Subgroups — #18075474

Order 4864.

454 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926621	C₂	1	1	yes	maximal, characteristic
2	#14563624	C₂	1	1	yes	maximal, characteristic
3	#14563624	C₂	1	1	yes	maximal, characteristic
4	#14562393	C₂	1	1	yes	maximal, characteristic, complemented
5	#14562390	C₂	1	1	yes	maximal, characteristic, complemented
6	#14561510	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1292	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#64515	C₄	1	2	yes	—
9	C₂×C₄×D₇₆	C₂²	1	1	yes	characteristic
10	#11926621	—	2	1	no	—
11	#11926621	—	2	1	no	—
12	#11926621	—	2	1	no	—
13	#11926621	—	2	1	no	—
14	#11926568	C₂²	1	1	yes	characteristic
15	#11926568	C₂²	1	1	yes	characteristic
16	C₂×#10597280	—	2	1	no	—
17	C₂×#10597280	—	2	1	no	—
18	C₂×(C₂×D₁₉)⋊C₈	C₄	1	2	yes	—
19	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
20	#11926312	—	2	1	no	—
21	#11926312	—	2	1	no	—
22	#11926296	—	2	1	no	—
23	#11926296	—	2	1	no	—
24	#11926294	—	2	1	no	—
25	#11926294	—	2	1	no	—
26	C₁₉×#423	C₂²	1	1	yes	characteristic
27	C₁₉×#423	C₂²	1	1	yes	characteristic
28	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
29	C₂²×D₇₆	D₄	1	1	yes	characteristic
30	C₂²×C₄×D₁₉	C₂×C₄	1	2	yes	—
31	C₂×M₁₆×C₁₉	—	2	1	no	—
32	C₂×M₁₆×C₁₉	—	2	1	no	—
33	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
35	C₂×(C₂×D₁₉)⋊C₄	Q₈	1	2	yes	—
36	C₂×C₄⋊Dic₁₉	D₄	1	1	yes	characteristic
37	C₂×#64520	—	2	1	no	—
38	C₂×#64520	—	2	1	no	—
39	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
41	C₂×#64515	—	2	2	no	—
42	C₂×#64515	—	2	2	no	—
43	C₂×#64515	—	4	4	no	—
44	C₄×D₇₆	—	2	1	no	—
45	C₄×D₇₆	—	2	1	no	—
46	C₄×D₇₆	—	2	1	no	—
47	C₄×D₇₆	—	2	1	no	—
48	C₁₉×C₄⋊C₈	—	2	1	no	—
49	C₁₉×C₄⋊C₈	—	2	1	no	—
50	C₁₉×C₄⋊C₈	—	2	1	no	—
51	C₁₉×C₄⋊C₈	—	2	1	no	—
52	C₁₉×C₄⋊C₈	—	2	1	no	—
53	C₁₉×C₄⋊C₈	—	2	1	no	—
54	#10597280	—	4	2	no	—
55	#10597280	—	4	2	no	—
56	(C₂×D₁₉)⋊C₈	—	2	2	no	—
57	(C₂×D₁₉)⋊C₈	—	2	2	no	—
58	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
59	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
60	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
61	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
62	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
63	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
64	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
65	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
66	C₂³×D₁₉	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
70	C₂²×Dic₁₉	C₄⋊C₄	1	2	yes	—
71	C₂×D₇₆	—	2	1	no	—
72	C₂×D₇₆	—	2	1	no	—
73	C₂×D₇₆	—	2	1	no	—
74	C₂×D₇₆	—	2	1	no	—
75	C₂×D₇₆	—	2	1	no	—
76	C₂×D₇₆	—	2	1	no	—
77	C₂×C₄×D₁₉	—	2	2	no	—
78	C₂×C₄×D₁₉	—	2	2	no	—
79	C₂×C₄×D₁₉	—	4	2	no	—
80	C₂×C₄×D₁₉	—	4	2	no	—
81	M₁₆×C₁₉	—	4	1	no	—
82	M₁₆×C₁₉	—	4	1	no	—
83	M₁₆×C₁₉	—	4	1	no	—
84	M₁₆×C₁₉	—	4	1	no	—
85	C₂×C₈×C₁₉	—	2	2	no	—
86	C₂×C₈×C₁₉	—	4	2	no	—
87	C₂×C₈×C₁₉	—	4	1	no	—
88	C₂×C₈×C₁₉	—	4	1	no	—
89	C₄²×C₁₉	C₂×D₄	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	(C₂×D₁₉)⋊C₄	—	2	2	no	—
94	(C₂×D₁₉)⋊C₄	—	2	2	no	—
95	C₄⋊Dic₁₉	—	2	1	no	—
96	C₄⋊Dic₁₉	—	2	1	no	—
97	#64520	—	4	1	no	—
98	#64520	—	4	1	no	—
99	#64520	—	4	1	no	—
100	#64520	—	4	1	no	—
101	C₂×C₁₉⋊C₈	—	2	2	no	—
102	C₂×C₁₉⋊C₈	—	4	1	no	—
103	C₂×C₁₉⋊C₈	—	4	1	no	—
104	C₂×C₁₉⋊C₈	—	4	1	no	—
105	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
109	#64515	—	4	4	no	—
110	#64515	—	4	4	no	—
111	#8618	—	19	1	no	maximal, Sylow
112	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂²×D₁₉	—	2	2	no	—
114	C₂²×D₁₉	—	2	2	no	—
115	C₂²×D₁₉	—	2	2	no	—
116	C₂²×D₁₉	—	4	2	no	—
117	C₂²×D₁₉	—	4	2	no	—
118	C₂×C₄×C₁₉	—	2	1	no	—
119	C₂×C₄×C₁₉	—	2	1	no	—
120	C₂×C₄×C₁₉	—	4	1	no	—
121	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
123	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
124	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
125	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
126	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
127	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
128	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
129	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
130	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
131	C₂×Dic₁₉	—	2	2	no	—
132	C₂×Dic₁₉	—	4	2	no	—
133	D₇₆	—	4	1	no	—
134	D₇₆	—	4	1	no	—
135	D₇₆	—	4	1	no	—
136	D₇₆	—	4	1	no	—
137	C₄×D₁₉	—	4	4	no	—
138	C₄×D₁₉	—	4	2	no	—
139	C₄×D₁₉	—	4	2	no	—
140	C₈×C₁₉	—	4	2	no	—
141	C₈×C₁₉	—	8	1	no	—
142	C₈×C₁₉	—	8	1	no	—
143	C₁₉⋊C₈	—	2	1	no	—
144	C₁₉⋊C₈	—	2	1	no	—
145	C₁₉⋊C₈	—	2	1	no	—
146	C₁₉⋊C₈	—	8	1	no	—
147	C₁₉⋊C₈	—	8	1	no	—
148	C₁₉⋊C₈	#100	1	2	yes	complemented
149	C₂×#436	—	19	1	no	—
150	#1800	—	19	1	no	—
151	#1800	—	19	1	no	—
152	#1567	—	19	1	no	—
153	#1566	—	19	1	no	—
154	#1292	—	19	1	no	—
155	#1292	D₁₉	1	1	yes	characteristic, complemented
156	C₂²×C₁₉	—	2	1	no	—
157	C₂²×C₁₉	—	2	1	no	—
158	C₂²×C₁₉	#380	1	1	yes	characteristic
159	C₂²×C₁₉	#428	1	1	yes	characteristic
160	C₂²×C₁₉	#428	1	1	yes	characteristic
161	C₂×D₁₉	—	4	2	no	—
162	C₂×D₁₉	—	4	2	no	—
163	C₂×D₁₉	—	4	2	no	—
164	C₂×D₁₉	—	8	2	no	—
165	C₂×D₁₉	—	8	2	no	—
166	C₄×C₁₉	—	2	1	no	—
167	C₄×C₁₉	—	2	1	no	—
168	C₄×C₁₉	—	2	1	no	—
169	C₄×C₁₉	C₂×#100	1	1	yes	characteristic
170	C₄×C₁₉	#413	1	1	yes	characteristic
171	C₄×C₁₉	#468	1	1	yes	characteristic
172	C₄×C₁₉	#468	1	1	yes	characteristic
173	C₄×C₁₉	#469	1	1	yes	characteristic
174	C₄×C₁₉	#470	1	1	yes	characteristic
175	Dic₁₉	—	4	2	no	—
176	C₂²×M₁₆	—	19	2	no	—
177	C₂×C₄×D₄	—	19	1	no	—
178	#436	—	38	1	no	—
179	#436	—	38	1	no	—
180	#436	—	38	1	no	—
181	#436	—	38	1	no	—
182	#423	—	19	1	no	—
183	#423	—	19	1	no	—
184	#423	C₂×D₁₉	1	1	yes	characteristic
185	#423	C₂×D₁₉	1	1	yes	characteristic
186	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
187	C₂×#100	—	38	1	no	—
188	C₂×#100	—	38	1	no	—
189	C₂×C₂²⋊C₈	—	19	2	no	—
190	C₂×C₄×C₈	—	19	1	no	—
191	#343	—	38	1	no	—
192	#343	—	38	1	no	—
193	#335	—	38	1	no	—
194	#335	—	38	1	no	—
195	#334	—	38	1	no	—
196	#334	—	38	1	no	—
197	C₂×C₁₉	—	2	1	no	—
198	C₂×C₁₉	—	2	1	no	—
199	C₂×C₁₉	#1800	1	1	yes	characteristic
200	C₂×C₁₉	#1809	1	1	yes	characteristic
201	C₂×C₁₉	#1813	1	1	yes	characteristic
202	D₁₉	—	8	2	no	—
203	C₂²×D₄	—	19	1	no	—
204	C₂³×C₄	—	19	2	no	—
205	C₂×M₁₆	—	2	1	no	—
206	C₂×M₁₆	—	2	1	no	—
207	C₂×M₁₆	—	38	2	no	—
208	C₂×M₁₆	—	38	2	no	—
209	C₂×M₁₆	—	38	1	no	—
210	C₂×M₁₆	—	38	1	no	—
211	C₂×M₁₆	—	76	4	no	—
212	C₂²×C₈	—	19	1	no	—
213	C₂²×C₈	—	19	1	no	—
214	C₂²×C₈	C₄×D₁₉	1	2	yes	—
215	C₄×D₄	—	38	1	no	—
216	C₄×D₄	—	38	1	no	—
217	C₄×D₄	—	38	1	no	—
218	C₄×D₄	—	38	1	no	—
219	C₂×C₄⋊C₄	—	19	1	no	—
220	C₂×C₂²⋊C₄	—	19	2	no	—
221	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
222	C₄⋊C₈	—	2	1	no	—
223	C₄⋊C₈	—	2	1	no	—
224	C₄⋊C₈	—	2	1	no	—
225	C₄⋊C₈	—	2	1	no	—
226	C₄⋊C₈	—	2	1	no	—
227	C₄⋊C₈	—	2	1	no	—
228	C₄⋊C₈	—	38	1	no	—
229	C₄⋊C₈	—	38	1	no	—
230	C₄⋊C₈	—	38	1	no	—
231	C₄⋊C₈	—	38	1	no	—
232	#100	—	76	2	no	—
233	#100	—	76	2	no	—
234	C₂²⋊C₈	—	38	2	no	—
235	C₂²⋊C₈	—	38	2	no	—
236	C₄×C₈	—	19	1	no	—
237	C₄×C₈	—	19	1	no	—
238	C₄×C₈	—	19	1	no	—
239	C₄×C₈	—	19	1	no	—
240	C₁₉	#8618	1	1	yes	characteristic, complemented, Sylow
241	C₂⁴	—	19	2	no	—
242	C₂×D₄	—	38	1	no	—
243	C₂×D₄	—	38	1	no	—
244	C₂×D₄	—	38	1	no	—
245	C₂×D₄	—	38	1	no	—
246	C₂×D₄	—	38	1	no	—
247	C₂×D₄	—	38	1	no	—
248	C₂²×C₄	—	19	2	no	—
249	C₂²×C₄	—	38	2	no	—
250	C₂²×C₄	—	38	2	no	—
251	C₂²×C₄	—	76	2	no	—
252	C₂²×C₄	—	76	2	no	—
253	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
254	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
255	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
256	M₁₆	—	4	1	no	—
257	M₁₆	—	4	1	no	—
258	M₁₆	—	4	1	no	—
259	M₁₆	—	4	1	no	—
260	M₁₆	—	76	4	no	—
261	M₁₆	—	76	4	no	—
262	M₁₆	—	76	1	no	—
263	M₁₆	—	76	1	no	—
264	M₁₆	—	76	1	no	—
265	M₁₆	—	76	1	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	4	2	no	—
268	C₂×C₈	—	4	1	no	—
269	C₂×C₈	—	4	1	no	—
270	C₂×C₈	—	19	1	no	—
271	C₂×C₈	—	19	1	no	—
272	C₂×C₈	—	19	1	no	—
273	C₂×C₈	—	19	1	no	—
274	C₂×C₈	—	38	2	no	—
275	C₂×C₈	—	76	1	no	—
276	C₂×C₈	—	76	1	no	—
277	C₂×C₈	—	76	1	no	—
278	C₄⋊C₄	—	38	1	no	—
279	C₄⋊C₄	—	38	1	no	—
280	C₂²⋊C₄	—	38	2	no	—
281	C₂²⋊C₄	—	38	2	no	—
282	C₄²	#64541	1	1	yes	characteristic
283	C₄²	#64541	1	1	yes	characteristic
284	C₄²	D₄×D₁₉	1	1	yes	characteristic
285	C₄²	D₄×D₁₉	1	1	yes	characteristic
286	C₂³	—	38	2	no	—
287	C₂³	—	38	2	no	—
288	C₂³	—	38	2	no	—
289	C₂³	—	76	2	no	—
290	C₂³	—	76	2	no	—
291	C₂³	C₄⋊C₄×D₁₉	1	1	yes	characteristic
292	D₄	—	76	1	no	—
293	D₄	—	76	1	no	—
294	D₄	—	76	1	no	—
295	D₄	—	76	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	38	2	no	—
300	C₂×C₄	—	76	4	no	—
301	C₂×C₄	—	76	2	no	—
302	C₂×C₄	—	76	2	no	—
303	C₂×C₄	—	76	2	no	—
304	C₂×C₄	#10597318	1	1	yes	characteristic
305	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
306	C₂×C₄	#10597330	1	1	yes	characteristic
307	C₂×C₄	#10597330	1	1	yes	characteristic
308	C₂×C₄	#10597342	1	1	yes	characteristic
309	C₂×C₄	#10597342	1	1	yes	characteristic
310	C₂×C₄	#10597357	1	1	yes	characteristic
311	C₂×C₄	#10597360	1	1	yes	characteristic
312	C₂×C₄	#10597361	1	1	yes	characteristic
313	C₂×C₄	#10597364	1	1	yes	characteristic
314	C₈	—	4	2	no	—
315	C₈	—	8	1	no	—
316	C₈	—	8	1	no	—
317	C₈	—	19	2	no	—
318	C₈	—	38	1	no	—
319	C₈	—	38	1	no	—
320	C₈	—	38	1	no	—
321	C₈	—	152	1	no	—
322	C₈	—	152	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	76	2	no	—
326	C₂²	—	76	2	no	—
327	C₂²	—	76	2	no	—
328	C₂²	—	152	2	no	—
329	C₂²	—	152	2	no	—
330	C₂²	#11926403	1	1	yes	characteristic
331	C₂²	#11926583	1	1	yes	characteristic
332	C₂²	#11926585	1	1	yes	characteristic
333	C₄	—	2	1	no	—
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	76	2	no	—
337	C₄	#100×D₁₉	1	1	yes	characteristic
338	C₄	#11926527	1	1	yes	characteristic
339	C₄	#11926789	1	1	yes	characteristic
340	C₄	#11926793	1	1	yes	characteristic
341	C₄	#11926796	1	1	yes	characteristic
342	C₄	#11926801	1	1	yes	characteristic
343	C₂	—	2	1	no	—
344	C₂	—	2	1	no	—
345	C₂	—	152	2	no	—
346	C₂	#14563626	1	1	yes	characteristic
347	C₂	#14563667	1	1	yes	characteristic
348	C₂	#14563684	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11926403
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1292	C₂
socle	C₂²×C₁₉	#380
prime radical	C₂²×D₇₆	D₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1292	D₁₉
hypocenter	C₁₉	#8618
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1292

Complements

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

N	H ≅ G/N	type
4. #14562393	C₂	semidirect
5. #14562390	C₂	semidirect
6. #14561510	C₂	semidirect
7. C₁₉×#1292	C₂	semidirect
148. C₁₉⋊C₈	#100	semidirect
155. #1292	D₁₉	semidirect
240. C₁₉	#8618	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926621	2
#14563624	2
#14562393	2
#14562390	2
#14561510	2
C₁₉×#1292	2
#8618	19