Subgroups — #18077992

Order 4864.

536 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926654	C₂	1	1	yes	maximal, characteristic
2	#14562027	C₂	1	1	yes	maximal, characteristic, complemented
3	#14562026	C₂	1	1	yes	maximal, characteristic, complemented
4	#14561955	C₂	1	1	yes	maximal, characteristic, complemented
5	#14561479	C₂	1	1	yes	maximal, characteristic, complemented
6	#14561465	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#2975	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#64525	C₄	1	1	yes	characteristic
9	C₂×C₄×D₇₆	C₂²	1	1	yes	characteristic
10	#11926654	—	2	2	no	—
11	#11926654	—	2	2	no	—
12	#11926597	C₂²	1	1	yes	characteristic
13	#11926597	C₂²	1	1	yes	characteristic
14	C₂×#10597265	C₄	1	1	yes	characteristic
15	C₂×#10597289	C₄	1	1	yes	characteristic
16	C₂×#10597266	C₄	1	1	yes	characteristic
17	C₂×#10597260	C₂²	1	1	yes	characteristic
18	#11926273	—	2	2	no	—
19	D₄⋊(C₁₉⋊C₈)	—	2	2	no	—
20	#11926267	—	2	2	no	—
21	C₂×C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
22	C₁₉×#436	—	2	2	no	—
23	C₁₉×#436	—	2	2	no	—
24	C₁₉×#431	C₂²	1	1	yes	characteristic
25	C₁₉×#431	C₂²	1	1	yes	characteristic
26	C₁₉×#407	—	2	2	no	—
27	C₂²×D₄×C₁₉	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₁₉	—	2	1	no	—
29	C₂²×D₇₆	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×D₁₉	—	2	1	no	—
31	C₂×M₁₆×C₁₉	—	2	4	no	—
32	C₄×D₄×C₁₉	—	2	1	no	—
33	C₄×D₄×C₁₉	—	2	1	no	—
34	C₄×D₄×C₁₉	D₄	1	2	yes	—
35	C₄×D₄×C₁₉	D₄	1	2	yes	—
36	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
38	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
39	C₂×#64525	—	2	2	no	—
40	C₂×#64525	—	2	2	no	—
41	C₂×#64525	—	2	2	no	—
42	C₂×(C₂×D₁₉)⋊C₄	—	2	1	no	—
43	C₂×C₄⋊Dic₁₉	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
46	C₄×D₇₆	—	2	1	no	—
47	C₄×D₇₆	—	2	1	no	—
48	C₄×D₇₆	—	2	1	no	—
49	C₄×D₇₆	—	2	1	no	—
50	C₁₉×C₄⋊C₈	—	2	1	no	—
51	C₁₉×C₄⋊C₈	—	2	1	no	—
52	C₁₉×C₂²⋊C₈	—	2	4	no	—
53	C₁₉×C₂²⋊C₈	—	2	1	no	—
54	C₁₉×C₂²⋊C₈	D₄	1	2	yes	—
55	C₄×C₈×C₁₉	—	2	1	no	—
56	C₄×C₈×C₁₉	D₄	1	2	yes	—
57	#10597289	—	2	2	no	—
58	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
59	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
60	#10597266	—	2	2	no	—
61	#10597265	—	2	2	no	—
62	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
63	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
64	#10597260	—	2	1	no	—
65	#10597260	—	2	1	no	—
66	C₄×C₁₉⋊C₈	—	2	1	no	—
67	C₄×C₁₉⋊C₈	—	2	1	no	—
68	C₂⁴×C₁₉	—	2	1	no	—
69	C₂³×D₁₉	—	2	1	no	—
70	C₂×D₄×C₁₉	—	2	1	no	—
71	C₂×D₄×C₁₉	—	2	1	no	—
72	C₂×D₄×C₁₉	—	2	1	no	—
73	C₂×D₄×C₁₉	—	2	1	no	—
74	C₂×D₄×C₁₉	—	2	1	no	—
75	C₂×D₄×C₁₉	C₂²⋊C₄	1	2	yes	—
76	C₂²×C₄×C₁₉	—	2	1	no	—
77	C₂²×C₄×C₁₉	—	2	2	no	—
78	C₂²×C₄×C₁₉	—	2	2	no	—
79	C₂²×C₄×C₁₉	—	8	2	no	—
80	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
83	C₂²×Dic₁₉	—	2	1	no	—
84	C₂×D₇₆	—	2	1	no	—
85	C₂×D₇₆	—	2	1	no	—
86	C₂×D₇₆	—	2	1	no	—
87	C₂×D₇₆	—	2	1	no	—
88	C₂×D₇₆	—	2	1	no	—
89	C₂×D₇₆	M₁₆	1	2	yes	complemented
90	C₂×C₄×D₁₉	—	4	1	no	—
91	C₂×C₄×D₁₉	—	4	1	no	—
92	C₂×C₄×D₁₉	—	8	2	no	—
93	M₁₆×C₁₉	—	8	4	no	—
94	C₂×C₈×C₁₉	—	2	2	no	—
95	C₂×C₈×C₁₉	—	2	2	no	—
96	C₄⋊C₄×C₁₉	—	2	1	no	—
97	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
98	C₂²⋊C₄×C₁₉	—	2	2	no	—
99	C₂²⋊C₄×C₁₉	—	2	2	no	—
100	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₁₉	#41	1	1	yes	characteristic
104	#64525	—	2	8	no	—
105	(C₂×D₁₉)⋊C₄	—	4	1	no	—
106	(C₂×D₁₉)⋊C₄	—	4	1	no	—
107	C₄⋊Dic₁₉	—	2	1	no	—
108	C₄⋊Dic₁₉	—	2	1	no	—
109	C₂×C₁₉⋊C₈	—	2	1	no	—
110	C₂×C₁₉⋊C₈	—	2	1	no	—
111	C₂×C₁₉⋊C₈	—	2	1	no	—
112	C₂×C₁₉⋊C₈	—	2	1	no	—
113	C₂×C₁₉⋊C₈	—	2	1	no	—
114	C₂×C₁₉⋊C₈	—	4	1	no	—
115	C₂×C₁₉⋊C₈	—	4	1	no	—
116	C₂×C₁₉⋊C₈	M₁₆	1	2	yes	complemented
117	#9093	—	19	1	no	maximal, Sylow
118	C₂³×C₁₉	—	2	2	no	—
119	C₂³×C₁₉	—	4	1	no	—
120	C₂³×C₁₉	—	4	1	no	—
121	C₂³×C₁₉	—	8	1	no	—
122	C₂³×C₁₉	—	8	1	no	—
123	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
124	C₂²×D₁₉	—	2	2	no	—
125	C₂²×D₁₉	—	4	1	no	—
126	C₂²×D₁₉	—	4	1	no	—
127	C₂²×D₁₉	—	8	1	no	—
128	C₂²×D₁₉	—	8	1	no	—
129	D₄×C₁₉	—	2	2	no	—
130	D₄×C₁₉	—	2	2	no	—
131	D₄×C₁₉	—	4	1	no	—
132	D₄×C₁₉	—	4	1	no	—
133	C₂×C₄×C₁₉	—	2	2	no	—
134	C₂×C₄×C₁₉	—	4	1	no	—
135	C₂×C₄×C₁₉	—	4	1	no	—
136	C₂×C₄×C₁₉	—	4	4	no	—
137	C₂×C₄×C₁₉	—	8	2	no	—
138	C₂×C₄×C₁₉	—	8	1	no	—
139	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
142	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
143	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
144	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
145	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
146	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
147	C₂×C₄×C₁₉	#131	1	1	yes	characteristic
148	C₂×C₄×C₁₉	#131	1	1	yes	characteristic
149	C₂×Dic₁₉	—	4	1	no	—
150	C₂×Dic₁₉	—	8	1	no	—
151	D₇₆	—	2	2	no	—
152	D₇₆	—	2	2	no	—
153	D₇₆	—	4	1	no	—
154	D₇₆	—	4	1	no	—
155	C₄×D₁₉	—	8	2	no	—
156	C₄×D₁₉	—	8	2	no	—
157	C₈×C₁₉	—	4	2	no	—
158	C₈×C₁₉	—	4	2	no	—
159	C₁₉⋊C₈	—	2	2	no	—
160	C₁₉⋊C₈	—	4	1	no	—
161	C₁₉⋊C₈	—	8	1	no	—
162	C₁₉⋊C₈	—	8	1	no	—
163	#2975	D₁₉	1	1	yes	characteristic, complemented
164	C₂×#442	—	19	1	no	—
165	#1478	—	19	1	no	—
166	#1478	—	19	1	no	—
167	#1460	—	19	1	no	—
168	#1281	—	19	1	no	—
169	#1275	—	19	1	no	—
170	C₂²×C₁₉	—	2	1	no	—
171	C₂²×C₁₉	—	2	1	no	—
172	C₂²×C₁₉	—	4	2	no	—
173	C₂²×C₁₉	—	4	2	no	—
174	C₂²×C₁₉	—	4	2	no	—
175	C₂²×C₁₉	—	8	2	no	—
176	C₂²×C₁₉	—	8	2	no	—
177	C₂²×C₁₉	#386	1	1	yes	characteristic
178	C₂²×C₁₉	#435	1	1	yes	characteristic
179	C₂²×C₁₉	#436	1	1	yes	characteristic
180	C₂×D₁₉	—	4	2	no	—
181	C₂×D₁₉	—	8	2	no	—
182	C₂×D₁₉	—	8	1	no	—
183	C₂×D₁₉	—	8	1	no	—
184	C₂×D₁₉	—	8	2	no	—
185	C₄×C₁₉	—	2	1	no	—
186	C₄×C₁₉	—	2	1	no	—
187	C₄×C₁₉	—	2	1	no	—
188	C₄×C₁₉	—	2	1	no	—
189	C₄×C₁₉	—	4	2	no	—
190	C₄×C₁₉	#407	1	1	yes	characteristic
191	C₄×C₁₉	#408	1	1	yes	characteristic
192	C₄×C₁₉	#435	1	1	yes	characteristic
193	C₄×C₁₉	#436	1	1	yes	characteristic
194	Dic₁₉	—	8	1	no	—
195	C₂²×D₈	—	19	1	no	—
196	C₂×C₄×D₄	—	19	1	no	—
197	C₂×C₄×D₄	C₂×D₁₉	1	1	yes	characteristic
198	#442	—	38	2	no	—
199	#442	—	38	2	no	—
200	#436	—	2	2	no	—
201	#436	—	2	2	no	—
202	#431	—	19	1	no	—
203	#431	—	19	1	no	—
204	#431	C₂×D₁₉	1	1	yes	characteristic
205	#431	C₂×D₁₉	1	1	yes	characteristic
206	C₂×#106	—	19	1	no	—
207	C₂×#102	—	19	1	no	—
208	C₂×#102	—	19	1	no	—
209	#407	—	2	2	no	—
210	C₂×#97	—	19	1	no	—
211	D₄⋊C₈	—	38	2	no	—
212	D₄⋊C₈	—	38	2	no	—
213	#323	—	38	2	no	—
214	C₂×C₁₉	—	2	1	no	—
215	C₂×C₁₉	—	2	1	no	—
216	C₂×C₁₉	—	8	2	no	—
217	C₂×C₁₉	#1870	1	1	yes	characteristic
218	C₂×C₁₉	#1903	1	1	yes	characteristic
219	C₂×C₁₉	#1904	1	1	yes	characteristic
220	D₁₉	—	8	2	no	—
221	C₂²×D₄	—	19	1	no	—
222	C₂²×D₄	C₄×D₁₉	1	1	yes	characteristic
223	C₂³×C₄	—	2	1	no	—
224	C₂³×C₄	—	38	1	no	—
225	C₂×D₈	—	38	2	no	—
226	C₂×D₈	—	38	2	no	—
227	C₂×D₈	—	38	2	no	—
228	C₂×M₁₆	—	2	4	no	—
229	C₂²×C₈	—	19	1	no	—
230	C₂²×C₈	—	19	1	no	—
231	C₄×D₄	—	2	1	no	—
232	C₄×D₄	—	2	1	no	—
233	C₄×D₄	—	38	1	no	—
234	C₄×D₄	—	38	1	no	—
235	C₄×D₄	—	38	1	no	—
236	C₄×D₄	—	38	1	no	—
237	C₄×D₄	D₇₆	1	2	yes	—
238	C₄×D₄	C₂²⋊D₁₉	1	2	yes	—
239	C₂×C₄⋊C₄	—	19	1	no	—
240	C₂×C₄⋊C₄	C₄×D₁₉	1	1	yes	characteristic
241	C₂×C₂²⋊C₄	—	2	1	no	—
242	C₂×C₂²⋊C₄	—	38	1	no	—
243	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
244	#106	—	38	2	no	—
245	C₄⋊C₈	—	2	1	no	—
246	C₄⋊C₈	—	2	1	no	—
247	C₄⋊C₈	—	38	1	no	—
248	C₄⋊C₈	—	38	1	no	—
249	#102	—	38	2	no	—
250	#102	—	38	2	no	—
251	C₂²⋊C₈	—	2	4	no	—
252	C₂²⋊C₈	—	2	1	no	—
253	C₂²⋊C₈	—	38	1	no	—
254	C₂²⋊C₈	—	38	1	no	—
255	C₂²⋊C₈	C₂²⋊D₁₉	1	2	yes	—
256	#97	—	38	1	no	—
257	#97	—	38	1	no	—
258	C₄×C₈	—	2	1	no	—
259	C₄×C₈	—	38	1	no	—
260	C₄×C₈	—	38	1	no	—
261	C₄×C₈	C₂²⋊D₁₉	1	2	yes	—
262	C₁₉	#9093	1	1	yes	characteristic, complemented, Sylow
263	C₂⁴	—	2	1	no	—
264	C₂⁴	—	38	1	no	—
265	C₂×D₄	—	2	1	no	—
266	C₂×D₄	—	2	1	no	—
267	C₂×D₄	—	2	1	no	—
268	C₂×D₄	—	2	1	no	—
269	C₂×D₄	—	2	1	no	—
270	C₂×D₄	—	19	2	no	—
271	C₂×D₄	—	38	1	no	—
272	C₂×D₄	—	38	1	no	—
273	C₂×D₄	—	38	1	no	—
274	C₂×D₄	—	38	1	no	—
275	C₂×D₄	—	38	1	no	—
276	C₂×D₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
277	C₂²×C₄	—	2	1	no	—
278	C₂²×C₄	—	2	2	no	—
279	C₂²×C₄	—	2	2	no	—
280	C₂²×C₄	—	8	2	no	—
281	C₂²×C₄	—	38	1	no	—
282	C₂²×C₄	—	76	1	no	—
283	C₂²×C₄	—	76	1	no	—
284	C₂²×C₄	—	152	2	no	—
285	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
286	C₂²×C₄	#64541	1	1	yes	characteristic
287	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
288	D₈	—	38	8	no	—
289	M₁₆	—	8	4	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	19	2	no	—
293	C₂×C₈	—	38	1	no	—
294	C₂×C₈	—	38	1	no	—
295	C₂×C₈	—	38	1	no	—
296	C₂×C₈	—	38	1	no	—
297	C₂×C₈	—	38	1	no	—
298	C₂×C₈	—	76	1	no	—
299	C₂×C₈	—	76	1	no	—
300	C₄⋊C₄	—	2	1	no	—
301	C₄⋊C₄	—	38	1	no	—
302	C₄⋊C₄	—	38	1	no	—
303	C₄⋊C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
304	C₂²⋊C₄	—	2	2	no	—
305	C₂²⋊C₄	—	2	2	no	—
306	C₂²⋊C₄	—	76	1	no	—
307	C₂²⋊C₄	—	76	1	no	—
308	C₄²	C₂×D₇₆	1	1	yes	characteristic
309	C₄²	#64541	1	1	yes	characteristic
310	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
311	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
312	C₂³	—	2	2	no	—
313	C₂³	—	4	1	no	—
314	C₂³	—	4	1	no	—
315	C₂³	—	8	1	no	—
316	C₂³	—	8	1	no	—
317	C₂³	—	38	2	no	—
318	C₂³	—	76	1	no	—
319	C₂³	—	76	1	no	—
320	C₂³	—	152	1	no	—
321	C₂³	—	152	1	no	—
322	C₂³	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
323	D₄	—	2	2	no	—
324	D₄	—	2	2	no	—
325	D₄	—	4	1	no	—
326	D₄	—	4	1	no	—
327	D₄	—	38	2	no	—
328	D₄	—	38	2	no	—
329	D₄	—	76	1	no	—
330	D₄	—	76	1	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	—	4	4	no	—
335	C₂×C₄	—	8	2	no	—
336	C₂×C₄	—	8	1	no	—
337	C₂×C₄	—	76	1	no	—
338	C₂×C₄	—	152	2	no	—
339	C₂×C₄	—	152	2	no	—
340	C₂×C₄	—	152	1	no	—
341	C₂×C₄	M₁₆×D₁₉	1	1	yes	characteristic
342	C₂×C₄	M₁₆×D₁₉	1	1	yes	characteristic
343	C₂×C₄	#10597353	1	1	yes	characteristic
344	C₂×C₄	#10597353	1	1	yes	characteristic
345	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
346	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
347	C₂×C₄	#10597375	1	1	yes	characteristic
348	C₂×C₄	#10597376	1	1	yes	characteristic
349	C₂×C₄	#10597376	1	1	yes	characteristic
350	C₂×C₄	#10597399	1	1	yes	characteristic
351	C₈	—	4	2	no	—
352	C₈	—	4	2	no	—
353	C₈	—	38	2	no	—
354	C₈	—	76	1	no	—
355	C₈	—	152	1	no	—
356	C₈	—	152	1	no	—
357	C₂²	—	2	1	no	—
358	C₂²	—	2	1	no	—
359	C₂²	—	4	2	no	—
360	C₂²	—	4	2	no	—
361	C₂²	—	4	2	no	—
362	C₂²	—	8	2	no	—
363	C₂²	—	8	2	no	—
364	C₂²	—	76	2	no	—
365	C₂²	—	152	2	no	—
366	C₂²	—	152	2	no	—
367	C₂²	—	152	1	no	—
368	C₂²	—	152	1	no	—
369	C₂²	#11926423	1	1	yes	characteristic
370	C₂²	#11926613	1	1	yes	characteristic
371	C₂²	#11926620	1	1	yes	characteristic
372	C₄	—	2	1	no	—
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	4	2	no	—
377	C₄	—	152	1	no	—
378	C₄	#11926507	1	1	yes	characteristic
379	C₄	#11926512	1	1	yes	characteristic
380	C₄	#11926613	1	1	yes	characteristic
381	C₄	#11926620	1	1	yes	characteristic
382	C₂	—	2	1	no	—
383	C₂	—	2	1	no	—
384	C₂	—	8	2	no	—
385	C₂	—	152	2	no	—
386	C₂	#14563932	1	1	yes	characteristic
387	C₂	#14564115	1	1	yes	characteristic
388	C₂	#14564122	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11926423
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#2975	C₂
socle	C₂²×C₁₉	#386
prime radical	C₂²×#64525	C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#2975	D₁₉
hypocenter	C₁₉	#9093
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2975

Complements

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

N	H ≅ G/N	type
2. #14562027	C₂	semidirect
3. #14562026	C₂	semidirect
4. #14561955	C₂	semidirect
5. #14561479	C₂	semidirect
6. #14561465	C₂	semidirect
7. C₁₉×#2975	C₂	semidirect
89. C₂×D₇₆	M₁₆	semidirect
116. C₂×C₁₉⋊C₈	M₁₆	semidirect
163. #2975	D₁₉	semidirect
262. C₁₉	#9093	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926654	2
#14562027	2
#14562026	2
#14561955	2
#14561479	2
#14561465	2
C₁₉×#2975	2
#9093	19