Subgroups — #18073375

Order 4864.

538 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926623	C₂	1	1	yes	maximal, characteristic
2	C₄×#10597266	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926271	C₂	1	1	yes	maximal, characteristic
4	C₂×#11926266	C₂	1	1	yes	maximal, characteristic, complemented
5	#14561476	C₂	1	1	yes	maximal, characteristic, complemented
6	#14561469	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1770	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#64515	—	2	1	no	—
9	C₂×C₄×D₇₆	C₂²	1	1	yes	characteristic
10	#11926623	—	2	2	no	—
11	#11926623	C₄	1	4	yes	complemented
12	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×#10597266	C₄	1	2	yes	—
14	C₂×(C₂×D₁₉)⋊C₈	—	2	1	no	—
15	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
17	#11926306	C₄	1	2	yes	—
18	#11926271	C₄	1	4	yes	complemented
19	#11926266	—	2	2	no	—
20	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₁₉×#97	—	2	1	no	—
22	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
23	C₄×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
24	C₁₉×#336	—	2	1	no	—
25	C₂²×D₇₆	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×D₁₉	—	2	1	no	—
27	C₂²×C₈×C₁₉	—	2	1	no	—
28	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₁₉	—	2	1	no	—
32	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
33	C₂×(C₂×D₁₉)⋊C₄	—	2	1	no	—
34	C₂×C₄⋊Dic₁₉	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₁₉⋊C₈	—	2	1	no	—
36	C₂²×C₁₉⋊C₈	—	2	1	no	—
37	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
38	C₂×#64515	—	2	4	no	—
39	C₂×#64515	—	4	2	no	—
40	C₂×#64515	—	4	2	no	—
41	C₄×D₇₆	—	2	1	no	—
42	C₄×D₇₆	—	2	1	no	—
43	C₄×D₇₆	C₂×C₄	1	2	yes	—
44	C₄×D₇₆	C₂×C₄	1	2	yes	—
45	C₁₉×C₄⋊C₈	—	2	1	no	—
46	C₁₉×C₄⋊C₈	C₂×C₄	1	2	yes	—
47	C₁₉×#97	—	4	2	no	—
48	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
49	C₄×C₈×C₁₉	D₄	1	2	yes	—
50	C₁₉×#95	—	2	1	no	—
51	(C₂×D₁₉)⋊C₈	—	2	4	no	—
52	#10597266	—	2	4	no	—
53	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
54	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	2	yes	—
55	C₄×C₁₉⋊C₈	—	2	1	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	C₂³×D₁₉	—	2	1	no	—
60	C₂²×C₄×C₁₉	—	2	2	no	—
61	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
64	C₂²×Dic₁₉	—	2	1	no	—
65	C₂×D₇₆	—	2	2	no	—
66	C₂×D₇₆	—	2	2	no	—
67	C₂×D₇₆	—	2	1	no	—
68	C₂×D₇₆	C₄²	1	2	yes	—
69	C₂×C₄×D₁₉	—	2	2	no	—
70	C₂×C₄×D₁₉	—	2	2	no	—
71	C₂×C₄×D₁₉	—	4	1	no	—
72	C₂×C₄×D₁₉	—	4	1	no	—
73	C₂×C₄×D₁₉	—	4	1	no	—
74	C₂×C₄×D₁₉	—	4	1	no	—
75	C₂×C₈×C₁₉	—	2	1	no	—
76	C₂×C₈×C₁₉	—	2	1	no	—
77	C₂×C₈×C₁₉	—	2	1	no	—
78	C₂×C₈×C₁₉	—	2	1	no	—
79	C₂×C₈×C₁₉	—	2	2	no	—
80	C₂×C₈×C₁₉	—	4	1	no	—
81	C₂×C₈×C₁₉	—	4	1	no	—
82	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
83	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
84	C₄⋊C₄×C₁₉	—	2	2	no	—
85	C₄⋊C₄×C₁₉	—	2	2	no	—
86	C₄²×C₁₉	—	4	1	no	—
87	C₄²×C₁₉	—	4	1	no	—
88	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
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	(C₂×D₁₉)⋊C₄	—	2	2	no	—
93	(C₂×D₁₉)⋊C₄	—	2	2	no	—
94	C₄⋊Dic₁₉	—	2	1	no	—
95	C₄⋊Dic₁₉	C₄²	1	2	yes	—
96	C₂×C₁₉⋊C₈	—	2	2	no	—
97	C₂×C₁₉⋊C₈	—	2	2	no	—
98	C₂×C₁₉⋊C₈	—	2	2	no	—
99	C₂×C₁₉⋊C₈	—	2	2	no	—
100	C₂×C₁₉⋊C₈	—	4	2	no	—
101	C₂×C₁₉⋊C₈	—	4	1	no	—
102	C₂×C₁₉⋊C₈	—	4	1	no	—
103	C₂×C₁₉⋊C₈	M₁₆	1	4	yes	complemented
104	#64515	—	4	8	no	—
105	#8208	—	19	1	no	maximal, Sylow
106	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
107	C₂²×D₁₉	—	2	2	no	—
108	C₂²×D₁₉	—	4	2	no	—
109	C₂²×D₁₉	—	4	1	no	—
110	C₂²×D₁₉	—	4	1	no	—
111	C₂²×D₁₉	—	4	2	no	—
112	C₂×C₄×C₁₉	—	2	2	no	—
113	C₂×C₄×C₁₉	—	4	2	no	—
114	C₂×C₄×C₁₉	—	4	2	no	—
115	C₂×C₄×C₁₉	—	4	2	no	—
116	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
117	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
119	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
123	C₂×C₄×C₁₉	#131	1	2	yes	—
124	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
126	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
128	C₂×Dic₁₉	—	2	2	no	—
129	C₂×Dic₁₉	—	4	1	no	—
130	C₂×Dic₁₉	—	4	1	no	—
131	D₇₆	—	2	2	no	—
132	D₇₆	—	2	2	no	—
133	D₇₆	—	4	2	no	—
134	C₄×D₁₉	—	4	2	no	—
135	C₄×D₁₉	—	4	2	no	—
136	C₄×D₁₉	—	4	2	no	—
137	C₄×D₁₉	—	4	2	no	—
138	C₈×C₁₉	—	2	2	no	—
139	C₈×C₁₉	—	2	2	no	—
140	C₈×C₁₉	—	4	2	no	—
141	C₁₉⋊C₈	—	2	4	no	—
142	C₁₉⋊C₈	—	4	2	no	—
143	C₁₉⋊C₈	—	4	2	no	—
144	C₂×#436	—	19	1	no	—
145	#1770	D₁₉	1	1	yes	characteristic, complemented
146	C₄×#102	—	19	1	no	—
147	C₂×D₄⋊C₈	—	19	1	no	—
148	C₂×#322	—	19	1	no	—
149	#1279	—	19	1	no	—
150	#1277	—	19	1	no	—
151	C₂²×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
152	C₂²×C₁₉	#436	1	2	yes	—
153	C₂²×C₁₉	#440	1	1	yes	characteristic
154	C₂²×C₁₉	#442	1	1	yes	characteristic
155	C₂²×C₁₉	#444	1	2	yes	—
156	C₂×D₁₉	—	4	2	no	—
157	C₂×D₁₉	—	4	2	no	—
158	C₂×D₁₉	—	4	2	no	—
159	C₂×D₁₉	—	8	2	no	—
160	C₂×D₁₉	—	8	2	no	—
161	C₄×C₁₉	—	2	1	no	—
162	C₄×C₁₉	—	2	1	no	—
163	C₄×C₁₉	—	8	2	no	—
164	C₄×C₁₉	C₄×M₁₆	1	1	yes	characteristic
165	C₄×C₁₉	#405	1	1	yes	characteristic
166	C₄×C₁₉	#417	1	1	yes	characteristic
167	C₄×C₁₉	#418	1	1	yes	characteristic
168	C₄×C₁₉	#435	1	2	yes	—
169	C₄×C₁₉	#440	1	1	yes	characteristic
170	C₄×C₁₉	#442	1	1	yes	characteristic
171	Dic₁₉	—	4	2	no	—
172	C₂²×M₁₆	—	38	1	no	—
173	C₂×C₄×D₄	—	19	1	no	—
174	#436	—	19	4	no	—
175	#436	—	38	2	no	—
176	C₂×C₄⋊C₈	—	19	1	no	—
177	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
178	C₂×#102	—	19	2	no	—
179	C₂×C₂²⋊C₈	—	38	1	no	—
180	C₂×#97	—	2	1	no	—
181	C₂×C₄×C₈	—	19	1	no	—
182	C₂×C₄×C₈	—	19	1	no	—
183	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
184	C₄×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
185	#340	—	19	2	no	—
186	#336	—	2	1	no	—
187	D₄⋊C₈	—	19	4	no	—
188	#322	—	38	2	no	—
189	C₂×C₁₉	#1592	1	2	yes	—
190	C₂×C₁₉	#1594	1	2	yes	—
191	C₂×C₁₉	#1753	1	1	yes	characteristic
192	C₂×C₁₉	#1762	1	1	yes	characteristic
193	C₂×C₁₉	#1765	1	1	yes	characteristic
194	D₁₉	—	8	2	no	—
195	C₂²×D₄	—	19	1	no	—
196	C₂³×C₄	—	38	1	no	—
197	C₂×M₁₆	—	38	4	no	—
198	C₂×M₁₆	—	76	2	no	—
199	C₂×M₁₆	—	76	2	no	—
200	C₂²×C₈	—	2	1	no	—
201	C₂²×C₈	—	19	2	no	—
202	C₂²×C₈	—	38	1	no	—
203	C₂²×C₈	—	38	1	no	—
204	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
205	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
206	C₄×D₄	—	19	2	no	—
207	C₄×D₄	—	19	2	no	—
208	C₄×D₄	—	38	1	no	—
209	C₄×D₄	—	38	1	no	—
210	C₂×C₄⋊C₄	—	19	1	no	—
211	C₂×C₄⋊C₄	C₄×D₁₉	1	2	yes	—
212	C₂×C₂²⋊C₄	—	38	1	no	—
213	C₂×C₄²	—	2	1	no	—
214	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
215	C₄⋊C₈	—	2	1	no	—
216	C₄⋊C₈	—	19	2	no	—
217	C₄⋊C₈	—	38	1	no	—
218	C₄⋊C₈	C₄×D₁₉	1	2	yes	—
219	#102	—	38	4	no	—
220	C₂²⋊C₈	—	38	4	no	—
221	#97	—	4	2	no	—
222	C₄×C₈	—	19	2	no	—
223	C₄×C₈	—	38	1	no	—
224	C₄×C₈	—	38	1	no	—
225	C₄×C₈	—	38	1	no	—
226	C₄×C₈	C₄×D₁₉	1	2	yes	—
227	C₄×C₈	D₇₆	1	2	yes	—
228	#95	—	2	1	no	—
229	C₁₉	#8208	1	1	yes	characteristic, complemented, Sylow
230	C₂⁴	—	38	1	no	—
231	C₂×D₄	—	19	2	no	—
232	C₂×D₄	—	38	1	no	—
233	C₂×D₄	—	38	2	no	—
234	C₂×D₄	—	38	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	38	1	no	—
237	C₂²×C₄	—	38	2	no	—
238	C₂²×C₄	—	38	2	no	—
239	C₂²×C₄	—	76	1	no	—
240	C₂²×C₄	—	76	1	no	—
241	C₂²×C₄	—	76	1	no	—
242	C₂²×C₄	—	76	1	no	—
243	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
244	C₂²×C₄	#64541	1	1	yes	characteristic
245	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
246	M₁₆	—	76	8	no	—
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	4	1	no	—
253	C₂×C₈	—	4	1	no	—
254	C₂×C₈	—	19	4	no	—
255	C₂×C₈	—	38	2	no	—
256	C₂×C₈	—	38	2	no	—
257	C₂×C₈	—	38	2	no	—
258	C₂×C₈	—	38	2	no	—
259	C₂×C₈	—	76	2	no	—
260	C₂×C₈	—	76	1	no	—
261	C₂×C₈	—	76	1	no	—
262	C₂×C₈	(C₂×D₁₉)⋊C₄	1	2	yes	—
263	C₂×C₈	(C₂×D₁₉)⋊C₄	1	2	yes	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	19	2	no	—
267	C₄⋊C₄	—	38	1	no	—
268	C₂²⋊C₄	—	38	2	no	—
269	C₂²⋊C₄	—	38	2	no	—
270	C₄²	—	4	1	no	—
271	C₄²	—	4	1	no	—
272	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
273	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
274	C₄²	C₂×D₇₆	1	1	yes	characteristic
275	C₄²	#64541	1	1	yes	characteristic
276	C₂³	—	38	2	no	—
277	C₂³	—	76	2	no	—
278	C₂³	—	76	1	no	—
279	C₂³	—	76	1	no	—
280	C₂³	—	76	2	no	—
281	C₂³	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
282	D₄	—	38	2	no	—
283	D₄	—	38	2	no	—
284	D₄	—	76	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	38	2	no	—
290	C₂×C₄	—	76	1	no	—
291	C₂×C₄	—	76	1	no	—
292	C₂×C₄	—	76	2	no	—
293	C₂×C₄	—	76	2	no	—
294	C₂×C₄	—	76	2	no	—
295	C₂×C₄	—	76	2	no	—
296	C₂×C₄	C₄²×D₁₉	1	1	yes	characteristic
297	C₂×C₄	#10597318	1	1	yes	characteristic
298	C₂×C₄	C₄×D₇₆	1	1	yes	characteristic
299	C₂×C₄	C₄×D₇₆	1	1	yes	characteristic
300	C₂×C₄	M₁₆×D₁₉	1	2	yes	—
301	C₂×C₄	#10597353	1	2	yes	—
302	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
303	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
304	C₂×C₄	#10597378	1	1	yes	characteristic
305	C₂×C₄	#10597388	1	1	yes	characteristic
306	C₂×C₄	#10597395	1	1	yes	characteristic
307	C₂×C₄	#10597395	1	1	yes	characteristic
308	C₈	—	2	2	no	—
309	C₈	—	2	2	no	—
310	C₈	—	4	2	no	—
311	C₈	—	38	4	no	—
312	C₈	—	76	2	no	—
313	C₈	—	76	2	no	—
314	C₂²	—	76	2	no	—
315	C₂²	—	76	2	no	—
316	C₂²	—	76	2	no	—
317	C₂²	—	152	2	no	—
318	C₂²	—	152	2	no	—
319	C₂²	C₄×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
320	C₂²	#11926620	1	2	yes	—
321	C₂²	#11926642	1	1	yes	characteristic
322	C₂²	#11926654	1	1	yes	characteristic
323	C₂²	#11926664	1	2	yes	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	8	2	no	—
327	C₄	—	76	2	no	—
328	C₄	#11926496	1	1	yes	characteristic
329	C₄	#11926501	1	1	yes	characteristic
330	C₄	#11926543	1	1	yes	characteristic
331	C₄	#11926548	1	1	yes	characteristic
332	C₄	#11926617	1	2	yes	—
333	C₄	#11926642	1	1	yes	characteristic
334	C₄	#11926654	1	1	yes	characteristic
335	C₂	—	152	2	no	—
336	C₂	#14562535	1	2	yes	—
337	C₂	#14562547	1	2	yes	—
338	C₂	#14563420	1	1	yes	characteristic
339	C₂	#14563463	1	1	yes	characteristic
340	C₂	#14563481	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₉
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1770	C₂
socle	C₂³×C₁₉	C₄×D₄
prime radical	C₂²×D₇₆	C₂×C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1770	D₁₉
hypocenter	C₁₉	#8208
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1770

Complements

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

N	H ≅ G/N	type
4. C₂×#11926266	C₂	semidirect
5. #14561476	C₂	semidirect
6. #14561469	C₂	semidirect
7. C₁₉×#1770	C₂	semidirect
11. #11926623	C₄	semidirect
18. #11926271	C₄	semidirect
103. C₂×C₁₉⋊C₈	M₁₆	semidirect
145. #1770	D₁₉	semidirect
229. C₁₉	#8208	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926623	2
C₄×#10597266	2
C₂×#11926271	2
C₂×#11926266	2
#14561476	2
#14561469	2
C₁₉×#1770	2
#8208	19