Subgroups — #18075358

Order 4864.

644 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#10597320	C₂	1	1	yes	maximal, characteristic
2	#14563752	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11926287	C₂	1	2	yes	maximal
4	#14561519	C₂	1	2	yes	maximal, complemented
5	C₁₉×#1831	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10597320	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₇₆	—	2	1	no	—
8	C₂×C₄×D₇₆	—	2	1	no	—
9	C₂×C₄×D₇₆	C₄	1	2	yes	—
10	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
11	#11926446	—	2	1	no	—
12	#11926446	—	2	1	no	—
13	#11926446	C₄	1	2	yes	—
14	#11926414	C₂²	1	1	yes	characteristic
15	#11926299	—	2	1	no	—
16	#11926299	—	2	1	no	—
17	#11926287	—	2	4	no	—
18	C₂×C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
19	C₄³×C₁₉	C₂²	1	1	yes	characteristic
20	C₁₉×#336	—	2	1	no	—
21	C₁₉×#336	—	2	1	no	—
22	C₂²×D₇₆	—	2	1	no	—
23	C₂²×D₇₆	—	2	1	no	—
24	C₂²×D₇₆	C₂×C₄	1	2	yes	—
25	C₂²×C₄×D₁₉	—	2	2	no	—
26	C₂²×C₈×C₁₉	—	2	2	no	—
27	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
34	C₂×(C₂×D₁₉)⋊C₄	—	2	2	no	—
35	C₂×(C₂×D₁₉)⋊C₄	—	2	2	no	—
36	C₂×(C₂×D₁₉)⋊C₄	—	2	2	no	—
37	C₂×C₄⋊Dic₁₉	—	2	1	no	—
38	C₂×C₄⋊Dic₁₉	—	2	1	no	—
39	C₂×C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
40	C₂²×C₁₉⋊C₈	—	2	2	no	—
41	#10597320	—	2	1	no	—
42	#10597320	—	2	1	no	—
43	#10597320	D₄	1	2	yes	—
44	#10597320	Q₈	1	2	yes	—
45	C₄×D₇₆	—	4	4	no	—
46	C₄×D₇₆	—	4	4	no	—
47	C₄×D₇₆	—	4	4	no	—
48	C₁₉×C₄⋊C₈	—	2	2	no	—
49	C₁₉×C₄⋊C₈	—	2	2	no	—
50	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
51	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
52	C₂³×D₁₉	—	2	2	no	—
53	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
60	C₂²×Dic₁₉	—	2	2	no	—
61	C₂×D₇₆	—	2	2	no	—
62	C₂×D₇₆	—	2	2	no	—
63	C₂×D₇₆	—	2	2	no	—
64	C₂×D₇₆	—	2	2	no	—
65	C₂×D₇₆	—	2	2	no	—
66	C₂×D₇₆	—	4	2	no	—
67	C₂×D₇₆	—	4	2	no	—
68	C₂×D₇₆	—	4	2	no	—
69	C₂×D₇₆	—	4	2	no	—
70	C₂×D₇₆	—	4	2	no	—
71	C₂×D₇₆	—	4	2	no	—
72	C₂×D₇₆	C₄⋊C₄	1	4	yes	—
73	C₂×C₄×D₁₉	—	4	4	no	—
74	C₂×C₄×D₁₉	—	4	4	no	—
75	C₂×C₄×D₁₉	—	4	4	no	—
76	C₂×C₈×C₁₉	—	4	2	no	—
77	C₂×C₈×C₁₉	—	4	2	no	—
78	C₂×C₈×C₁₉	—	4	2	no	—
79	C₄²×C₁₉	—	2	2	no	—
80	C₄²×C₁₉	—	2	2	no	—
81	C₄²×C₁₉	—	2	2	no	—
82	C₄²×C₁₉	—	2	2	no	—
83	C₄²×C₁₉	—	2	2	no	—
84	C₄²×C₁₉	—	4	1	no	—
85	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
87	C₄²×C₁₉	#41	1	1	yes	characteristic
88	C₄²×C₁₉	#41	1	1	yes	characteristic
89	(C₂×D₁₉)⋊C₄	—	4	4	no	—
90	(C₂×D₁₉)⋊C₄	—	4	4	no	—
91	(C₂×D₁₉)⋊C₄	—	4	4	no	—
92	C₄⋊Dic₁₉	—	4	2	no	—
93	C₄⋊Dic₁₉	—	4	2	no	—
94	C₄⋊Dic₁₉	—	4	2	no	—
95	C₂×C₁₉⋊C₈	—	4	2	no	—
96	C₂×C₁₉⋊C₈	—	4	2	no	—
97	C₂×C₁₉⋊C₈	—	4	2	no	—
98	#8592	—	19	1	no	maximal, Sylow
99	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂²×D₁₉	—	2	4	no	—
101	C₂²×D₁₉	—	4	2	no	—
102	C₂²×D₁₉	—	4	2	no	—
103	C₂²×D₁₉	—	4	2	no	—
104	C₂²×D₁₉	—	4	2	no	—
105	C₂²×D₁₉	—	4	2	no	—
106	C₂²×D₁₉	—	4	2	no	—
107	C₂×C₄×C₁₉	—	2	1	no	—
108	C₂×C₄×C₁₉	—	2	1	no	—
109	C₂×C₄×C₁₉	—	2	1	no	—
110	C₂×C₄×C₁₉	—	2	1	no	—
111	C₂×C₄×C₁₉	—	2	1	no	—
112	C₂×C₄×C₁₉	—	2	1	no	—
113	C₂×C₄×C₁₉	—	2	1	no	—
114	C₂×C₄×C₁₉	—	2	1	no	—
115	C₂×C₄×C₁₉	—	2	1	no	—
116	C₂×C₄×C₁₉	—	2	1	no	—
117	C₂×C₄×C₁₉	—	2	1	no	—
118	C₂×C₄×C₁₉	—	2	1	no	—
119	C₂×C₄×C₁₉	—	2	1	no	—
120	C₂×C₄×C₁₉	#100	1	2	yes	—
121	C₂×C₄×C₁₉	C₂×C₄⋊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	#135	1	2	yes	—
128	C₂×C₄×C₁₉	#135	1	2	yes	—
129	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
130	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
131	C₂×Dic₁₉	—	4	2	no	—
132	C₂×Dic₁₉	—	4	2	no	—
133	C₂×Dic₁₉	—	4	2	no	—
134	D₇₆	—	4	4	no	—
135	D₇₆	—	4	4	no	—
136	D₇₆	—	4	4	no	—
137	D₇₆	—	4	4	no	—
138	D₇₆	—	4	4	no	—
139	D₇₆	—	4	4	no	—
140	C₄×D₁₉	—	8	8	no	—
141	C₈×C₁₉	—	8	2	no	—
142	C₁₉⋊C₈	—	8	2	no	—
143	C₄×#127	—	19	1	no	—
144	#1831	—	19	1	no	—
145	#1831	D₁₉	1	1	yes	characteristic, complemented
146	C₂×#331	—	19	2	no	—
147	#1296	—	19	2	no	—
148	C₂²×C₁₉	#380	1	1	yes	characteristic
149	C₂²×C₁₉	C₂×#100	1	1	yes	characteristic
150	C₂²×C₁₉	#413	1	1	yes	characteristic
151	C₂²×C₁₉	#416	1	1	yes	characteristic
152	C₂²×C₁₉	#416	1	1	yes	characteristic
153	C₂²×C₁₉	#418	1	1	yes	characteristic
154	C₂²×C₁₉	#418	1	1	yes	characteristic
155	C₂×D₁₉	—	4	4	no	—
156	C₂×D₁₉	—	4	4	no	—
157	C₂×D₁₉	—	4	4	no	—
158	C₂×D₁₉	—	8	2	no	—
159	C₂×D₁₉	—	8	2	no	—
160	C₂×D₁₉	—	8	2	no	—
161	C₂×D₁₉	—	8	2	no	—
162	C₄×C₁₉	—	2	1	no	—
163	C₄×C₁₉	—	2	1	no	—
164	C₄×C₁₉	—	2	2	no	—
165	C₄×C₁₉	—	4	1	no	—
166	C₄×C₁₉	—	4	1	no	—
167	C₄×C₁₉	—	4	1	no	—
168	C₄×C₁₉	#463	1	2	yes	—
169	C₄×C₁₉	#463	1	2	yes	—
170	C₄×C₁₉	#478	1	2	yes	—
171	C₄×C₁₉	#478	1	2	yes	—
172	Dic₁₉	—	8	2	no	—
173	C₂×#127	—	19	1	no	—
174	C₂×C₄×D₄	—	19	2	no	—
175	C₂×C₄×D₄	—	38	1	no	—
176	C₂×C₄×D₄	—	38	1	no	—
177	C₂×C₄⋊C₈	—	19	2	no	—
178	C₂×C₄⋊C₈	C₂×D₁₉	1	2	yes	—
179	#390	—	19	2	no	—
180	#390	—	38	1	no	—
181	#390	—	38	1	no	—
182	#384	—	19	1	no	—
183	C₄³	C₂×D₁₉	1	1	yes	characteristic
184	#336	—	2	1	no	—
185	#336	—	2	1	no	—
186	#336	—	38	1	no	—
187	#336	—	38	1	no	—
188	#331	—	38	4	no	—
189	C₂×C₁₉	#1543	1	1	yes	characteristic
190	C₂×C₁₉	#1553	1	1	yes	characteristic
191	C₂×C₁₉	#1553	1	1	yes	characteristic
192	C₂×C₁₉	#1555	1	1	yes	characteristic
193	C₂×C₁₉	#1800	1	1	yes	characteristic
194	C₂×C₁₉	#1804	1	1	yes	characteristic
195	C₂×C₁₉	#1804	1	1	yes	characteristic
196	D₁₉	—	8	4	no	—
197	C₂²×D₄	—	19	2	no	—
198	C₂²×D₄	—	38	1	no	—
199	C₂²×D₄	—	38	1	no	—
200	C₂³×C₄	—	38	2	no	—
201	C₂²×C₈	—	2	2	no	—
202	C₂²×C₈	—	38	2	no	—
203	#127	—	19	2	no	—
204	#127	—	19	2	no	—
205	#127	—	38	1	no	—
206	#127	—	38	1	no	—
207	C₄×D₄	—	76	4	no	—
208	C₄×D₄	—	76	4	no	—
209	C₄×D₄	—	76	4	no	—
210	C₂×C₄⋊C₄	—	19	2	no	—
211	C₂×C₄⋊C₄	—	38	1	no	—
212	C₂×C₄⋊C₄	—	38	1	no	—
213	C₂×C₂²⋊C₄	—	38	2	no	—
214	C₂×C₂²⋊C₄	—	38	2	no	—
215	C₂×C₂²⋊C₄	—	38	2	no	—
216	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
217	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
218	C₂×C₄²	D₇₆	1	1	yes	characteristic
219	C₂×C₄²	D₇₆	1	1	yes	characteristic
220	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
221	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
222	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
223	C₄⋊C₈	—	2	2	no	—
224	C₄⋊C₈	—	2	2	no	—
225	C₄⋊C₈	—	38	2	no	—
226	C₄⋊C₈	—	38	2	no	—
227	C₁₉	#8592	1	1	yes	characteristic, complemented, Sylow
228	C₂⁴	—	38	2	no	—
229	C₂×D₄	—	19	4	no	—
230	C₂×D₄	—	38	2	no	—
231	C₂×D₄	—	38	2	no	—
232	C₂×D₄	—	38	2	no	—
233	C₂×D₄	—	38	2	no	—
234	C₂×D₄	—	38	2	no	—
235	C₂×D₄	—	76	2	no	—
236	C₂×D₄	—	76	2	no	—
237	C₂×D₄	—	76	2	no	—
238	C₂×D₄	—	76	2	no	—
239	C₂×D₄	—	76	2	no	—
240	C₂×D₄	—	76	2	no	—
241	C₂²×C₄	—	38	2	no	—
242	C₂²×C₄	—	76	4	no	—
243	C₂²×C₄	—	76	4	no	—
244	C₂²×C₄	—	76	4	no	—
245	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
246	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
247	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
248	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
249	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
250	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
251	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
252	C₂×C₈	—	4	2	no	—
253	C₂×C₈	—	4	2	no	—
254	C₂×C₈	—	4	2	no	—
255	C₂×C₈	—	76	2	no	—
256	C₂×C₈	—	76	2	no	—
257	C₂×C₈	—	76	2	no	—
258	C₄⋊C₄	—	76	2	no	—
259	C₄⋊C₄	—	76	2	no	—
260	C₄⋊C₄	—	76	2	no	—
261	C₂²⋊C₄	—	76	4	no	—
262	C₂²⋊C₄	—	76	4	no	—
263	C₂²⋊C₄	—	76	4	no	—
264	C₄²	—	2	2	no	—
265	C₄²	—	2	2	no	—
266	C₄²	—	2	2	no	—
267	C₄²	—	2	2	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	4	1	no	—
270	C₄²	D₄×D₁₉	1	1	yes	characteristic
271	C₄²	#64543	1	1	yes	characteristic
272	C₄²	Q₈×D₁₉	1	1	yes	characteristic
273	C₄²	#64545	1	1	yes	characteristic
274	C₂³	—	38	4	no	—
275	C₂³	—	76	2	no	—
276	C₂³	—	76	2	no	—
277	C₂³	—	76	2	no	—
278	C₂³	—	76	2	no	—
279	C₂³	—	76	2	no	—
280	C₂³	—	76	2	no	—
281	C₂³	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
282	D₄	—	76	4	no	—
283	D₄	—	76	4	no	—
284	D₄	—	76	4	no	—
285	D₄	—	76	4	no	—
286	D₄	—	76	4	no	—
287	D₄	—	76	4	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	76	2	no	—
302	C₂×C₄	—	76	2	no	—
303	C₂×C₄	—	76	2	no	—
304	C₂×C₄	—	152	8	no	—
305	C₂×C₄	#10597280	1	2	yes	—
306	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
307	C₂×C₄	#10597338	1	1	yes	characteristic
308	C₂×C₄	#10597341	1	1	yes	characteristic
309	C₂×C₄	#10597343	1	1	yes	characteristic
310	C₂×C₄	#10597350	1	2	yes	—
311	C₂×C₄	#10597354	1	2	yes	—
312	C₂×C₄	#10597384	1	1	yes	characteristic
313	C₂×C₄	#10597392	1	1	yes	characteristic
314	C₂×C₄	#10597395	1	2	yes	—
315	C₂×C₄	#10597396	1	2	yes	—
316	C₈	—	8	2	no	—
317	C₈	—	152	2	no	—
318	C₂²	—	76	4	no	—
319	C₂²	—	76	4	no	—
320	C₂²	—	76	4	no	—
321	C₂²	—	152	2	no	—
322	C₂²	—	152	2	no	—
323	C₂²	—	152	2	no	—
324	C₂²	—	152	2	no	—
325	C₂²	#11926401	1	1	yes	characteristic
326	C₂²	C₂×#10597280	1	1	yes	characteristic
327	C₂²	#11926526	1	1	yes	characteristic
328	C₂²	#11926538	1	1	yes	characteristic
329	C₂²	#11926541	1	1	yes	characteristic
330	C₂²	#11926548	1	1	yes	characteristic
331	C₂²	#11926550	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	2	2	no	—
335	C₄	—	4	1	no	—
336	C₄	—	4	1	no	—
337	C₄	—	4	1	no	—
338	C₄	—	152	2	no	—
339	C₄	#11926756	1	2	yes	—
340	C₄	#11926760	1	2	yes	—
341	C₄	#11926841	1	2	yes	—
342	C₄	#11926845	1	2	yes	—
343	C₂	—	152	4	no	—
344	C₂	#14562295	1	1	yes	characteristic
345	C₂	#14562338	1	1	yes	characteristic
346	C₂	#14562341	1	1	yes	characteristic
347	C₂	#14562348	1	1	yes	characteristic
348	C₂	#14563624	1	1	yes	characteristic
349	C₂	#14563642	1	1	yes	characteristic
350	C₂	#14563646	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₁₉)⋊C₄
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1831	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂×#10597320	C₂²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1831	D₁₉
hypocenter	C₁₉	#8592
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

Complements

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

N	H ≅ G/N	type
2. #14563752	C₂	semidirect
4. #14561519	C₂	semidirect
5. C₁₉×#1831	C₂	semidirect
145. #1831	D₁₉	semidirect
227. C₁₉	#8592	semidirect

Maximal subgroups

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

subgroup	index
C₄×#10597320	2
#14563752	2
C₂×#11926287	2
#14561519	2
C₁₉×#1831	2
#8592	19