Subgroups — #18079086

Order 4864.

560 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14572379	C₂	1	1	yes	maximal, characteristic, complemented
2	#14564476	C₂	1	1	yes	maximal, characteristic, complemented
3	#14562312	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11926288	C₂	1	1	yes	maximal, characteristic, complemented
5	#14561531	C₂	1	1	yes	maximal, characteristic, complemented
6	#14561484	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#3205	C₂	1	1	yes	maximal, characteristic
8	C₂×#10597394	—	2	1	no	—
9	#11926622	—	2	1	no	—
10	#11926616	—	2	2	no	—
11	D₄×C₁₉⋊C₈	—	2	1	no	—
12	#11926568	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
15	C₂×#10597290	—	2	1	no	—
16	C₂×#10597290	—	2	1	no	—
17	#11926511	—	2	1	no	—
18	C₄×#64520	C₂²	1	1	yes	characteristic
19	#11926290	—	2	1	no	—
20	#11926288	—	2	1	no	—
21	#11926288	—	2	1	no	—
22	#11926288	—	2	1	no	—
23	C₁₉×#532	C₂²	1	1	yes	characteristic
24	C₂×C₁₉×#127	C₂²	1	1	yes	characteristic
25	C₁₉×#493	—	2	1	no	—
26	C₁₉×#486	—	2	1	no	—
27	C₁₉×#459	—	2	2	no	—
28	C₁₉×#447	C₄	1	4	yes	—
29	C₁₉×#431	C₂²	1	1	yes	characteristic
30	C₂×#41×C₁₉	—	2	1	no	—
31	C₂²×D₄×C₁₉	—	2	1	no	—
32	C₂²×D₄×C₁₉	—	2	1	no	—
33	C₂²×D₄×C₁₉	C₂×C₄	1	2	yes	—
34	C₂×D₈×C₁₉	—	2	4	no	—
35	C₁₉×C₄⋊Q₈	D₄	1	1	yes	characteristic
36	C₁₉×#127	—	2	1	no	—
37	C₁₉×#127	—	2	1	no	—
38	C₁₉×#127	—	2	1	no	—
39	C₁₉×#127	—	2	1	no	—
40	C₁₉×#127	D₄	1	1	yes	characteristic
41	C₁₉×#124	—	2	1	no	—
42	C₁₉×#121	—	2	1	no	—
43	C₁₉×#121	C₂×C₄	1	2	yes	—
44	C₄×D₄×C₁₉	—	2	1	no	—
45	C₄×D₄×C₁₉	D₄	1	2	yes	—
46	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
47	#10597394	—	4	2	no	—
48	#10597394	—	4	2	no	—
49	C₂×#64520	—	2	2	no	—
50	C₂×#64520	—	2	1	no	—
51	C₂×#64520	D₄	1	1	yes	characteristic
52	C₂×#64520	D₄	1	1	yes	characteristic
53	C₂²×C₁₉⋊C₈	—	2	2	no	—
54	C₂²×C₁₉⋊C₈	—	2	1	no	—
55	C₂²×C₁₉⋊C₈	—	2	1	no	—
56	C₁₉×C₄⋊C₈	—	2	1	no	—
57	C₁₉×#102	—	2	4	no	—
58	C₁₉×C₂²⋊C₈	C₂×C₄	1	2	yes	—
59	C₄×C₈×C₁₉	—	2	1	no	—
60	#10597290	—	4	2	no	—
61	#10597290	—	4	2	no	—
62	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
63	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
64	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
65	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
66	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
67	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
68	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
69	C₄⋊(C₁₉⋊C₈)	D₄	1	1	yes	characteristic
70	C₄⋊(C₁₉⋊C₈)	D₄	1	1	yes	characteristic
71	#10597260	—	2	1	no	—
72	C₄×C₁₉⋊C₈	—	2	1	no	—
73	C₂⁴×C₁₉	—	2	2	no	—
74	#41×C₁₉	—	4	1	no	—
75	#41×C₁₉	—	4	1	no	—
76	#41×C₁₉	—	4	1	no	—
77	#41×C₁₉	—	4	1	no	—
78	C₂×Q₈×C₁₉	—	2	1	no	—
79	C₂×D₄×C₁₉	—	2	2	no	—
80	C₂×D₄×C₁₉	—	2	2	no	—
81	C₂×D₄×C₁₉	—	2	2	no	—
82	C₂×D₄×C₁₉	—	2	1	no	—
83	C₂×D₄×C₁₉	—	2	1	no	—
84	C₂×D₄×C₁₉	—	4	1	no	—
85	C₂×D₄×C₁₉	—	4	1	no	—
86	C₂×D₄×C₁₉	—	4	1	no	—
87	C₂×D₄×C₁₉	—	4	1	no	—
88	C₂×D₄×C₁₉	—	4	1	no	—
89	C₂×D₄×C₁₉	—	4	1	no	—
90	C₂×D₄×C₁₉	—	4	1	no	—
91	C₂×D₄×C₁₉	—	4	1	no	—
92	C₂×D₄×C₁₉	—	8	1	no	—
93	C₂×D₄×C₁₉	—	8	1	no	—
94	C₂×D₄×C₁₉	—	8	2	no	—
95	C₂×D₄×C₁₉	C₂²⋊C₄	1	2	yes	—
96	C₂²×C₄×C₁₉	—	2	2	no	—
97	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
100	D₈×C₁₉	—	8	4	no	—
101	C₂×C₈×C₁₉	—	2	2	no	—
102	C₄⋊C₄×C₁₉	—	2	1	no	—
103	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
104	C₂²⋊C₄×C₁₉	—	2	2	no	—
105	C₂²⋊C₄×C₁₉	—	2	2	no	—
106	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₁₉	#41	1	1	yes	characteristic
110	#64520	—	2	1	no	—
111	#64520	—	2	1	no	—
112	#64520	—	2	1	no	—
113	#64520	—	4	4	no	—
114	#64520	—	4	1	no	—
115	#64520	—	4	1	no	—
116	#64520	D₈	1	2	yes	complemented
117	C₂×C₁₉⋊C₈	—	2	1	no	—
118	C₂×C₁₉⋊C₈	—	2	1	no	—
119	C₂×C₁₉⋊C₈	—	2	1	no	—
120	C₂×C₁₉⋊C₈	—	2	1	no	—
121	C₂×C₁₉⋊C₈	—	4	4	no	—
122	C₂×C₁₉⋊C₈	—	4	1	no	—
123	C₂×C₁₉⋊C₈	—	4	1	no	—
124	C₂×C₁₉⋊C₈	—	4	1	no	—
125	C₂×C₁₉⋊C₈	—	4	1	no	—
126	C₂×C₁₉⋊C₈	—	8	1	no	—
127	#9273	—	19	1	no	maximal, Sylow
128	C₂³×C₁₉	—	2	2	no	—
129	C₂³×C₁₉	—	4	2	no	—
130	C₂³×C₁₉	—	4	2	no	—
131	C₂³×C₁₉	—	4	2	no	—
132	C₂³×C₁₉	—	8	2	no	—
133	C₂³×C₁₉	—	8	2	no	—
134	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
135	Q₈×C₁₉	—	4	1	no	—
136	Q₈×C₁₉	—	4	1	no	—
137	D₄×C₁₉	—	4	2	no	—
138	D₄×C₁₉	—	4	4	no	—
139	D₄×C₁₉	—	4	2	no	—
140	D₄×C₁₉	—	4	2	no	—
141	D₄×C₁₉	—	4	1	no	—
142	D₄×C₁₉	—	4	1	no	—
143	D₄×C₁₉	—	4	1	no	—
144	D₄×C₁₉	—	4	1	no	—
145	D₄×C₁₉	—	8	2	no	—
146	D₄×C₁₉	—	8	2	no	—
147	D₄×C₁₉	—	8	1	no	—
148	D₄×C₁₉	—	8	1	no	—
149	D₄×C₁₉	—	8	1	no	—
150	D₄×C₁₉	—	8	1	no	—
151	D₄×C₁₉	—	8	1	no	—
152	D₄×C₁₉	—	8	1	no	—
153	C₂×C₄×C₁₉	—	2	1	no	—
154	C₂×C₄×C₁₉	—	2	1	no	—
155	C₂×C₄×C₁₉	—	2	2	no	—
156	C₂×C₄×C₁₉	—	4	1	no	—
157	C₂×C₄×C₁₉	—	4	2	no	—
158	C₂×C₄×C₁₉	—	4	2	no	—
159	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
160	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
161	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
162	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
163	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
164	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
165	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
166	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
167	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
168	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
169	C₈×C₁₉	—	4	2	no	—
170	C₁₉⋊C₈	—	2	2	no	—
171	C₁₉⋊C₈	—	4	1	no	—
172	C₁₉⋊C₈	—	4	1	no	—
173	C₁₉⋊C₈	—	4	1	no	—
174	C₁₉⋊C₈	—	8	1	no	—
175	C₁₉⋊C₈	—	8	1	no	—
176	#3205	D₁₉	1	1	yes	characteristic
177	#2935	—	19	1	no	—
178	#1965	—	19	1	no	—
179	#1547	—	19	1	no	—
180	C₂×#331	—	19	1	no	—
181	#1301	—	19	1	no	—
182	#1283	—	19	1	no	—
183	C₂²×C₁₉	—	2	1	no	—
184	C₂²×C₁₉	—	2	1	no	—
185	C₂²×C₁₉	—	4	2	no	—
186	C₂²×C₁₉	—	4	2	no	—
187	C₂²×C₁₉	—	4	2	no	—
188	C₂²×C₁₉	—	8	2	no	—
189	C₂²×C₁₉	—	8	2	no	—
190	C₂²×C₁₉	—	8	2	no	—
191	C₂²×C₁₉	—	16	2	no	—
192	C₂²×C₁₉	—	16	2	no	—
193	C₂²×C₁₉	#386	1	1	yes	characteristic
194	C₂²×C₁₉	C₄×D₈	1	1	yes	characteristic
195	C₂²×C₁₉	#442	1	1	yes	characteristic
196	C₄×C₁₉	—	2	1	no	—
197	C₄×C₁₉	—	2	1	no	—
198	C₄×C₁₉	—	2	1	no	—
199	C₄×C₁₉	—	4	2	no	—
200	C₄×C₁₉	#447	1	1	yes	characteristic
201	C₄×C₁₉	#449	1	1	yes	characteristic
202	C₄×C₁₉	#459	1	1	yes	characteristic
203	C₄×C₁₉	#463	1	1	yes	characteristic
204	C₄×C₁₉	#471	1	1	yes	characteristic
205	C₄×C₁₉	#472	1	1	yes	characteristic
206	C₂×#131	—	38	1	no	—
207	#532	C₂×D₁₉	1	1	yes	characteristic
208	C₂×#127	C₂×D₁₉	1	1	yes	characteristic
209	#493	—	2	1	no	—
210	#486	—	2	1	no	—
211	#459	—	2	2	no	—
212	#447	Dic₁₉	1	4	yes	—
213	#436	—	38	1	no	—
214	#435	—	38	2	no	—
215	C₈×D₄	—	38	1	no	—
216	#431	C₂×D₁₉	1	1	yes	characteristic
217	#423	—	19	1	no	—
218	C₂×C₄⋊C₈	—	19	1	no	—
219	C₂×C₄⋊C₈	—	19	1	no	—
220	C₂×#100	—	38	1	no	—
221	C₂×#100	—	38	1	no	—
222	#408	—	38	1	no	—
223	C₄×M₁₆	—	19	1	no	—
224	#332	—	38	1	no	—
225	#331	—	38	1	no	—
226	#331	—	38	1	no	—
227	#331	—	38	1	no	—
228	C₂×C₁₉	—	2	1	no	—
229	C₂×C₁₉	—	2	1	no	—
230	C₂×C₁₉	—	8	2	no	—
231	C₂×C₁₉	—	16	2	no	—
232	C₂×C₁₉	#1879	1	1	yes	characteristic
233	C₂×C₁₉	#1907	1	1	yes	characteristic
234	C₂×C₁₉	#1914	1	1	yes	characteristic
235	C₂×#41	—	2	1	no	—
236	C₂²×D₄	—	2	1	no	—
237	C₂²×D₄	—	2	1	no	—
238	C₂²×D₄	C₂×Dic₁₉	1	2	yes	—
239	C₂×D₈	—	2	4	no	—
240	#131	—	76	2	no	—
241	#131	—	76	2	no	—
242	C₂×M₁₆	—	19	1	no	—
243	C₂×M₁₆	—	19	1	no	—
244	C₂×M₁₆	—	38	2	no	—
245	C₂×M₁₆	—	38	1	no	—
246	C₂²×C₈	—	38	2	no	—
247	C₂²×C₈	—	38	1	no	—
248	C₂²×C₈	—	38	1	no	—
249	C₄⋊Q₈	C₂²⋊D₁₉	1	1	yes	characteristic
250	#127	—	2	1	no	—
251	#127	—	2	1	no	—
252	#127	—	2	1	no	—
253	#127	—	2	1	no	—
254	#127	C₂²⋊D₁₉	1	1	yes	characteristic
255	#124	—	2	1	no	—
256	#121	—	2	1	no	—
257	#121	C₂×Dic₁₉	1	2	yes	—
258	C₄×D₄	—	2	1	no	—
259	C₄×D₄	C₂²⋊D₁₉	1	2	yes	—
260	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
261	C₄⋊C₈	—	2	1	no	—
262	C₄⋊C₈	—	19	1	no	—
263	C₄⋊C₈	—	19	1	no	—
264	C₄⋊C₈	—	38	1	no	—
265	C₄⋊C₈	—	38	1	no	—
266	C₄⋊C₈	—	38	1	no	—
267	C₄⋊C₈	—	38	1	no	—
268	C₄⋊C₈	—	38	1	no	—
269	#102	—	2	4	no	—
270	#100	—	76	2	no	—
271	#100	—	76	2	no	—
272	C₂²⋊C₈	—	38	2	no	—
273	C₂²⋊C₈	—	38	2	no	—
274	C₂²⋊C₈	C₂×Dic₁₉	1	2	yes	—
275	#97	—	38	1	no	—
276	C₄×C₈	—	2	1	no	—
277	C₄×C₈	—	38	1	no	—
278	C₁₉	#9273	1	1	yes	characteristic, complemented, Sylow
279	C₂⁴	—	2	2	no	—
280	#41	—	4	1	no	—
281	#41	—	4	1	no	—
282	#41	—	4	1	no	—
283	#41	—	4	1	no	—
284	C₂×Q₈	—	2	1	no	—
285	C₂×D₄	—	2	2	no	—
286	C₂×D₄	—	2	2	no	—
287	C₂×D₄	—	2	1	no	—
288	C₂×D₄	—	2	1	no	—
289	C₂×D₄	—	2	2	no	—
290	C₂×D₄	—	4	1	no	—
291	C₂×D₄	—	4	1	no	—
292	C₂×D₄	—	4	1	no	—
293	C₂×D₄	—	4	1	no	—
294	C₂×D₄	—	4	1	no	—
295	C₂×D₄	—	4	1	no	—
296	C₂×D₄	—	4	1	no	—
297	C₂×D₄	—	4	1	no	—
298	C₂×D₄	—	8	1	no	—
299	C₂×D₄	—	8	1	no	—
300	C₂×D₄	—	8	2	no	—
301	C₂×D₄	C₂²⋊Dic₁₉	1	2	yes	—
302	C₂²×C₄	—	2	2	no	—
303	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
304	C₂²×C₄	#64543	1	1	yes	characteristic
305	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
306	D₈	—	8	4	no	—
307	M₁₆	—	19	2	no	—
308	M₁₆	—	38	1	no	—
309	M₁₆	—	38	1	no	—
310	M₁₆	—	38	1	no	—
311	M₁₆	—	76	4	no	—
312	M₁₆	—	76	1	no	—
313	M₁₆	—	76	1	no	—
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	—	38	1	no	—
316	C₂×C₈	—	38	1	no	—
317	C₂×C₈	—	38	1	no	—
318	C₂×C₈	—	38	1	no	—
319	C₂×C₈	—	76	4	no	—
320	C₂×C₈	—	76	1	no	—
321	C₂×C₈	—	76	1	no	—
322	C₂×C₈	—	76	1	no	—
323	C₂×C₈	—	76	1	no	—
324	C₂×C₈	—	152	1	no	—
325	C₄⋊C₄	—	2	1	no	—
326	C₄⋊C₄	C₂²⋊Dic₁₉	1	2	yes	—
327	C₂²⋊C₄	—	2	2	no	—
328	C₂²⋊C₄	—	2	2	no	—
329	C₄²	D₄×D₁₉	1	1	yes	characteristic
330	C₄²	#64543	1	1	yes	characteristic
331	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
332	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
333	C₂³	—	2	2	no	—
334	C₂³	—	4	2	no	—
335	C₂³	—	4	2	no	—
336	C₂³	—	4	2	no	—
337	C₂³	—	8	2	no	—
338	C₂³	—	8	2	no	—
339	C₂³	D₄×Dic₁₉	1	1	yes	characteristic
340	Q₈	—	4	1	no	—
341	Q₈	—	4	1	no	—
342	D₄	—	4	2	no	—
343	D₄	—	4	4	no	—
344	D₄	—	4	2	no	—
345	D₄	—	4	2	no	—
346	D₄	—	4	1	no	—
347	D₄	—	4	1	no	—
348	D₄	—	4	1	no	—
349	D₄	—	4	1	no	—
350	D₄	—	8	2	no	—
351	D₄	—	8	2	no	—
352	D₄	—	8	1	no	—
353	D₄	—	8	1	no	—
354	D₄	—	8	1	no	—
355	D₄	—	8	1	no	—
356	D₄	—	8	1	no	—
357	D₄	—	8	1	no	—
358	C₂×C₄	—	2	1	no	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	4	1	no	—
362	C₂×C₄	—	4	2	no	—
363	C₂×C₄	—	4	2	no	—
364	C₂×C₄	D₈×D₁₉	1	1	yes	characteristic
365	C₂×C₄	#10597357	1	1	yes	characteristic
366	C₂×C₄	#10597357	1	1	yes	characteristic
367	C₂×C₄	#10597358	1	1	yes	characteristic
368	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
369	C₂×C₄	#10597381	1	1	yes	characteristic
370	C₂×C₄	#10597383	1	1	yes	characteristic
371	C₂×C₄	#10597385	1	1	yes	characteristic
372	C₂×C₄	#10597385	1	1	yes	characteristic
373	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
374	C₈	—	4	2	no	—
375	C₈	—	38	2	no	—
376	C₈	—	76	1	no	—
377	C₈	—	76	1	no	—
378	C₈	—	76	1	no	—
379	C₈	—	152	1	no	—
380	C₈	—	152	1	no	—
381	C₂²	—	2	1	no	—
382	C₂²	—	2	1	no	—
383	C₂²	—	4	2	no	—
384	C₂²	—	4	2	no	—
385	C₂²	—	4	2	no	—
386	C₂²	—	8	2	no	—
387	C₂²	—	8	2	no	—
388	C₂²	—	8	2	no	—
389	C₂²	—	16	2	no	—
390	C₂²	—	16	2	no	—
391	C₂²	#11926424	1	1	yes	characteristic
392	C₂²	D₈×Dic₁₉	1	1	yes	characteristic
393	C₂²	#11926656	1	1	yes	characteristic
394	C₄	—	2	1	no	—
395	C₄	—	2	1	no	—
396	C₄	—	2	1	no	—
397	C₄	—	4	2	no	—
398	C₄	#11926676	1	1	yes	characteristic
399	C₄	#11926686	1	1	yes	characteristic
400	C₄	#11926737	1	1	yes	characteristic
401	C₄	#11926757	1	1	yes	characteristic
402	C₄	#11926805	1	1	yes	characteristic
403	C₄	#11926812	1	1	yes	characteristic
404	C₂	—	2	1	no	—
405	C₂	—	2	1	no	—
406	C₂	—	8	2	no	—
407	C₂	—	16	2	no	—
408	C₂	#14563985	1	1	yes	characteristic
409	C₂	#14564145	1	1	yes	characteristic
410	C₂	#14564183	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11926424
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting, prime radical	C₁₉×#3205	C₂
socle	C₂²×C₁₉	#386
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#3205	D₁₉
hypocenter	C₁₉	#9273
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3205

Complements

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

N	H ≅ G/N	type
1. #14572379	C₂	semidirect
2. #14564476	C₂	semidirect
3. #14562312	C₂	semidirect
4. C₂×#11926288	C₂	semidirect
5. #14561531	C₂	semidirect
6. #14561484	C₂	semidirect
116. #64520	D₈	semidirect
278. C₁₉	#9273	semidirect

Maximal subgroups

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

subgroup	index
#14572379	2
#14564476	2
#14562312	2
C₂×#11926288	2
#14561531	2
#14561484	2
C₁₉×#3205	2
#9273	19