Subgroups — #18085334

Order 4864.

618 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926947	C₂	1	1	yes	maximal, characteristic
2	C₂×#11926923	C₂	1	1	yes	maximal, characteristic
3	#14564597	C₂	1	1	yes	maximal, characteristic
4	#14564596	C₂	1	1	yes	maximal, characteristic
5	#14564438	C₂	1	1	yes	maximal, characteristic
6	C₁₉⋊#1273	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₉×#500	C₂	1	1	yes	maximal, characteristic
8	C₂²×#64528	—	2	2	no	—
9	C₂×#10597315	C₂²	1	1	yes	characteristic
10	#11926947	—	2	2	no	—
11	#11926947	—	2	2	no	—
12	#11926923	—	2	2	no	—
13	#11926923	—	2	2	no	—
14	C₂×#10597264	—	2	2	no	—
15	C₂×#10597292	—	2	2	no	—
16	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
18	#11926414	C₂²	1	1	yes	characteristic
19	#11926307	—	2	2	no	—
20	#11926306	—	2	2	no	—
21	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
22	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
23	C₁₉×#500	—	2	2	no	—
24	C₁₉×#500	—	2	2	no	—
25	C₂×C₁₉×#107	—	2	2	no	—
26	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
27	C₂²×Q₈×C₁₉	—	2	1	no	—
28	C₂²×Q₈×C₁₉	—	2	1	no	—
29	C₂²×Dic₃₈	—	2	1	no	—
30	C₂²×C₈×C₁₉	D₄	1	2	yes	—
31	C₁₉×C₄⋊Q₈	—	2	1	no	—
32	C₁₉×C₄⋊Q₈	—	2	1	no	—
33	C₁₉×C₄⋊Q₈	—	2	1	no	—
34	C₁₉×C₄⋊Q₈	—	2	1	no	—
35	C₁₉×C₄⋊Q₈	—	2	1	no	—
36	C₁₉×C₄⋊Q₈	—	2	1	no	—
37	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
38	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
39	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
40	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
41	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
42	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
43	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
44	C₂×#64528	—	2	8	no	—
45	C₂×#64528	—	4	4	no	—
46	C₂×#64528	—	4	4	no	—
47	C₂×C₄⋊Dic₁₉	—	2	1	no	—
48	C₂×C₄⋊Dic₁₉	—	2	1	no	—
49	C₂×C₄⋊Dic₁₉	—	2	1	no	—
50	C₂×C₄⋊Dic₁₉	—	2	1	no	—
51	C₂×C₄⋊Dic₁₉	—	2	1	no	—
52	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
53	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
54	#10597315	—	2	1	no	—
55	#10597315	—	2	1	no	—
56	#10597315	—	2	1	no	—
57	#10597315	D₄	1	2	yes	—
58	C₁₉×#107	—	2	8	no	—
59	C₄×C₈×C₁₉	—	2	1	no	—
60	C₄×C₈×C₁₉	—	2	1	no	—
61	#10597292	—	4	4	no	—
62	#10597264	—	2	8	no	—
63	C₄×C₁₉⋊C₈	—	2	1	no	—
64	C₄×C₁₉⋊C₈	—	2	1	no	—
65	C₄×C₁₉⋊C₈	—	2	1	no	—
66	C₄×C₁₉⋊C₈	—	2	1	no	—
67	C₂×Q₈×C₁₉	—	2	2	no	—
68	C₂×Q₈×C₁₉	—	2	2	no	—
69	C₂×Q₈×C₁₉	—	4	1	no	—
70	C₂×Q₈×C₁₉	—	4	1	no	—
71	C₂×Q₈×C₁₉	—	4	2	no	—
72	C₂×Q₈×C₁₉	—	4	2	no	—
73	C₂×Q₈×C₁₉	—	4	2	no	—
74	C₂×Q₈×C₁₉	—	4	2	no	—
75	C₂²×C₄×C₁₉	—	4	1	no	—
76	C₂²×C₄×C₁₉	—	4	1	no	—
77	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
80	C₂²×Dic₁₉	—	4	1	no	—
81	C₂²×Dic₁₉	—	4	1	no	—
82	C₂×Dic₃₈	—	2	2	no	—
83	C₂×Dic₃₈	—	4	2	no	—
84	C₂×Dic₃₈	—	4	2	no	—
85	C₂×Dic₃₈	—	4	1	no	—
86	C₂×C₈×C₁₉	—	2	2	no	—
87	C₂×C₈×C₁₉	—	2	2	no	—
88	C₂×C₈×C₁₉	Q₁₆	1	4	yes	complemented
89	C₄⋊C₄×C₁₉	—	2	2	no	—
90	C₄⋊C₄×C₁₉	—	2	2	no	—
91	C₄⋊C₄×C₁₉	—	2	2	no	—
92	C₄⋊C₄×C₁₉	—	2	2	no	—
93	C₄⋊C₄×C₁₉	—	4	1	no	—
94	C₄⋊C₄×C₁₉	—	4	1	no	—
95	C₄⋊C₄×C₁₉	—	4	1	no	—
96	C₄⋊C₄×C₁₉	—	4	1	no	—
97	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₁₉	#41	1	1	yes	characteristic
101	#64528	—	4	16	no	—
102	C₄⋊Dic₁₉	—	2	2	no	—
103	C₄⋊Dic₁₉	—	2	2	no	—
104	C₄⋊Dic₁₉	—	4	1	no	—
105	C₄⋊Dic₁₉	—	4	1	no	—
106	C₄⋊Dic₁₉	—	4	2	no	—
107	C₄⋊Dic₁₉	—	4	2	no	—
108	C₄⋊Dic₁₉	—	4	2	no	—
109	C₂×C₁₉⋊C₈	—	2	2	no	—
110	C₂×C₁₉⋊C₈	—	2	2	no	—
111	C₂×C₁₉⋊C₈	—	2	2	no	—
112	C₂×C₁₉⋊C₈	—	2	2	no	—
113	C₂×C₁₉⋊C₈	D₈	1	4	yes	—
114	C₂×C₁₉⋊C₈	Q₁₆	1	4	yes	complemented
115	#10309	—	19	1	no	maximal, Sylow
116	C₂³×C₁₉	#127	1	1	yes	characteristic
117	Q₈×C₁₉	—	4	4	no	—
118	Q₈×C₁₉	—	4	4	no	—
119	Q₈×C₁₉	—	8	2	no	—
120	Q₈×C₁₉	—	8	2	no	—
121	C₂×C₄×C₁₉	—	4	2	no	—
122	C₂×C₄×C₁₉	—	4	2	no	—
123	C₂×C₄×C₁₉	—	8	1	no	—
124	C₂×C₄×C₁₉	—	8	1	no	—
125	C₂×C₄×C₁₉	—	8	1	no	—
126	C₂×C₄×C₁₉	—	8	1	no	—
127	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
128	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
129	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
130	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
131	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
132	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
133	C₂×C₄×C₁₉	C₂×D₈	1	2	yes	—
134	C₂×C₄×C₁₉	C₂×Q₁₆	1	2	yes	—
135	C₂×C₄×C₁₉	C₂×Q₁₆	1	2	yes	—
136	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
137	C₂×C₄×C₁₉	#137	1	2	yes	—
138	C₂×C₄×C₁₉	#137	1	2	yes	—
139	C₂×Dic₁₉	—	4	2	no	—
140	C₂×Dic₁₉	—	8	1	no	—
141	C₂×Dic₁₉	—	8	1	no	—
142	C₂×Dic₁₉	—	8	1	no	—
143	C₂×Dic₁₉	—	8	1	no	—
144	C₂×Dic₁₉	—	8	1	no	—
145	Dic₃₈	—	4	4	no	—
146	Dic₃₈	—	8	2	no	—
147	C₈×C₁₉	—	2	4	no	—
148	C₁₉⋊C₈	—	2	4	no	—
149	C₁₉⋊C₈	—	2	4	no	—
150	C₂×#500	—	19	1	no	—
151	C₂×#500	D₁₉	1	1	yes	characteristic
152	C₂×#494	—	19	1	no	—
153	#1990	—	19	1	no	—
154	#1990	—	19	1	no	—
155	#1956	—	19	1	no	—
156	#1273	—	19	1	no	—
157	C₂²×C₁₉	#392	1	1	yes	characteristic
158	C₂²×C₁₉	#493	1	1	yes	characteristic
159	C₂²×C₁₉	#494	1	1	yes	characteristic
160	C₂²×C₁₉	#494	1	1	yes	characteristic
161	C₂²×C₁₉	#496	1	1	yes	characteristic
162	C₂²×C₁₉	#497	1	1	yes	characteristic
163	C₂²×C₁₉	#497	1	1	yes	characteristic
164	C₄×C₁₉	—	8	2	no	—
165	C₄×C₁₉	—	8	2	no	—
166	C₄×C₁₉	#447	1	2	yes	—
167	C₄×C₁₉	#451	1	2	yes	—
168	C₄×C₁₉	#451	1	2	yes	—
169	C₄×C₁₉	#459	1	2	yes	—
170	C₄×C₁₉	#462	1	2	yes	—
171	C₄×C₁₉	#462	1	2	yes	—
172	Dic₁₉	—	8	2	no	—
173	Dic₁₉	—	16	1	no	—
174	C₂²×Q₁₆	—	38	2	no	—
175	C₂×C₄⋊Q₈	—	19	1	no	—
176	C₂×C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
177	C₂×C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
178	#500	—	2	2	no	—
179	#500	—	2	2	no	—
180	#500	—	38	2	no	—
181	#500	—	38	2	no	—
182	#494	—	38	2	no	—
183	#494	—	38	2	no	—
184	C₂×#107	—	2	2	no	—
185	C₂×#107	—	38	2	no	—
186	C₂×#103	—	38	2	no	—
187	C₂×C₄×C₈	—	19	1	no	—
188	C₂×C₄×C₈	—	19	1	no	—
189	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
190	#384	—	19	1	no	—
191	#340	—	38	2	no	—
192	#340	—	38	2	no	—
193	C₂×C₁₉	#1713	1	1	yes	characteristic
194	C₂×C₁₉	#1722	1	1	yes	characteristic
195	C₂×C₁₉	#1723	1	1	yes	characteristic
196	C₂×C₁₉	#1723	1	1	yes	characteristic
197	C₂×C₁₉	#2011	1	1	yes	characteristic
198	C₂×C₁₉	#2016	1	1	yes	characteristic
199	C₂×C₁₉	#2016	1	1	yes	characteristic
200	C₂²×Q₈	—	2	1	no	—
201	C₂²×Q₈	—	2	1	no	—
202	C₂²×Q₈	—	38	1	no	—
203	C₂×Q₁₆	—	38	8	no	—
204	C₂×Q₁₆	—	76	4	no	—
205	C₂×Q₁₆	—	76	4	no	—
206	C₂²×C₈	—	19	2	no	—
207	C₂²×C₈	—	19	2	no	—
208	C₂²×C₈	C₂²⋊D₁₉	1	2	yes	—
209	C₄⋊Q₈	—	2	1	no	—
210	C₄⋊Q₈	—	2	1	no	—
211	C₄⋊Q₈	—	2	1	no	—
212	C₄⋊Q₈	—	2	1	no	—
213	C₄⋊Q₈	—	2	1	no	—
214	C₄⋊Q₈	—	2	1	no	—
215	C₄⋊Q₈	—	19	2	no	—
216	C₄⋊Q₈	—	38	1	no	—
217	C₄⋊Q₈	—	38	1	no	—
218	C₄⋊Q₈	—	38	1	no	—
219	C₄⋊Q₈	C₂²⋊D₁₉	1	2	yes	—
220	C₄⋊Q₈	C₂²⋊D₁₉	1	2	yes	—
221	C₂×C₄⋊C₄	—	2	1	no	—
222	C₂×C₄⋊C₄	—	2	1	no	—
223	C₂×C₄⋊C₄	—	2	1	no	—
224	C₂×C₄⋊C₄	—	2	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×C₄⋊C₄	—	38	1	no	—
229	C₂×C₄⋊C₄	—	38	1	no	—
230	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
231	#107	—	2	8	no	—
232	#107	—	38	8	no	—
233	#103	—	76	4	no	—
234	C₄×C₈	—	2	1	no	—
235	C₄×C₈	—	2	1	no	—
236	C₄×C₈	—	38	1	no	—
237	C₄×C₈	—	38	1	no	—
238	C₄×C₈	—	38	1	no	—
239	C₄×C₈	—	38	1	no	—
240	C₁₉	#10309	1	1	yes	characteristic, complemented, Sylow
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	2	2	no	—
243	C₂×Q₈	—	4	1	no	—
244	C₂×Q₈	—	4	1	no	—
245	C₂×Q₈	—	4	2	no	—
246	C₂×Q₈	—	4	2	no	—
247	C₂×Q₈	—	4	2	no	—
248	C₂×Q₈	—	4	2	no	—
249	C₂×Q₈	—	38	2	no	—
250	C₂×Q₈	—	76	1	no	—
251	C₂×Q₈	—	76	2	no	—
252	C₂×Q₈	—	76	2	no	—
253	C₂²×C₄	—	4	1	no	—
254	C₂²×C₄	—	4	1	no	—
255	C₂²×C₄	—	76	1	no	—
256	C₂²×C₄	—	76	1	no	—
257	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
258	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
259	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
260	Q₁₆	—	76	16	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	19	4	no	—
264	C₂×C₈	—	19	4	no	—
265	C₂×C₈	—	38	2	no	—
266	C₂×C₈	—	38	2	no	—
267	C₂×C₈	—	38	2	no	—
268	C₂×C₈	—	38	2	no	—
269	C₂×C₈	#64528	1	4	yes	complemented
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₄⋊C₄	—	38	2	no	—
279	C₄⋊C₄	—	38	2	no	—
280	C₄⋊C₄	—	76	1	no	—
281	C₄⋊C₄	—	76	1	no	—
282	C₄⋊C₄	—	76	2	no	—
283	C₄⋊C₄	—	76	2	no	—
284	C₄⋊C₄	—	76	2	no	—
285	C₄²	D₄×D₁₉	1	1	yes	characteristic
286	C₄²	#64543	1	1	yes	characteristic
287	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
288	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
289	C₂³	#10597386	1	1	yes	characteristic
290	Q₈	—	4	4	no	—
291	Q₈	—	4	4	no	—
292	Q₈	—	8	2	no	—
293	Q₈	—	8	2	no	—
294	Q₈	—	76	4	no	—
295	Q₈	—	152	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	8	1	no	—
299	C₂×C₄	—	8	1	no	—
300	C₂×C₄	—	8	1	no	—
301	C₂×C₄	—	8	1	no	—
302	C₂×C₄	—	76	2	no	—
303	C₂×C₄	—	152	1	no	—
304	C₂×C₄	—	152	1	no	—
305	C₂×C₄	—	152	1	no	—
306	C₂×C₄	—	152	1	no	—
307	C₂×C₄	—	152	1	no	—
308	C₂×C₄	D₈×D₁₉	1	2	yes	—
309	C₂×C₄	#10597357	1	2	yes	—
310	C₂×C₄	Q₁₆×D₁₉	1	2	yes	—
311	C₂×C₄	#10597364	1	2	yes	—
312	C₂×C₄	#10597383	1	1	yes	characteristic
313	C₂×C₄	#10597383	1	1	yes	characteristic
314	C₂×C₄	#10597384	1	1	yes	characteristic
315	C₂×C₄	#10597385	1	1	yes	characteristic
316	C₂×C₄	#10597385	1	1	yes	characteristic
317	C₂×C₄	#10597388	1	2	yes	—
318	C₂×C₄	C₂×#64528	1	2	yes	—
319	C₂×C₄	#10597399	1	1	yes	characteristic
320	C₈	—	2	4	no	—
321	C₈	—	38	4	no	—
322	C₈	—	38	4	no	—
323	C₂²	#11926452	1	1	yes	characteristic
324	C₂²	#11926919	1	1	yes	characteristic
325	C₂²	#11926922	1	1	yes	characteristic
326	C₂²	#11926923	1	1	yes	characteristic
327	C₂²	#11926932	1	1	yes	characteristic
328	C₂²	#11926936	1	1	yes	characteristic
329	C₂²	#11926937	1	1	yes	characteristic
330	C₄	—	8	2	no	—
331	C₄	—	8	2	no	—
332	C₄	—	304	1	no	—
333	C₄	—	152	2	no	—
334	C₄	#11926676	1	2	yes	—
335	C₄	#11926698	1	2	yes	—
336	C₄	#11926700	1	2	yes	—
337	C₄	#11926737	1	2	yes	—
338	C₄	#11926751	1	2	yes	—
339	C₄	#11926752	1	2	yes	—
340	C₂	#14563252	1	1	yes	characteristic
341	C₂	#14563305	1	1	yes	characteristic
342	C₂	#14563311	1	1	yes	characteristic
343	C₂	#14563313	1	1	yes	characteristic
344	C₂	#14564704	1	1	yes	characteristic
345	C₂	#14564733	1	1	yes	characteristic
346	C₂	#14564735	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10597386
commutator, cosocle	C₂×C₄²×C₁₉	C₂³
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₁₉×#500	C₂
socle, prime radical	C₂³×C₁₉	#127
hypercenter	C₂×#500	D₁₉
hypocenter	C₁₉	#10309
LCS	G ▷ C₂×C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
88. C₂×C₈×C₁₉	Q₁₆	semidirect
114. C₂×C₁₉⋊C₈	Q₁₆	semidirect
240. C₁₉	#10309	semidirect
269. C₂×C₈	#64528	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926947	2
C₂×#11926923	2
#14564597	2
#14564596	2
#14564438	2
C₁₉⋊#1273	2
C₂×C₁₉×#500	2
#10309	19