Subgroups — #18071008

Order 4864.

702 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂²⋊(C₁₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
2	#14563329	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	C₂²×#64520	—	2	1	no	—
6	C₂×C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
7	C₂×C₄×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	4	no	—
32	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
33	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
34	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
35	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
36	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
37	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
38	C₂×C₄²×C₁₉	—	4	2	no	—
39	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
43	C₂×#64520	—	2	4	no	—
44	C₂×#64520	—	4	2	no	—
45	C₂×#64520	—	4	2	no	—
46	C₂²×C₁₉⋊C₈	—	2	2	no	—
47	C₂²×C₁₉⋊C₈	—	2	2	no	—
48	C₂²⋊(C₁₉⋊C₈)	—	2	4	no	—
49	C₄×C₁₉⋊C₈	—	4	2	no	—
50	C₂⁴×C₁₉	—	2	2	no	—
51	C₂⁴×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×D₄×C₁₉	—	2	4	no	—
53	C₂×D₄×C₁₉	—	4	2	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	4	no	—
58	C₂×D₄×C₁₉	—	4	4	no	—
59	C₂×D₄×C₁₉	—	4	4	no	—
60	C₂×D₄×C₁₉	—	4	2	no	—
61	C₂×D₄×C₁₉	—	4	2	no	—
62	C₂×D₄×C₁₉	—	4	2	no	—
63	C₂×D₄×C₁₉	—	4	2	no	—
64	C₂²×C₄×C₁₉	—	2	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	—	2	2	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	SD₁₆	1	2	yes	—
78	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
81	C₄⋊C₄×C₁₉	—	4	2	no	—
82	C₄⋊C₄×C₁₉	—	4	2	no	—
83	C₄⋊C₄×C₁₉	—	4	2	no	—
84	C₂²⋊C₄×C₁₉	—	4	4	no	—
85	C₂²⋊C₄×C₁₉	—	4	2	no	—
86	C₂²⋊C₄×C₁₉	—	4	2	no	—
87	C₂²⋊C₄×C₁₉	—	4	4	no	—
88	C₄²×C₁₉	—	2	2	no	—
89	C₄²×C₁₉	—	2	2	no	—
90	C₄²×C₁₉	—	4	1	no	—
91	C₄²×C₁₉	—	4	1	no	—
92	#64520	—	4	8	no	—
93	C₂×C₁₉⋊C₈	—	2	4	no	—
94	C₂×C₁₉⋊C₈	—	2	4	no	—
95	C₂×C₁₉⋊C₈	—	4	2	no	—
96	C₂×C₁₉⋊C₈	—	4	2	no	—
97	C₂×C₁₉⋊C₈	—	4	2	no	—
98	C₂×C₁₉⋊C₈	—	4	2	no	—
99	#7684	—	19	1	no	maximal, Sylow
100	C₂³×C₁₉	—	2	1	no	—
101	C₂³×C₁₉	—	2	1	no	—
102	C₂³×C₁₉	—	2	1	no	—
103	C₂³×C₁₉	—	2	1	no	—
104	C₂³×C₁₉	—	2	1	no	—
105	C₂³×C₁₉	—	2	1	no	—
106	C₂³×C₁₉	—	4	2	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	#102	1	2	yes	—
114	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	D₄×C₁₉	—	8	4	no	—
116	D₄×C₁₉	—	8	4	no	—
117	D₄×C₁₉	—	8	8	no	—
118	D₄×C₁₉	—	8	4	no	—
119	C₂×C₄×C₁₉	—	2	1	no	—
120	C₂×C₄×C₁₉	—	2	1	no	—
121	C₂×C₄×C₁₉	—	2	1	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	4	no	—
130	C₂×C₄×C₁₉	—	2	4	no	—
131	C₂×C₄×C₁₉	—	4	4	no	—
132	C₂×C₄×C₁₉	—	4	4	no	—
133	C₂×C₄×C₁₉	—	4	2	no	—
134	C₂×C₄×C₁₉	—	4	2	no	—
135	C₂×C₄×C₁₉	—	4	2	no	—
136	C₂×C₄×C₁₉	#99	1	2	yes	—
137	C₂×C₄×C₁₉	#100	1	2	yes	—
138	C₂×C₄×C₁₉	#102	1	2	yes	—
139	C₂×C₄×C₁₉	#104	1	2	yes	—
140	C₂×C₄×C₁₉	#104	1	2	yes	—
141	C₂×C₄×C₁₉	#131	1	2	yes	—
142	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
144	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
145	C₁₉⋊C₈	—	4	4	no	—
146	C₁₉⋊C₈	—	4	4	no	—
147	#2856	D₁₉	1	1	yes	characteristic
148	C₄×C₂²⋊C₈	—	19	1	no	—
149	#1728	—	19	1	no	—
150	#1270	—	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	—	8	2	no	—
159	C₂²×C₁₉	—	8	2	no	—
160	C₂²×C₁₉	—	8	2	no	—
161	C₂²×C₁₉	—	8	2	no	—
162	C₂²×C₁₉	—	8	2	no	—
163	C₂²×C₁₉	—	8	2	no	—
164	C₂²×C₁₉	—	8	2	no	—
165	C₂²×C₁₉	#408	1	1	yes	characteristic
166	C₂²×C₁₉	C₂×#99	1	1	yes	characteristic
167	C₂²×C₁₉	C₂×#100	1	1	yes	characteristic
168	C₂²×C₁₉	C₂×#102	1	1	yes	characteristic
169	C₂²×C₁₉	#418	1	1	yes	characteristic
170	C₂²×C₁₉	C₂×#104	1	1	yes	characteristic
171	C₂²×C₁₉	C₂×#104	1	1	yes	characteristic
172	C₄×C₁₉	—	2	2	no	—
173	C₄×C₁₉	—	2	2	no	—
174	C₄×C₁₉	—	2	1	no	—
175	C₄×C₁₉	—	2	1	no	—
176	C₄×C₁₉	—	2	2	no	—
177	C₄×C₁₉	—	2	2	no	—
178	C₄×C₁₉	—	8	2	no	—
179	C₄×C₁₉	#327	1	4	yes	—
180	C₂²×M₁₆	—	38	1	no	—
181	C₂×#127	Dic₁₉	1	1	yes	characteristic
182	C₂×#121	—	2	1	no	—
183	C₂×#121	—	2	1	no	—
184	C₂×#121	Dic₁₉	1	2	yes	—
185	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
186	C₂×C₂²⋊C₈	—	38	1	no	—
187	C₂×C₄×C₈	—	38	1	no	—
188	#390	—	2	1	no	—
189	#390	—	2	1	no	—
190	#390	C₂×D₁₉	1	2	yes	—
191	#384	Dic₁₉	1	1	yes	characteristic
192	#336	—	19	2	no	—
193	#336	—	19	2	no	—
194	#336	—	38	1	no	—
195	C₂×C₁₉	—	4	2	no	—
196	C₂×C₁₉	—	16	2	no	—
197	C₂×C₁₉	#1471	1	1	yes	characteristic
198	C₂×C₁₉	#1484	1	1	yes	characteristic
199	C₂×C₁₉	#1492	1	1	yes	characteristic
200	C₂×C₁₉	C₂×#327	1	1	yes	characteristic
201	C₂×C₁₉	#1507	1	1	yes	characteristic
202	C₂×C₁₉	#1525	1	1	yes	characteristic
203	C₂×C₁₉	#1545	1	1	yes	characteristic
204	C₂²×D₄	—	2	2	no	—
205	C₂²×D₄	—	2	1	no	—
206	C₂²×D₄	—	2	1	no	—
207	C₂²×D₄	C₂×Dic₁₉	1	2	yes	—
208	C₂³×C₄	C₂×Dic₁₉	1	1	yes	characteristic
209	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
210	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
211	C₂×M₁₆	—	38	4	no	—
212	C₂×M₁₆	—	76	2	no	—
213	C₂×M₁₆	—	76	2	no	—
214	C₂²×C₈	—	38	2	no	—
215	C₂²×C₈	—	38	2	no	—
216	#127	—	4	2	no	—
217	#121	—	4	4	no	—
218	#121	—	4	2	no	—
219	#121	—	4	2	no	—
220	#121	—	4	4	no	—
221	C₂×C₄⋊C₄	—	2	1	no	—
222	C₂×C₄⋊C₄	—	2	1	no	—
223	C₂×C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
224	C₂×C₂²⋊C₄	—	2	2	no	—
225	C₂×C₂²⋊C₄	—	2	2	no	—
226	C₂×C₂²⋊C₄	—	2	2	no	—
227	C₂×C₄²	—	4	2	no	—
228	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
229	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
230	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
231	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
232	C₂²⋊C₈	—	38	4	no	—
233	C₄×C₈	—	76	2	no	—
234	C₁₉	#7684	1	1	yes	characteristic, complemented, Sylow
235	C₂⁴	—	2	2	no	—
236	C₂⁴	C₂²⋊Dic₁₉	1	1	yes	characteristic
237	C₂×D₄	—	2	4	no	—
238	C₂×D₄	—	4	2	no	—
239	C₂×D₄	—	4	2	no	—
240	C₂×D₄	—	4	2	no	—
241	C₂×D₄	—	4	2	no	—
242	C₂×D₄	—	4	4	no	—
243	C₂×D₄	—	4	4	no	—
244	C₂×D₄	—	4	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	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	—	2	2	no	—
256	C₂²×C₄	—	4	1	no	—
257	C₂²×C₄	—	4	1	no	—
258	C₂²×C₄	#64525	1	2	yes	—
259	C₂²×C₄	#64526	1	2	yes	—
260	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
261	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
262	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
263	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
264	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
265	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
266	M₁₆	—	76	8	no	—
267	C₂×C₈	—	38	4	no	—
268	C₂×C₈	—	38	4	no	—
269	C₂×C₈	—	76	2	no	—
270	C₂×C₈	—	76	2	no	—
271	C₂×C₈	—	76	2	no	—
272	C₂×C₈	—	76	2	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₄⋊C₄	—	4	2	no	—
275	C₄⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₂²⋊C₄	—	4	4	no	—
278	C₂²⋊C₄	—	4	2	no	—
279	C₂²⋊C₄	—	4	4	no	—
280	C₄²	—	2	2	no	—
281	C₄²	—	2	2	no	—
282	C₄²	—	4	1	no	—
283	C₄²	—	4	1	no	—
284	C₂³	—	2	1	no	—
285	C₂³	—	2	1	no	—
286	C₂³	—	2	1	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	2	1	no	—
289	C₂³	—	2	1	no	—
290	C₂³	—	4	2	no	—
291	C₂³	—	4	2	no	—
292	C₂³	—	4	2	no	—
293	C₂³	—	4	2	no	—
294	C₂³	—	4	2	no	—
295	C₂³	—	4	2	no	—
296	C₂³	—	4	2	no	—
297	C₂³	#10597289	1	2	yes	—
298	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
299	D₄	—	8	4	no	—
300	D₄	—	8	4	no	—
301	D₄	—	8	8	no	—
302	D₄	—	8	4	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	4	no	—
314	C₂×C₄	—	2	4	no	—
315	C₂×C₄	—	4	4	no	—
316	C₂×C₄	—	4	4	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	#10597289	1	2	yes	—
321	C₂×C₄	#10597290	1	2	yes	—
322	C₂×C₄	#10597291	1	2	yes	—
323	C₂×C₄	#10597294	1	2	yes	—
324	C₂×C₄	#10597294	1	2	yes	—
325	C₂×C₄	C₂×#64525	1	1	yes	characteristic
326	C₂×C₄	#10597378	1	2	yes	—
327	C₂×C₄	C₂×#64526	1	1	yes	characteristic
328	C₂×C₄	#10597394	1	2	yes	—
329	C₈	—	76	4	no	—
330	C₈	—	76	4	no	—
331	C₂²	—	2	2	no	—
332	C₂²	—	2	2	no	—
333	C₂²	—	2	2	no	—
334	C₂²	—	4	1	no	—
335	C₂²	—	4	1	no	—
336	C₂²	—	4	1	no	—
337	C₂²	—	4	1	no	—
338	C₂²	—	8	2	no	—
339	C₂²	—	8	2	no	—
340	C₂²	—	8	2	no	—
341	C₂²	—	8	2	no	—
342	C₂²	—	8	2	no	—
343	C₂²	—	8	2	no	—
344	C₂²	—	8	2	no	—
345	C₂²	#11926511	1	1	yes	characteristic
346	C₂²	C₂×#10597291	1	1	yes	characteristic
347	C₂²	C₂×#10597290	1	1	yes	characteristic
348	C₂²	C₂×#10597289	1	1	yes	characteristic
349	C₂²	#11926549	1	1	yes	characteristic
350	C₂²	C₂×#10597294	1	1	yes	characteristic
351	C₂²	C₂×#10597294	1	1	yes	characteristic
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	2	no	—
355	C₄	—	2	2	no	—
356	C₄	—	2	2	no	—
357	C₄	—	2	2	no	—
358	C₄	—	8	2	no	—
359	C₄	#11926278	1	4	yes	—
360	C₂	—	4	2	no	—
361	C₂	—	16	2	no	—
362	C₂	#14561995	1	1	yes	characteristic
363	C₂	#14562052	1	1	yes	characteristic
364	C₂	#14562086	1	1	yes	characteristic
365	C₂	C₂×#11926278	1	1	yes	characteristic
366	C₂	#14562151	1	1	yes	characteristic
367	C₂	#14562227	1	1	yes	characteristic
368	C₂	#14562303	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	#7684
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. C₄×C₂²⋊(C₁₉⋊C₈)	C₂	semidirect
2. #14563329	C₂	semidirect
3. #14561456	C₂	semidirect
234. C₁₉	#7684	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₂²⋊(C₁₉⋊C₈)	2
#14563329	2
#14561456	2
C₁₉×#2856	2
#7684	19