Subgroups — #18073291

Order 4864.

500 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14572601	C₂	1	1	yes	maximal, characteristic
2	C₄×#10597282	C₂	1	1	yes	maximal, characteristic
3	C₄×#10597272	C₂	1	1	yes	maximal, characteristic, complemented
4	#14562015	C₂	1	1	yes	maximal, characteristic
5	#14561473	C₂	1	1	yes	maximal, characteristic, complemented
6	#14561467	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×M₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#64541	C₂²	1	1	yes	characteristic
9	#11926674	C₄	1	2	yes	complemented
10	#11926623	C₄	1	2	yes	complemented
11	#11926615	—	2	2	no	—
12	#11926597	C₂²	1	1	yes	characteristic
13	#11926595	—	2	1	no	—
14	C₂×#10597282	C₄	1	2	yes	—
15	C₄×#64515	—	2	1	no	—
16	C₂×#10597272	—	2	2	no	—
17	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
18	C₄²×Dic₁₉	C₂²	1	1	yes	characteristic
19	#11926305	C₄	1	2	yes	—
20	#11926274	C₄	1	2	yes	complemented
21	#11926271	C₄	1	2	yes	complemented
22	C₁₉×#431	C₂²	1	1	yes	characteristic
23	C₄×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
25	C₁₉×#322	—	2	1	no	—
26	C₈²×C₁₉	—	2	1	no	—
27	C₂×#64541	C₂×C₄	1	1	yes	characteristic
28	C₂×M₁₆×C₁₉	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₁₉	D₄	1	2	yes	—
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	C₂×C₄×Dic₁₉	—	2	1	no	—
35	C₂×C₄×Dic₁₉	—	2	1	no	—
36	C₂×C₄×Dic₁₉	C₂×C₄	1	2	yes	—
37	C₂²×C₁₉⋊C₈	—	2	1	no	—
38	C₂×#64515	—	2	2	no	—
39	C₄×D₇₆	C₂×C₄	1	1	yes	characteristic
40	C₄×D₇₆	C₂×C₄	1	1	yes	characteristic
41	#10597318	—	2	1	no	—
42	C₄²×D₁₉	—	2	1	no	—
43	C₄×Dic₃₈	C₂×C₄	1	1	yes	characteristic
44	C₄×Dic₃₈	C₂×C₄	1	1	yes	characteristic
45	C₁₉×C₄⋊C₈	C₂×C₄	1	2	yes	—
46	C₁₉×C₂²⋊C₈	—	2	1	no	—
47	C₁₉×#97	—	2	1	no	—
48	C₄×C₈×C₁₉	—	2	1	no	—
49	C₄×C₈×C₁₉	—	2	1	no	—
50	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
51	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
52	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
53	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
54	#10597282	—	2	4	no	—
55	(C₂×D₁₉)⋊C₈	—	2	2	no	—
56	#10597272	—	2	8	no	—
57	#10597272	—	2	2	no	—
58	Dic₁₉⋊C₈	—	2	2	no	—
59	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	2	yes	—
60	C₄×C₁₉⋊C₈	—	2	1	no	—
61	C₄×C₁₉⋊C₈	—	2	1	no	—
62	C₄×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
63	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
66	C₂×C₂²⋊D₁₉	—	2	1	no	—
67	C₂²×Dic₁₉	—	2	2	no	—
68	#64541	—	2	2	no	—
69	#64541	—	2	2	no	—
70	C₂×D₇₆	C₄²	1	1	yes	characteristic
71	C₂×C₄×D₁₉	—	2	1	no	—
72	C₂×C₄×D₁₉	—	2	1	no	—
73	C₂×C₄×D₁₉	—	2	1	no	—
74	C₂×Dic₃₈	C₄²	1	1	yes	characteristic
75	M₁₆×C₁₉	—	2	2	no	—
76	M₁₆×C₁₉	M₁₆	1	4	yes	complemented
77	C₂×C₈×C₁₉	—	2	2	no	—
78	C₂×C₈×C₁₉	—	2	4	no	—
79	C₂×C₈×C₁₉	—	2	2	no	—
80	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
81	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
82	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₁₉	#41	1	1	yes	characteristic
86	C₂²⋊Dic₁₉	—	2	1	no	—
87	(C₂×D₁₉)⋊C₄	—	2	1	no	—
88	(C₂×D₁₉)⋊C₄	—	2	1	no	—
89	C₄⋊Dic₁₉	C₄²	1	1	yes	characteristic
90	C₄⋊Dic₁₉	C₄²	1	1	yes	characteristic
91	Dic₁₉⋊C₄	—	2	1	no	—
92	Dic₁₉⋊C₄	—	2	1	no	—
93	C₄×Dic₁₉	—	2	4	no	—
94	C₄×Dic₁₉	—	2	1	no	—
95	C₄×Dic₁₉	—	2	1	no	—
96	C₄×Dic₁₉	—	2	1	no	—
97	C₄×Dic₁₉	—	2	2	no	—
98	C₄×Dic₁₉	—	2	2	no	—
99	C₄×Dic₁₉	—	4	2	no	—
100	C₂×C₁₉⋊C₈	—	2	4	no	—
101	C₂×C₁₉⋊C₈	—	2	2	no	—
102	C₂×C₁₉⋊C₈	—	2	2	no	—
103	#64515	—	4	4	no	—
104	#8196	—	19	1	no	maximal, Sylow
105	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
106	C₂²×D₁₉	—	2	1	no	—
107	C₂×C₄×C₁₉	—	2	2	no	—
108	C₂×C₄×C₁₉	—	2	2	no	—
109	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
110	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
112	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
116	C₂×C₄×C₁₉	#131	1	2	yes	—
117	C₂²⋊D₁₉	—	4	2	no	—
118	C₂×Dic₁₉	—	2	4	no	—
119	C₂×Dic₁₉	—	2	1	no	—
120	C₂×Dic₁₉	—	2	1	no	—
121	C₂×Dic₁₉	—	2	1	no	—
122	C₂×Dic₁₉	—	4	2	no	—
123	C₂×Dic₁₉	—	4	2	no	—
124	D₇₆	—	2	1	no	—
125	D₇₆	—	2	1	no	—
126	C₄×D₁₉	—	4	2	no	—
127	C₄×D₁₉	—	4	2	no	—
128	C₄×D₁₉	—	4	2	no	—
129	Dic₃₈	—	2	1	no	—
130	Dic₃₈	—	2	1	no	—
131	C₈×C₁₉	—	2	2	no	—
132	C₈×C₁₉	—	2	4	no	—
133	C₈×C₁₉	—	4	2	no	—
134	C₈×C₁₉	#104	1	4	yes	complemented
135	C₁₉⋊C₈	—	2	4	no	—
136	C₁₉⋊C₈	—	4	2	no	—
137	#2965	—	19	1	no	—
138	C₄×#104	—	19	1	no	—
139	C₄×#97	—	19	1	no	—
140	#1476	—	19	1	no	—
141	C₈×M₁₆	D₁₉	1	1	yes	characteristic, complemented
142	#1278	—	19	1	no	—
143	#1276	—	19	1	no	—
144	C₂²×C₁₉	—	2	1	no	—
145	C₂²×C₁₉	—	2	1	no	—
146	C₂²×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
147	C₂²×C₁₉	#436	1	2	yes	—
148	C₂×D₁₉	—	4	1	no	—
149	C₂×D₁₉	—	4	1	no	—
150	C₂×D₁₉	—	4	1	no	—
151	C₄×C₁₉	—	2	1	no	—
152	C₄×C₁₉	—	2	1	no	—
153	C₄×C₁₉	C₄×M₁₆	1	1	yes	characteristic
154	C₄×C₁₉	#405	1	1	yes	characteristic
155	C₄×C₁₉	C₂×#104	1	2	yes	—
156	C₄×C₁₉	#435	1	2	yes	—
157	C₄×C₁₉	#444	1	2	yes	—
158	Dic₁₉	—	4	4	no	—
159	Dic₁₉	—	4	1	no	—
160	Dic₁₉	—	4	1	no	—
161	Dic₁₉	—	4	1	no	—
162	C₄×#41	—	19	1	no	—
163	#446	—	19	2	no	—
164	#436	—	19	2	no	—
165	#435	—	38	2	no	—
166	#431	—	19	1	no	—
167	#431	—	38	1	no	—
168	#431	C₂×D₁₉	1	1	yes	characteristic
169	C₂×#104	—	19	2	no	—
170	C₄×M₁₆	—	38	1	no	—
171	C₄×M₁₆	C₂×D₁₉	1	1	yes	characteristic
172	C₂×#97	—	38	2	no	—
173	C₂×C₄×C₈	—	19	1	no	—
174	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
175	C₄³	—	19	1	no	—
176	#339	—	19	2	no	—
177	Q₈⋊C₈	—	19	2	no	—
178	D₄⋊C₈	—	19	2	no	—
179	#322	—	2	1	no	—
180	C₈²	—	2	1	no	—
181	C₂×C₁₉	—	2	2	no	—
182	C₂×C₁₉	#1753	1	1	yes	characteristic
183	C₂×C₁₉	C₄×#104	1	1	yes	characteristic
184	C₂×C₁₉	#1765	1	1	yes	characteristic
185	D₁₉	—	8	1	no	—
186	C₂×#41	—	19	1	no	—
187	C₂×M₁₆	—	38	2	no	—
188	C₂×M₁₆	C₄×D₁₉	1	2	yes	—
189	C₂²×C₈	—	38	1	no	—
190	C₂²×C₈	D₇₆	1	2	yes	—
191	C₄×Q₈	—	19	1	no	—
192	C₄×Q₈	—	19	1	no	—
193	C₄×D₄	—	19	1	no	—
194	C₄×D₄	—	19	1	no	—
195	C₄×D₄	—	38	1	no	—
196	C₄×D₄	—	38	1	no	—
197	#117	—	19	1	no	—
198	#117	—	38	1	no	—
199	C₂×C₄²	—	19	2	no	—
200	C₂×C₄²	—	38	1	no	—
201	C₂×C₄²	—	38	1	no	—
202	C₂×C₄²	—	38	1	no	—
203	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
204	C₄⋊C₈	—	19	2	no	—
205	C₄⋊C₈	—	38	2	no	—
206	C₄⋊C₈	C₄×D₁₉	1	2	yes	—
207	#104	—	38	4	no	—
208	C₂²⋊C₈	—	2	1	no	—
209	C₂²⋊C₈	—	38	1	no	—
210	C₂²⋊C₈	—	38	2	no	—
211	#97	—	2	1	no	—
212	#97	—	38	8	no	—
213	#97	—	38	2	no	—
214	C₄×C₈	—	2	1	no	—
215	C₄×C₈	—	2	1	no	—
216	C₄×C₈	—	19	2	no	—
217	C₄×C₈	—	38	1	no	—
218	C₄×C₈	—	38	1	no	—
219	C₄×C₈	C₄×D₁₉	1	2	yes	—
220	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
221	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
222	C₁₉	#8196	1	1	yes	characteristic, complemented, Sylow
223	#41	—	38	2	no	—
224	#41	—	38	2	no	—
225	C₂×Q₈	—	19	1	no	—
226	C₂×D₄	—	19	1	no	—
227	C₂×D₄	—	38	1	no	—
228	C₂²×C₄	—	38	2	no	—
229	C₂²×C₄	—	38	1	no	—
230	C₂²×C₄	—	38	1	no	—
231	C₂²×C₄	—	38	1	no	—
232	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
234	C₂²×C₄	#64541	1	1	yes	characteristic
235	M₁₆	—	2	2	no	—
236	M₁₆	—	76	4	no	—
237	M₁₆	#64515	1	4	yes	complemented
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	38	4	no	—
242	C₂×C₈	—	38	2	no	—
243	C₂×C₈	—	38	2	no	—
244	C₂×C₈	(C₂×D₁₉)⋊C₄	1	2	yes	—
245	C₂×C₈	(C₂×D₁₉)⋊C₄	1	2	yes	—
246	C₄⋊C₄	—	19	1	no	—
247	C₄⋊C₄	—	19	1	no	—
248	C₄⋊C₄	—	38	1	no	—
249	C₄⋊C₄	—	38	1	no	—
250	C₂²⋊C₄	—	38	1	no	—
251	C₂²⋊C₄	—	38	1	no	—
252	C₂²⋊C₄	—	38	1	no	—
253	C₄²	—	38	4	no	—
254	C₄²	—	38	1	no	—
255	C₄²	—	38	1	no	—
256	C₄²	—	38	1	no	—
257	C₄²	—	38	2	no	—
258	C₄²	—	38	2	no	—
259	C₄²	—	76	2	no	—
260	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
261	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
262	C₄²	#64541	1	1	yes	characteristic
263	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
264	C₂³	—	38	1	no	—
265	C₂³	C₄×D₇₆	1	1	yes	characteristic
266	Q₈	—	38	1	no	—
267	Q₈	—	38	1	no	—
268	D₄	—	38	1	no	—
269	D₄	—	38	1	no	—
270	D₄	—	76	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	38	4	no	—
274	C₂×C₄	—	38	1	no	—
275	C₂×C₄	—	38	1	no	—
276	C₂×C₄	—	38	1	no	—
277	C₂×C₄	—	76	2	no	—
278	C₂×C₄	—	76	2	no	—
279	C₂×C₄	—	76	2	no	—
280	C₂×C₄	—	76	2	no	—
281	C₂×C₄	—	76	2	no	—
282	C₂×C₄	C₄²×D₁₉	1	1	yes	characteristic
283	C₂×C₄	#10597318	1	1	yes	characteristic
284	C₂×C₄	C₄×D₇₆	1	1	yes	characteristic
285	C₂×C₄	C₂×#64515	1	2	yes	—
286	C₂×C₄	#10597347	1	2	yes	—
287	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
288	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
289	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
290	C₈	—	2	2	no	—
291	C₈	—	2	4	no	—
292	C₈	—	4	2	no	—
293	C₈	—	38	4	no	—
294	C₈	—	76	2	no	—
295	C₈	#10597282	1	4	yes	complemented
296	C₂²	—	2	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	76	1	no	—
299	C₂²	—	76	1	no	—
300	C₂²	—	76	1	no	—
301	C₂²	C₄×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
302	C₂²	#11926623	1	2	yes	—
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	76	4	no	—
306	C₄	—	76	1	no	—
307	C₄	—	76	1	no	—
308	C₄	—	76	1	no	—
309	C₄	C₄×#64515	1	1	yes	characteristic
310	C₄	#11926498	1	1	yes	characteristic
311	C₄	C₂×#10597282	1	2	yes	—
312	C₄	#11926615	1	2	yes	—
313	C₄	#11926664	1	2	yes	—
314	C₂	—	2	2	no	—
315	C₂	—	152	1	no	—
316	C₂	#14563418	1	1	yes	characteristic
317	C₂	C₄×#10597282	1	1	yes	characteristic
318	C₂	#14563481	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×C₂²⋊D₁₉
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₈×M₁₆×C₁₉	C₂
socle	C₂²×C₁₉	C₄×C₂²⋊C₄
prime radical	C₂×#64541	C₂×C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₈×M₁₆	D₁₉
hypocenter	C₁₉	#8196
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
3. C₄×#10597272	C₂	semidirect
5. #14561473	C₂	semidirect
6. #14561467	C₂	semidirect
7. C₈×M₁₆×C₁₉	C₂	semidirect
9. #11926674	C₄	semidirect
10. #11926623	C₄	semidirect
20. #11926274	C₄	semidirect
21. #11926271	C₄	semidirect
76. M₁₆×C₁₉	M₁₆	semidirect
134. C₈×C₁₉	#104	semidirect
141. C₈×M₁₆	D₁₉	semidirect
222. C₁₉	#8196	semidirect
237. M₁₆	#64515	semidirect
295. C₈	#10597282	semidirect

Maximal subgroups

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

subgroup	index
#14572601	2
C₄×#10597282	2
C₄×#10597272	2
#14562015	2
#14561473	2
#14561467	2
C₈×M₁₆×C₁₉	2
#8196	19