Subgroups — #18073168

Order 4864.

578 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Q₈×Dic₁₉	C₂	1	1	yes	maximal, characteristic
2	#14563490	C₂	1	1	yes	maximal, characteristic
3	#14563380	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₈⋊(C₁₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
5	#14561475	C₂	1	1	yes	maximal, characteristic
6	#14561474	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₉×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₈×Dic₁₉	C₄	1	2	yes	—
9	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂×#10597272	—	2	1	no	—
11	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×Dic₁₉	—	2	2	no	—
13	C₄⋊C₄×Dic₁₉	C₄	1	2	yes	—
14	C₄×C₄⋊Dic₁₉	—	2	1	no	—
15	C₄×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
16	C₄²×Dic₁₉	—	2	1	no	—
17	C₄²×Dic₁₉	C₂²	1	1	yes	characteristic
18	#11926298	—	2	2	no	—
19	#11926298	—	2	1	no	—
20	Q₈⋊(C₁₉⋊C₈)	C₄	1	4	yes	complemented
21	C₂×C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
22	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
24	C₁₉×Q₈⋊C₈	C₄	1	4	yes	complemented
25	C₂²×Q₈×C₁₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₉	—	2	1	no	—
27	C₂²×C₈×C₁₉	—	2	1	no	—
28	C₄×Q₈×C₁₉	—	2	1	no	—
29	C₄×Q₈×C₁₉	—	2	1	no	—
30	C₄×Q₈×C₁₉	C₂×C₄	1	2	yes	—
31	C₄×Q₈×C₁₉	C₂×C₄	1	2	yes	—
32	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
33	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₉	—	2	1	no	—
35	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
36	Q₈×Dic₁₉	—	2	4	no	—
37	Q₈×Dic₁₉	—	2	4	no	—
38	C₂×C₄⋊Dic₁₉	—	2	2	no	—
39	C₂×C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
40	C₂×C₄×Dic₁₉	—	2	2	no	—
41	C₂×C₄×Dic₁₉	—	2	1	no	—
42	C₂×C₄×Dic₁₉	—	2	2	no	—
43	C₂×C₄×Dic₁₉	C₂×C₄	1	2	yes	—
44	C₂×C₄×Dic₁₉	D₄	1	1	yes	characteristic
45	C₂×C₄×Dic₁₉	D₄	1	2	yes	—
46	C₂×C₄×Dic₁₉	D₄	1	1	yes	characteristic
47	C₂²×C₁₉⋊C₈	—	2	1	no	—
48	C₂²×C₁₉⋊C₈	—	2	1	no	—
49	C₁₉×C₄⋊C₈	—	2	1	no	—
50	C₁₉×C₄⋊C₈	C₂×C₄	1	2	yes	—
51	C₄×C₈×C₁₉	—	2	1	no	—
52	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
53	C₁₉⋊#95	—	2	1	no	—
54	C₁₉⋊#95	—	2	2	no	—
55	#10597272	—	4	2	no	—
56	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
57	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	2	yes	—
58	C₄×C₁₉⋊C₈	—	2	1	no	—
59	C₄×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
60	C₂×Q₈×C₁₉	—	2	1	no	—
61	C₂×Q₈×C₁₉	—	2	2	no	—
62	C₂×Q₈×C₁₉	C₄²	1	2	yes	—
63	C₂×Q₈×C₁₉	M₁₆	1	4	yes	—
64	C₂²×C₄×C₁₉	—	2	1	no	—
65	C₂²×C₄×C₁₉	—	2	1	no	—
66	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
69	C₂²×Dic₁₉	—	2	2	no	—
70	C₂²×Dic₁₉	C₂²⋊C₄	1	2	yes	—
71	C₂²×Dic₁₉	C₂²⋊C₄	1	2	yes	—
72	C₂×C₈×C₁₉	—	2	2	no	—
73	C₂×C₈×C₁₉	—	2	2	no	—
74	C₂×C₈×C₁₉	—	4	2	no	—
75	C₂×C₈×C₁₉	—	4	1	no	—
76	C₂×C₈×C₁₉	—	4	1	no	—
77	C₄⋊C₄×C₁₉	—	2	1	no	—
78	C₄⋊C₄×C₁₉	—	2	2	no	—
79	C₄⋊C₄×C₁₉	—	2	2	no	—
80	C₄⋊C₄×C₁₉	C₄²	1	2	yes	—
81	C₄²×C₁₉	—	2	2	no	—
82	C₄²×C₁₉	—	2	2	no	—
83	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₁₉	#41	1	1	yes	characteristic
86	C₄²×C₁₉	#41	1	1	yes	characteristic
87	C₄⋊Dic₁₉	—	2	8	no	—
88	C₄⋊Dic₁₉	—	2	2	no	—
89	C₄⋊Dic₁₉	M₁₆	1	4	yes	—
90	C₄×Dic₁₉	—	2	8	no	—
91	C₄×Dic₁₉	—	2	2	no	—
92	C₄×Dic₁₉	—	2	2	no	—
93	C₄×Dic₁₉	—	2	1	no	—
94	C₄×Dic₁₉	—	2	1	no	—
95	C₄×Dic₁₉	—	4	2	no	—
96	C₄×Dic₁₉	—	4	4	no	—
97	C₄×Dic₁₉	—	4	1	no	—
98	C₄×Dic₁₉	—	4	1	no	—
99	C₄×Dic₁₉	SD₁₆	1	2	yes	—
100	C₄×Dic₁₉	Q₁₆	1	2	yes	—
101	C₂×C₁₉⋊C₈	—	2	2	no	—
102	C₂×C₁₉⋊C₈	—	2	2	no	—
103	C₂×C₁₉⋊C₈	—	4	2	no	—
104	C₂×C₁₉⋊C₈	—	4	1	no	—
105	C₂×C₁₉⋊C₈	—	4	1	no	—
106	#8175	—	19	1	no	maximal, Sylow
107	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	Q₈×C₁₉	—	2	2	no	—
109	Q₈×C₁₉	—	2	4	no	—
110	Q₈×C₁₉	#97	1	4	yes	complemented
111	C₂×C₄×C₁₉	—	2	2	no	—
112	C₂×C₄×C₁₉	—	2	4	no	—
113	C₂×C₄×C₁₉	—	2	2	no	—
114	C₂×C₄×C₁₉	—	2	2	no	—
115	C₂×C₄×C₁₉	—	4	2	no	—
116	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
117	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
118	C₂×C₄×C₁₉	C₄×D₄	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	C₂×M₁₆	1	2	yes	—
124	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
126	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
127	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
128	C₂×Dic₁₉	—	2	2	no	—
129	C₂×Dic₁₉	—	2	2	no	—
130	C₂×Dic₁₉	—	2	2	no	—
131	C₂×Dic₁₉	—	2	4	no	—
132	C₂×Dic₁₉	—	2	2	no	—
133	C₂×Dic₁₉	—	2	2	no	—
134	C₂×Dic₁₉	—	4	2	no	—
135	C₂×Dic₁₉	—	4	2	no	—
136	C₂×Dic₁₉	#103	1	4	yes	—
137	C₈×C₁₉	—	4	2	no	—
138	C₈×C₁₉	—	4	2	no	—
139	C₁₉⋊C₈	—	4	2	no	—
140	C₁₉⋊C₈	—	4	2	no	—
141	C₄²×Q₈	—	19	1	no	—
142	#1767	—	19	1	no	—
143	#1744	—	19	1	no	—
144	C₂×Q₈⋊C₈	—	19	1	no	—
145	C₂×Q₈⋊C₈	D₁₉	1	1	yes	characteristic
146	#1279	—	19	1	no	—
147	#1279	—	19	1	no	—
148	C₂²×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
149	C₂²×C₁₉	#407	1	2	yes	—
150	C₂²×C₁₉	C₂×#103	1	1	yes	characteristic
151	C₂²×C₁₉	#416	1	1	yes	characteristic
152	C₂²×C₁₉	#421	1	2	yes	—
153	C₄×C₁₉	—	2	4	no	—
154	C₄×C₁₉	—	2	1	no	—
155	C₄×C₁₉	—	2	1	no	—
156	C₄×C₁₉	—	4	2	no	—
157	C₄×C₁₉	C₂×#97	1	1	yes	characteristic
158	C₄×C₁₉	C₄×M₁₆	1	1	yes	characteristic
159	C₄×C₁₉	#432	1	2	yes	—
160	C₄×C₁₉	C₄×SD₁₆	1	1	yes	characteristic
161	C₄×C₁₉	C₄×SD₁₆	1	1	yes	characteristic
162	C₄×C₁₉	C₄×Q₁₆	1	1	yes	characteristic
163	C₄×C₁₉	C₄×Q₁₆	1	1	yes	characteristic
164	Dic₁₉	—	2	4	no	—
165	Dic₁₉	—	4	2	no	—
166	Dic₁₉	—	4	4	no	—
167	C₂×C₄×Q₈	—	19	2	no	—
168	C₂×C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic
169	C₂×C₄⋊C₈	—	19	1	no	—
170	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
171	C₂×#97	—	38	1	no	—
172	C₂×C₄×C₈	—	19	1	no	—
173	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
174	C₄×C₄⋊C₄	—	19	2	no	—
175	C₄×C₄⋊C₄	—	19	1	no	—
176	C₄×C₄⋊C₄	—	38	1	no	—
177	C₄×C₄⋊C₄	—	38	2	no	—
178	C₄³	—	19	1	no	—
179	C₄³	—	38	1	no	—
180	#336	—	38	2	no	—
181	#336	—	38	1	no	—
182	Q₈⋊C₈	—	19	4	no	—
183	Q₈⋊C₈	Dic₁₉	1	4	yes	complemented
184	C₂×C₁₉	#1487	1	2	yes	—
185	C₂×C₁₉	#1488	1	2	yes	—
186	C₂×C₁₉	#1749	1	1	yes	characteristic
187	C₂×C₁₉	#1759	1	1	yes	characteristic
188	C₂×C₁₉	C₄×#103	1	1	yes	characteristic
189	C₂²×Q₈	C₄×D₁₉	1	1	yes	characteristic
190	C₂²×C₈	—	2	1	no	—
191	C₂²×C₈	—	2	1	no	—
192	C₂²×C₈	—	38	1	no	—
193	C₂²×C₈	—	38	1	no	—
194	C₄×Q₈	—	2	1	no	—
195	C₄×Q₈	—	2	1	no	—
196	C₄×Q₈	—	38	4	no	—
197	C₄×Q₈	—	38	4	no	—
198	C₄×Q₈	C₄×D₁₉	1	2	yes	—
199	C₄×Q₈	C₂×Dic₁₉	1	2	yes	—
200	C₂×C₄⋊C₄	—	2	1	no	—
201	C₂×C₄⋊C₄	—	19	2	no	—
202	C₂×C₄⋊C₄	—	38	2	no	—
203	C₂×C₄⋊C₄	C₄×D₁₉	1	1	yes	characteristic
204	C₂×C₄²	—	2	1	no	—
205	C₂×C₄²	—	19	1	no	—
206	C₂×C₄²	—	19	2	no	—
207	C₂×C₄²	—	19	1	no	—
208	C₂×C₄²	—	19	2	no	—
209	C₂×C₄²	—	38	2	no	—
210	C₂×C₄²	—	38	2	no	—
211	C₂×C₄²	—	38	1	no	—
212	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
213	C₄⋊C₈	—	2	1	no	—
214	C₄⋊C₈	—	19	2	no	—
215	C₄⋊C₈	—	38	1	no	—
216	C₄⋊C₈	C₂×Dic₁₉	1	2	yes	—
217	#97	—	76	2	no	—
218	C₄×C₈	—	2	1	no	—
219	C₄×C₈	—	19	2	no	—
220	C₄×C₈	—	38	1	no	—
221	C₄×C₈	C₂×Dic₁₉	1	2	yes	—
222	#95	—	38	2	no	—
223	#95	—	38	1	no	—
224	C₁₉	#8175	1	1	yes	characteristic, complemented, Sylow
225	C₂×Q₈	—	2	1	no	—
226	C₂×Q₈	—	2	2	no	—
227	C₂×Q₈	#64515	1	4	yes	—
228	C₂×Q₈	C₄×Dic₁₉	1	2	yes	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	—	19	2	no	—
232	C₂²×C₄	—	19	2	no	—
233	C₂²×C₄	—	38	2	no	—
234	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
235	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
236	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	4	2	no	—
240	C₂×C₈	—	4	1	no	—
241	C₂×C₈	—	4	1	no	—
242	C₂×C₈	—	38	2	no	—
243	C₂×C₈	—	38	2	no	—
244	C₂×C₈	—	76	2	no	—
245	C₂×C₈	—	76	1	no	—
246	C₂×C₈	—	76	1	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	19	4	no	—
251	C₄⋊C₄	—	38	8	no	—
252	C₄⋊C₄	—	38	2	no	—
253	C₄⋊C₄	C₄×Dic₁₉	1	2	yes	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	19	2	no	—
257	C₄²	—	19	2	no	—
258	C₄²	—	38	8	no	—
259	C₄²	—	38	2	no	—
260	C₄²	—	38	2	no	—
261	C₄²	—	38	1	no	—
262	C₄²	—	38	1	no	—
263	C₄²	—	76	2	no	—
264	C₄²	—	76	4	no	—
265	C₄²	—	76	1	no	—
266	C₄²	—	76	1	no	—
267	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
268	C₄²	#64543	1	1	yes	characteristic
269	C₄²	#64543	1	1	yes	characteristic
270	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
271	C₂³	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
272	Q₈	—	2	4	no	—
273	Q₈	—	2	2	no	—
274	Q₈	#10597272	1	4	yes	complemented
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	19	4	no	—
281	C₂×C₄	—	38	2	no	—
282	C₂×C₄	—	38	2	no	—
283	C₂×C₄	—	38	2	no	—
284	C₂×C₄	—	38	2	no	—
285	C₂×C₄	—	38	4	no	—
286	C₂×C₄	—	38	2	no	—
287	C₂×C₄	—	76	2	no	—
288	C₂×C₄	—	76	2	no	—
289	C₂×C₄	#10597323	1	1	yes	characteristic
290	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
291	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
292	C₂×C₄	C₂×#64515	1	2	yes	—
293	C₂×C₄	M₁₆×D₁₉	1	2	yes	—
294	C₂×C₄	SD₁₆×D₁₉	1	1	yes	characteristic
295	C₂×C₄	#10597362	1	1	yes	characteristic
296	C₂×C₄	Q₁₆×D₁₉	1	1	yes	characteristic
297	C₂×C₄	#10597365	1	1	yes	characteristic
298	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
299	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
300	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
301	C₈	—	4	2	no	—
302	C₈	—	4	2	no	—
303	C₈	—	76	2	no	—
304	C₈	—	76	2	no	—
305	C₂²	C₂²⋊C₄×Dic₁₉	1	1	yes	characteristic
306	C₂²	#11926506	1	2	yes	—
307	C₂²	#103×D₁₉	1	1	yes	characteristic
308	C₂²	#11926539	1	1	yes	characteristic
309	C₂²	#11926559	1	2	yes	—
310	C₄	—	2	4	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	4	2	no	—
314	C₄	—	38	4	no	—
315	C₄	—	76	2	no	—
316	C₄	—	76	4	no	—
317	C₄	C₂×#10597272	1	1	yes	characteristic
318	C₄	M₁₆×Dic₁₉	1	1	yes	characteristic
319	C₄	#11926598	1	2	yes	—
320	C₄	#11926629	1	1	yes	characteristic
321	C₄	SD₁₆×Dic₁₉	1	1	yes	characteristic
322	C₄	#11926636	1	1	yes	characteristic
323	C₄	Q₁₆×Dic₁₉	1	1	yes	characteristic
324	C₂	#14562062	1	2	yes	—
325	C₂	#14562068	1	2	yes	—
326	C₂	#14563399	1	1	yes	characteristic
327	C₂	#14563447	1	1	yes	characteristic
328	C₂	#103×Dic₁₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₉
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₁₉×Q₈⋊C₈	C₂
socle, prime radical	C₂³×C₁₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×Q₈⋊C₈	D₁₉
hypocenter	C₁₉	#8175
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×Q₈⋊C₈

Complements

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

N	H ≅ G/N	type
20. Q₈⋊(C₁₉⋊C₈)	C₄	semidirect
24. C₁₉×Q₈⋊C₈	C₄	semidirect
110. Q₈×C₁₉	#97	semidirect
183. Q₈⋊C₈	Dic₁₉	semidirect
224. C₁₉	#8175	semidirect
274. Q₈	#10597272	semidirect

Maximal subgroups

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

subgroup	index
C₄×Q₈×Dic₁₉	2
#14563490	2
#14563380	2
C₂×Q₈⋊(C₁₉⋊C₈)	2
#14561475	2
#14561474	2
C₂×C₁₉×Q₈⋊C₈	2
#8175	19