Subgroups — #18085386

Order 4864.

602 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926942	C₂	1	1	yes	maximal, characteristic
2	C₂×#11926923	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926896	C₂	1	1	yes	maximal, characteristic
4	#14564595	C₂	1	1	yes	maximal, characteristic
5	#14564595	C₂	1	1	yes	maximal, characteristic
6	C₁₉⋊#1273	C₂	1	1	yes	maximal, characteristic
7	C₁₉×#1990	C₂	1	1	yes	maximal, characteristic
8	C₂²×#64528	—	2	2	no	—
9	C₂×#10597315	C₂²	1	1	yes	characteristic
10	C₂×#10597315	C₂²	1	1	yes	characteristic
11	#11926942	—	2	2	no	—
12	#11926942	—	2	2	no	—
13	#11926923	—	2	2	no	—
14	#11926923	—	2	2	no	—
15	#11926896	—	2	2	no	—
16	#11926896	—	2	2	no	—
17	C₂×#10597274	—	2	2	no	—
18	C₂×#10597267	—	2	2	no	—
19	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
21	#11926306	—	2	2	no	—
22	#11926306	—	2	2	no	—
23	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
25	C₁₉×#384	C₂²	1	1	yes	characteristic
26	C₁₉×#340	—	2	2	no	—
27	C₂²×Q₈×C₁₉	—	2	1	no	—
28	C₂²×Dic₃₈	—	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	D₄	1	2	yes	—
35	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
36	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
37	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
38	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
39	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
40	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
41	C₂×#64528	—	2	8	no	—
42	C₂×#64528	—	4	4	no	—
43	C₂×#64528	—	4	4	no	—
44	C₂×C₄⋊Dic₁₉	—	2	1	no	—
45	C₂×C₄⋊Dic₁₉	—	2	1	no	—
46	C₂×C₄⋊Dic₁₉	—	2	1	no	—
47	C₂×C₄⋊Dic₁₉	—	2	1	no	—
48	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
49	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
50	#10597315	—	2	1	no	—
51	#10597315	—	2	1	no	—
52	#10597315	—	2	1	no	—
53	#10597315	—	2	1	no	—
54	#10597315	—	2	1	no	—
55	#10597315	—	2	1	no	—
56	#10597315	D₄	1	2	yes	—
57	#10597315	D₄	1	2	yes	—
58	C₄×C₈×C₁₉	—	2	1	no	—
59	C₄×C₈×C₁₉	—	2	1	no	—
60	#10597274	—	2	8	no	—
61	#10597267	—	2	8	no	—
62	C₄×C₁₉⋊C₈	—	2	1	no	—
63	C₄×C₁₉⋊C₈	—	2	1	no	—
64	C₄×C₁₉⋊C₈	—	2	1	no	—
65	C₄×C₁₉⋊C₈	—	2	1	no	—
66	C₂×Q₈×C₁₉	—	2	2	no	—
67	C₂×Q₈×C₁₉	—	4	1	no	—
68	C₂×Q₈×C₁₉	—	4	2	no	—
69	C₂×Q₈×C₁₉	—	4	2	no	—
70	C₂²×C₄×C₁₉	—	4	1	no	—
71	C₂²×C₄×C₁₉	—	4	1	no	—
72	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
75	C₂²×Dic₁₉	—	4	1	no	—
76	C₂²×Dic₁₉	—	4	1	no	—
77	C₂×Dic₃₈	—	2	2	no	—
78	C₂×Dic₃₈	—	2	2	no	—
79	C₂×Dic₃₈	—	4	2	no	—
80	C₂×Dic₃₈	—	4	2	no	—
81	C₂×Dic₃₈	—	4	2	no	—
82	C₂×Dic₃₈	—	4	2	no	—
83	C₂×Dic₃₈	—	4	1	no	—
84	C₂×Dic₃₈	—	4	1	no	—
85	C₂×C₈×C₁₉	—	2	2	no	—
86	C₂×C₈×C₁₉	—	2	2	no	—
87	C₂×C₈×C₁₉	Q₁₆	1	4	yes	complemented
88	C₄⋊C₄×C₁₉	—	2	2	no	—
89	C₄⋊C₄×C₁₉	—	2	2	no	—
90	C₄⋊C₄×C₁₉	—	4	1	no	—
91	C₄⋊C₄×C₁₉	—	4	1	no	—
92	C₄⋊C₄×C₁₉	—	4	2	no	—
93	C₄⋊C₄×C₁₉	—	4	2	no	—
94	C₄⋊C₄×C₁₉	—	4	2	no	—
95	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₁₉	#41	1	1	yes	characteristic
99	#64528	—	4	16	no	—
100	C₄⋊Dic₁₉	—	2	2	no	—
101	C₄⋊Dic₁₉	—	2	2	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	1	no	—
107	C₄⋊Dic₁₉	—	4	1	no	—
108	C₂×C₁₉⋊C₈	—	2	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	SD₁₆	1	4	yes	—
114	#10318	—	19	1	no	maximal, Sylow
115	C₂³×C₁₉	#127	1	1	yes	characteristic
116	Q₈×C₁₉	—	4	4	no	—
117	Q₈×C₁₉	—	8	2	no	—
118	C₂×C₄×C₁₉	—	4	2	no	—
119	C₂×C₄×C₁₉	—	8	1	no	—
120	C₂×C₄×C₁₉	—	8	1	no	—
121	C₂×C₄×C₁₉	—	8	1	no	—
122	C₂×C₄×C₁₉	—	8	1	no	—
123	C₂×C₄×C₁₉	—	8	1	no	—
124	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
125	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
126	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
127	C₂×C₄×C₁₉	#121	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	C₂×SD₁₆	1	2	yes	—
131	C₂×C₄×C₁₉	C₂×Q₁₆	1	2	yes	—
132	C₂×C₄×C₁₉	#135	1	2	yes	—
133	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
134	C₂×C₄×C₁₉	#137	1	2	yes	—
135	C₂×C₄×C₁₉	#137	1	2	yes	—
136	C₂×Dic₁₉	—	4	2	no	—
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	Dic₃₈	—	4	4	no	—
143	Dic₃₈	—	4	4	no	—
144	Dic₃₈	—	8	2	no	—
145	Dic₃₈	—	8	2	no	—
146	C₈×C₁₉	—	2	4	no	—
147	C₁₉⋊C₈	—	2	4	no	—
148	C₁₉⋊C₈	—	4	2	no	—
149	C₂×#498	—	19	1	no	—
150	C₂×#494	—	19	1	no	—
151	C₂×#487	—	19	1	no	—
152	#1990	—	19	1	no	—
153	#1990	—	19	1	no	—
154	#1990	D₁₉	1	1	yes	characteristic
155	#1273	—	19	1	no	—
156	C₂²×C₁₉	#392	1	1	yes	characteristic
157	C₂²×C₁₉	#492	1	1	yes	characteristic
158	C₂²×C₁₉	#494	1	1	yes	characteristic
159	C₂²×C₁₉	#495	1	1	yes	characteristic
160	C₂²×C₁₉	#496	1	1	yes	characteristic
161	C₂²×C₁₉	#497	1	1	yes	characteristic
162	C₂²×C₁₉	#497	1	1	yes	characteristic
163	C₄×C₁₉	—	8	2	no	—
164	C₄×C₁₉	—	16	1	no	—
165	C₄×C₁₉	#450	1	2	yes	—
166	C₄×C₁₉	#451	1	2	yes	—
167	C₄×C₁₉	#452	1	2	yes	—
168	C₄×C₁₉	#460	1	2	yes	—
169	C₄×C₁₉	#462	1	2	yes	—
170	C₄×C₁₉	#464	1	2	yes	—
171	Dic₁₉	—	8	2	no	—
172	Dic₁₉	—	8	2	no	—
173	C₂²×Q₁₆	—	38	2	no	—
174	C₂×C₄⋊Q₈	—	19	1	no	—
175	C₂×C₄⋊Q₈	—	19	1	no	—
176	C₂×C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
177	#498	—	38	2	no	—
178	#498	—	38	2	no	—
179	#494	—	38	2	no	—
180	#494	—	38	2	no	—
181	#487	—	38	2	no	—
182	#487	—	38	2	no	—
183	C₂×#106	—	38	2	no	—
184	C₂×#103	—	38	2	no	—
185	C₂×C₄×C₈	—	19	1	no	—
186	C₂×C₄×C₈	—	19	1	no	—
187	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
188	#384	C₂×D₁₉	1	1	yes	characteristic
189	#340	—	2	2	no	—
190	#340	—	38	2	no	—
191	#340	—	38	2	no	—
192	C₂×C₁₉	#1707	1	1	yes	characteristic
193	C₂×C₁₉	#1716	1	1	yes	characteristic
194	C₂×C₁₉	#1720	1	1	yes	characteristic
195	C₂×C₁₉	#1723	1	1	yes	characteristic
196	C₂×C₁₉	#2014	1	1	yes	characteristic
197	C₂×C₁₉	#2015	1	1	yes	characteristic
198	C₂×C₁₉	#2016	1	1	yes	characteristic
199	C₂²×Q₈	—	2	1	no	—
200	C₂²×Q₈	—	38	1	no	—
201	C₂²×Q₈	—	38	1	no	—
202	C₂×Q₁₆	—	38	8	no	—
203	C₂×Q₁₆	—	76	4	no	—
204	C₂×Q₁₆	—	76	4	no	—
205	C₂²×C₈	—	19	2	no	—
206	C₂²×C₈	—	19	2	no	—
207	C₂²×C₈	C₂²⋊D₁₉	1	2	yes	—
208	C₄⋊Q₈	—	2	1	no	—
209	C₄⋊Q₈	—	2	1	no	—
210	C₄⋊Q₈	—	2	1	no	—
211	C₄⋊Q₈	—	19	2	no	—
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	C₄⋊Q₈	—	38	1	no	—
217	C₄⋊Q₈	—	38	1	no	—
218	C₄⋊Q₈	—	38	1	no	—
219	C₄⋊Q₈	D₇₆	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	—	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×C₄²	C₂²×D₁₉	1	1	yes	characteristic
230	#106	—	38	8	no	—
231	#103	—	38	8	no	—
232	C₄×C₈	—	2	1	no	—
233	C₄×C₈	—	2	1	no	—
234	C₄×C₈	—	38	1	no	—
235	C₄×C₈	—	38	1	no	—
236	C₄×C₈	—	38	1	no	—
237	C₄×C₈	—	38	1	no	—
238	C₁₉	#10318	1	1	yes	characteristic, complemented, Sylow
239	C₂×Q₈	—	2	2	no	—
240	C₂×Q₈	—	4	1	no	—
241	C₂×Q₈	—	4	2	no	—
242	C₂×Q₈	—	4	2	no	—
243	C₂×Q₈	—	38	2	no	—
244	C₂×Q₈	—	38	2	no	—
245	C₂×Q₈	—	76	1	no	—
246	C₂×Q₈	—	76	1	no	—
247	C₂×Q₈	—	76	2	no	—
248	C₂×Q₈	—	76	2	no	—
249	C₂×Q₈	—	76	2	no	—
250	C₂×Q₈	—	76	2	no	—
251	C₂²×C₄	—	4	1	no	—
252	C₂²×C₄	—	4	1	no	—
253	C₂²×C₄	—	76	1	no	—
254	C₂²×C₄	—	76	1	no	—
255	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
256	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
257	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
258	Q₁₆	—	76	16	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	19	4	no	—
262	C₂×C₈	—	38	2	no	—
263	C₂×C₈	—	38	2	no	—
264	C₂×C₈	—	38	2	no	—
265	C₂×C₈	—	38	2	no	—
266	C₂×C₈	—	38	2	no	—
267	C₂×C₈	#64528	1	4	yes	complemented
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₄⋊C₄	—	4	1	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	—	38	2	no	—
278	C₄⋊C₄	—	38	2	no	—
279	C₄⋊C₄	—	76	1	no	—
280	C₄⋊C₄	—	76	1	no	—
281	C₄⋊C₄	—	76	1	no	—
282	C₄⋊C₄	—	76	1	no	—
283	C₄²	C₂×D₇₆	1	1	yes	characteristic
284	C₄²	#64541	1	1	yes	characteristic
285	C₄²	D₄×D₁₉	1	1	yes	characteristic
286	C₄²	D₄×D₁₉	1	1	yes	characteristic
287	C₂³	#10597386	1	1	yes	characteristic
288	Q₈	—	4	4	no	—
289	Q₈	—	8	2	no	—
290	Q₈	—	76	4	no	—
291	Q₈	—	76	4	no	—
292	Q₈	—	152	2	no	—
293	Q₈	—	152	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	8	1	no	—
296	C₂×C₄	—	8	1	no	—
297	C₂×C₄	—	8	1	no	—
298	C₂×C₄	—	8	1	no	—
299	C₂×C₄	—	8	1	no	—
300	C₂×C₄	—	76	2	no	—
301	C₂×C₄	—	76	2	no	—
302	C₂×C₄	—	152	1	no	—
303	C₂×C₄	—	152	1	no	—
304	C₂×C₄	—	152	1	no	—
305	C₂×C₄	—	152	1	no	—
306	C₂×C₄	#10597328	1	1	yes	characteristic
307	C₂×C₄	#10597328	1	1	yes	characteristic
308	C₂×C₄	#10597330	1	1	yes	characteristic
309	C₂×C₄	#10597330	1	1	yes	characteristic
310	C₂×C₄	#10597357	1	2	yes	—
311	C₂×C₄	SD₁₆×D₁₉	1	2	yes	—
312	C₂×C₄	#10597362	1	2	yes	—
313	C₂×C₄	#10597364	1	2	yes	—
314	C₂×C₄	#10597376	1	1	yes	characteristic
315	C₂×C₄	#10597383	1	1	yes	characteristic
316	C₂×C₄	C₂×#64528	1	2	yes	—
317	C₂×C₄	#10597397	1	2	yes	—
318	C₈	—	2	4	no	—
319	C₈	—	38	4	no	—
320	C₈	—	76	2	no	—
321	C₂²	#11926453	1	1	yes	characteristic
322	C₂²	#11926915	1	1	yes	characteristic
323	C₂²	#11926923	1	1	yes	characteristic
324	C₂²	#11926925	1	1	yes	characteristic
325	C₂²	#11926932	1	1	yes	characteristic
326	C₂²	#11926937	1	1	yes	characteristic
327	C₂²	#11926939	1	1	yes	characteristic
328	C₄	—	8	2	no	—
329	C₄	—	16	1	no	—
330	C₄	—	152	2	no	—
331	C₄	—	152	2	no	—
332	C₄	#11926696	1	2	yes	—
333	C₄	#11926699	1	2	yes	—
334	C₄	#11926709	1	2	yes	—
335	C₄	#11926743	1	2	yes	—
336	C₄	#11926754	1	2	yes	—
337	C₄	#11926769	1	2	yes	—
338	C₂	#14563230	1	1	yes	characteristic
339	C₂	#14563270	1	1	yes	characteristic
340	C₂	#14563297	1	1	yes	characteristic
341	C₂	#14563313	1	1	yes	characteristic
342	C₂	#14564724	1	1	yes	characteristic
343	C₂	#14564730	1	1	yes	characteristic
344	C₂	#14564734	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10597386
commutator, cosocle	C₂×C₄²×C₁₉	C₂³
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1990	C₂
socle, prime radical	C₂³×C₁₉	#127
hypercenter	#1990	D₁₉
hypocenter	C₁₉	#10318
LCS	G ▷ C₂×C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1990

Complements

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

N	H ≅ G/N	type
87. C₂×C₈×C₁₉	Q₁₆	semidirect
238. C₁₉	#10318	semidirect
267. 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×#11926942	2
C₂×#11926923	2
C₂×#11926896	2
#14564595	2
C₁₉⋊#1273	2
C₁₉×#1990	2
#10318	19