Subgroups — #18074282

Order 4864.

966 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14563627	C₂	1	1	yes	maximal, characteristic
2	#14562316	C₂	1	2	yes	maximal, complemented
3	#14562064	C₂	1	2	yes	maximal, complemented
4	#14561956	C₂	1	1	yes	maximal, characteristic
5	C₂×D₄×Q₈×C₁₉	C₂	1	1	yes	maximal, characteristic
6	#11926597	C₂²	1	2	yes	—
7	#11926568	C₂²	1	2	yes	—
8	C₂×#10597290	—	2	1	no	—
9	#11926313	—	2	1	no	—
10	#11926293	—	2	4	no	—
11	Q₈⋊(C₁₉⋊C₈)	—	2	4	no	—
12	#11926268	—	2	2	no	—
13	C₂³×Q₈×C₁₉	C₄	1	2	yes	—
14	D₄×Q₈×C₁₉	—	2	4	no	—
15	D₄×Q₈×C₁₉	—	2	4	no	—
16	C₂×C₁₉×C₄⋊Q₈	—	2	1	no	—
17	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₂×C₁₉×C₂²⋊Q₈	—	2	2	no	—
19	C₂×C₁₉×C₂²⋊Q₈	C₄	1	2	yes	—
20	C₂×C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₁₉	—	2	1	no	—
22	C₂×C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
23	C₂²×Q₈×C₁₉	—	2	4	no	—
24	C₂²×Q₈×C₁₉	—	2	4	no	—
25	C₂²×Q₈×C₁₉	—	2	2	no	—
26	C₂²×Q₈×C₁₉	—	4	8	no	—
27	C₂²×Q₈×C₁₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×Q₈×C₁₉	C₂×C₄	1	2	yes	—
29	C₂²×D₄×C₁₉	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₁₉	—	2	2	no	—
31	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
32	C₁₉×C₄⋊Q₈	—	2	8	no	—
33	C₁₉×C₄⋊Q₈	—	2	2	no	—
34	C₁₉×C₄⋊Q₈	D₄	1	4	yes	—
35	C₁₉×C₂²⋊Q₈	—	2	16	no	—
36	C₁₉×C₂²⋊Q₈	—	2	4	no	—
37	C₁₉×C₂²⋊Q₈	—	2	4	no	—
38	C₄×Q₈×C₁₉	—	2	2	no	—
39	C₄×Q₈×C₁₉	D₄	1	4	yes	—
40	C₄×D₄×C₁₉	—	2	8	no	—
41	C₄×D₄×C₁₉	—	2	2	no	—
42	C₄×D₄×C₁₉	—	2	2	no	—
43	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
44	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
45	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
46	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
47	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
49	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
50	C₂×C₂²⋊C₄×C₁₉	D₄	1	2	yes	—
51	C₂×C₄²×C₁₉	—	2	1	no	—
52	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
53	C₂×#64520	—	2	2	no	—
54	#10597290	—	4	2	no	—
55	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
56	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
57	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
58	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
59	C₄×C₁₉⋊C₈	—	2	2	no	—
60	C₂⁴×C₁₉	C₂²⋊C₄	1	2	yes	—
61	C₂×Q₈×C₁₉	—	2	4	no	—
62	C₂×Q₈×C₁₉	—	2	8	no	—
63	C₂×Q₈×C₁₉	—	2	1	no	—
64	C₂×Q₈×C₁₉	—	2	2	no	—
65	C₂×Q₈×C₁₉	—	4	4	no	—
66	C₂×Q₈×C₁₉	—	4	8	no	—
67	C₂×Q₈×C₁₉	—	4	16	no	—
68	C₂×Q₈×C₁₉	—	4	8	no	—
69	C₂×Q₈×C₁₉	—	4	8	no	—
70	C₂×Q₈×C₁₉	C₂²⋊C₄	1	2	yes	—
71	C₂×Q₈×C₁₉	C₂²⋊C₄	1	4	yes	—
72	C₂×D₄×C₁₉	—	2	2	no	—
73	C₂×D₄×C₁₉	—	2	2	no	—
74	C₂×D₄×C₁₉	SD₁₆	1	2	yes	—
75	C₂×D₄×C₁₉	Q₁₆	1	2	yes	—
76	C₂²×C₄×C₁₉	—	2	1	no	—
77	C₂²×C₄×C₁₉	—	2	8	no	—
78	C₂²×C₄×C₁₉	—	2	1	no	—
79	C₂²×C₄×C₁₉	—	2	2	no	—
80	C₂²×C₄×C₁₉	—	2	2	no	—
81	C₂²×C₄×C₁₉	—	2	2	no	—
82	C₂²×C₄×C₁₉	—	4	8	no	—
83	C₂²×C₄×C₁₉	—	4	4	no	—
84	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
85	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
88	C₄⋊C₄×C₁₉	—	2	8	no	—
89	C₄⋊C₄×C₁₉	—	2	1	no	—
90	C₄⋊C₄×C₁₉	—	2	4	no	—
91	C₄⋊C₄×C₁₉	—	2	4	no	—
92	C₄⋊C₄×C₁₉	—	2	4	no	—
93	C₄⋊C₄×C₁₉	—	2	2	no	—
94	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
95	C₂²⋊C₄×C₁₉	—	2	8	no	—
96	C₂²⋊C₄×C₁₉	—	2	4	no	—
97	C₄²×C₁₉	—	2	4	no	—
98	C₄²×C₁₉	C₂×D₄	1	2	yes	—
99	C₄²×C₁₉	C₂×D₄	1	2	yes	—
100	#64520	—	4	2	no	—
101	#64520	—	4	2	no	—
102	C₂×C₁₉⋊C₈	—	4	2	no	—
103	C₂×C₁₉⋊C₈	—	4	2	no	—
104	#8391	—	19	1	no	maximal, Sylow
105	C₂³×C₁₉	—	2	2	no	—
106	C₂³×C₁₉	—	2	2	no	—
107	C₂³×C₁₉	—	4	2	no	—
108	C₂³×C₁₉	—	4	2	no	—
109	C₂³×C₁₉	#103	1	4	yes	—
110	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	Q₈×C₁₉	—	2	4	no	—
112	Q₈×C₁₉	—	2	2	no	—
113	Q₈×C₁₉	—	4	4	no	—
114	Q₈×C₁₉	—	4	16	no	—
115	Q₈×C₁₉	—	4	4	no	—
116	Q₈×C₁₉	#100	1	4	yes	complemented
117	D₄×C₁₉	—	2	4	no	—
118	D₄×C₁₉	—	2	4	no	—
119	C₂×C₄×C₁₉	—	2	4	no	—
120	C₂×C₄×C₁₉	—	2	2	no	—
121	C₂×C₄×C₁₉	—	2	2	no	—
122	C₂×C₄×C₁₉	—	2	2	no	—
123	C₂×C₄×C₁₉	—	2	4	no	—
124	C₂×C₄×C₁₉	—	2	2	no	—
125	C₂×C₄×C₁₉	—	4	1	no	—
126	C₂×C₄×C₁₉	—	4	16	no	—
127	C₂×C₄×C₁₉	—	4	4	no	—
128	C₂×C₄×C₁₉	—	4	2	no	—
129	C₂×C₄×C₁₉	—	4	4	no	—
130	C₂×C₄×C₁₉	—	4	4	no	—
131	C₂×C₄×C₁₉	—	4	2	no	—
132	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₁₉	#120	1	2	yes	—
135	C₂×C₄×C₁₉	#120	1	2	yes	—
136	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
137	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
138	C₂×C₄×C₁₉	#137	1	2	yes	—
139	C₁₉⋊C₈	—	8	2	no	—
140	C₁₉⋊C₈	—	8	2	no	—
141	C₂×D₄×Q₈	D₁₉	1	1	yes	characteristic
142	#1800	—	19	1	no	—
143	#1548	—	19	2	no	—
144	#1487	—	19	2	no	—
145	#1460	—	19	1	no	—
146	C₂²×C₁₉	—	2	1	no	—
147	C₂²×C₁₉	—	2	1	no	—
148	C₂²×C₁₉	—	2	4	no	—
149	C₂²×C₁₉	—	4	4	no	—
150	C₂²×C₁₉	—	4	4	no	—
151	C₂²×C₁₉	—	4	2	no	—
152	C₂²×C₁₉	—	4	2	no	—
153	C₂²×C₁₉	#379	1	1	yes	characteristic
154	C₂²×C₁₉	C₂×#103	1	1	yes	characteristic
155	C₂²×C₁₉	#419	1	1	yes	characteristic
156	C₄×C₁₉	—	2	2	no	—
157	C₄×C₁₉	—	2	4	no	—
158	C₄×C₁₉	—	2	1	no	—
159	C₄×C₁₉	—	4	2	no	—
160	C₄×C₁₉	—	4	4	no	—
161	C₄×C₁₉	—	4	2	no	—
162	C₄×C₁₉	C₂×#100	1	2	yes	—
163	C₄×C₁₉	#448	1	2	yes	—
164	C₄×C₁₉	#451	1	2	yes	—
165	C₂³×Q₈	Dic₁₉	1	2	yes	—
166	D₄×Q₈	—	2	4	no	—
167	D₄×Q₈	—	2	4	no	—
168	C₂×C₄⋊Q₈	—	2	1	no	—
169	C₂×C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
170	C₂×C₂²⋊Q₈	—	2	2	no	—
171	C₂×C₂²⋊Q₈	Dic₁₉	1	2	yes	—
172	C₂×C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic
173	C₂×C₄×D₄	—	2	1	no	—
174	C₂×C₄×D₄	C₂×D₁₉	1	1	yes	characteristic
175	#431	—	19	2	no	—
176	#423	—	19	2	no	—
177	C₂×#100	—	38	1	no	—
178	#343	—	38	1	no	—
179	#333	—	38	4	no	—
180	Q₈⋊C₈	—	38	4	no	—
181	#323	—	38	2	no	—
182	C₂×C₁₉	—	2	1	no	—
183	C₂×C₁₉	—	2	1	no	—
184	C₂×C₁₉	—	4	4	no	—
185	C₂×C₁₉	#1783	1	1	yes	characteristic
186	C₂×C₁₉	#1787	1	1	yes	characteristic
187	C₂×C₁₉	#1796	1	1	yes	characteristic
188	C₂²×Q₈	—	2	4	no	—
189	C₂²×Q₈	—	2	4	no	—
190	C₂²×Q₈	—	2	2	no	—
191	C₂²×Q₈	—	4	8	no	—
192	C₂²×Q₈	C₂×Dic₁₉	1	1	yes	characteristic
193	C₂²×Q₈	C₂×Dic₁₉	1	2	yes	—
194	C₂²×D₄	C₂²⋊D₁₉	1	1	yes	characteristic
195	C₂³×C₄	—	2	2	no	—
196	C₂³×C₄	C₂×Dic₁₉	1	2	yes	—
197	C₂×M₁₆	—	38	2	no	—
198	C₄⋊Q₈	—	2	8	no	—
199	C₄⋊Q₈	—	2	2	no	—
200	C₄⋊Q₈	C₂²⋊D₁₉	1	4	yes	—
201	C₂²⋊Q₈	—	2	16	no	—
202	C₂²⋊Q₈	—	2	4	no	—
203	C₂²⋊Q₈	—	2	4	no	—
204	C₄×Q₈	—	2	2	no	—
205	C₄×Q₈	C₂²⋊D₁₉	1	4	yes	—
206	C₄×D₄	—	2	8	no	—
207	C₄×D₄	—	2	2	no	—
208	C₄×D₄	—	2	2	no	—
209	C₂×C₄⋊C₄	—	2	2	no	—
210	C₂×C₄⋊C₄	—	2	1	no	—
211	C₂×C₄⋊C₄	—	2	1	no	—
212	C₂×C₄⋊C₄	—	2	1	no	—
213	C₂×C₄⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
214	C₂×C₄⋊C₄	C₂²⋊D₁₉	1	1	yes	characteristic
215	C₂×C₂²⋊C₄	—	2	2	no	—
216	C₂×C₂²⋊C₄	C₂²⋊D₁₉	1	2	yes	—
217	C₂×C₄²	—	2	1	no	—
218	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
219	C₄⋊C₈	—	38	2	no	—
220	C₄⋊C₈	—	38	2	no	—
221	C₄⋊C₈	—	38	2	no	—
222	#100	—	76	2	no	—
223	C₂²⋊C₈	—	38	2	no	—
224	C₄×C₈	—	38	2	no	—
225	C₁₉	#8391	1	1	yes	characteristic, complemented, Sylow
226	C₂⁴	C₂²⋊Dic₁₉	1	2	yes	—
227	C₂×Q₈	—	2	4	no	—
228	C₂×Q₈	—	2	8	no	—
229	C₂×Q₈	—	2	1	no	—
230	C₂×Q₈	—	2	2	no	—
231	C₂×Q₈	—	4	4	no	—
232	C₂×Q₈	—	4	8	no	—
233	C₂×Q₈	—	4	16	no	—
234	C₂×Q₈	—	4	8	no	—
235	C₂×Q₈	—	4	8	no	—
236	C₂×Q₈	C₂²⋊Dic₁₉	1	2	yes	—
237	C₂×Q₈	C₂²⋊Dic₁₉	1	4	yes	—
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	#64527	1	2	yes	—
241	C₂×D₄	#64528	1	2	yes	—
242	C₂²×C₄	—	2	1	no	—
243	C₂²×C₄	—	2	8	no	—
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	4	8	no	—
249	C₂²×C₄	—	4	4	no	—
250	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
251	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
252	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
253	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
254	M₁₆	—	76	2	no	—
255	M₁₆	—	76	2	no	—
256	C₂×C₈	—	76	2	no	—
257	C₂×C₈	—	76	2	no	—
258	C₄⋊C₄	—	2	8	no	—
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	2	4	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	C₂²⋊Dic₁₉	1	2	yes	—
265	C₂²⋊C₄	—	2	8	no	—
266	C₂²⋊C₄	—	2	4	no	—
267	C₄²	—	2	4	no	—
268	C₄²	C₂×C₂²⋊D₁₉	1	2	yes	—
269	C₄²	C₂×C₂²⋊D₁₉	1	2	yes	—
270	C₂³	—	2	2	no	—
271	C₂³	—	2	2	no	—
272	C₂³	—	4	2	no	—
273	C₂³	—	4	2	no	—
274	C₂³	#10597292	1	4	yes	—
275	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
276	Q₈	—	2	4	no	—
277	Q₈	—	2	2	no	—
278	Q₈	—	4	4	no	—
279	Q₈	—	4	16	no	—
280	Q₈	—	4	4	no	—
281	Q₈	#10597290	1	4	yes	complemented
282	D₄	—	2	4	no	—
283	D₄	—	2	4	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	4	16	no	—
292	C₂×C₄	—	4	4	no	—
293	C₂×C₄	—	4	4	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	4	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	C₂×#64527	1	1	yes	characteristic
298	C₂×C₄	#10597388	1	2	yes	—
299	C₂×C₄	C₂×#64528	1	1	yes	characteristic
300	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
301	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
302	C₂×C₄	#10597399	1	2	yes	—
303	C₂×C₄	#10597399	1	2	yes	—
304	C₈	—	152	2	no	—
305	C₈	—	152	2	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	—	2	4	no	—
309	C₂²	—	4	4	no	—
310	C₂²	—	4	4	no	—
311	C₂²	—	4	2	no	—
312	C₂²	—	4	2	no	—
313	C₂²	#11926400	1	1	yes	characteristic
314	C₂²	C₂×#10597292	1	1	yes	characteristic
315	C₂²	#11926553	1	1	yes	characteristic
316	C₄	—	2	2	no	—
317	C₄	—	2	4	no	—
318	C₄	—	2	1	no	—
319	C₄	—	4	2	no	—
320	C₄	—	4	4	no	—
321	C₄	—	4	2	no	—
322	C₄	C₂×#10597290	1	2	yes	—
323	C₄	#11926683	1	2	yes	—
324	C₄	#11926700	1	2	yes	—
325	C₂	—	2	1	no	—
326	C₂	—	2	1	no	—
327	C₂	—	4	4	no	—
328	C₂	#14563556	1	1	yes	characteristic
329	C₂	#14563570	1	1	yes	characteristic
330	C₂	#14563610	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11926400
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×D₄×Q₈×C₁₉	C₂
socle	C₂²×C₁₉	#379
prime radical	C₂²×D₄×C₁₉	D₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×D₄×Q₈	D₁₉
hypocenter	C₁₉	#8391
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×D₄×Q₈

Complements

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

N	H ≅ G/N	type
2. #14562316	C₂	semidirect
3. #14562064	C₂	semidirect
116. Q₈×C₁₉	#100	semidirect
225. C₁₉	#8391	semidirect
281. Q₈	#10597290	semidirect

Maximal subgroups

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

subgroup	index
#14563627	2
#14562316	2
#14562064	2
#14561956	2
C₂×D₄×Q₈×C₁₉	2
#8391	19