Subgroups — #18082402

Order 4864.

626 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926916	C₂	1	1	yes	maximal, characteristic
2	#14564596	C₂	1	1	yes	maximal, characteristic, complemented
3	#14564595	C₂	1	1	yes	maximal, characteristic, complemented
4	#14564442	C₂	1	1	yes	maximal, characteristic
5	#14563794	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11926266	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1957	C₂	1	1	yes	maximal, characteristic
8	C₂²×#64526	—	2	2	no	—
9	C₂×#10597315	C₂²	1	1	yes	characteristic
10	#11926916	—	2	2	no	—
11	#11926916	C₄	1	4	yes	complemented
12	C₂×#10597264	—	2	2	no	—
13	C₂×#10597289	—	2	2	no	—
14	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
16	#11926414	C₂²	1	1	yes	characteristic
17	#11926307	—	2	2	no	—
18	#11926306	—	2	2	no	—
19	#11926266	—	2	2	no	—
20	C₂×C₁₉×#127	C₂²	1	1	yes	characteristic
21	C₂×C₁₉×#102	—	2	2	no	—
22	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
23	C₁₉×#384	C₂²	1	1	yes	characteristic
24	C₂²×D₄×C₁₉	—	2	1	no	—
25	C₂²×D₄×C₁₉	—	2	1	no	—
26	C₂²×D₄×C₁₉	—	2	1	no	—
27	C₂²×Dic₃₈	—	2	1	no	—
28	C₂²×C₈×C₁₉	D₄	1	2	yes	—
29	C₁₉×#127	—	2	1	no	—
30	C₁₉×#127	—	2	1	no	—
31	C₁₉×#127	—	2	1	no	—
32	C₁₉×#127	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
34	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
35	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
36	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
37	C₂×#64526	—	2	8	no	—
38	C₂×#64526	—	4	4	no	—
39	C₂×#64526	—	4	4	no	—
40	C₂×C₄⋊Dic₁₉	—	2	1	no	—
41	C₂×C₄⋊Dic₁₉	—	2	1	no	—
42	C₂×C₄⋊Dic₁₉	—	2	1	no	—
43	C₂×C₄⋊Dic₁₉	—	2	1	no	—
44	C₂×C₄⋊Dic₁₉	—	2	1	no	—
45	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
46	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
47	#10597315	—	2	1	no	—
48	#10597315	—	2	1	no	—
49	#10597315	—	2	1	no	—
50	#10597315	C₂×C₄	1	2	yes	—
51	C₁₉×#102	—	4	4	no	—
52	C₄×C₈×C₁₉	—	2	1	no	—
53	C₄×C₈×C₁₉	—	2	1	no	—
54	#10597289	—	4	4	no	—
55	#10597264	—	4	4	no	—
56	C₄×C₁₉⋊C₈	—	2	1	no	—
57	C₄×C₁₉⋊C₈	—	2	1	no	—
58	C₄×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
59	C₄×C₁₉⋊C₈	D₄	1	2	yes	—
60	C₂⁴×C₁₉	—	4	1	no	—
61	C₂×D₄×C₁₉	—	2	2	no	—
62	C₂×D₄×C₁₉	—	2	2	no	—
63	C₂×D₄×C₁₉	—	2	2	no	—
64	C₂×D₄×C₁₉	—	4	1	no	—
65	C₂×D₄×C₁₉	—	4	1	no	—
66	C₂×D₄×C₁₉	—	4	1	no	—
67	C₂×D₄×C₁₉	—	4	2	no	—
68	C₂×D₄×C₁₉	—	4	2	no	—
69	C₂×D₄×C₁₉	—	4	2	no	—
70	C₂×D₄×C₁₉	—	4	2	no	—
71	C₂×D₄×C₁₉	—	4	2	no	—
72	C₂×D₄×C₁₉	—	4	2	no	—
73	C₂²×C₄×C₁₉	—	4	1	no	—
74	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×Dic₁₉	—	4	1	no	—
78	C₂²×Dic₁₉	—	4	1	no	—
79	C₂×Dic₃₈	—	2	2	no	—
80	C₂×Dic₃₈	—	4	2	no	—
81	C₂×Dic₃₈	—	4	2	no	—
82	C₂×Dic₃₈	—	4	1	no	—
83	C₂×C₈×C₁₉	—	2	2	no	—
84	C₂×C₈×C₁₉	—	2	2	no	—
85	C₂×C₈×C₁₉	—	2	2	no	—
86	C₄⋊C₄×C₁₉	—	4	2	no	—
87	C₄⋊C₄×C₁₉	—	4	2	no	—
88	C₄⋊C₄×C₁₉	—	4	2	no	—
89	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₁₉	#41	1	1	yes	characteristic
92	C₄²×C₁₉	#41	1	1	yes	characteristic
93	#64526	—	4	16	no	—
94	C₄⋊Dic₁₉	—	2	2	no	—
95	C₄⋊Dic₁₉	—	2	2	no	—
96	C₄⋊Dic₁₉	—	4	1	no	—
97	C₄⋊Dic₁₉	—	4	1	no	—
98	C₄⋊Dic₁₉	—	4	2	no	—
99	C₄⋊Dic₁₉	—	4	2	no	—
100	C₄⋊Dic₁₉	—	4	2	no	—
101	C₂×C₁₉⋊C₈	—	2	2	no	—
102	C₂×C₁₉⋊C₈	—	2	2	no	—
103	C₂×C₁₉⋊C₈	—	2	2	no	—
104	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	4	yes	—
105	C₂×C₁₉⋊C₈	D₈	1	4	yes	—
106	C₂×C₁₉⋊C₈	SD₁₆	1	4	yes	—
107	#9842	—	19	1	no	maximal, Sylow
108	C₂³×C₁₉	—	4	2	no	—
109	C₂³×C₁₉	—	8	1	no	—
110	C₂³×C₁₉	—	8	1	no	—
111	C₂³×C₁₉	—	8	1	no	—
112	C₂³×C₁₉	—	8	1	no	—
113	C₂³×C₁₉	—	8	1	no	—
114	C₂³×C₁₉	—	8	1	no	—
115	C₂³×C₁₉	#127	1	1	yes	characteristic
116	D₄×C₁₉	—	4	4	no	—
117	D₄×C₁₉	—	4	4	no	—
118	D₄×C₁₉	—	4	4	no	—
119	D₄×C₁₉	—	8	2	no	—
120	D₄×C₁₉	—	8	2	no	—
121	D₄×C₁₉	—	8	2	no	—
122	C₂×C₄×C₁₉	—	8	1	no	—
123	C₂×C₄×C₁₉	—	8	1	no	—
124	C₂×C₄×C₁₉	—	8	1	no	—
125	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
129	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
130	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
131	C₂×C₄×C₁₉	C₂×D₈	1	2	yes	—
132	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
133	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
134	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
135	C₂×C₄×C₁₉	#137	1	2	yes	—
136	C₂×C₄×C₁₉	#137	1	2	yes	—
137	C₂×Dic₁₉	—	4	2	no	—
138	C₂×Dic₁₉	—	8	1	no	—
139	C₂×Dic₁₉	—	8	1	no	—
140	C₂×Dic₁₉	—	8	1	no	—
141	C₂×Dic₁₉	—	8	1	no	—
142	C₂×Dic₁₉	—	8	1	no	—
143	Dic₃₈	—	4	4	no	—
144	Dic₃₈	—	8	2	no	—
145	C₈×C₁₉	—	4	2	no	—
146	C₁₉⋊C₈	—	4	2	no	—
147	C₁₉⋊C₈	#102	1	8	yes	complemented
148	C₂×#492	—	19	1	no	—
149	#1990	—	19	1	no	—
150	#1990	—	19	1	no	—
151	#1957	—	19	1	no	—
152	#1957	D₁₉	1	1	yes	characteristic
153	#1845	—	19	1	no	—
154	C₂×#322	—	19	1	no	—
155	C₂²×C₁₉	—	8	2	no	—
156	C₂²×C₁₉	—	8	2	no	—
157	C₂²×C₁₉	—	8	2	no	—
158	C₂²×C₁₉	—	16	1	no	—
159	C₂²×C₁₉	—	16	1	no	—
160	C₂²×C₁₉	—	16	1	no	—
161	C₂²×C₁₉	—	16	1	no	—
162	C₂²×C₁₉	#390	1	1	yes	characteristic
163	C₂²×C₁₉	#492	1	1	yes	characteristic
164	C₂²×C₁₉	#493	1	1	yes	characteristic
165	C₂²×C₁₉	#496	1	1	yes	characteristic
166	C₂²×C₁₉	#496	1	1	yes	characteristic
167	C₂²×C₁₉	#497	1	1	yes	characteristic
168	C₂²×C₁₉	#497	1	1	yes	characteristic
169	C₄×C₁₉	—	16	1	no	—
170	C₄×C₁₉	C₂×#102	1	2	yes	—
171	C₄×C₁₉	#440	1	2	yes	—
172	C₄×C₁₉	#440	1	2	yes	—
173	C₄×C₁₉	#468	1	2	yes	—
174	C₄×C₁₉	#468	1	2	yes	—
175	C₄×C₁₉	#486	1	2	yes	—
176	Dic₁₉	—	8	2	no	—
177	Dic₁₉	—	16	1	no	—
178	C₂²×SD₁₆	—	38	2	no	—
179	C₂×C₄⋊Q₈	—	19	1	no	—
180	C₂×#127	C₂×D₁₉	1	1	yes	characteristic
181	#492	—	19	4	no	—
182	#492	—	38	2	no	—
183	C₂×#107	—	38	2	no	—
184	C₂×#102	—	2	2	no	—
185	C₂×#102	—	38	2	no	—
186	C₂×C₄×C₈	—	19	1	no	—
187	C₂×C₄×C₈	—	19	1	no	—
188	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
189	#384	—	19	1	no	—
190	#384	C₂×D₁₉	1	1	yes	characteristic
191	#340	—	38	2	no	—
192	#340	—	38	2	no	—
193	#322	—	38	2	no	—
194	C₂×C₁₉	—	16	2	no	—
195	C₂×C₁₉	#1714	1	1	yes	characteristic
196	C₂×C₁₉	#1714	1	1	yes	characteristic
197	C₂×C₁₉	#1721	1	1	yes	characteristic
198	C₂×C₁₉	#1722	1	1	yes	characteristic
199	C₂×C₁₉	#1957	1	1	yes	characteristic
200	C₂×C₁₉	#1973	1	1	yes	characteristic
201	C₂×C₁₉	#1973	1	1	yes	characteristic
202	C₂²×Q₈	—	38	1	no	—
203	C₂²×D₄	—	2	1	no	—
204	C₂²×D₄	—	2	1	no	—
205	C₂²×D₄	—	2	1	no	—
206	C₂×SD₁₆	—	38	8	no	—
207	C₂×SD₁₆	—	76	4	no	—
208	C₂×SD₁₆	—	76	4	no	—
209	C₂²×C₈	—	19	2	no	—
210	C₂²×C₈	—	19	2	no	—
211	C₂²×C₈	C₂²⋊D₁₉	1	2	yes	—
212	C₄⋊Q₈	—	19	2	no	—
213	C₄⋊Q₈	—	38	1	no	—
214	C₄⋊Q₈	—	38	1	no	—
215	C₄⋊Q₈	—	38	1	no	—
216	#127	—	2	1	no	—
217	#127	—	2	1	no	—
218	#127	—	2	1	no	—
219	#127	C₄×D₁₉	1	2	yes	—
220	C₂×C₄⋊C₄	—	2	1	no	—
221	C₂×C₄⋊C₄	—	2	1	no	—
222	C₂×C₄⋊C₄	—	2	1	no	—
223	C₂×C₄⋊C₄	—	38	1	no	—
224	C₂×C₄⋊C₄	—	38	1	no	—
225	C₂×C₄⋊C₄	—	38	1	no	—
226	C₂×C₄⋊C₄	—	38	1	no	—
227	C₂×C₄⋊C₄	—	38	1	no	—
228	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
229	#107	—	76	4	no	—
230	#102	—	4	4	no	—
231	#102	—	76	4	no	—
232	C₄×C₈	—	2	1	no	—
233	C₄×C₈	—	2	1	no	—
234	C₄×C₈	—	19	2	no	—
235	C₄×C₈	—	19	2	no	—
236	C₄×C₈	—	38	1	no	—
237	C₄×C₈	—	38	1	no	—
238	C₁₉	#9842	1	1	yes	characteristic, complemented, Sylow
239	C₂⁴	—	4	1	no	—
240	C₂×Q₈	—	38	2	no	—
241	C₂×Q₈	—	76	1	no	—
242	C₂×Q₈	—	76	2	no	—
243	C₂×Q₈	—	76	2	no	—
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	—	2	2	no	—
246	C₂×D₄	—	2	2	no	—
247	C₂×D₄	—	4	2	no	—
248	C₂×D₄	—	4	1	no	—
249	C₂×D₄	—	4	2	no	—
250	C₂×D₄	—	4	1	no	—
251	C₂×D₄	—	4	1	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	—	4	1	no	—
257	C₂²×C₄	—	76	1	no	—
258	C₂²×C₄	—	76	1	no	—
259	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
260	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
261	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
262	SD₁₆	—	76	16	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	19	4	no	—
267	C₂×C₈	—	19	4	no	—
268	C₂×C₈	—	19	4	no	—
269	C₂×C₈	—	38	2	no	—
270	C₂×C₈	—	38	2	no	—
271	C₂×C₈	—	38	2	no	—
272	C₄⋊C₄	—	4	2	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₄⋊C₄	—	4	2	no	—
275	C₄⋊C₄	—	38	2	no	—
276	C₄⋊C₄	—	38	2	no	—
277	C₄⋊C₄	—	76	1	no	—
278	C₄⋊C₄	—	76	1	no	—
279	C₄⋊C₄	—	76	2	no	—
280	C₄⋊C₄	—	76	2	no	—
281	C₄⋊C₄	—	76	2	no	—
282	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
283	C₄²	#64541	1	1	yes	characteristic
284	C₄²	D₄×D₁₉	1	1	yes	characteristic
285	C₄²	#64543	1	1	yes	characteristic
286	C₂³	—	4	2	no	—
287	C₂³	—	8	1	no	—
288	C₂³	—	8	1	no	—
289	C₂³	—	8	1	no	—
290	C₂³	—	8	1	no	—
291	C₂³	—	8	1	no	—
292	C₂³	—	8	1	no	—
293	C₂³	#10597386	1	1	yes	characteristic
294	Q₈	—	76	4	no	—
295	Q₈	—	152	2	no	—
296	D₄	—	4	4	no	—
297	D₄	—	4	4	no	—
298	D₄	—	4	4	no	—
299	D₄	—	8	2	no	—
300	D₄	—	8	2	no	—
301	D₄	—	8	2	no	—
302	C₂×C₄	—	8	1	no	—
303	C₂×C₄	—	8	1	no	—
304	C₂×C₄	—	8	1	no	—
305	C₂×C₄	—	76	2	no	—
306	C₂×C₄	—	152	1	no	—
307	C₂×C₄	—	152	1	no	—
308	C₂×C₄	—	152	1	no	—
309	C₂×C₄	—	152	1	no	—
310	C₂×C₄	—	152	1	no	—
311	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
312	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
313	C₂×C₄	#10597330	1	1	yes	characteristic
314	C₂×C₄	#10597331	1	1	yes	characteristic
315	C₂×C₄	D₈×D₁₉	1	2	yes	—
316	C₂×C₄	#10597357	1	2	yes	—
317	C₂×C₄	SD₁₆×D₁₉	1	2	yes	—
318	C₂×C₄	#10597361	1	2	yes	—
319	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
320	C₂×C₄	#10597378	1	2	yes	—
321	C₂×C₄	#10597385	1	1	yes	characteristic
322	C₂×C₄	#10597397	1	2	yes	—
323	C₈	—	4	2	no	—
324	C₈	—	19	8	no	—
325	C₈	—	76	2	no	—
326	C₂²	—	8	2	no	—
327	C₂²	—	8	2	no	—
328	C₂²	—	8	2	no	—
329	C₂²	—	16	1	no	—
330	C₂²	—	16	1	no	—
331	C₂²	—	16	1	no	—
332	C₂²	—	16	1	no	—
333	C₂²	#11926444	1	1	yes	characteristic
334	C₂²	#11926915	1	1	yes	characteristic
335	C₂²	#11926919	1	1	yes	characteristic
336	C₂²	#11926931	1	1	yes	characteristic
337	C₂²	#11926932	1	1	yes	characteristic
338	C₂²	#11926935	1	1	yes	characteristic
339	C₂²	#11926939	1	1	yes	characteristic
340	C₄	—	16	1	no	—
341	C₄	—	304	1	no	—
342	C₄	—	152	2	no	—
343	C₄	#102×D₁₉	1	2	yes	—
344	C₄	#11926641	1	2	yes	—
345	C₄	#11926644	1	2	yes	—
346	C₄	#11926788	1	2	yes	—
347	C₄	#11926789	1	2	yes	—
348	C₄	#11926889	1	2	yes	—
349	C₂	—	16	2	no	—
350	C₂	#14563255	1	1	yes	characteristic
351	C₂	#14563256	1	1	yes	characteristic
352	C₂	#14563300	1	1	yes	characteristic
353	C₂	#14563308	1	1	yes	characteristic
354	C₂	#14564439	1	1	yes	characteristic
355	C₂	#14564515	1	1	yes	characteristic
356	C₂	#14564517	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10597386
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1957	C₂
socle	C₂³×C₁₉	#127
prime radical	C₂×C₁₉×#127	C₂²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1957	D₁₉
hypocenter	C₁₉	#9842
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1957

Complements

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

N	H ≅ G/N	type
2. #14564596	C₂	semidirect
3. #14564595	C₂	semidirect
5. #14563794	C₂	semidirect
6. C₂×#11926266	C₂	semidirect
11. #11926916	C₄	semidirect
147. C₁₉⋊C₈	#102	semidirect
238. C₁₉	#9842	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926916	2
#14564596	2
#14564595	2
#14564442	2
#14563794	2
C₂×#11926266	2
C₁₉×#1957	2
#9842	19