Subgroups — #18088300

Order 4864.

686 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	C₂×#11926312	C₂	1	2	yes	maximal
2	C₂×#11926309	C₂	1	1	yes	maximal, characteristic
3	#14563356	C₂	1	2	yes	maximal
4	C₂×#11926307	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₉×#405	C₂	1	1	yes	maximal, characteristic
6	C₂²×#64520	C₂²	1	2	yes	—
7	C₂²×C₄⋊Dic₁₉	C₂²	1	2	yes	—
8	#11926312	C₄	1	8	yes	complemented
9	#11926309	—	2	2	no	—
10	#11926307	—	2	1	no	—
11	#11926307	—	2	1	no	—
12	#11926306	—	2	2	no	—
13	#11926306	—	2	2	no	—
14	C₂²×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
16	C₂×C₁₉×#117	C₂²	1	1	yes	characteristic
17	C₁₉×#405	—	2	2	no	—
18	C₁₉×#405	C₄	1	4	yes	complemented
19	C₂×C₁₉×#97	—	2	1	no	—
20	C₂×C₄×C₈×C₁₉	—	2	1	no	—
21	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
22	C₂³×Dic₁₉	—	2	2	no	—
23	C₂×M₁₆×C₁₉	—	2	1	no	—
24	C₂×M₁₆×C₁₉	—	2	2	no	—
25	C₂×M₁₆×C₁₉	C₂×C₄	1	4	yes	—
26	C₂×M₁₆×C₁₉	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₁₉	—	2	1	no	—
28	C₂²×C₈×C₁₉	—	2	2	no	—
29	C₂²×C₈×C₁₉	—	2	2	no	—
30	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₉	D₄	1	2	yes	—
34	C₁₉×#117	—	2	1	no	—
35	C₁₉×#117	—	2	1	no	—
36	C₁₉×#117	C₂×C₄	1	2	yes	—
37	C₁₉×#117	Q₈	1	2	yes	—
38	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
40	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
41	C₂×C₄²×C₁₉	—	2	1	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	—	2	2	no	—
47	C₂×C₄⋊Dic₁₉	C₂×C₄	1	4	yes	—
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	—	2	2	no	—
53	C₂×#64520	C₂×C₄	1	4	yes	—
54	C₂²×C₁₉⋊C₈	—	2	2	no	—
55	C₁₉×#97	—	2	4	no	—
56	C₄×C₈×C₁₉	—	2	4	no	—
57	C₂⁴×C₁₉	#41	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	—	2	1	no	—
59	C₂²×C₄×C₁₉	—	2	1	no	—
60	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
67	C₂²×C₄×C₁₉	#41	1	2	yes	—
68	C₂²×C₄×C₁₉	#41	1	2	yes	—
69	C₂²×C₄×C₁₉	#41	1	2	yes	—
70	C₂²×Dic₁₉	—	2	4	no	—
71	C₂²×Dic₁₉	—	4	2	no	—
72	C₂²×Dic₁₉	—	4	2	no	—
73	C₂²×Dic₁₉	—	8	2	no	—
74	C₂²×Dic₁₉	—	8	2	no	—
75	M₁₆×C₁₉	—	2	2	no	—
76	M₁₆×C₁₉	—	2	4	no	—
77	M₁₆×C₁₉	C₄²	1	4	yes	—
78	C₂×C₈×C₁₉	—	2	2	no	—
79	C₂×C₈×C₁₉	—	2	2	no	—
80	C₂×C₈×C₁₉	—	2	2	no	—
81	C₂×C₈×C₁₉	—	2	4	no	—
82	C₂×C₈×C₁₉	—	2	2	no	—
83	C₂×C₈×C₁₉	—	2	2	no	—
84	C₂×C₈×C₁₉	—	2	4	no	—
85	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
86	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
87	C₄⋊C₄×C₁₉	—	2	1	no	—
88	C₄⋊C₄×C₁₉	—	2	1	no	—
89	C₄⋊C₄×C₁₉	C₄⋊C₄	1	2	yes	—
90	C₄⋊C₄×C₁₉	C₄⋊C₄	1	2	yes	—
91	C₂²⋊C₄×C₁₉	—	2	2	no	—
92	C₂²⋊C₄×C₁₉	—	2	2	no	—
93	C₄²×C₁₉	—	2	2	no	—
94	C₄²×C₁₉	—	2	2	no	—
95	C₄⋊Dic₁₉	—	2	4	no	—
96	C₄⋊Dic₁₉	—	2	4	no	—
97	C₄⋊Dic₁₉	—	4	2	no	—
98	C₄⋊Dic₁₉	—	4	2	no	—
99	#64520	—	2	4	no	—
100	#64520	—	2	4	no	—
101	#64520	—	4	2	no	—
102	#64520	—	4	2	no	—
103	C₂×C₁₉⋊C₈	—	2	4	no	—
104	C₂×C₁₉⋊C₈	—	4	2	no	—
105	C₂×C₁₉⋊C₈	—	4	2	no	—
106	#10827	—	19	1	no	maximal, Sylow
107	C₂³×C₁₉	—	2	2	no	—
108	C₂³×C₁₉	—	2	1	no	—
109	C₂³×C₁₉	—	2	1	no	—
110	C₂³×C₁₉	#117	1	2	yes	—
111	C₂³×C₁₉	#117	1	1	yes	characteristic
112	C₂³×C₁₉	#121	1	1	yes	characteristic
113	C₂³×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
114	C₂³×C₁₉	#126	1	2	yes	—
115	C₂×C₄×C₁₉	—	2	2	no	—
116	C₂×C₄×C₁₉	—	2	2	no	—
117	C₂×C₄×C₁₉	—	4	2	no	—
118	C₂×C₄×C₁₉	—	4	1	no	—
119	C₂×C₄×C₁₉	—	4	1	no	—
120	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
121	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₉	#117	1	2	yes	—
124	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
125	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
126	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
127	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
128	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
129	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
130	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
131	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
132	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
133	C₂×C₄×C₁₉	#123	1	2	yes	—
134	C₂×C₄×C₁₉	#123	1	2	yes	—
135	C₂×C₄×C₁₉	#124	1	2	yes	—
136	C₂×C₄×C₁₉	#125	1	2	yes	—
137	C₂×C₄×C₁₉	#126	1	2	yes	—
138	C₂×Dic₁₉	—	4	4	no	—
139	C₂×Dic₁₉	—	8	4	no	—
140	C₂×Dic₁₉	—	8	2	no	—
141	C₂×Dic₁₉	—	8	2	no	—
142	C₂×Dic₁₉	—	8	4	no	—
143	C₈×C₁₉	—	2	4	no	—
144	C₈×C₁₉	—	2	2	no	—
145	C₈×C₁₉	—	2	2	no	—
146	C₁₉⋊C₈	—	4	4	no	—
147	C₂×#405	D₁₉	1	1	yes	characteristic
148	C₂×#343	—	19	2	no	—
149	C₂×#341	—	19	1	no	—
150	#1736	—	19	2	no	—
151	C₂×#340	—	19	1	no	—
152	C₂²×C₁₉	—	2	1	no	—
153	C₂²×C₁₉	—	2	1	no	—
154	C₂²×C₁₉	—	2	1	no	—
155	C₂²×C₁₉	—	2	1	no	—
156	C₂²×C₁₉	—	2	2	no	—
157	C₂²×C₁₉	—	2	2	no	—
158	C₂²×C₁₉	—	2	2	no	—
159	C₂²×C₁₉	—	2	2	no	—
160	C₂²×C₁₉	#376	1	1	yes	characteristic
161	C₂²×C₁₉	#380	1	1	yes	characteristic
162	C₂²×C₁₉	#383	1	1	yes	characteristic
163	C₂²×C₁₉	#387	1	2	yes	—
164	C₂²×C₁₉	#388	1	2	yes	—
165	C₂²×C₁₉	#471	1	2	yes	—
166	C₂²×C₁₉	#472	1	1	yes	characteristic
167	C₂²×C₁₉	#473	1	1	yes	characteristic
168	C₄×C₁₉	—	4	2	no	—
169	C₄×C₁₉	—	4	2	no	—
170	C₄×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
171	C₄×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
172	C₄×C₁₉	#381	1	1	yes	characteristic
173	C₄×C₁₉	#382	1	1	yes	characteristic
174	C₄×C₁₉	#387	1	2	yes	—
175	C₄×C₁₉	#388	1	2	yes	—
176	Dic₁₉	—	8	4	no	—
177	C₂²×M₁₆	—	19	2	no	—
178	C₂²×M₁₆	C₂×D₁₉	1	1	yes	characteristic
179	C₂³×C₈	C₂×D₁₉	1	1	yes	characteristic
180	C₂×#117	C₂×D₁₉	1	1	yes	characteristic
181	C₂²×C₄⋊C₄	—	19	2	no	—
182	#405	—	2	2	no	—
183	#405	Dic₁₉	1	4	yes	complemented
184	C₂×#97	—	2	1	no	—
185	C₂×C₄×C₈	—	2	1	no	—
186	#343	—	19	8	no	—
187	#341	—	38	2	no	—
188	#340	—	38	1	no	—
189	#340	—	38	1	no	—
190	#340	—	38	2	no	—
191	#340	—	38	2	no	—
192	C₂×C₁₉	—	2	1	no	—
193	C₂×C₁₉	—	2	1	no	—
194	C₂×C₁₉	—	2	2	no	—
195	C₂×C₁₉	#1434	1	1	yes	characteristic
196	C₂×C₁₉	#1815	1	1	yes	characteristic
197	C₂×C₁₉	#1816	1	1	yes	characteristic
198	C₂×C₁₉	#1930	1	2	yes	—
199	C₂×C₁₉	#1931	1	2	yes	—
200	C₂³×C₄	—	38	2	no	—
201	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
202	C₂×M₁₆	—	2	2	no	—
203	C₂×M₁₆	—	2	1	no	—
204	C₂×M₁₆	—	19	4	no	—
205	C₂×M₁₆	—	38	2	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	C₂×M₁₆	C₄×D₁₉	1	4	yes	—
211	C₂×M₁₆	C₂×Dic₁₉	1	2	yes	—
212	C₂²×C₈	—	2	2	no	—
213	C₂²×C₈	—	2	2	no	—
214	C₂²×C₈	—	2	1	no	—
215	C₂²×C₈	—	38	2	no	—
216	C₂²×C₈	C₂×Dic₁₉	1	2	yes	—
217	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
218	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
219	C₂²×C₈	C₂²⋊D₁₉	1	2	yes	—
220	#117	—	2	1	no	—
221	#117	—	2	1	no	—
222	#117	Dic₃₈	1	2	yes	—
223	#117	C₂×Dic₁₉	1	2	yes	—
224	C₂×C₄⋊C₄	—	19	4	no	—
225	C₂×C₄⋊C₄	—	38	2	no	—
226	C₂×C₄⋊C₄	—	38	2	no	—
227	C₂×C₄⋊C₄	—	38	2	no	—
228	C₂×C₄⋊C₄	—	38	2	no	—
229	C₂×C₄⋊C₄	—	38	2	no	—
230	C₂×C₄⋊C₄	D₇₆	1	1	yes	characteristic
231	C₂×C₄⋊C₄	D₇₆	1	1	yes	characteristic
232	C₂×C₂²⋊C₄	—	2	1	no	—
233	C₂×C₄²	—	2	1	no	—
234	#97	—	2	4	no	—
235	C₄×C₈	—	2	4	no	—
236	C₁₉	#10827	1	1	yes	characteristic, complemented, Sylow
237	C₂⁴	#64541	1	1	yes	characteristic
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	2	1	no	—
240	C₂²×C₄	—	38	4	no	—
241	C₂²×C₄	—	76	2	no	—
242	C₂²×C₄	—	76	2	no	—
243	C₂²×C₄	—	152	2	no	—
244	C₂²×C₄	—	152	2	no	—
245	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
246	C₂²×C₄	C₂×C₄×D₁₉	1	2	yes	—
247	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
248	C₂²×C₄	#64541	1	1	yes	characteristic
249	C₂²×C₄	#64541	1	2	yes	—
250	C₂²×C₄	#64541	1	2	yes	—
251	C₂²×C₄	#64541	1	2	yes	—
252	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
253	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
254	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
255	M₁₆	—	2	2	no	—
256	M₁₆	—	2	4	no	—
257	M₁₆	—	38	4	no	—
258	M₁₆	—	38	4	no	—
259	M₁₆	—	76	2	no	—
260	M₁₆	—	76	2	no	—
261	M₁₆	C₄×Dic₁₉	1	4	yes	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	4	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	4	no	—
269	C₂×C₈	—	38	4	no	—
270	C₂×C₈	—	76	2	no	—
271	C₂×C₈	—	76	2	no	—
272	C₂×C₈	C₂²⋊Dic₁₉	1	2	yes	—
273	C₂×C₈	C₂²⋊Dic₁₉	1	2	yes	—
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	38	4	no	—
277	C₄⋊C₄	—	38	4	no	—
278	C₄⋊C₄	—	76	2	no	—
279	C₄⋊C₄	—	76	2	no	—
280	C₄⋊C₄	C₄⋊Dic₁₉	1	2	yes	—
281	C₄⋊C₄	C₄⋊Dic₁₉	1	2	yes	—
282	C₂²⋊C₄	—	2	2	no	—
283	C₂²⋊C₄	—	2	2	no	—
284	C₄²	—	2	2	no	—
285	C₄²	—	2	2	no	—
286	C₂³	—	2	2	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	2	1	no	—
289	C₂³	#10597318	1	2	yes	—
290	C₂³	#10597322	1	2	yes	—
291	C₂³	#10597370	1	1	yes	characteristic
292	C₂³	#10597372	1	1	yes	characteristic
293	C₂³	#10597376	1	1	yes	characteristic
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	76	4	no	—
300	C₂×C₄	—	152	4	no	—
301	C₂×C₄	—	152	2	no	—
302	C₂×C₄	—	152	2	no	—
303	C₂×C₄	—	152	4	no	—
304	C₂×C₄	C₄×Dic₃₈	1	2	yes	—
305	C₂×C₄	#10597316	1	2	yes	—
306	C₂×C₄	#10597318	1	2	yes	—
307	C₂×C₄	C₄×D₇₆	1	2	yes	—
308	C₂×C₄	#10597321	1	2	yes	—
309	C₂×C₄	#10597322	1	2	yes	—
310	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
311	C₂×C₄	#10597370	1	1	yes	characteristic
312	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
313	C₂×C₄	#10597372	1	1	yes	characteristic
314	C₂×C₄	#10597372	1	1	yes	characteristic
315	C₂×C₄	#10597372	1	1	yes	characteristic
316	C₂×C₄	C₄×C₂²⋊D₁₉	1	2	yes	—
317	C₂×C₄	C₄×C₂²⋊D₁₉	1	2	yes	—
318	C₂×C₄	#10597375	1	2	yes	—
319	C₂×C₄	#10597375	1	2	yes	—
320	C₂×C₄	#10597376	1	1	yes	characteristic
321	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
322	C₈	—	2	4	no	—
323	C₈	—	2	2	no	—
324	C₈	—	2	2	no	—
325	C₈	—	76	4	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	2	no	—
331	C₂²	—	2	2	no	—
332	C₂²	—	2	2	no	—
333	C₂²	—	2	2	no	—
334	C₂²	#11926390	1	1	yes	characteristic
335	C₂²	#11926404	1	1	yes	characteristic
336	C₂²	#11926412	1	1	yes	characteristic
337	C₂²	#11926428	1	2	yes	—
338	C₂²	#11926435	1	2	yes	—
339	C₂²	#11926807	1	2	yes	—
340	C₂²	#11926814	1	1	yes	characteristic
341	C₂²	#11926819	1	1	yes	characteristic
342	C₄	—	4	2	no	—
343	C₄	—	4	2	no	—
344	C₄	—	152	4	no	—
345	C₄	C₄×C₂²⋊Dic₁₉	1	1	yes	characteristic
346	C₄	C₄×C₄⋊Dic₁₉	1	1	yes	characteristic
347	C₄	#11926407	1	1	yes	characteristic
348	C₄	#11926410	1	1	yes	characteristic
349	C₄	#11926428	1	2	yes	—
350	C₄	#11926435	1	2	yes	—
351	C₂	—	2	1	no	—
352	C₂	—	2	1	no	—
353	C₂	—	2	2	no	—
354	C₂	#14561862	1	1	yes	characteristic
355	C₂	#14563694	1	1	yes	characteristic
356	C₂	#14563699	1	1	yes	characteristic
357	C₂	#14564290	1	2	yes	—
358	C₂	#14564297	1	2	yes	—

	subgroup	G/H
center	C₂³	#10597376
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₂×C₁₉×#405	C₂
socle	C₂³×C₁₉	#121
prime radical	C₂⁴×C₁₉	#41
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂×#405	D₁₉
hypocenter	C₁₉	#10827
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
8. #11926312	C₄	semidirect
18. C₁₉×#405	C₄	semidirect
183. #405	Dic₁₉	semidirect
236. C₁₉	#10827	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926312	2
C₂×#11926309	2
#14563356	2
C₂×#11926307	2
C₂×C₁₉×#405	2
#10827	19