Subgroups — #18070916

Order 4864.

708 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14563615	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11926299	C₂	1	1	yes	maximal, characteristic, complemented
3	#14561456	C₂	1	4	yes	maximal, complemented
4	C₁₉×#2856	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₁₉⋊C₈	—	2	1	no	—
6	C₂×C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
7	C₂×C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
8	#11926299	—	2	1	no	—
9	#11926299	C₂²	1	2	yes	—
10	#11926299	C₂²	1	2	yes	—
11	C₂×C₁₉×#127	C₄	1	1	yes	characteristic
12	C₂×C₁₉×#121	—	2	1	no	—
13	C₂×C₁₉×#121	—	2	1	no	—
14	C₂×C₁₉×#121	C₄	1	2	yes	—
15	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
16	C₁₉×#390	—	2	1	no	—
17	C₁₉×#390	—	2	1	no	—
18	C₁₉×#390	C₂²	1	2	yes	—
19	C₁₉×#384	C₄	1	1	yes	characteristic
20	C₂²×D₄×C₁₉	—	2	2	no	—
21	C₂²×D₄×C₁₉	—	2	1	no	—
22	C₂²×D₄×C₁₉	—	2	1	no	—
23	C₂²×D₄×C₁₉	C₂×C₄	1	2	yes	—
24	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
27	C₁₉×#127	—	4	2	no	—
28	C₁₉×#121	—	4	4	no	—
29	C₁₉×#121	—	4	2	no	—
30	C₁₉×#121	—	4	2	no	—
31	C₁₉×#121	—	4	2	no	—
32	C₁₉×#121	—	4	2	no	—
33	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
34	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
35	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
36	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
37	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
38	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
39	C₂×C₄²×C₁₉	—	4	2	no	—
40	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
44	C₂²×C₁₉⋊C₈	—	2	4	no	—
45	C₂²×C₁₉⋊C₈	—	2	2	no	—
46	C₂²×C₁₉⋊C₈	—	2	2	no	—
47	C₂²×C₁₉⋊C₈	—	4	2	no	—
48	C₂²×C₁₉⋊C₈	—	4	2	no	—
49	C₂²⋊(C₁₉⋊C₈)	—	4	2	no	—
50	C₄⋊(C₁₉⋊C₈)	—	4	2	no	—
51	C₂⁴×C₁₉	—	2	2	no	—
52	C₂⁴×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×D₄×C₁₉	—	2	4	no	—
54	C₂×D₄×C₁₉	—	4	2	no	—
55	C₂×D₄×C₁₉	—	4	2	no	—
56	C₂×D₄×C₁₉	—	4	2	no	—
57	C₂×D₄×C₁₉	—	4	2	no	—
58	C₂×D₄×C₁₉	—	4	4	no	—
59	C₂×D₄×C₁₉	—	4	4	no	—
60	C₂×D₄×C₁₉	—	4	4	no	—
61	C₂×D₄×C₁₉	—	4	2	no	—
62	C₂×D₄×C₁₉	—	4	2	no	—
63	C₂×D₄×C₁₉	—	4	2	no	—
64	C₂×D₄×C₁₉	—	4	2	no	—
65	C₂²×C₄×C₁₉	—	2	2	no	—
66	C₂²×C₄×C₁₉	—	2	2	no	—
67	C₂²×C₄×C₁₉	—	2	2	no	—
68	C₂²×C₄×C₁₉	—	2	2	no	—
69	C₂²×C₄×C₁₉	—	2	2	no	—
70	C₂²×C₄×C₁₉	—	4	1	no	—
71	C₂²×C₄×C₁₉	—	4	1	no	—
72	C₂²×C₄×C₁₉	—	4	1	no	—
73	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₉	D₈	1	2	yes	—
77	C₂²×C₄×C₁₉	D₈	1	2	yes	—
78	C₂²×C₄×C₁₉	SD₁₆	1	2	yes	—
79	C₂²×C₄×C₁₉	SD₁₆	1	2	yes	—
80	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
83	C₄⋊C₄×C₁₉	—	4	2	no	—
84	C₄⋊C₄×C₁₉	—	4	2	no	—
85	C₄⋊C₄×C₁₉	—	4	2	no	—
86	C₂²⋊C₄×C₁₉	—	4	4	no	—
87	C₂²⋊C₄×C₁₉	—	4	2	no	—
88	C₂²⋊C₄×C₁₉	—	4	2	no	—
89	C₂²⋊C₄×C₁₉	—	4	2	no	—
90	C₂²⋊C₄×C₁₉	—	4	2	no	—
91	C₄²×C₁₉	—	2	2	no	—
92	C₄²×C₁₉	—	2	2	no	—
93	C₄²×C₁₉	—	4	1	no	—
94	C₄²×C₁₉	—	4	1	no	—
95	C₂×C₁₉⋊C₈	—	2	4	no	—
96	C₂×C₁₉⋊C₈	—	4	2	no	—
97	C₂×C₁₉⋊C₈	—	4	8	no	—
98	C₂×C₁₉⋊C₈	—	4	2	no	—
99	C₂×C₁₉⋊C₈	—	4	2	no	—
100	C₂×C₁₉⋊C₈	—	4	2	no	—
101	C₂×C₁₉⋊C₈	—	4	2	no	—
102	#7664	—	19	1	no	maximal, Sylow
103	C₂³×C₁₉	—	2	1	no	—
104	C₂³×C₁₉	—	2	1	no	—
105	C₂³×C₁₉	—	2	1	no	—
106	C₂³×C₁₉	—	2	1	no	—
107	C₂³×C₁₉	—	4	2	no	—
108	C₂³×C₁₉	—	4	2	no	—
109	C₂³×C₁₉	—	4	2	no	—
110	C₂³×C₁₉	—	4	2	no	—
111	C₂³×C₁₉	—	4	2	no	—
112	C₂³×C₁₉	—	4	2	no	—
113	C₂³×C₁₉	—	4	2	no	—
114	C₂³×C₁₉	#100	1	2	yes	—
115	C₂³×C₁₉	#102	1	2	yes	—
116	C₂³×C₁₉	#102	1	2	yes	—
117	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	D₄×C₁₉	—	8	4	no	—
119	D₄×C₁₉	—	8	4	no	—
120	D₄×C₁₉	—	8	8	no	—
121	D₄×C₁₉	—	8	4	no	—
122	C₂×C₄×C₁₉	—	2	1	no	—
123	C₂×C₄×C₁₉	—	2	1	no	—
124	C₂×C₄×C₁₉	—	2	1	no	—
125	C₂×C₄×C₁₉	—	2	1	no	—
126	C₂×C₄×C₁₉	—	2	1	no	—
127	C₂×C₄×C₁₉	—	2	1	no	—
128	C₂×C₄×C₁₉	—	2	1	no	—
129	C₂×C₄×C₁₉	—	2	1	no	—
130	C₂×C₄×C₁₉	—	2	4	no	—
131	C₂×C₄×C₁₉	—	2	4	no	—
132	C₂×C₄×C₁₉	—	2	1	no	—
133	C₂×C₄×C₁₉	—	2	1	no	—
134	C₂×C₄×C₁₉	—	2	1	no	—
135	C₂×C₄×C₁₉	—	4	2	no	—
136	C₂×C₄×C₁₉	—	4	2	no	—
137	C₂×C₄×C₁₉	—	4	4	no	—
138	C₂×C₄×C₁₉	—	4	2	no	—
139	C₂×C₄×C₁₉	—	4	2	no	—
140	C₂×C₄×C₁₉	—	4	2	no	—
141	C₂×C₄×C₁₉	#99	1	2	yes	—
142	C₂×C₄×C₁₉	#102	1	2	yes	—
143	C₂×C₄×C₁₉	#102	1	2	yes	—
144	C₂×C₄×C₁₉	#104	1	2	yes	—
145	C₂×C₄×C₁₉	#131	1	2	yes	—
146	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
148	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
149	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
150	C₁₉⋊C₈	—	4	4	no	—
151	C₁₉⋊C₈	—	8	2	no	—
152	#2856	D₁₉	1	1	yes	characteristic
153	#1797	—	19	1	no	—
154	C₂×#336	—	19	1	no	—
155	#1270	—	19	4	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	—	2	2	no	—
161	C₂²×C₁₉	—	2	2	no	—
162	C₂²×C₁₉	—	8	2	no	—
163	C₂²×C₁₉	—	8	2	no	—
164	C₂²×C₁₉	—	8	2	no	—
165	C₂²×C₁₉	—	8	2	no	—
166	C₂²×C₁₉	—	8	2	no	—
167	C₂²×C₁₉	—	8	2	no	—
168	C₂²×C₁₉	—	8	2	no	—
169	C₂²×C₁₉	#331	1	4	yes	—
170	C₂²×C₁₉	#408	1	1	yes	characteristic
171	C₂²×C₁₉	C₂×#99	1	1	yes	characteristic
172	C₂²×C₁₉	C₂×#100	1	1	yes	characteristic
173	C₂²×C₁₉	C₂×#102	1	1	yes	characteristic
174	C₂²×C₁₉	C₂×#102	1	1	yes	characteristic
175	C₂²×C₁₉	C₂×#104	1	1	yes	characteristic
176	C₂²×C₁₉	#421	1	1	yes	characteristic
177	C₄×C₁₉	—	2	2	no	—
178	C₄×C₁₉	—	2	1	no	—
179	C₄×C₁₉	—	2	1	no	—
180	C₄×C₁₉	—	2	1	no	—
181	C₄×C₁₉	—	2	2	no	—
182	C₄×C₁₉	—	2	1	no	—
183	C₄×C₁₉	—	4	1	no	—
184	C₄×C₁₉	—	8	2	no	—
185	C₄×C₁₉	#327	1	4	yes	—
186	C₂³×C₈	—	38	1	no	—
187	C₂×#127	Dic₁₉	1	1	yes	characteristic
188	C₂×#121	—	2	1	no	—
189	C₂×#121	—	2	1	no	—
190	C₂×#121	Dic₁₉	1	2	yes	—
191	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
192	C₂×C₄⋊C₈	—	38	1	no	—
193	C₂×C₂²⋊C₈	—	38	1	no	—
194	#390	—	2	1	no	—
195	#390	—	2	1	no	—
196	#390	C₂×D₁₉	1	2	yes	—
197	#384	Dic₁₉	1	1	yes	characteristic
198	#336	—	19	2	no	—
199	#336	—	19	2	no	—
200	#336	—	38	1	no	—
201	C₂×C₁₉	—	2	4	no	—
202	C₂×C₁₉	—	16	2	no	—
203	C₂×C₁₉	#1471	1	1	yes	characteristic
204	C₂×C₁₉	#1480	1	1	yes	characteristic
205	C₂×C₁₉	#1492	1	1	yes	characteristic
206	C₂×C₁₉	C₂×#327	1	1	yes	characteristic
207	C₂×C₁₉	#1519	1	1	yes	characteristic
208	C₂×C₁₉	#1530	1	1	yes	characteristic
209	C₂×C₁₉	C₂×#331	1	1	yes	characteristic
210	C₂²×D₄	—	2	2	no	—
211	C₂²×D₄	—	2	1	no	—
212	C₂²×D₄	—	2	1	no	—
213	C₂²×D₄	C₂×Dic₁₉	1	2	yes	—
214	C₂³×C₄	C₂×Dic₁₉	1	1	yes	characteristic
215	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
216	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
217	C₂²×C₈	—	38	4	no	—
218	C₂²×C₈	—	38	2	no	—
219	C₂²×C₈	—	38	2	no	—
220	C₂²×C₈	—	76	2	no	—
221	C₂²×C₈	—	76	2	no	—
222	#127	—	4	2	no	—
223	#121	—	4	4	no	—
224	#121	—	4	2	no	—
225	#121	—	4	2	no	—
226	#121	—	4	2	no	—
227	#121	—	4	2	no	—
228	C₂×C₄⋊C₄	—	2	1	no	—
229	C₂×C₄⋊C₄	—	2	1	no	—
230	C₂×C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
231	C₂×C₂²⋊C₄	—	2	2	no	—
232	C₂×C₂²⋊C₄	—	2	2	no	—
233	C₂×C₂²⋊C₄	—	2	2	no	—
234	C₂×C₄²	—	4	2	no	—
235	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
236	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
237	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
238	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
239	C₄⋊C₈	—	76	2	no	—
240	C₂²⋊C₈	—	76	2	no	—
241	C₁₉	#7664	1	1	yes	characteristic, complemented, Sylow
242	C₂⁴	—	2	2	no	—
243	C₂⁴	C₂²⋊Dic₁₉	1	1	yes	characteristic
244	C₂×D₄	—	2	4	no	—
245	C₂×D₄	—	4	2	no	—
246	C₂×D₄	—	4	2	no	—
247	C₂×D₄	—	4	2	no	—
248	C₂×D₄	—	4	2	no	—
249	C₂×D₄	—	4	4	no	—
250	C₂×D₄	—	4	4	no	—
251	C₂×D₄	—	4	4	no	—
252	C₂×D₄	—	4	2	no	—
253	C₂×D₄	—	4	2	no	—
254	C₂×D₄	—	4	2	no	—
255	C₂×D₄	—	4	2	no	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	—	2	2	no	—
258	C₂²×C₄	—	2	2	no	—
259	C₂²×C₄	—	2	2	no	—
260	C₂²×C₄	—	2	2	no	—
261	C₂²×C₄	—	4	1	no	—
262	C₂²×C₄	—	4	1	no	—
263	C₂²×C₄	—	4	1	no	—
264	C₂²×C₄	#64525	1	2	yes	—
265	C₂²×C₄	#64525	1	2	yes	—
266	C₂²×C₄	#64526	1	2	yes	—
267	C₂²×C₄	#64526	1	2	yes	—
268	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
269	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
270	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
271	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
272	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
273	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
274	C₂×C₈	—	38	4	no	—
275	C₂×C₈	—	76	2	no	—
276	C₂×C₈	—	76	8	no	—
277	C₂×C₈	—	76	2	no	—
278	C₂×C₈	—	76	2	no	—
279	C₂×C₈	—	76	2	no	—
280	C₂×C₈	—	76	2	no	—
281	C₄⋊C₄	—	4	2	no	—
282	C₄⋊C₄	—	4	2	no	—
283	C₄⋊C₄	—	4	2	no	—
284	C₂²⋊C₄	—	4	4	no	—
285	C₂²⋊C₄	—	4	2	no	—
286	C₂²⋊C₄	—	4	2	no	—
287	C₂²⋊C₄	—	4	2	no	—
288	C₂²⋊C₄	—	4	2	no	—
289	C₄²	—	2	2	no	—
290	C₄²	—	2	2	no	—
291	C₄²	—	4	1	no	—
292	C₄²	—	4	1	no	—
293	C₂³	—	2	1	no	—
294	C₂³	—	2	1	no	—
295	C₂³	—	2	1	no	—
296	C₂³	—	2	1	no	—
297	C₂³	—	4	2	no	—
298	C₂³	—	4	2	no	—
299	C₂³	—	4	2	no	—
300	C₂³	—	4	2	no	—
301	C₂³	—	4	2	no	—
302	C₂³	—	4	2	no	—
303	C₂³	—	4	2	no	—
304	C₂³	#10597289	1	2	yes	—
305	C₂³	#10597289	1	2	yes	—
306	C₂³	#10597290	1	2	yes	—
307	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
308	D₄	—	8	4	no	—
309	D₄	—	8	4	no	—
310	D₄	—	8	8	no	—
311	D₄	—	8	4	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	4	no	—
321	C₂×C₄	—	2	4	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	4	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	#10597289	1	2	yes	—
332	C₂×C₄	#10597289	1	2	yes	—
333	C₂×C₄	#10597291	1	2	yes	—
334	C₂×C₄	#10597294	1	2	yes	—
335	C₂×C₄	C₂×#64525	1	1	yes	characteristic
336	C₂×C₄	C₂×#64525	1	1	yes	characteristic
337	C₂×C₄	C₂×#64526	1	1	yes	characteristic
338	C₂×C₄	C₂×#64526	1	1	yes	characteristic
339	C₂×C₄	#10597394	1	2	yes	—
340	C₈	—	76	4	no	—
341	C₈	—	152	2	no	—
342	C₂²	—	2	2	no	—
343	C₂²	—	2	2	no	—
344	C₂²	—	2	2	no	—
345	C₂²	—	2	2	no	—
346	C₂²	—	2	2	no	—
347	C₂²	—	2	2	no	—
348	C₂²	—	8	2	no	—
349	C₂²	—	8	2	no	—
350	C₂²	—	8	2	no	—
351	C₂²	—	8	2	no	—
352	C₂²	—	8	2	no	—
353	C₂²	—	8	2	no	—
354	C₂²	—	8	2	no	—
355	C₂²	#11926288	1	4	yes	—
356	C₂²	#11926511	1	1	yes	characteristic
357	C₂²	C₂×#10597291	1	1	yes	characteristic
358	C₂²	C₂×#10597290	1	1	yes	characteristic
359	C₂²	C₂×#10597289	1	1	yes	characteristic
360	C₂²	C₂×#10597289	1	1	yes	characteristic
361	C₂²	C₂×#10597294	1	1	yes	characteristic
362	C₂²	#11926561	1	1	yes	characteristic
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	2	1	no	—
367	C₄	—	2	2	no	—
368	C₄	—	2	2	no	—
369	C₄	—	4	1	no	—
370	C₄	—	8	2	no	—
371	C₄	#11926278	1	4	yes	—
372	C₂	—	2	4	no	—
373	C₂	—	16	2	no	—
374	C₂	#14561995	1	1	yes	characteristic
375	C₂	#14562034	1	1	yes	characteristic
376	C₂	#14562086	1	1	yes	characteristic
377	C₂	C₂×#11926278	1	1	yes	characteristic
378	C₂	#14562203	1	1	yes	characteristic
379	C₂	#14562246	1	1	yes	characteristic
380	C₂	C₂×#11926288	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₁₉
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting, prime radical	C₁₉×#2856	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#2856	D₁₉
hypocenter	C₁₉	#7664
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #2856

Complements

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

N	H ≅ G/N	type
1. #14563615	C₂	semidirect
2. C₂×#11926299	C₂	semidirect
3. #14561456	C₂	semidirect
241. C₁₉	#7664	semidirect

Maximal subgroups

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

subgroup	index
#14563615	2
C₂×#11926299	2
#14561456	2
C₁₉×#2856	2
#7664	19