	yes	characteristic
287	C₂³	#10597381	1	1	yes	characteristic
288	C₂³	#10597383	1	1	yes	characteristic
289	C₂³	#10597390	1	1	yes	characteristic
290	C₂³	#10597393	1	1	yes	characteristic
291	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	8	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	8	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	—	4	4	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	8	2	no	—
303	C₂×C₄	—	8	2	no	—
304	C₂×C₄	—	8	2	no	—
305	C₂×C₄	—	76	4	no	—
306	C₂×C₄	—	76	4	no	—
307	C₂×C₄	—	76	4	no	—
308	C₂×C₄	#10597334	1	2	yes	—
309	C₂×C₄	#10597335	1	2	yes	—
310	C₂×C₄	#10597336	1	2	yes	—
311	C₂×C₄	#10597340	1	2	yes	—
312	C₂×C₄	#10597341	1	2	yes	—
313	C₂×C₄	#10597342	1	2	yes	—
314	C₂×C₄	#10597343	1	2	yes	—
315	C₂×C₄	#10597344	1	2	yes	—
316	C₂×C₄	#10597370	1	1	yes	characteristic
317	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
318	C₂×C₄	#10597372	1	1	yes	characteristic
319	C₂×C₄	#10597376	1	1	yes	characteristic
320	C₂×C₄	D₄×Dic₁₉	1	2	yes	—
321	C₂×C₄	#10597385	1	2	yes	—
322	C₂×C₄	Q₈×Dic₁₉	1	2	yes	—
323	C₂×C₄	#10597392	1	2	yes	—
324	C₈	—	76	4	no	—
325	C₂²	—	2	2	no	—
326	C₂²	—	2	2	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	2	no	—
332	C₂²	—	2	2	no	—
333	C₂²	#11926404	1	1	yes	characteristic
334	C₂²	#11926424	1	1	yes	characteristic
335	C₂²	#11926451	1	1	yes	characteristic
336	C₂²	#11926461	1	1	yes	characteristic
337	C₂²	#11926464	1	1	yes	characteristic
338	C₂²	#11926474	1	1	yes	characteristic
339	C₂²	#11926483	1	1	yes	characteristic
340	C₂²	#11926711	1	2	yes	—
341	C₂²	#11926726	1	2	yes	—
342	C₄	—	2	8	no	—
343	C₄	—	4	4	no	—
344	C₄	—	8	2	no	—
345	C₄	—	152	4	no	—
346	C₄	#11926430	1	2	yes	—
347	C₄	#11926443	1	2	yes	—
348	C₄	#11926466	1	2	yes	—
349	C₄	#11926477	1	2	yes	—
350	C₂	—	2	1	no	—
351	C₂	—	2	1	no	—
352	C₂	—	2	1	no	—
353	C₂	—	2	1	no	—
354	C₂	#14561888	1	1	yes	characteristic
355	C₂	#14564029	1	1	yes	characteristic
356	C₂	#14564034	1	1	yes	characteristic
357	C₂	#14564859	1	2	yes	—
358	C₂	#14564980	1	2	yes	—

	subgroup	G/H
center	C₂³	#10597383
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₂×C₁₉×#384	C₂
socle	C₂³×C₁₉	#120
prime radical	C₂⁴×C₁₉	C₂×D₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂×#384	D₁₉
hypocenter	C₁₉	#11210
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#384

Complements

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

N	H ≅ G/N	type
18. C₁₉×#384	C₄	semidirect
88. C₄²×C₁₉	C₄⋊C₄	semidirect
184. #384	Dic₁₉	semidirect
234. C₁₉	#11210	semidirect
282. C₄²	C₄⋊Dic₁₉	semidirect

Maximal subgroups

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

subgroup	index
#14566253	2
C₂×#11926313	2
#14563354	2
C₂×#11926304	2
C₂×C₁₉×#384	2
#11210	19