Subgroups — #18072072

Order 4864.

770 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926597	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊(C₁₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#14561476	C₂	1	4	yes	maximal, complemented
4	C₄×C₁₉×#117	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₁₉⋊C₈	—	2	1	no	—
6	#11926597	—	2	2	no	—
7	#11926597	C₄	1	4	yes	complemented
8	C₂×C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
9	C₂×C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
10	C₂×C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
11	C₂×C₄⋊(C₁₉⋊C₈)	C₄	1	4	yes	complemented
12	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
13	C₂×C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
14	C₂×C₄×C₁₉⋊C₈	C₂²	1	2	yes	—
15	C₂×C₁₉×#117	C₄	1	2	yes	—
16	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
17	C₄×C₄⋊C₄×C₁₉	C₄	1	2	yes	—
18	C₄×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
19	C₄×C₂²⋊C₄×C₁₉	—	2	1	no	—
20	C₄×C₂²⋊C₄×C₁₉	—	2	1	no	—
21	C₁₉×#376	—	2	1	no	—
22	C₄³×C₁₉	—	2	1	no	—
23	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
26	C₁₉×#117	—	2	4	no	—
27	C₁₉×#117	C₈	1	8	yes	complemented
28	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	4	yes	—
29	C₂×C₂²⋊C₄×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	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₉	D₄	1	2	yes	—
39	C₂×C₄²×C₁₉	Q₈	1	2	yes	—
40	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
41	C₂²×C₁₉⋊C₈	—	2	4	no	—
42	C₂²×C₁₉⋊C₈	—	2	2	no	—
43	C₂²×C₁₉⋊C₈	—	2	2	no	—
44	C₂²×C₁₉⋊C₈	—	4	4	no	—
45	C₁₉×#95	—	2	2	no	—
46	C₁₉×#95	—	2	2	no	—
47	C₂²⋊(C₁₉⋊C₈)	—	2	4	no	—
48	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
49	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
50	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
51	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	4	yes	—
52	C₄×C₁₉⋊C₈	—	2	2	no	—
53	C₄×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
54	C₂⁴×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₉	—	2	2	no	—
56	C₂²×C₄×C₁₉	—	2	2	no	—
57	C₂²×C₄×C₁₉	—	2	4	no	—
58	C₂²×C₄×C₁₉	—	4	2	no	—
59	C₂²×C₄×C₁₉	C₄²	1	2	yes	—
60	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
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	2	yes	—
66	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₁₉	C₂×C₈	1	4	yes	—
69	C₂²×C₄×C₁₉	M₁₆	1	4	yes	—
70	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
73	C₄⋊C₄×C₁₉	—	2	4	no	—
74	C₄⋊C₄×C₁₉	C₂×C₈	1	8	yes	complemented
75	C₂²⋊C₄×C₁₉	—	2	8	no	—
76	C₄²×C₁₉	—	2	8	no	—
77	C₄²×C₁₉	—	4	4	no	—
78	C₄²×C₁₉	—	4	2	no	—
79	C₄²×C₁₉	—	4	2	no	—
80	C₄²×C₁₉	C₄²	1	2	yes	—
81	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
82	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
83	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
84	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
85	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
86	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
90	C₂×C₁₉⋊C₈	—	2	4	no	—
91	C₂×C₁₉⋊C₈	—	2	4	no	—
92	C₂×C₁₉⋊C₈	—	4	4	no	—
93	C₂×C₁₉⋊C₈	—	4	8	no	—
94	C₂×C₁₉⋊C₈	—	4	4	no	—
95	#7891	—	19	1	no	maximal, Sylow
96	C₂³×C₁₉	—	2	2	no	—
97	C₂³×C₁₉	—	2	2	no	—
98	C₂³×C₁₉	#95	1	2	yes	—
99	C₂³×C₁₉	C₂²⋊C₈	1	4	yes	—
100	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₁₉	—	2	2	no	—
102	C₂×C₄×C₁₉	—	2	2	no	—
103	C₂×C₄×C₁₉	—	2	2	no	—
104	C₂×C₄×C₁₉	—	2	2	no	—
105	C₂×C₄×C₁₉	—	2	2	no	—
106	C₂×C₄×C₁₉	—	2	4	no	—
107	C₂×C₄×C₁₉	—	2	8	no	—
108	C₂×C₄×C₁₉	—	4	2	no	—
109	C₂×C₄×C₁₉	—	4	2	no	—
110	C₂×C₄×C₁₉	—	4	4	no	—
111	C₂×C₄×C₁₉	—	4	4	no	—
112	C₂×C₄×C₁₉	#95	1	2	yes	—
113	C₂×C₄×C₁₉	#95	1	2	yes	—
114	C₂×C₄×C₁₉	#95	1	2	yes	—
115	C₂×C₄×C₁₉	C₄×C₈	1	4	yes	—
116	C₂×C₄×C₁₉	#97	1	4	yes	—
117	C₂×C₄×C₁₉	C₂²⋊C₈	1	4	yes	—
118	C₂×C₄×C₁₉	C₄⋊C₈	1	8	yes	—
119	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
120	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₉	C₂²×C₈	1	2	yes	—
126	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
127	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
128	C₂×C₄×C₁₉	#137	1	2	yes	—
129	C₁₉⋊C₈	—	4	4	no	—
130	C₁₉⋊C₈	—	4	4	no	—
131	C₂×#431	—	19	1	no	—
132	C₂²×C₄⋊C₈	—	19	1	no	—
133	C₄×#117	D₁₉	1	1	yes	characteristic
134	#1279	—	19	4	no	—
135	C₂²×C₁₉	—	2	8	no	—
136	C₂²×C₁₉	—	2	2	no	—
137	C₂²×C₁₉	—	2	2	no	—
138	C₂²×C₁₉	#336	1	4	yes	—
139	C₂²×C₁₉	C₂×#95	1	1	yes	characteristic
140	C₂²×C₁₉	C₂×C₂²⋊C₈	1	2	yes	—
141	C₂²×C₁₉	#417	1	2	yes	—
142	C₂²×C₁₉	#421	1	2	yes	—
143	C₄×C₁₉	—	2	1	no	—
144	C₄×C₁₉	—	2	2	no	—
145	C₄×C₁₉	—	2	1	no	—
146	C₄×C₁₉	—	2	2	no	—
147	C₄×C₁₉	—	4	8	no	—
148	C₄×C₁₉	#336	1	4	yes	—
149	C₄×C₁₉	C₂×C₄×C₈	1	1	yes	characteristic
150	C₄×C₁₉	C₂×#97	1	1	yes	characteristic
151	C₄×C₁₉	#418	1	1	yes	characteristic
152	C₄×C₁₉	#419	1	1	yes	characteristic
153	C₄×C₁₉	C₂×C₄⋊C₈	1	2	yes	—
154	C₄×C₁₉	#428	1	2	yes	—
155	C₂³×C₈	—	38	1	no	—
156	C₂×#117	Dic₁₉	1	2	yes	—
157	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
158	#431	—	19	4	no	—
159	#431	—	38	2	no	—
160	C₂×C₄⋊C₈	—	19	4	no	—
161	C₂×C₄⋊C₈	—	19	2	no	—
162	C₂×C₄⋊C₈	—	38	2	no	—
163	C₂×C₄⋊C₈	—	38	1	no	—
164	C₂×C₄⋊C₈	—	38	1	no	—
165	C₂×C₂²⋊C₈	—	38	1	no	—
166	C₂×C₄×C₈	—	19	2	no	—
167	C₄×C₄⋊C₄	Dic₁₉	1	2	yes	—
168	C₄×C₄⋊C₄	C₂×D₁₉	1	2	yes	—
169	C₄×C₂²⋊C₄	—	2	1	no	—
170	C₄×C₂²⋊C₄	—	2	1	no	—
171	#376	—	2	1	no	—
172	C₄³	—	2	1	no	—
173	C₂×C₁₉	—	2	4	no	—
174	C₂×C₁₉	#1477	1	4	yes	—
175	C₂×C₁₉	C₂×#336	1	1	yes	characteristic
176	C₂×C₁₉	#1733	1	1	yes	characteristic
177	C₂×C₁₉	#1736	1	1	yes	characteristic
178	C₂³×C₄	C₂×Dic₁₉	1	1	yes	characteristic
179	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
180	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
181	C₂²×C₈	—	38	4	no	—
182	C₂²×C₈	—	38	2	no	—
183	C₂²×C₈	—	38	2	no	—
184	C₂²×C₈	—	76	4	no	—
185	#117	—	2	4	no	—
186	#117	C₁₉⋊C₈	1	8	yes	complemented
187	C₂×C₄⋊C₄	C₂×Dic₁₉	1	4	yes	—
188	C₂×C₂²⋊C₄	—	2	2	no	—
189	C₂×C₄²	—	2	2	no	—
190	C₂×C₄²	—	2	2	no	—
191	C₂×C₄²	—	2	2	no	—
192	C₂×C₄²	Dic₃₈	1	2	yes	—
193	C₂×C₄²	C₄×D₁₉	1	2	yes	—
194	C₂×C₄²	C₄×D₁₉	1	2	yes	—
195	C₂×C₄²	D₇₆	1	2	yes	—
196	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
197	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
198	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
199	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
200	C₄⋊C₈	—	19	4	no	—
201	C₄⋊C₈	—	38	4	no	—
202	C₄⋊C₈	—	38	4	no	—
203	C₄⋊C₈	—	38	2	no	—
204	C₂²⋊C₈	—	38	4	no	—
205	C₄×C₈	—	19	4	no	—
206	C₄×C₈	—	38	2	no	—
207	#95	—	2	2	no	—
208	#95	—	2	2	no	—
209	C₁₉	#7891	1	1	yes	characteristic, complemented, Sylow
210	C₂⁴	C₂²⋊Dic₁₉	1	1	yes	characteristic
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	—	4	2	no	—
215	C₂²×C₄	C₂×C₁₉⋊C₈	1	4	yes	—
216	C₂²×C₄	#64520	1	4	yes	—
217	C₂²×C₄	C₄×Dic₁₉	1	2	yes	—
218	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
219	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
220	C₂²×C₄	C₄⋊Dic₁₉	1	2	yes	—
221	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
222	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
223	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
224	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
225	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
226	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
229	C₂×C₈	—	38	4	no	—
230	C₂×C₈	—	38	4	no	—
231	C₂×C₈	—	76	4	no	—
232	C₂×C₈	—	76	8	no	—
233	C₂×C₈	—	76	4	no	—
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	C₂×C₁₉⋊C₈	1	8	yes	complemented
236	C₂²⋊C₄	—	2	8	no	—
237	C₄²	—	2	8	no	—
238	C₄²	—	4	4	no	—
239	C₄²	—	4	2	no	—
240	C₄²	—	4	2	no	—
241	C₄²	C₄×Dic₁₉	1	2	yes	—
242	C₄²	Dic₁₉⋊C₄	1	2	yes	—
243	C₄²	Dic₁₉⋊C₄	1	2	yes	—
244	C₄²	C₄⋊Dic₁₉	1	2	yes	—
245	C₄²	(C₂×D₁₉)⋊C₄	1	2	yes	—
246	C₄²	(C₂×D₁₉)⋊C₄	1	2	yes	—
247	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
248	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
249	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
250	C₄²	C₂×D₇₆	1	1	yes	characteristic
251	C₂³	—	2	2	no	—
252	C₂³	—	2	2	no	—
253	C₂³	C₂²⋊(C₁₉⋊C₈)	1	4	yes	—
254	C₂³	C₁₉⋊#95	1	2	yes	—
255	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	C₄×C₁₉⋊C₈	1	4	yes	—
268	C₂×C₄	#10597260	1	4	yes	—
269	C₂×C₄	C₄⋊(C₁₉⋊C₈)	1	8	yes	—
270	C₂×C₄	C₂²⋊(C₁₉⋊C₈)	1	4	yes	—
271	C₂×C₄	C₁₉⋊#95	1	2	yes	—
272	C₂×C₄	C₁₉⋊#95	1	2	yes	—
273	C₂×C₄	C₁₉⋊#95	1	2	yes	—
274	C₂×C₄	C₂²×C₁₉⋊C₈	1	2	yes	—
275	C₂×C₄	C₂×#64520	1	2	yes	—
276	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
277	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
278	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
279	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
280	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
281	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
282	C₂×C₄	#10597395	1	2	yes	—
283	C₂×C₄	#10597397	1	2	yes	—
284	C₈	—	76	4	no	—
285	C₈	—	76	4	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	8	no	—
289	C₂²	#11926299	1	4	yes	—
290	C₂²	C₂×C₁₉⋊#95	1	1	yes	characteristic
291	C₂²	C₂×C₂²⋊(C₁₉⋊C₈)	1	2	yes	—
292	C₂²	#11926545	1	2	yes	—
293	C₂²	#11926561	1	2	yes	—
294	C₄	—	2	1	no	—
295	C₄	—	2	2	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	2	no	—
298	C₄	—	4	8	no	—
299	C₄	#11926299	1	4	yes	—
300	C₄	C₂×C₄×C₁₉⋊C₈	1	1	yes	characteristic
301	C₄	C₂×#10597260	1	1	yes	characteristic
302	C₄	#11926548	1	1	yes	characteristic
303	C₄	#11926552	1	1	yes	characteristic
304	C₄	C₂×C₄⋊(C₁₉⋊C₈)	1	2	yes	—
305	C₄	#11926587	1	2	yes	—
306	C₂	—	2	4	no	—
307	C₂	#14562022	1	4	yes	—
308	C₂	C₂×#11926299	1	1	yes	characteristic
309	C₂	#14563344	1	1	yes	characteristic
310	C₂	#14563356	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₉
commutator	C₄×C₁₉	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₄×C₁₉×#117	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₄×#117	D₁₉
hypocenter	C₁₉	#7891
LCS	G ▷ C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
3. #14561476	C₂	semidirect
7. #11926597	C₄	semidirect
11. C₂×C₄⋊(C₁₉⋊C₈)	C₄	semidirect
27. C₁₉×#117	C₈	semidirect
74. C₄⋊C₄×C₁₉	C₂×C₈	semidirect
186. #117	C₁₉⋊C₈	semidirect
209. C₁₉	#7891	semidirect
235. C₄⋊C₄	C₂×C₁₉⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926597	2
C₂²×C₄⋊(C₁₉⋊C₈)	2
#14561476	2
C₄×C₁₉×#117	2
#7891	19