Subgroups — #18088176

Order 4864.

802 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926313	C₂	1	2	yes	maximal
2	C₂×#11926313	C₂	1	2	yes	maximal
3	#14563329	C₂	1	2	yes	maximal
4	C₂×C₄×C₄⋊C₄×C₁₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#64520	C₂²	1	4	yes	—
6	#11926313	—	2	4	no	—
7	#11926313	C₄	1	8	yes	complemented
8	#11926299	—	2	2	no	—
9	#11926299	—	2	2	no	—
10	C₂²×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
11	C₂²×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₁₉	C₄	1	2	yes	—
13	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
14	C₄×C₄⋊C₄×C₁₉	—	2	2	no	—
15	C₄×C₄⋊C₄×C₁₉	—	2	2	no	—
16	C₂×C₁₉×#95	C₄	1	2	yes	—
17	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
19	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
21	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
23	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
24	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
25	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
26	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	4	yes	—
27	C₂×C₄⋊C₄×C₁₉	D₄	1	2	yes	—
28	C₂×C₄⋊C₄×C₁₉	Q₈	1	2	yes	—
29	C₂×C₄²×C₁₉	—	2	2	no	—
30	C₂×C₄²×C₁₉	—	2	2	no	—
31	C₂×C₄²×C₁₉	—	2	2	no	—
32	C₂×C₄²×C₁₉	—	2	2	no	—
33	C₂×C₄²×C₁₉	—	4	4	no	—
34	C₂×C₄²×C₁₉	D₄	1	2	yes	—
35	C₂×C₄²×C₁₉	D₄	1	2	yes	—
36	C₂×#64520	—	2	4	no	—
37	C₂×#64520	—	2	4	no	—
38	C₂×#64520	—	2	4	no	—
39	C₂×#64520	—	2	4	no	—
40	C₂×#64520	—	2	4	no	—
41	C₂×#64520	C₂×C₄	1	8	yes	—
42	C₂²×C₁₉⋊C₈	—	2	4	no	—
43	C₁₉×#95	—	2	4	no	—
44	C₂⁴×C₁₉	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₉	—	2	1	no	—
46	C₂²×C₄×C₁₉	—	2	1	no	—
47	C₂²×C₄×C₁₉	—	2	2	no	—
48	C₂²×C₄×C₁₉	—	2	2	no	—
49	C₂²×C₄×C₁₉	—	2	2	no	—
50	C₂²×C₄×C₁₉	—	2	2	no	—
51	C₂²×C₄×C₁₉	—	2	1	no	—
52	C₂²×C₄×C₁₉	—	2	1	no	—
53	C₂²×C₄×C₁₉	—	2	2	no	—
54	C₂²×C₄×C₁₉	—	2	2	no	—
55	C₂²×C₄×C₁₉	—	4	1	no	—
56	C₂²×C₄×C₁₉	—	4	1	no	—
57	C₂²×C₄×C₁₉	—	4	2	no	—
58	C₂²×C₄×C₁₉	—	4	2	no	—
59	C₂²×C₄×C₁₉	—	4	2	no	—
60	C₂²×C₄×C₁₉	—	4	2	no	—
61	C₂²×C₄×C₁₉	C₄²	1	4	yes	—
62	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	C₄⋊C₄	1	4	yes	—
66	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
70	C₂²×C₄×C₁₉	#41	1	2	yes	—
71	C₂²×C₄×C₁₉	#41	1	2	yes	—
72	C₂²×C₄×C₁₉	#41	1	2	yes	—
73	C₂²×C₄×C₁₉	#41	1	2	yes	—
74	C₄⋊C₄×C₁₉	—	2	8	no	—
75	C₄⋊C₄×C₁₉	—	2	4	no	—
76	C₄⋊C₄×C₁₉	—	2	4	no	—
77	C₄²×C₁₉	—	2	4	no	—
78	C₄²×C₁₉	—	2	4	no	—
79	C₄²×C₁₉	—	4	4	no	—
80	C₄²×C₁₉	—	4	4	no	—
81	#64520	—	2	8	no	—
82	#64520	—	2	8	no	—
83	#64520	—	4	4	no	—
84	#64520	—	4	4	no	—
85	C₂×C₁₉⋊C₈	—	2	8	no	—
86	C₂×C₁₉⋊C₈	—	4	4	no	—
87	C₂×C₁₉⋊C₈	—	4	4	no	—
88	#10802	—	19	1	no	maximal, Sylow
89	C₂³×C₁₉	—	2	1	no	—
90	C₂³×C₁₉	—	2	1	no	—
91	C₂³×C₁₉	—	2	1	no	—
92	C₂³×C₁₉	—	2	1	no	—
93	C₂³×C₁₉	C₂×C₄²	1	1	yes	characteristic
94	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂³×C₁₉	#117	1	1	yes	characteristic
97	C₂³×C₁₉	#117	1	1	yes	characteristic
98	C₂³×C₁₉	#117	1	1	yes	characteristic
99	C₂³×C₁₉	#117	1	1	yes	characteristic
100	C₂×C₄×C₁₉	—	2	4	no	—
101	C₂×C₄×C₁₉	—	2	4	no	—
102	C₂×C₄×C₁₉	—	2	1	no	—
103	C₂×C₄×C₁₉	—	2	1	no	—
104	C₂×C₄×C₁₉	—	2	1	no	—
105	C₂×C₄×C₁₉	—	2	1	no	—
106	C₂×C₄×C₁₉	—	2	4	no	—
107	C₂×C₄×C₁₉	—	2	4	no	—
108	C₂×C₄×C₁₉	—	2	2	no	—
109	C₂×C₄×C₁₉	—	2	2	no	—
110	C₂×C₄×C₁₉	—	2	2	no	—
111	C₂×C₄×C₁₉	—	4	4	no	—
112	C₂×C₄×C₁₉	—	4	2	no	—
113	C₂×C₄×C₁₉	—	4	2	no	—
114	C₂×C₄×C₁₉	—	4	4	no	—
115	C₂×C₄×C₁₉	—	4	2	no	—
116	C₂×C₄×C₁₉	—	4	2	no	—
117	C₂×C₄×C₁₉	—	4	2	no	—
118	C₂×C₄×C₁₉	—	4	2	no	—
119	C₂×C₄×C₁₉	—	4	4	no	—
120	C₂×C₄×C₁₉	—	4	4	no	—
121	C₂×C₄×C₁₉	#100	1	2	yes	—
122	C₂×C₄×C₁₉	#101	1	2	yes	—
123	C₂×C₄×C₁₉	#117	1	4	yes	—
124	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
125	C₂×C₄×C₁₉	C₄×D₄	1	4	yes	—
126	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
127	C₂×C₄×C₁₉	#121	1	2	yes	—
128	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
129	C₂×C₄×C₁₉	#123	1	4	yes	—
130	C₂×C₄×C₁₉	#124	1	2	yes	—
131	C₂×C₄×C₁₉	#125	1	2	yes	—
132	C₂×C₄×C₁₉	#126	1	4	yes	—
133	C₁₉⋊C₈	—	4	8	no	—
134	C₂×C₄×C₄⋊C₄	D₁₉	1	1	yes	characteristic
135	C₂×#343	—	19	2	no	—
136	C₂×#343	—	19	2	no	—
137	#1728	—	19	2	no	—
138	C₂²×C₁₉	—	2	1	no	—
139	C₂²×C₁₉	—	2	1	no	—
140	C₂²×C₁₉	—	2	1	no	—
141	C₂²×C₁₉	—	2	1	no	—
142	C₂²×C₁₉	—	2	1	no	—
143	C₂²×C₁₉	—	2	1	no	—
144	C₂²×C₁₉	—	2	1	no	—
145	C₂²×C₁₉	—	2	1	no	—
146	C₂²×C₁₉	—	2	1	no	—
147	C₂²×C₁₉	—	2	1	no	—
148	C₂²×C₁₉	—	2	1	no	—
149	C₂²×C₁₉	—	2	1	no	—
150	C₂²×C₁₉	#376	1	1	yes	characteristic
151	C₂²×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
152	C₂²×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
153	C₂²×C₁₉	#380	1	1	yes	characteristic
154	C₂²×C₁₉	#381	1	1	yes	characteristic
155	C₂²×C₁₉	#382	1	1	yes	characteristic
156	C₂²×C₁₉	#383	1	1	yes	characteristic
157	C₂²×C₁₉	C₂×#100	1	1	yes	characteristic
158	C₂²×C₁₉	C₂×#101	1	1	yes	characteristic
159	C₂²×C₁₉	#413	1	1	yes	characteristic
160	C₂²×C₁₉	#413	1	1	yes	characteristic
161	C₄×C₁₉	—	2	2	no	—
162	C₄×C₁₉	—	2	2	no	—
163	C₄×C₁₉	—	4	4	no	—
164	C₄×C₁₉	—	4	2	no	—
165	C₄×C₁₉	—	4	4	no	—
166	C₄×C₁₉	#387	1	4	yes	—
167	C₄×C₁₉	#388	1	4	yes	—
168	C₂²×M₁₆	—	19	4	no	—
169	C₂²×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
170	C₂²×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
171	C₂²×C₄²	Dic₁₉	1	2	yes	—
172	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
173	C₄×C₄⋊C₄	—	2	2	no	—
174	C₄×C₄⋊C₄	—	2	2	no	—
175	C₂×#95	Dic₁₉	1	2	yes	—
176	#343	—	19	8	no	—
177	#343	—	38	4	no	—
178	#336	—	38	2	no	—
179	#336	—	38	2	no	—
180	C₂×C₁₉	—	2	1	no	—
181	C₂×C₁₉	—	2	1	no	—
182	C₂×C₁₉	—	2	1	no	—
183	C₂×C₁₉	—	2	1	no	—
184	C₂×C₁₉	#1434	1	1	yes	characteristic
185	C₂×C₁₉	C₄×#100	1	1	yes	characteristic
186	C₂×C₁₉	C₄×#101	1	1	yes	characteristic
187	C₂×C₁₉	#1800	1	1	yes	characteristic
188	C₂×C₁₉	#1801	1	1	yes	characteristic
189	C₂×C₁₉	#1821	1	1	yes	characteristic
190	C₂×C₁₉	#1822	1	1	yes	characteristic
191	C₂³×C₄	C₂×Dic₁₉	1	1	yes	characteristic
192	C₂³×C₄	C₂×Dic₁₉	1	2	yes	—
193	C₂³×C₄	C₂×Dic₁₉	1	2	yes	—
194	C₂³×C₄	C₂×Dic₁₉	1	1	yes	characteristic
195	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
196	C₂×M₁₆	—	19	8	no	—
197	C₂×M₁₆	—	38	4	no	—
198	C₂×M₁₆	—	38	4	no	—
199	C₂×M₁₆	—	38	4	no	—
200	C₂×M₁₆	—	38	4	no	—
201	C₂×M₁₆	—	38	4	no	—
202	C₂²×C₈	—	38	4	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	Dic₃₈	1	2	yes	—
208	C₂×C₄⋊C₄	C₄×D₁₉	1	4	yes	—
209	C₂×C₄⋊C₄	D₇₆	1	2	yes	—
210	C₂×C₄²	—	2	2	no	—
211	C₂×C₄²	—	2	2	no	—
212	C₂×C₄²	—	2	2	no	—
213	C₂×C₄²	—	2	2	no	—
214	C₂×C₄²	—	4	4	no	—
215	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
216	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
217	#95	—	2	4	no	—
218	C₁₉	#10802	1	1	yes	characteristic, complemented, Sylow
219	C₂⁴	C₂²×Dic₁₉	1	1	yes	characteristic
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	2	1	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	4	1	no	—
231	C₂²×C₄	—	4	1	no	—
232	C₂²×C₄	—	4	2	no	—
233	C₂²×C₄	—	4	2	no	—
234	C₂²×C₄	—	4	2	no	—
235	C₂²×C₄	—	4	2	no	—
236	C₂²×C₄	C₄×Dic₁₉	1	4	yes	—
237	C₂²×C₄	C₄⋊Dic₁₉	1	4	yes	—
238	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
239	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
240	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
241	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
242	C₂²×C₄	C₂×C₄×D₁₉	1	2	yes	—
243	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
244	C₂²×C₄	#64541	1	2	yes	—
245	C₂²×C₄	#64541	1	2	yes	—
246	C₂²×C₄	#64541	1	2	yes	—
247	C₂²×C₄	#64541	1	2	yes	—
248	C₂²×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
249	M₁₆	—	38	8	no	—
250	M₁₆	—	38	8	no	—
251	M₁₆	—	76	4	no	—
252	M₁₆	—	76	4	no	—
253	C₂×C₈	—	38	8	no	—
254	C₂×C₈	—	76	4	no	—
255	C₂×C₈	—	76	4	no	—
256	C₄⋊C₄	—	2	4	no	—
257	C₄⋊C₄	—	2	4	no	—
258	C₄⋊C₄	—	2	8	no	—
259	C₄²	—	2	4	no	—
260	C₄²	—	2	4	no	—
261	C₄²	—	4	4	no	—
262	C₄²	—	4	4	no	—
263	C₂³	—	2	1	no	—
264	C₂³	—	2	1	no	—
265	C₂³	—	2	1	no	—
266	C₂³	—	2	1	no	—
267	C₂³	C₂×C₄×Dic₁₉	1	1	yes	characteristic
268	C₂³	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
269	C₂³	#10597372	1	1	yes	characteristic
270	C₂³	#10597372	1	1	yes	characteristic
271	C₂³	#10597372	1	1	yes	characteristic
272	C₂³	#10597372	1	1	yes	characteristic
273	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	4	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	#10597290	1	2	yes	—
296	C₂×C₄	#10597293	1	2	yes	—
297	C₂×C₄	C₄×Dic₃₈	1	2	yes	—
298	C₂×C₄	#10597316	1	2	yes	—
299	C₂×C₄	#10597318	1	4	yes	—
300	C₂×C₄	C₄×D₇₆	1	2	yes	—
301	C₂×C₄	#10597321	1	2	yes	—
302	C₂×C₄	#10597322	1	4	yes	—
303	C₂×C₄	#10597370	1	2	yes	—
304	C₂×C₄	C₄×C₂²⋊D₁₉	1	4	yes	—
305	C₂×C₄	#10597375	1	4	yes	—
306	C₂×C₄	#10597376	1	2	yes	—
307	C₈	—	76	8	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	#11926390	1	1	yes	characteristic
321	C₂²	C₄×C₂²⋊Dic₁₉	1	1	yes	characteristic
322	C₂²	C₄×C₄⋊Dic₁₉	1	1	yes	characteristic
323	C₂²	#11926404	1	1	yes	characteristic
324	C₂²	#11926407	1	1	yes	characteristic
325	C₂²	#11926410	1	1	yes	characteristic
326	C₂²	#11926412	1	1	yes	characteristic
327	C₂²	C₂×#10597290	1	1	yes	characteristic
328	C₂²	C₂×#10597293	1	1	yes	characteristic
329	C₂²	#11926528	1	1	yes	characteristic
330	C₂²	#11926528	1	1	yes	characteristic
331	C₄	—	2	2	no	—
332	C₄	—	2	2	no	—
333	C₄	—	4	4	no	—
334	C₄	—	4	4	no	—
335	C₄	—	4	2	no	—
336	C₄	#11926428	1	4	yes	—
337	C₄	#11926435	1	4	yes	—
338	C₂	—	2	1	no	—
339	C₂	—	2	1	no	—
340	C₂	—	2	1	no	—
341	C₂	—	2	1	no	—
342	C₂	#14561862	1	1	yes	characteristic
343	C₂	C₄×#10597290	1	1	yes	characteristic
344	C₂	C₄×#10597293	1	1	yes	characteristic
345	C₂	#14563627	1	1	yes	characteristic
346	C₂	#14563631	1	1	yes	characteristic
347	C₂	#14563715	1	1	yes	characteristic
348	C₂	#14563718	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₁₉
commutator	C₂³×C₁₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₂×C₄×C₄⋊C₄×C₁₉	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₉	C₂²×C₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂×C₄×C₄⋊C₄	D₁₉
hypocenter	C₁₉	#10802
LCS	G ▷ C₂³×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
7. #11926313	C₄	semidirect
218. C₁₉	#10802	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926313	2
#14563329	2
C₂×C₄×C₄⋊C₄×C₁₉	2
#10802	19