Subgroups — #18073237

Order 4864.

420 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14572280	C₂	1	1	yes	maximal, characteristic
2	#14562018	C₂	1	1	yes	maximal, characteristic
3	#14562015	C₂	1	1	yes	maximal, characteristic
4	#14561919	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11926265	C₂	1	1	yes	maximal, characteristic, complemented
6	#14561473	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1278	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#64541	C₂²	1	1	yes	characteristic
9	#11926606	—	2	2	no	—
10	#11926597	C₂²	1	1	yes	characteristic
11	#11926595	—	2	1	no	—
12	#11926595	—	2	1	no	—
13	#11926572	C₄	1	2	yes	—
14	#11926512	C₄	1	2	yes	—
15	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
17	#11926276	C₄	1	2	yes	—
18	#11926274	C₄	1	2	yes	—
19	#11926273	C₄	1	2	yes	—
20	#11926271	C₄	1	2	yes	—
21	#11926266	—	2	1	no	—
22	#11926265	—	2	2	no	—
23	C₈×C₁₉⋊C₈	—	2	1	no	—
24	C₁₉×#431	C₂²	1	1	yes	characteristic
25	C₄×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
27	C₂×#64541	C₂×C₄	1	1	yes	characteristic
28	C₂×M₁₆×C₁₉	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₁₉	—	2	1	no	—
30	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
31	C₄×C₂²⋊D₁₉	—	2	1	no	—
32	C₄×C₂²⋊D₁₉	—	2	1	no	—
33	#10597372	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₁₉⋊C₈	—	2	1	no	—
35	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
36	C₄×D₇₆	C₂×C₄	1	1	yes	characteristic
37	C₄×D₇₆	C₂×C₄	1	1	yes	characteristic
38	#10597318	—	2	1	no	—
39	C₄²×D₁₉	—	2	1	no	—
40	C₄×Dic₃₈	C₂×C₄	1	1	yes	characteristic
41	C₄×Dic₃₈	C₂×C₄	1	1	yes	characteristic
42	C₁₉×C₄⋊C₈	C₂×C₄	1	2	yes	—
43	C₁₉×C₂²⋊C₈	—	2	1	no	—
44	C₁₉×#97	—	2	1	no	—
45	C₄×C₈×C₁₉	—	2	1	no	—
46	C₄×C₈×C₁₉	—	2	1	no	—
47	C₄×C₈×C₁₉	—	2	1	no	—
48	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
49	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
50	(C₂×D₁₉)⋊C₈	—	2	2	no	—
51	(C₂×D₁₉)⋊C₈	—	2	2	no	—
52	Dic₁₉⋊C₈	—	2	2	no	—
53	Dic₁₉⋊C₈	—	2	2	no	—
54	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	2	yes	—
55	C₄×C₁₉⋊C₈	—	2	1	no	—
56	C₄×C₁₉⋊C₈	—	2	1	no	—
57	C₄×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
58	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
59	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
60	C₄×C₁₉⋊C₈	D₄	1	2	yes	—
61	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
64	C₂×C₂²⋊D₁₉	—	2	1	no	—
65	#64541	—	2	2	no	—
66	#64541	—	2	2	no	—
67	C₂×D₇₆	C₄²	1	1	yes	characteristic
68	C₂×C₄×D₁₉	—	2	1	no	—
69	C₂×C₄×D₁₉	—	2	1	no	—
70	C₂×C₄×D₁₉	—	2	1	no	—
71	C₂×Dic₃₈	C₄²	1	1	yes	characteristic
72	M₁₆×C₁₉	—	2	2	no	—
73	M₁₆×C₁₉	—	2	2	no	—
74	C₂×C₈×C₁₉	—	2	4	no	—
75	C₂×C₈×C₁₉	—	2	2	no	—
76	C₂×C₈×C₁₉	—	2	2	no	—
77	C₂×C₈×C₁₉	—	2	2	no	—
78	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
79	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
82	C₂²⋊Dic₁₉	—	2	1	no	—
83	(C₂×D₁₉)⋊C₄	—	2	1	no	—
84	(C₂×D₁₉)⋊C₄	—	2	1	no	—
85	C₄⋊Dic₁₉	C₄²	1	1	yes	characteristic
86	C₄⋊Dic₁₉	C₄²	1	1	yes	characteristic
87	Dic₁₉⋊C₄	—	2	1	no	—
88	Dic₁₉⋊C₄	—	2	1	no	—
89	C₄×Dic₁₉	—	2	1	no	—
90	C₄×Dic₁₉	—	2	1	no	—
91	C₄×Dic₁₉	—	2	1	no	—
92	C₂×C₁₉⋊C₈	—	2	2	no	—
93	C₂×C₁₉⋊C₈	—	2	2	no	—
94	C₂×C₁₉⋊C₈	—	2	2	no	—
95	C₂×C₁₉⋊C₈	—	2	2	no	—
96	C₂×C₁₉⋊C₈	—	4	2	no	—
97	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	2	yes	—
98	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	2	yes	—
99	#8189	—	19	1	no	maximal, Sylow
100	C₂³×C₁₉	#117	1	1	yes	characteristic
101	C₂²×D₁₉	—	2	1	no	—
102	C₂×C₄×C₁₉	—	2	2	no	—
103	C₂×C₄×C₁₉	—	2	2	no	—
104	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
105	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₉	#131	1	2	yes	—
111	C₂×C₄×C₁₉	#131	1	2	yes	—
112	C₂²⋊D₁₉	—	4	2	no	—
113	C₂×Dic₁₉	—	2	1	no	—
114	C₂×Dic₁₉	—	2	1	no	—
115	C₂×Dic₁₉	—	2	1	no	—
116	D₇₆	—	2	1	no	—
117	D₇₆	—	2	1	no	—
118	C₄×D₁₉	—	4	2	no	—
119	C₄×D₁₉	—	4	2	no	—
120	C₄×D₁₉	—	4	2	no	—
121	Dic₃₈	—	2	1	no	—
122	Dic₃₈	—	2	1	no	—
123	C₈×C₁₉	—	2	4	no	—
124	C₈×C₁₉	—	4	2	no	—
125	C₈×C₁₉	—	4	2	no	—
126	C₁₉⋊C₈	—	2	2	no	—
127	C₁₉⋊C₈	—	2	2	no	—
128	C₁₉⋊C₈	—	4	2	no	—
129	C₁₉⋊C₈	—	4	2	no	—
130	#2922	—	19	1	no	—
131	#1476	—	19	1	no	—
132	#1476	—	19	1	no	—
133	#1450	—	19	1	no	—
134	C₂×#322	—	19	1	no	—
135	#1278	—	19	1	no	—
136	#1278	D₁₉	1	1	yes	characteristic, complemented
137	C₂²×C₁₉	—	2	1	no	—
138	C₂²×C₁₉	—	2	1	no	—
139	C₂²×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
140	C₂²×C₁₉	#433	1	2	yes	—
141	C₂×D₁₉	—	4	1	no	—
142	C₂×D₁₉	—	4	1	no	—
143	C₂×D₁₉	—	4	1	no	—
144	C₄×C₁₉	—	2	1	no	—
145	C₄×C₁₉	—	2	1	no	—
146	C₄×C₁₉	#405	1	1	yes	characteristic
147	C₄×C₁₉	#405	1	1	yes	characteristic
148	C₄×C₁₉	#408	1	2	yes	—
149	C₄×C₁₉	#443	1	2	yes	—
150	C₄×C₁₉	#444	1	2	yes	—
151	Dic₁₉	—	4	1	no	—
152	Dic₁₉	—	4	1	no	—
153	Dic₁₉	—	4	1	no	—
154	C₄×#41	—	19	1	no	—
155	#433	—	38	2	no	—
156	#431	—	19	1	no	—
157	#431	—	38	1	no	—
158	#431	—	38	1	no	—
159	#431	C₂×D₁₉	1	1	yes	characteristic
160	#424	—	19	2	no	—
161	#408	—	19	2	no	—
162	C₄×M₁₆	C₂×D₁₉	1	1	yes	characteristic
163	C₂×C₄×C₈	—	19	1	no	—
164	C₂×C₄×C₈	—	19	1	no	—
165	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
166	Q₈⋊C₈	—	19	2	no	—
167	Q₈⋊C₈	—	19	2	no	—
168	D₄⋊C₈	—	19	2	no	—
169	D₄⋊C₈	—	19	2	no	—
170	#322	—	38	2	no	—
171	#322	—	38	1	no	—
172	C₈²	—	38	1	no	—
173	C₂×C₁₉	—	2	2	no	—
174	C₂×C₁₉	#1750	1	1	yes	characteristic
175	C₂×C₁₉	#1764	1	1	yes	characteristic
176	C₂×C₁₉	#1765	1	1	yes	characteristic
177	D₁₉	—	8	1	no	—
178	C₂×#41	—	19	1	no	—
179	C₂×M₁₆	C₄×D₁₉	1	2	yes	—
180	C₂²×C₈	—	2	1	no	—
181	C₂²×C₈	—	19	2	no	—
182	C₂²×C₈	—	38	1	no	—
183	C₄×Q₈	—	19	1	no	—
184	C₄×Q₈	—	19	1	no	—
185	C₄×D₄	—	19	1	no	—
186	C₄×D₄	—	19	1	no	—
187	C₄×D₄	—	38	1	no	—
188	C₄×D₄	—	38	1	no	—
189	#117	—	19	1	no	—
190	#117	—	38	1	no	—
191	C₂×C₄²	—	38	1	no	—
192	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
193	C₄⋊C₈	—	19	2	no	—
194	C₄⋊C₈	—	38	2	no	—
195	C₄⋊C₈	—	38	2	no	—
196	C₄⋊C₈	C₄×D₁₉	1	2	yes	—
197	C₂²⋊C₈	—	2	1	no	—
198	C₂²⋊C₈	—	38	2	no	—
199	C₂²⋊C₈	—	38	2	no	—
200	C₂²⋊C₈	—	38	1	no	—
201	#97	—	2	1	no	—
202	C₄×C₈	—	2	1	no	—
203	C₄×C₈	—	2	1	no	—
204	C₄×C₈	—	2	1	no	—
205	C₄×C₈	—	19	2	no	—
206	C₄×C₈	—	19	2	no	—
207	C₄×C₈	—	19	1	no	—
208	C₄×C₈	—	19	1	no	—
209	C₄×C₈	—	38	1	no	—
210	C₄×C₈	—	38	1	no	—
211	C₄×C₈	C₄×D₁₉	1	2	yes	—
212	C₁₉	#8189	1	1	yes	characteristic, complemented, Sylow
213	#41	—	38	2	no	—
214	#41	—	38	2	no	—
215	C₂×Q₈	—	19	1	no	—
216	C₂×D₄	—	19	1	no	—
217	C₂×D₄	—	38	1	no	—
218	C₂²×C₄	—	38	1	no	—
219	C₂²×C₄	—	38	1	no	—
220	C₂²×C₄	—	38	1	no	—
221	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
222	C₂²×C₄	#64543	1	1	yes	characteristic
223	C₂²×C₄	#64545	1	1	yes	characteristic
224	M₁₆	—	2	2	no	—
225	M₁₆	—	2	2	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	19	2	no	—
231	C₂×C₈	—	19	2	no	—
232	C₂×C₈	—	38	2	no	—
233	C₂×C₈	—	38	2	no	—
234	C₂×C₈	—	38	2	no	—
235	C₂×C₈	—	38	2	no	—
236	C₂×C₈	—	76	2	no	—
237	C₄⋊C₄	—	19	1	no	—
238	C₄⋊C₄	—	19	1	no	—
239	C₄⋊C₄	—	38	1	no	—
240	C₄⋊C₄	—	38	1	no	—
241	C₂²⋊C₄	—	38	1	no	—
242	C₂²⋊C₄	—	38	1	no	—
243	C₂²⋊C₄	—	38	1	no	—
244	C₄²	—	38	1	no	—
245	C₄²	—	38	1	no	—
246	C₄²	—	38	1	no	—
247	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
248	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
249	C₄²	D₄×D₁₉	1	1	yes	characteristic
250	C₄²	D₄×D₁₉	1	1	yes	characteristic
251	C₂³	—	38	1	no	—
252	C₂³	#10597338	1	1	yes	characteristic
253	Q₈	—	38	1	no	—
254	Q₈	—	38	1	no	—
255	D₄	—	38	1	no	—
256	D₄	—	38	1	no	—
257	D₄	—	76	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	38	1	no	—
261	C₂×C₄	—	38	1	no	—
262	C₂×C₄	—	38	1	no	—
263	C₂×C₄	—	76	2	no	—
264	C₂×C₄	—	76	2	no	—
265	C₂×C₄	—	76	2	no	—
266	C₂×C₄	C₄²×D₁₉	1	1	yes	characteristic
267	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
268	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
269	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
270	C₂×C₄	#10597339	1	1	yes	characteristic
271	C₂×C₄	#10597339	1	1	yes	characteristic
272	C₂×C₄	#10597347	1	2	yes	—
273	C₂×C₄	#10597353	1	2	yes	—
274	C₈	—	2	4	no	—
275	C₈	—	4	2	no	—
276	C₈	—	4	2	no	—
277	C₈	—	38	2	no	—
278	C₈	—	38	2	no	—
279	C₈	—	76	2	no	—
280	C₈	—	76	2	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	76	1	no	—
284	C₂²	—	76	1	no	—
285	C₂²	—	76	1	no	—
286	C₂²	#11926391	1	1	yes	characteristic
287	C₂²	#11926604	1	2	yes	—
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	76	1	no	—
291	C₄	—	76	1	no	—
292	C₄	—	76	1	no	—
293	C₄	#11926498	1	1	yes	characteristic
294	C₄	#11926501	1	1	yes	characteristic
295	C₄	#11926509	1	2	yes	—
296	C₄	#11926658	1	2	yes	—
297	C₄	#11926665	1	2	yes	—
298	C₂	—	2	2	no	—
299	C₂	—	152	1	no	—
300	C₂	#14563400	1	1	yes	characteristic
301	C₂	#14563469	1	1	yes	characteristic
302	C₂	#14563478	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₁₉
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1278	C₂
socle	C₂²×C₁₉	C₄×C₂²⋊C₄
prime radical	C₂×#64541	C₂×C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1278	D₁₉
hypocenter	C₁₉	#8189
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1278

Complements

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

N	H ≅ G/N	type
4. #14561919	C₂	semidirect
5. C₂×#11926265	C₂	semidirect
6. #14561473	C₂	semidirect
7. C₁₉×#1278	C₂	semidirect
136. #1278	D₁₉	semidirect
212. C₁₉	#8189	semidirect

Maximal subgroups

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

subgroup	index
#14572280	2
#14562018	2
#14562015	2
#14561919	2
C₂×#11926265	2
#14561473	2
C₁₉×#1278	2
#8189	19