Subgroups — #18087757

Order 4864.

2038 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂²⋊(C₁₉⋊C₈)	C₂	1	2	yes	maximal
2	C₂×#11926299	C₂	1	4	yes	maximal, complemented
3	C₂³×C₄²×C₁₉	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₁₉⋊C₈	C₂²	1	4	yes	—
5	C₂×C₂²⋊(C₁₉⋊C₈)	—	2	4	no	—
6	C₂×C₂²⋊(C₁₉⋊C₈)	—	2	4	no	—
7	C₂×C₂²⋊(C₁₉⋊C₈)	C₄	1	8	yes	complemented
8	#11926299	—	2	8	no	—
9	C₂⁴×C₄×C₁₉	C₄	1	2	yes	—
10	C₂⁴×C₄×C₁₉	C₂²	1	1	yes	characteristic
11	C₂²×C₄²×C₁₉	—	2	4	no	—
12	C₂²×C₄²×C₁₉	—	2	8	no	—
13	C₂²×C₄²×C₁₉	C₄	1	2	yes	—
14	C₂²×C₄²×C₁₉	C₂²	1	2	yes	—
15	C₂⁵×C₁₉	C₂×C₄	1	1	yes	characteristic
16	C₂³×C₄×C₁₉	—	2	4	no	—
17	C₂³×C₄×C₁₉	—	2	4	no	—
18	C₂³×C₄×C₁₉	—	2	8	no	—
19	C₂³×C₄×C₁₉	—	2	8	no	—
20	C₂³×C₄×C₁₉	—	2	4	no	—
21	C₂³×C₄×C₁₉	—	2	4	no	—
22	C₂³×C₄×C₁₉	C₈	1	8	yes	complemented
23	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₉	C₂×C₄	1	4	yes	—
25	C₂³×C₄×C₁₉	C₂×C₄	1	4	yes	—
26	C₂³×C₄×C₁₉	D₄	1	2	yes	—
27	C₂³×C₄×C₁₉	D₄	1	4	yes	—
28	C₂³×C₄×C₁₉	Q₈	1	2	yes	—
29	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
30	C₂×C₄²×C₁₉	—	2	32	no	—
31	C₂×C₄²×C₁₉	—	2	4	no	—
32	C₂×C₄²×C₁₉	—	2	8	no	—
33	C₂×C₄²×C₁₉	—	2	4	no	—
34	C₂×C₄²×C₁₉	—	2	4	no	—
35	C₂×C₄²×C₁₉	D₄	1	8	yes	—
36	C₂²×C₁₉⋊C₈	—	2	8	no	—
37	C₂²×C₁₉⋊C₈	—	2	4	no	—
38	C₂²×C₁₉⋊C₈	—	2	4	no	—
39	C₂²×C₁₉⋊C₈	—	2	8	no	—
40	C₂²×C₁₉⋊C₈	C₂×C₄	1	8	yes	complemented
41	C₂²⋊(C₁₉⋊C₈)	—	2	16	no	—
42	C₂⁴×C₁₉	—	2	2	no	—
43	C₂⁴×C₁₉	—	2	2	no	—
44	C₂⁴×C₁₉	—	2	4	no	—
45	C₂⁴×C₁₉	C₄²	1	2	yes	—
46	C₂⁴×C₁₉	C₂²⋊C₄	1	2	yes	—
47	C₂⁴×C₁₉	C₄⋊C₄	1	2	yes	—
48	C₂⁴×C₁₉	C₂×C₈	1	4	yes	—
49	C₂⁴×C₁₉	M₁₆	1	4	yes	—
50	C₂⁴×C₁₉	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₉	—	2	16	no	—
52	C₂²×C₄×C₁₉	—	2	32	no	—
53	C₂²×C₄×C₁₉	—	2	16	no	—
54	C₂²×C₄×C₁₉	—	2	16	no	—
55	C₂²×C₄×C₁₉	—	2	2	no	—
56	C₂²×C₄×C₁₉	—	2	4	no	—
57	C₂²×C₄×C₁₉	—	2	4	no	—
58	C₂²×C₄×C₁₉	—	2	2	no	—
59	C₂²×C₄×C₁₉	—	2	4	no	—
60	C₂²×C₄×C₁₉	—	2	4	no	—
61	C₂²×C₄×C₁₉	—	2	32	no	—
62	C₂²×C₄×C₁₉	—	2	16	no	—
63	C₂²×C₄×C₁₉	—	2	8	no	—
64	C₂²×C₄×C₁₉	—	2	8	no	—
65	C₂²×C₄×C₁₉	—	2	8	no	—
66	C₂²×C₄×C₁₉	—	2	8	no	—
67	C₂²×C₄×C₁₉	C₄²	1	2	yes	—
68	C₂²×C₄×C₁₉	C₂²⋊C₄	1	8	yes	—
69	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
70	C₂²×C₄×C₁₉	C₂²⋊C₄	1	8	yes	—
71	C₂²×C₄×C₁₉	C₄⋊C₄	1	8	yes	—
72	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
73	C₂²×C₄×C₁₉	C₂×C₈	1	8	yes	—
74	C₂²×C₄×C₁₉	M₁₆	1	8	yes	—
75	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₁₉	C₂×D₄	1	4	yes	—
79	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
80	C₂²×C₄×C₁₉	#41	1	4	yes	—
81	C₄²×C₁₉	—	2	16	no	—
82	C₄²×C₁₉	—	2	16	no	—
83	C₂×C₁₉⋊C₈	—	2	8	no	—
84	C₂×C₁₉⋊C₈	—	2	16	no	—
85	C₂×C₁₉⋊C₈	—	4	4	no	—
86	C₂×C₁₉⋊C₈	—	4	8	no	—
87	C₂×C₁₉⋊C₈	—	4	4	no	—
88	#10712	—	19	1	no	maximal, Sylow
89	C₂³×C₁₉	—	2	4	no	—
90	C₂³×C₁₉	—	2	2	no	—
91	C₂³×C₁₉	—	2	4	no	—
92	C₂³×C₁₉	—	2	2	no	—
93	C₂³×C₁₉	—	2	4	no	—
94	C₂³×C₁₉	—	2	8	no	—
95	C₂³×C₁₉	—	2	4	no	—
96	C₂³×C₁₉	—	2	8	no	—
97	C₂³×C₁₉	—	2	4	no	—
98	C₂³×C₁₉	—	2	8	no	—
99	C₂³×C₁₉	—	2	8	no	—
100	C₂³×C₁₉	#95	1	4	yes	—
101	C₂³×C₁₉	C₄×C₈	1	4	yes	—
102	C₂³×C₁₉	#97	1	4	yes	—
103	C₂³×C₁₉	C₂²⋊C₈	1	8	yes	—
104	C₂³×C₁₉	C₄⋊C₈	1	8	yes	—
105	C₂³×C₁₉	C₂×C₄²	1	1	yes	characteristic
106	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂³×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
108	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂³×C₁₉	#117	1	2	yes	—
110	C₂³×C₁₉	C₂²×C₈	1	2	yes	—
111	C₂³×C₁₉	C₂×M₁₆	1	2	yes	—
112	C₂³×C₁₉	C₂×M₁₆	1	4	yes	—
113	C₂×C₄×C₁₉	—	2	64	no	—
114	C₂×C₄×C₁₉	—	2	4	no	—
115	C₂×C₄×C₁₉	—	2	2	no	—
116	C₂×C₄×C₁₉	—	2	8	no	—
117	C₂×C₄×C₁₉	—	2	4	no	—
118	C₂×C₄×C₁₉	—	2	2	no	—
119	C₂×C₄×C₁₉	—	2	8	no	—
120	C₂×C₄×C₁₉	—	2	8	no	—
121	C₂×C₄×C₁₉	—	2	8	no	—
122	C₂×C₄×C₁₉	—	2	8	no	—
123	C₂×C₄×C₁₉	—	2	8	no	—
124	C₂×C₄×C₁₉	—	2	16	no	—
125	C₂×C₄×C₁₉	—	2	16	no	—
126	C₂×C₄×C₁₉	#95	1	4	yes	—
127	C₂×C₄×C₁₉	C₂²⋊C₈	1	16	yes	complemented
128	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
129	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
130	C₂×C₄×C₁₉	C₄×D₄	1	8	yes	—
131	C₂×C₄×C₁₉	#120	1	4	yes	—
132	C₂×C₄×C₁₉	#121	1	4	yes	—
133	C₂×C₄×C₁₉	C₂²⋊Q₈	1	4	yes	—
134	C₂×C₄×C₁₉	#123	1	4	yes	—
135	C₁₉⋊C₈	—	4	8	no	—
136	C₂³×C₄²	D₁₉	1	1	yes	characteristic
137	C₂²×C₂²⋊C₈	—	19	2	no	—
138	C₂×#336	—	19	4	no	—
139	C₂²×C₁₉	—	2	16	no	—
140	C₂²×C₁₉	—	2	8	no	—
141	C₂²×C₁₉	—	2	4	no	—
142	C₂²×C₁₉	—	2	4	no	—
143	C₂²×C₁₉	—	2	8	no	—
144	C₂²×C₁₉	—	2	8	no	—
145	C₂²×C₁₉	—	2	8	no	—
146	C₂²×C₁₉	#336	1	8	yes	complemented
147	C₂²×C₁₉	C₂×#95	1	1	yes	characteristic
148	C₂²×C₁₉	C₄×C₂²⋊C₄	1	2	yes	—
149	C₂²×C₁₉	#379	1	1	yes	characteristic
150	C₂²×C₁₉	#380	1	2	yes	—
151	C₂²×C₁₉	#381	1	1	yes	characteristic
152	C₂²×C₁₉	C₂×C₄×C₈	1	1	yes	characteristic
153	C₂²×C₁₉	C₂×#97	1	1	yes	characteristic
154	C₂²×C₁₉	C₄×M₁₆	1	2	yes	—
155	C₂²×C₁₉	C₂×C₂²⋊C₈	1	4	yes	—
156	C₂²×C₁₉	C₂×C₂²⋊C₈	1	2	yes	—
157	C₂²×C₁₉	#407	1	4	yes	—
158	C₂²×C₁₉	#407	1	2	yes	—
159	C₂²×C₁₉	C₂×C₄⋊C₈	1	2	yes	—
160	C₂²×C₁₉	#423	1	2	yes	—
161	C₂²×C₁₉	#431	1	4	yes	—
162	C₂²×C₁₉	#432	1	4	yes	—
163	C₄×C₁₉	—	2	4	no	—
164	C₄×C₁₉	—	2	16	no	—
165	C₄×C₁₉	#385	1	4	yes	—
166	C₄×C₁₉	#386	1	4	yes	—
167	C₂⁴×C₄	Dic₁₉	1	2	yes	—
168	C₂⁴×C₄	C₂×D₁₉	1	1	yes	characteristic
169	C₂³×C₈	—	19	4	no	—
170	C₂²×C₄²	—	2	4	no	—
171	C₂²×C₄²	—	2	8	no	—
172	C₂²×C₄²	Dic₁₉	1	2	yes	—
173	C₂²×C₄²	C₂×D₁₉	1	2	yes	—
174	C₂×C₂²⋊C₈	—	19	8	no	—
175	C₂×C₂²⋊C₈	—	38	4	no	—
176	C₂×C₂²⋊C₈	—	38	4	no	—
177	#336	—	38	8	no	—
178	C₂×C₁₉	—	2	8	no	—
179	C₂×C₁₉	C₂²⋊#95	1	1	yes	characteristic
180	C₂×C₁₉	C₂×#336	1	1	yes	characteristic
181	C₂×C₁₉	#1728	1	1	yes	characteristic
182	C₂×C₁₉	C₄×C₂²⋊C₈	1	2	yes	—
183	C₂×C₁₉	#1749	1	2	yes	—
184	C₂×C₁₉	#1779	1	2	yes	—
185	C₂×C₁₉	#1797	1	4	yes	—
186	C₂×C₁₉	#1817	1	2	yes	—
187	C₂⁵	C₂×Dic₁₉	1	1	yes	characteristic
188	C₂³×C₄	—	2	4	no	—
189	C₂³×C₄	—	2	4	no	—
190	C₂³×C₄	—	2	8	no	—
191	C₂³×C₄	—	2	8	no	—
192	C₂³×C₄	—	2	4	no	—
193	C₂³×C₄	—	2	4	no	—
194	C₂³×C₄	C₁₉⋊C₈	1	8	yes	complemented
195	C₂³×C₄	Dic₃₈	1	2	yes	—
196	C₂³×C₄	C₄×D₁₉	1	4	yes	—
197	C₂³×C₄	D₇₆	1	2	yes	—
198	C₂³×C₄	C₂×Dic₁₉	1	4	yes	—
199	C₂³×C₄	C₂×Dic₁₉	1	1	yes	characteristic
200	C₂³×C₄	C₂²⋊D₁₉	1	4	yes	—
201	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
202	C₂²×C₈	—	19	8	no	—
203	C₂²×C₈	—	38	4	no	—
204	C₂²×C₈	—	38	8	no	—
205	C₂²×C₈	—	38	8	no	—
206	C₂²×C₈	—	38	4	no	—
207	C₂×C₄²	—	2	32	no	—
208	C₂×C₄²	—	2	8	no	—
209	C₂×C₄²	—	2	4	no	—
210	C₂×C₄²	—	2	4	no	—
211	C₂×C₄²	—	2	4	no	—
212	C₂×C₄²	C₂²⋊D₁₉	1	8	yes	—
213	C₂²⋊C₈	—	38	16	no	—
214	C₁₉	#10712	1	1	yes	characteristic, complemented, Sylow
215	C₂⁴	—	2	2	no	—
216	C₂⁴	—	2	4	no	—
217	C₂⁴	—	2	2	no	—
218	C₂⁴	C₂×C₁₉⋊C₈	1	4	yes	—
219	C₂⁴	#64520	1	4	yes	—
220	C₂⁴	C₄×Dic₁₉	1	2	yes	—
221	C₂⁴	C₄⋊Dic₁₉	1	2	yes	—
222	C₂⁴	C₂²⋊Dic₁₉	1	2	yes	—
223	C₂⁴	C₂²×Dic₁₉	1	1	yes	characteristic
224	C₂²×C₄	—	2	16	no	—
225	C₂²×C₄	—	2	16	no	—
226	C₂²×C₄	—	2	32	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	4	no	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	—	2	16	no	—
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	8	no	—
235	C₂²×C₄	—	2	8	no	—
236	C₂²×C₄	—	2	32	no	—
237	C₂²×C₄	—	2	8	no	—
238	C₂²×C₄	—	2	16	no	—
239	C₂²×C₄	—	2	8	no	—
240	C₂²×C₄	C₂×C₁₉⋊C₈	1	8	yes	—
241	C₂²×C₄	#64520	1	8	yes	—
242	C₂²×C₄	C₄×Dic₁₉	1	2	yes	—
243	C₂²×C₄	Dic₁₉⋊C₄	1	8	yes	—
244	C₂²×C₄	C₄⋊Dic₁₉	1	2	yes	—
245	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	8	yes	—
246	C₂²×C₄	C₂²⋊Dic₁₉	1	8	yes	—
247	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
248	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
249	C₂²×C₄	C₂×C₄×D₁₉	1	2	yes	—
250	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
251	C₂²×C₄	#64541	1	4	yes	—
252	C₂²×C₄	C₂×C₂²⋊D₁₉	1	4	yes	—
253	C₂²×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
254	C₂×C₈	—	38	8	no	—
255	C₂×C₈	—	38	16	no	—
256	C₂×C₈	—	76	8	no	—
257	C₂×C₈	—	76	4	no	—
258	C₂×C₈	—	76	4	no	—
259	C₄²	—	2	16	no	—
260	C₄²	—	2	16	no	—
261	C₂³	—	2	4	no	—
262	C₂³	—	2	8	no	—
263	C₂³	—	2	8	no	—
264	C₂³	—	2	4	no	—
265	C₂³	—	2	2	no	—
266	C₂³	—	2	4	no	—
267	C₂³	—	2	2	no	—
268	C₂³	—	2	4	no	—
269	C₂³	—	2	8	no	—
270	C₂³	—	2	8	no	—
271	C₂³	—	2	4	no	—
272	C₂³	C₄×C₁₉⋊C₈	1	4	yes	—
273	C₂³	#10597260	1	4	yes	—
274	C₂³	C₄⋊(C₁₉⋊C₈)	1	8	yes	—
275	C₂³	C₂²⋊(C₁₉⋊C₈)	1	8	yes	—
276	C₂³	C₁₉⋊#95	1	4	yes	—
277	C₂³	C₂²×C₁₉⋊C₈	1	2	yes	—
278	C₂³	C₂×#64520	1	2	yes	—
279	C₂³	C₂×#64520	1	4	yes	—
280	C₂³	C₂×C₄×Dic₁₉	1	1	yes	characteristic
281	C₂³	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
282	C₂³	#10597372	1	2	yes	—
283	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
284	C₂³	C₂×C₂²⋊Dic₁₉	1	2	yes	—
285	C₂×C₄	—	2	64	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	8	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	8	no	—
292	C₂×C₄	—	2	8	no	—
293	C₂×C₄	—	2	8	no	—
294	C₂×C₄	—	2	8	no	—
295	C₂×C₄	—	2	8	no	—
296	C₂×C₄	—	2	16	no	—
297	C₂×C₄	—	2	16	no	—
298	C₂×C₄	C₂²⋊(C₁₉⋊C₈)	1	16	yes	complemented
299	C₂×C₄	C₁₉⋊#95	1	4	yes	—
300	C₂×C₄	C₂×Dic₁₉⋊C₄	1	2	yes	—
301	C₂×C₄	#10597370	1	4	yes	—
302	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	2	yes	—
303	C₂×C₄	C₄×C₂²⋊D₁₉	1	8	yes	—
304	C₂×C₄	#10597375	1	4	yes	—
305	C₂×C₄	#10597376	1	4	yes	—
306	C₂×C₄	#10597399	1	4	yes	—
307	C₈	—	76	8	no	—
308	C₂²	—	2	4	no	—
309	C₂²	—	2	16	no	—
310	C₂²	—	2	8	no	—
311	C₂²	—	2	4	no	—
312	C₂²	—	2	8	no	—
313	C₂²	—	2	8	no	—
314	C₂²	—	2	8	no	—
315	C₂²	#11926299	1	8	yes	complemented
316	C₂²	C₂×C₁₉⋊#95	1	1	yes	characteristic
317	C₂²	C₄×C₂²⋊Dic₁₉	1	2	yes	—
318	C₂²	#11926400	1	1	yes	characteristic
319	C₂²	#11926404	1	2	yes	—
320	C₂²	#11926407	1	1	yes	characteristic
321	C₂²	C₂×C₄×C₁₉⋊C₈	1	1	yes	characteristic
322	C₂²	C₂×#10597260	1	1	yes	characteristic
323	C₂²	C₄×#64520	1	2	yes	—
324	C₂²	C₂×C₂²⋊(C₁₉⋊C₈)	1	4	yes	—
325	C₂²	C₂×C₂²⋊(C₁₉⋊C₈)	1	2	yes	—
326	C₂²	#11926508	1	2	yes	—
327	C₂²	#11926508	1	4	yes	—
328	C₂²	C₂×C₄⋊(C₁₉⋊C₈)	1	2	yes	—
329	C₂²	#11926568	1	2	yes	—
330	C₂²	#11926597	1	4	yes	—
331	C₂²	#11926601	1	4	yes	—
332	C₄	—	2	4	no	—
333	C₄	—	2	16	no	—
334	C₄	#11926417	1	4	yes	—
335	C₄	#11926423	1	4	yes	—
336	C₂	—	2	8	no	—
337	C₂	#14561858	1	1	yes	characteristic
338	C₂	C₂×#11926299	1	1	yes	characteristic
339	C₂	#14563329	1	1	yes	characteristic
340	C₂	C₄×C₂²⋊(C₁₉⋊C₈)	1	2	yes	—
341	C₂	#14563397	1	2	yes	—
342	C₂	#14563545	1	2	yes	—
343	C₂	#14563615	1	4	yes	—
344	C₂	#14563703	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³×C₄	C₂²×D₁₉
commutator	C₂²×C₁₉	C₂×C₄×C₈
Fitting	C₂³×C₄²×C₁₉	C₂
socle	C₂⁴×C₁₉	C₂²×C₄
prime radical	C₂⁵×C₁₉	C₂×C₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂³×C₄²	D₁₉
hypocenter	C₁₉	#10712
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×#11926299	C₂	semidirect
7. C₂×C₂²⋊(C₁₉⋊C₈)	C₄	semidirect
22. C₂³×C₄×C₁₉	C₈	semidirect
40. C₂²×C₁₉⋊C₈	C₂×C₄	semidirect
127. C₂×C₄×C₁₉	C₂²⋊C₈	semidirect
146. C₂²×C₁₉	#336	semidirect
194. C₂³×C₄	C₁₉⋊C₈	semidirect
214. C₁₉	#10712	semidirect
298. C₂×C₄	C₂²⋊(C₁₉⋊C₈)	semidirect
315. C₂²	#11926299	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₂²⋊(C₁₉⋊C₈)	2
C₂×#11926299	2
C₂³×C₄²×C₁₉	2
#10712	19