Subgroups — #18073161

Order 4864.

698 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14563488	C₂	1	1	yes	maximal, characteristic, complemented
2	#14563381	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₄⋊(C₁₉⋊C₈)	C₂	1	2	yes	maximal
4	#14561476	C₂	1	2	yes	maximal, complemented
5	C₄²×D₄×C₁₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
7	C₂×#10597260	—	2	1	no	—
8	C₂×C₄×C₁₉⋊C₈	C₂²	1	2	yes	—
9	#11926299	—	2	2	no	—
10	#11926299	—	2	1	no	—
11	D₄⋊(C₁₉⋊C₈)	C₄	1	8	yes	complemented
12	C₂×C₄×D₄×C₁₉	C₄	1	2	yes	—
13	C₂×C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
14	C₂²×C₄²×C₁₉	—	2	1	no	—
15	C₄×C₄⋊C₄×C₁₉	C₄	1	2	yes	—
16	C₄×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
17	C₄×C₂²⋊C₄×C₁₉	—	2	2	no	—
18	C₄×C₂²⋊C₄×C₁₉	—	2	1	no	—
19	C₄³×C₁₉	C₂²	1	1	yes	characteristic
20	C₂²×D₄×C₁₉	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₉	—	2	2	no	—
22	C₂³×C₄×C₁₉	—	2	1	no	—
23	C₄×D₄×C₁₉	—	2	4	no	—
24	C₄×D₄×C₁₉	—	2	4	no	—
25	C₄×D₄×C₁₉	—	2	2	no	—
26	C₄×D₄×C₁₉	C₂×C₄	1	4	yes	—
27	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
29	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
30	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
31	C₂×C₄²×C₁₉	—	2	2	no	—
32	C₂×C₄²×C₁₉	—	2	1	no	—
33	C₂×C₄²×C₁₉	—	4	2	no	—
34	C₂×C₄²×C₁₉	—	4	2	no	—
35	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₉	D₄	1	2	yes	—
38	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
40	C₂²×C₁₉⋊C₈	—	2	2	no	—
41	C₂²×C₁₉⋊C₈	—	2	2	no	—
42	C₁₉×#95	—	2	2	no	—
43	C₁₉×#95	—	2	1	no	—
44	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
45	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	4	yes	complemented
46	#10597260	—	4	2	no	—
47	C₄×C₁₉⋊C₈	—	2	2	no	—
48	C₄×C₁₉⋊C₈	C₂×C₄	1	4	yes	complemented
49	C₂⁴×C₁₉	—	2	1	no	—
50	C₂×D₄×C₁₉	—	2	1	no	—
51	C₂×D₄×C₁₉	—	2	2	no	—
52	C₂×D₄×C₁₉	C₄²	1	2	yes	—
53	C₂×D₄×C₁₉	M₁₆	1	4	yes	—
54	C₂²×C₄×C₁₉	—	2	8	no	—
55	C₂²×C₄×C₁₉	—	2	1	no	—
56	C₂²×C₄×C₁₉	—	2	2	no	—
57	C₂²×C₄×C₁₉	—	2	4	no	—
58	C₂²×C₄×C₁₉	—	4	4	no	—
59	C₂²×C₄×C₁₉	—	4	4	no	—
60	C₂²×C₄×C₁₉	—	4	1	no	—
61	C₂²×C₄×C₁₉	—	4	1	no	—
62	C₂²×C₄×C₁₉	—	4	2	no	—
63	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
68	C₄⋊C₄×C₁₉	—	2	1	no	—
69	C₄⋊C₄×C₁₉	—	2	2	no	—
70	C₄⋊C₄×C₁₉	C₄²	1	2	yes	—
71	C₄⋊C₄×C₁₉	M₁₆	1	4	yes	—
72	C₂²⋊C₄×C₁₉	—	2	8	no	—
73	C₂²⋊C₄×C₁₉	—	2	4	no	—
74	C₄²×C₁₉	—	2	2	no	—
75	C₄²×C₁₉	—	2	2	no	—
76	C₄²×C₁₉	—	2	2	no	—
77	C₄²×C₁₉	—	4	2	no	—
78	C₄²×C₁₉	—	4	4	no	—
79	C₄²×C₁₉	—	4	1	no	—
80	C₄²×C₁₉	—	4	1	no	—
81	C₄²×C₁₉	D₈	1	2	yes	—
82	C₄²×C₁₉	SD₁₆	1	2	yes	—
83	C₄²×C₁₉	C₂²×C₄	1	2	yes	—
84	C₄²×C₁₉	#41	1	2	yes	—
85	C₂×C₁₉⋊C₈	—	2	4	no	—
86	C₂×C₁₉⋊C₈	—	2	4	no	—
87	C₂×C₁₉⋊C₈	—	4	4	no	—
88	C₂×C₁₉⋊C₈	—	4	2	no	—
89	C₂×C₁₉⋊C₈	—	4	2	no	—
90	#8174	—	19	1	no	maximal, Sylow
91	C₂³×C₁₉	—	2	2	no	—
92	C₂³×C₁₉	—	2	4	no	—
93	C₂³×C₁₉	—	4	1	no	—
94	C₂³×C₁₉	—	4	1	no	—
95	C₂³×C₁₉	—	4	2	no	—
96	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	D₄×C₁₉	—	2	2	no	—
98	D₄×C₁₉	—	2	4	no	—
99	D₄×C₁₉	#97	1	4	yes	complemented
100	C₂×C₄×C₁₉	—	2	2	no	—
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	4	no	—
106	C₂×C₄×C₁₉	—	2	2	no	—
107	C₂×C₄×C₁₉	—	2	2	no	—
108	C₂×C₄×C₁₉	—	4	16	no	—
109	C₂×C₄×C₁₉	—	4	2	no	—
110	C₂×C₄×C₁₉	—	4	1	no	—
111	C₂×C₄×C₁₉	—	4	1	no	—
112	C₂×C₄×C₁₉	—	4	2	no	—
113	C₂×C₄×C₁₉	—	4	4	no	—
114	C₂×C₄×C₁₉	—	4	4	no	—
115	C₂×C₄×C₁₉	#102	1	4	yes	—
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	2	yes	—
119	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
121	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
122	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₄×C₁₉	#135	1	2	yes	—
125	C₁₉⋊C₈	—	4	4	no	—
126	C₁₉⋊C₈	—	4	4	no	—
127	C₄²×D₄	D₁₉	1	1	yes	characteristic
128	#1767	—	19	1	no	—
129	#1744	—	19	1	no	—
130	C₂×D₄⋊C₈	—	19	2	no	—
131	#1279	—	19	2	no	—
132	C₂²×C₁₉	—	2	4	no	—
133	C₂²×C₁₉	—	4	4	no	—
134	C₂²×C₁₉	—	4	4	no	—
135	C₂²×C₁₉	—	4	2	no	—
136	C₂²×C₁₉	—	4	2	no	—
137	C₂²×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
138	C₂²×C₁₉	#407	1	2	yes	—
139	C₂²×C₁₉	C₂×#102	1	1	yes	characteristic
140	C₂²×C₁₉	#416	1	1	yes	characteristic
141	C₂²×C₁₉	#421	1	2	yes	—
142	C₄×C₁₉	—	2	4	no	—
143	C₄×C₁₉	—	2	2	no	—
144	C₄×C₁₉	—	4	4	no	—
145	C₄×C₁₉	—	4	2	no	—
146	C₄×C₁₉	—	4	2	no	—
147	C₄×C₁₉	C₂×#97	1	1	yes	characteristic
148	C₄×C₁₉	C₄×M₁₆	1	1	yes	characteristic
149	C₄×C₁₉	#432	1	2	yes	—
150	C₄×C₁₉	C₄×D₈	1	2	yes	—
151	C₄×C₁₉	C₄×SD₁₆	1	2	yes	—
152	C₂×C₄×D₄	Dic₁₉	1	2	yes	—
153	C₂×C₄×D₄	C₂×D₁₉	1	1	yes	characteristic
154	C₂²×C₄²	—	2	1	no	—
155	C₂×C₄⋊C₈	—	19	2	no	—
156	C₂×#97	—	38	1	no	—
157	C₂×C₄×C₈	—	19	2	no	—
158	C₄×C₄⋊C₄	Dic₁₉	1	2	yes	—
159	C₄×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
160	C₄×C₂²⋊C₄	—	2	1	no	—
161	C₄×C₂²⋊C₄	—	2	2	no	—
162	C₄³	C₂×D₁₉	1	1	yes	characteristic
163	#336	—	38	2	no	—
164	#336	—	38	1	no	—
165	D₄⋊C₈	—	19	8	no	—
166	C₂×C₁₉	—	4	4	no	—
167	C₂×C₁₉	#1486	1	2	yes	—
168	C₂×C₁₉	#1488	1	2	yes	—
169	C₂×C₁₉	#1749	1	1	yes	characteristic
170	C₂×C₁₉	#1759	1	1	yes	characteristic
171	C₂×C₁₉	C₄×#102	1	1	yes	characteristic
172	C₂²×D₄	C₂×Dic₁₉	1	1	yes	characteristic
173	C₂³×C₄	—	2	2	no	—
174	C₂³×C₄	—	2	1	no	—
175	C₂²×C₈	—	38	2	no	—
176	C₂²×C₈	—	38	2	no	—
177	C₄×D₄	—	2	4	no	—
178	C₄×D₄	—	2	4	no	—
179	C₄×D₄	—	2	2	no	—
180	C₄×D₄	C₄×D₁₉	1	4	yes	—
181	C₂×C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
182	C₂×C₄⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
183	C₂×C₂²⋊C₄	—	2	2	no	—
184	C₂×C₂²⋊C₄	—	2	1	no	—
185	C₂×C₄²	—	2	2	no	—
186	C₂×C₄²	—	2	1	no	—
187	C₂×C₄²	—	4	2	no	—
188	C₂×C₄²	—	4	2	no	—
189	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
190	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
191	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
192	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
193	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
194	C₄⋊C₈	—	19	4	no	—
195	C₄⋊C₈	—	38	2	no	—
196	#97	—	76	2	no	—
197	C₄×C₈	—	19	4	no	—
198	C₄×C₈	—	38	2	no	—
199	#95	—	2	2	no	—
200	#95	—	2	1	no	—
201	C₁₉	#8174	1	1	yes	characteristic, complemented, Sylow
202	C₂⁴	—	2	1	no	—
203	C₂×D₄	—	2	2	no	—
204	C₂×D₄	—	2	1	no	—
205	C₂×D₄	#64520	1	4	yes	—
206	C₂×D₄	C₄×Dic₁₉	1	2	yes	—
207	C₂²×C₄	—	2	8	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	4	4	no	—
212	C₂²×C₄	—	4	1	no	—
213	C₂²×C₄	—	4	1	no	—
214	C₂²×C₄	—	4	2	no	—
215	C₂²×C₄	—	4	4	no	—
216	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
217	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
218	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
221	C₂×C₈	—	38	4	no	—
222	C₂×C₈	—	38	4	no	—
223	C₂×C₈	—	76	4	no	—
224	C₂×C₈	—	76	2	no	—
225	C₂×C₈	—	76	2	no	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	2	1	no	—
228	C₄⋊C₄	#64520	1	4	yes	—
229	C₄⋊C₄	C₄×Dic₁₉	1	2	yes	—
230	C₂²⋊C₄	—	2	8	no	—
231	C₂²⋊C₄	—	2	4	no	—
232	C₄²	—	2	2	no	—
233	C₄²	—	2	2	no	—
234	C₄²	—	2	2	no	—
235	C₄²	—	4	2	no	—
236	C₄²	—	4	1	no	—
237	C₄²	—	4	1	no	—
238	C₄²	—	4	4	no	—
239	C₄²	#64525	1	2	yes	—
240	C₄²	#64526	1	2	yes	—
241	C₄²	C₂×C₄×D₁₉	1	2	yes	—
242	C₄²	#64541	1	2	yes	—
243	C₂³	—	2	2	no	—
244	C₂³	—	2	4	no	—
245	C₂³	—	4	1	no	—
246	C₂³	—	4	1	no	—
247	C₂³	—	4	2	no	—
248	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
249	D₄	—	2	4	no	—
250	D₄	—	2	2	no	—
251	D₄	#10597260	1	4	yes	complemented
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	4	16	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	#10597289	1	4	yes	—
268	C₂×C₄	C₂×#64520	1	2	yes	—
269	C₂×C₄	C₂×#64520	1	2	yes	—
270	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
271	C₂×C₄	#10597372	1	1	yes	characteristic
272	C₂×C₄	C₄×C₂²⋊D₁₉	1	2	yes	—
273	C₂×C₄	C₄×C₂²⋊D₁₉	1	2	yes	—
274	C₂×C₄	C₂×#64525	1	1	yes	characteristic
275	C₂×C₄	C₂×#64526	1	1	yes	characteristic
276	C₂×C₄	#10597396	1	2	yes	—
277	C₈	—	76	4	no	—
278	C₈	—	76	4	no	—
279	C₂²	—	2	4	no	—
280	C₂²	—	4	4	no	—
281	C₂²	—	4	4	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	4	2	no	—
284	C₂²	C₄×C₂²⋊Dic₁₉	1	1	yes	characteristic
285	C₂²	#11926508	1	2	yes	—
286	C₂²	C₂×#10597289	1	1	yes	characteristic
287	C₂²	#11926540	1	1	yes	characteristic
288	C₂²	#11926561	1	2	yes	—
289	C₄	—	2	4	no	—
290	C₄	—	2	2	no	—
291	C₄	—	4	4	no	—
292	C₄	—	4	2	no	—
293	C₄	—	4	2	no	—
294	C₄	C₂×#10597260	1	1	yes	characteristic
295	C₄	C₄×#64520	1	1	yes	characteristic
296	C₄	#11926601	1	2	yes	—
297	C₄	C₄×#64525	1	2	yes	—
298	C₄	C₄×#64526	1	2	yes	—
299	C₂	—	4	4	no	—
300	C₂	#14562060	1	2	yes	—
301	C₂	#14562069	1	2	yes	—
302	C₂	#14563397	1	1	yes	characteristic
303	C₂	#14563445	1	1	yes	characteristic
304	C₂	C₄×#10597289	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₉
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₄²×D₄×C₁₉	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂²×D₄×C₁₉	C₂×C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₄²×D₄	D₁₉
hypocenter	C₁₉	#8174
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×D₄

Complements

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

N	H ≅ G/N	type
1. #14563488	C₂	semidirect
2. #14563381	C₂	semidirect
4. #14561476	C₂	semidirect
11. D₄⋊(C₁₉⋊C₈)	C₄	semidirect
45. C₄⋊(C₁₉⋊C₈)	C₂×C₄	semidirect
48. C₄×C₁₉⋊C₈	C₂×C₄	semidirect
99. D₄×C₁₉	#97	semidirect
201. C₁₉	#8174	semidirect
251. D₄	#10597260	semidirect

Maximal subgroups

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

subgroup	index
#14563488	2
#14563381	2
C₂×D₄⋊(C₁₉⋊C₈)	2
#14561476	2
C₄²×D₄×C₁₉	2
#8174	19