Subgroups — #18081427

Order 4864.

534 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#64527	C₂	1	1	yes	maximal, characteristic
2	#14564429	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926294	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×Q₈⋊(C₁₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11926271	C₂	1	1	yes	maximal, characteristic
6	C₂×#11926265	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1984	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#64527	C₄	1	1	yes	characteristic
9	C₂×C₄×D₇₆	C₂²	1	1	yes	characteristic
10	C₄×#64527	—	2	2	no	—
11	C₄×#64527	—	2	2	no	—
12	C₂×#10597265	C₄	1	1	yes	characteristic
13	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×#10597292	C₄	1	1	yes	characteristic
15	C₂×#10597266	C₄	1	1	yes	characteristic
16	C₂×(C₂×D₁₉)⋊C₈	—	2	2	no	—
17	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
19	#11926294	—	2	2	no	—
20	Q₈⋊(C₁₉⋊C₈)	—	2	2	no	—
21	#11926271	—	2	2	no	—
22	#11926265	—	2	2	no	—
23	C₂×C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
24	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
26	C₁₉×#336	—	2	2	no	—
27	C₂²×Q₈×C₁₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×D₇₆	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×D₁₉	—	2	1	no	—
30	C₂²×C₈×C₁₉	—	2	1	no	—
31	C₂²×C₈×C₁₉	D₄	1	2	yes	—
32	C₄×Q₈×C₁₉	—	2	1	no	—
33	C₄×Q₈×C₁₉	—	2	1	no	—
34	C₄×Q₈×C₁₉	—	2	1	no	—
35	C₄×Q₈×C₁₉	—	2	1	no	—
36	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
37	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₉	—	2	1	no	—
39	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
40	C₂×#64527	—	2	2	no	—
41	C₂×#64527	—	2	2	no	—
42	C₂×#64527	—	2	2	no	—
43	C₂×(C₂×D₁₉)⋊C₄	—	2	1	no	—
44	C₂×C₄⋊Dic₁₉	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₁₉⋊C₈	—	2	1	no	—
46	C₂²×C₁₉⋊C₈	—	2	1	no	—
47	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
49	C₄×D₇₆	—	2	1	no	—
50	C₄×D₇₆	—	2	1	no	—
51	C₄×D₇₆	—	2	1	no	—
52	C₄×D₇₆	—	2	1	no	—
53	C₁₉×C₄⋊C₈	—	2	1	no	—
54	C₁₉×C₄⋊C₈	—	2	1	no	—
55	C₄×C₈×C₁₉	—	2	1	no	—
56	C₄×C₈×C₁₉	—	2	1	no	—
57	#10597292	—	2	2	no	—
58	(C₂×D₁₉)⋊C₈	—	4	4	no	—
59	#10597266	—	2	2	no	—
60	#10597265	—	2	2	no	—
61	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
62	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
63	C₄×C₁₉⋊C₈	—	2	1	no	—
64	C₄×C₁₉⋊C₈	D₄	1	2	yes	—
65	C₄×C₁₉⋊C₈	D₄	1	2	yes	—
66	C₄×C₁₉⋊C₈	D₄	1	2	yes	—
67	C₂³×D₁₉	—	2	1	no	—
68	C₂×Q₈×C₁₉	—	2	1	no	—
69	C₂×Q₈×C₁₉	—	2	1	no	—
70	C₂×Q₈×C₁₉	—	2	1	no	—
71	C₂×Q₈×C₁₉	—	2	1	no	—
72	C₂×Q₈×C₁₉	—	2	1	no	—
73	C₂×Q₈×C₁₉	M₁₆	1	2	yes	—
74	C₂²×C₄×C₁₉	—	2	1	no	—
75	C₂²×C₄×C₁₉	—	2	1	no	—
76	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
79	C₂²×Dic₁₉	—	2	1	no	—
80	C₂×D₇₆	—	2	1	no	—
81	C₂×D₇₆	—	2	1	no	—
82	C₂×D₇₆	—	2	1	no	—
83	C₂×D₇₆	—	2	1	no	—
84	C₂×D₇₆	—	2	1	no	—
85	C₂×D₇₆	M₁₆	1	2	yes	—
86	C₂×C₄×D₁₉	—	4	1	no	—
87	C₂×C₄×D₁₉	—	4	1	no	—
88	C₂×C₄×D₁₉	—	4	2	no	—
89	C₂×C₄×D₁₉	—	4	2	no	—
90	C₂×C₈×C₁₉	—	2	2	no	—
91	C₂×C₈×C₁₉	—	2	2	no	—
92	C₂×C₈×C₁₉	—	4	1	no	—
93	C₂×C₈×C₁₉	—	4	2	no	—
94	C₂×C₈×C₁₉	SD₁₆	1	4	yes	complemented
95	C₄⋊C₄×C₁₉	—	2	1	no	—
96	C₄⋊C₄×C₁₉	—	2	1	no	—
97	C₄⋊C₄×C₁₉	—	4	1	no	—
98	C₄⋊C₄×C₁₉	—	4	1	no	—
99	C₄²×C₁₉	—	4	1	no	—
100	C₄²×C₁₉	—	4	1	no	—
101	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₁₉	#41	1	1	yes	characteristic
105	#64527	—	2	8	no	—
106	(C₂×D₁₉)⋊C₄	—	4	1	no	—
107	(C₂×D₁₉)⋊C₄	—	4	1	no	—
108	C₄⋊Dic₁₉	—	2	1	no	—
109	C₄⋊Dic₁₉	—	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	—	2	2	no	—
115	C₂×C₁₉⋊C₈	—	4	2	no	—
116	C₂×C₁₉⋊C₈	—	4	1	no	—
117	C₂×C₁₉⋊C₈	—	4	2	no	—
118	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	2	yes	—
119	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	2	yes	—
120	#9661	—	19	1	no	maximal, Sylow
121	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
122	C₂²×D₁₉	—	2	2	no	—
123	C₂²×D₁₉	—	4	1	no	—
124	C₂²×D₁₉	—	4	1	no	—
125	C₂²×D₁₉	—	4	2	no	—
126	C₂²×D₁₉	—	4	2	no	—
127	Q₈×C₁₉	—	2	2	no	—
128	Q₈×C₁₉	—	2	2	no	—
129	Q₈×C₁₉	—	4	1	no	—
130	Q₈×C₁₉	—	4	1	no	—
131	C₂×C₄×C₁₉	—	2	2	no	—
132	C₂×C₄×C₁₉	—	2	2	no	—
133	C₂×C₄×C₁₉	—	4	1	no	—
134	C₂×C₄×C₁₉	—	4	1	no	—
135	C₂×C₄×C₁₉	—	4	1	no	—
136	C₂×C₄×C₁₉	—	4	2	no	—
137	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
140	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
141	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
142	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
143	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
144	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
145	C₂×C₄×C₁₉	#131	1	1	yes	characteristic
146	C₂×C₄×C₁₉	#131	1	1	yes	characteristic
147	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
148	C₂×C₄×C₁₉	#135	1	2	yes	—
149	C₂×Dic₁₉	—	4	1	no	—
150	C₂×Dic₁₉	—	4	2	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	1	no	—
157	C₄×D₁₉	—	8	1	no	—
158	C₈×C₁₉	—	2	4	no	—
159	C₈×C₁₉	—	8	1	no	—
160	C₁₉⋊C₈	—	2	2	no	—
161	C₁₉⋊C₈	—	2	2	no	—
162	C₁₉⋊C₈	—	4	2	no	—
163	C₁₉⋊C₈	—	8	1	no	—
164	C₂×C₄×SD₁₆	—	19	1	no	—
165	#1984	D₁₉	1	1	yes	characteristic, complemented
166	#1954	—	19	1	no	—
167	C₂×#334	—	19	1	no	—
168	C₂×Q₈⋊C₈	—	19	1	no	—
169	C₂×D₄⋊C₈	—	19	1	no	—
170	C₂×#322	—	19	1	no	—
171	C₂²×C₁₉	#390	1	1	yes	characteristic
172	C₂²×C₁₉	#435	1	1	yes	characteristic
173	C₂²×C₁₉	#435	1	1	yes	characteristic
174	C₂²×C₁₉	C₄×SD₁₆	1	2	yes	—
175	C₂²×C₁₉	#444	1	2	yes	—
176	C₂×D₁₉	—	4	4	no	—
177	C₂×D₁₉	—	4	2	no	—
178	C₂×D₁₉	—	8	2	no	—
179	C₂×D₁₉	—	8	1	no	—
180	C₂×D₁₉	—	8	1	no	—
181	C₄×C₁₉	—	2	1	no	—
182	C₄×C₁₉	—	2	1	no	—
183	C₄×C₁₉	—	4	2	no	—
184	C₄×C₁₉	—	8	1	no	—
185	C₄×C₁₉	#407	1	1	yes	characteristic
186	C₄×C₁₉	#408	1	1	yes	characteristic
187	C₄×C₁₉	#435	1	1	yes	characteristic
188	C₄×C₁₉	#435	1	1	yes	characteristic
189	C₄×C₁₉	#465	1	2	yes	—
190	C₄×C₁₉	#492	1	1	yes	characteristic
191	C₄×C₁₉	#495	1	1	yes	characteristic
192	Dic₁₉	—	8	1	no	—
193	C₂²×SD₁₆	—	19	1	no	—
194	C₂×C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic
195	C₂×C₄×D₄	—	19	1	no	—
196	C₄×SD₁₆	—	38	2	no	—
197	C₄×SD₁₆	—	38	2	no	—
198	C₂×#106	—	19	1	no	—
199	C₂×C₄⋊C₈	—	19	1	no	—
200	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
201	C₂×#103	—	19	1	no	—
202	C₂×#102	—	19	1	no	—
203	C₂×C₂²⋊C₈	—	38	2	no	—
204	C₂×C₄×C₈	—	19	1	no	—
205	C₂×C₄×C₈	—	19	1	no	—
206	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
207	#336	—	2	2	no	—
208	#334	—	38	2	no	—
209	Q₈⋊C₈	—	38	2	no	—
210	D₄⋊C₈	—	38	2	no	—
211	#322	—	38	2	no	—
212	C₂×C₁₉	#1582	1	2	yes	—
213	C₂×C₁₉	#1583	1	2	yes	—
214	C₂×C₁₉	#1955	1	1	yes	characteristic
215	C₂×C₁₉	#1968	1	1	yes	characteristic
216	C₂×C₁₉	#1975	1	1	yes	characteristic
217	D₁₉	—	8	2	no	—
218	C₂²×Q₈	C₄×D₁₉	1	1	yes	characteristic
219	C₂²×D₄	—	19	1	no	—
220	C₂³×C₄	—	38	1	no	—
221	C₂×SD₁₆	—	38	2	no	—
222	C₂×SD₁₆	—	38	2	no	—
223	C₂×SD₁₆	—	38	2	no	—
224	C₂²×C₈	—	2	1	no	—
225	C₂²×C₈	—	19	1	no	—
226	C₂²×C₈	—	19	1	no	—
227	C₂²×C₈	—	38	1	no	—
228	C₂²×C₈	—	38	1	no	—
229	C₂²×C₈	C₂²⋊D₁₉	1	2	yes	—
230	C₄×Q₈	—	2	1	no	—
231	C₄×Q₈	—	2	1	no	—
232	C₄×Q₈	—	2	1	no	—
233	C₄×Q₈	—	2	1	no	—
234	C₄×D₄	—	38	1	no	—
235	C₄×D₄	—	38	1	no	—
236	C₄×D₄	—	38	1	no	—
237	C₄×D₄	—	38	1	no	—
238	C₂×C₄⋊C₄	—	2	1	no	—
239	C₂×C₄⋊C₄	—	19	1	no	—
240	C₂×C₄⋊C₄	C₄×D₁₉	1	1	yes	characteristic
241	C₂×C₂²⋊C₄	—	38	1	no	—
242	C₂×C₄²	—	2	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	#103	—	38	2	no	—
250	#102	—	38	2	no	—
251	C₂²⋊C₈	—	76	4	no	—
252	C₄×C₈	—	2	1	no	—
253	C₄×C₈	—	2	1	no	—
254	C₄×C₈	—	19	2	no	—
255	C₄×C₈	—	19	2	no	—
256	C₄×C₈	—	19	2	no	—
257	C₄×C₈	—	38	1	no	—
258	C₁₉	#9661	1	1	yes	characteristic, complemented, Sylow
259	C₂⁴	—	38	1	no	—
260	C₂×Q₈	—	2	1	no	—
261	C₂×Q₈	—	2	1	no	—
262	C₂×Q₈	—	2	1	no	—
263	C₂×Q₈	—	2	1	no	—
264	C₂×Q₈	—	2	1	no	—
265	C₂×Q₈	#64515	1	2	yes	—
266	C₂×D₄	—	19	2	no	—
267	C₂×D₄	—	38	1	no	—
268	C₂×D₄	—	38	1	no	—
269	C₂×D₄	—	38	1	no	—
270	C₂×D₄	—	38	1	no	—
271	C₂×D₄	—	38	1	no	—
272	C₂²×C₄	—	2	1	no	—
273	C₂²×C₄	—	2	1	no	—
274	C₂²×C₄	—	38	1	no	—
275	C₂²×C₄	—	76	1	no	—
276	C₂²×C₄	—	76	1	no	—
277	C₂²×C₄	—	76	2	no	—
278	C₂²×C₄	—	76	2	no	—
279	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
280	C₂²×C₄	#64541	1	1	yes	characteristic
281	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
282	SD₁₆	—	38	8	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	4	1	no	—
286	C₂×C₈	—	4	2	no	—
287	C₂×C₈	—	19	2	no	—
288	C₂×C₈	—	19	2	no	—
289	C₂×C₈	—	38	1	no	—
290	C₂×C₈	—	38	1	no	—
291	C₂×C₈	—	38	1	no	—
292	C₂×C₈	—	38	1	no	—
293	C₂×C₈	—	38	2	no	—
294	C₂×C₈	—	76	2	no	—
295	C₂×C₈	—	76	1	no	—
296	C₂×C₈	—	76	2	no	—
297	C₂×C₈	#64527	1	4	yes	complemented
298	C₄⋊C₄	—	2	1	no	—
299	C₄⋊C₄	—	2	1	no	—
300	C₄⋊C₄	—	4	1	no	—
301	C₄⋊C₄	—	4	1	no	—
302	C₄⋊C₄	—	38	1	no	—
303	C₄⋊C₄	—	38	1	no	—
304	C₂²⋊C₄	—	76	1	no	—
305	C₂²⋊C₄	—	76	1	no	—
306	C₄²	—	4	1	no	—
307	C₄²	—	4	1	no	—
308	C₄²	D₄×D₁₉	1	1	yes	characteristic
309	C₄²	D₄×D₁₉	1	1	yes	characteristic
310	C₄²	D₄×D₁₉	1	1	yes	characteristic
311	C₄²	#64543	1	1	yes	characteristic
312	C₂³	—	38	2	no	—
313	C₂³	—	76	1	no	—
314	C₂³	—	76	2	no	—
315	C₂³	—	76	2	no	—
316	C₂³	—	76	1	no	—
317	C₂³	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
318	Q₈	—	2	2	no	—
319	Q₈	—	2	2	no	—
320	Q₈	—	4	1	no	—
321	Q₈	—	4	1	no	—
322	D₄	—	38	2	no	—
323	D₄	—	38	2	no	—
324	D₄	—	76	1	no	—
325	D₄	—	76	1	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	4	1	no	—
329	C₂×C₄	—	4	1	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	76	2	no	—
333	C₂×C₄	—	76	1	no	—
334	C₂×C₄	—	152	1	no	—
335	C₂×C₄	—	152	1	no	—
336	C₂×C₄	—	152	2	no	—
337	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
338	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
339	C₂×C₄	#10597330	1	1	yes	characteristic
340	C₂×C₄	#10597331	1	1	yes	characteristic
341	C₂×C₄	C₂×#64515	1	1	yes	characteristic
342	C₂×C₄	#10597347	1	1	yes	characteristic
343	C₂×C₄	M₁₆×D₁₉	1	1	yes	characteristic
344	C₂×C₄	#10597353	1	1	yes	characteristic
345	C₂×C₄	#10597385	1	1	yes	characteristic
346	C₂×C₄	#10597386	1	1	yes	characteristic
347	C₂×C₄	C₂×#64527	1	2	yes	—
348	C₂×C₄	#10597396	1	2	yes	—
349	C₈	—	2	4	no	—
350	C₈	—	8	1	no	—
351	C₈	—	38	2	no	—
352	C₈	—	38	2	no	—
353	C₈	—	76	2	no	—
354	C₈	—	152	1	no	—
355	C₂²	—	76	2	no	—
356	C₂²	—	76	4	no	—
357	C₂²	—	152	1	no	—
358	C₂²	—	152	2	no	—
359	C₂²	—	152	1	no	—
360	C₂²	#11926444	1	1	yes	characteristic
361	C₂²	#11926613	1	1	yes	characteristic
362	C₂²	#11926615	1	1	yes	characteristic
363	C₂²	C₄×#64527	1	2	yes	—
364	C₂²	#11926664	1	2	yes	—
365	C₄	—	2	1	no	—
366	C₄	—	2	1	no	—
367	C₄	—	4	2	no	—
368	C₄	—	8	1	no	—
369	C₄	—	152	1	no	—
370	C₄	#11926506	1	1	yes	characteristic
371	C₄	#11926509	1	1	yes	characteristic
372	C₄	#11926612	1	1	yes	characteristic
373	C₄	#11926618	1	1	yes	characteristic
374	C₄	#11926775	1	2	yes	—
375	C₄	#11926918	1	1	yes	characteristic
376	C₄	#11926926	1	1	yes	characteristic
377	C₂	—	152	2	no	—
378	C₂	#14562469	1	2	yes	—
379	C₂	#14562478	1	2	yes	—
380	C₂	#14564431	1	1	yes	characteristic
381	C₂	#14564492	1	1	yes	characteristic
382	C₂	#14564527	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₉
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1984	C₂
socle	C₂³×C₁₉	C₄×D₄
prime radical	C₂²×D₇₆	C₂×C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1984	D₁₉
hypocenter	C₁₉	#9661
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1984

Complements

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

N	H ≅ G/N	type
3. C₂×#11926294	C₂	semidirect
4. C₂×Q₈⋊(C₁₉⋊C₈)	C₂	semidirect
6. C₂×#11926265	C₂	semidirect
7. C₁₉×#1984	C₂	semidirect
94. C₂×C₈×C₁₉	SD₁₆	semidirect
165. #1984	D₁₉	semidirect
258. C₁₉	#9661	semidirect
297. C₂×C₈	#64527	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#64527	2
#14564429	2
C₂×#11926294	2
C₂×Q₈⋊(C₁₉⋊C₈)	2
C₂×#11926271	2
C₂×#11926265	2
C₁₉×#1984	2
#9661	19