Subgroups — #18073290

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	#14572602	C₂	1	1	yes	maximal, characteristic
2	C₄×#10597260	C₂	1	1	yes	maximal, characteristic, complemented
3	#14562017	C₂	1	1	yes	maximal, characteristic
4	M₁₆×C₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#14561473	C₂	1	1	yes	maximal, characteristic, complemented
6	#14561468	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₁₉×#104	C₂	1	1	yes	maximal, characteristic
8	#11926675	C₄	1	2	yes	complemented
9	#11926622	C₄	1	2	yes	complemented
10	#11926616	—	2	2	no	—
11	#11926597	—	2	1	no	—
12	#11926597	C₂²	1	1	yes	characteristic
13	#11926597	C₂²	1	1	yes	characteristic
14	C₄×#64520	—	2	1	no	—
15	C₂×#10597260	—	2	2	no	—
16	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
18	Q₈⋊(C₁₉⋊C₈)	C₄	1	2	yes	complemented
19	D₄⋊(C₁₉⋊C₈)	C₄	1	2	yes	complemented
20	#11926265	—	2	1	no	—
21	C₈×C₁₉⋊C₈	—	2	1	no	—
22	C₄×#41×C₁₉	C₂²	1	1	yes	characteristic
23	C₂×C₁₉×#104	C₄	1	2	yes	—
24	C₄×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
25	C₄³×C₁₉	C₂²	1	1	yes	characteristic
26	C₁₉×#339	C₄	1	2	yes	—
27	C₂×#41×C₁₉	C₂×C₄	1	1	yes	characteristic
28	C₂×M₁₆×C₁₉	C₂×C₄	1	2	yes	—
29	C₄×Q₈×C₁₉	C₂×C₄	1	1	yes	characteristic
30	C₄×Q₈×C₁₉	C₂×C₄	1	1	yes	characteristic
31	C₄×D₄×C₁₉	—	2	1	no	—
32	C₄×D₄×C₁₉	—	2	1	no	—
33	C₄×D₄×C₁₉	C₂×C₄	1	1	yes	characteristic
34	C₄×D₄×C₁₉	C₂×C₄	1	1	yes	characteristic
35	C₁₉×#117	—	2	1	no	—
36	C₁₉×#117	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₉	—	2	1	no	—
38	C₂×C₄²×C₁₉	—	2	1	no	—
39	C₂×C₄²×C₁₉	—	2	1	no	—
40	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
42	C₂×#64520	—	2	2	no	—
43	C₂²×C₁₉⋊C₈	—	2	1	no	—
44	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
45	C₁₉×#104	—	2	4	no	—
46	C₁₉×#97	—	2	1	no	—
47	C₄×C₈×C₁₉	—	2	1	no	—
48	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
49	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
50	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
51	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
52	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	2	yes	—
53	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	2	yes	—
54	#10597260	—	2	8	no	—
55	#10597260	—	2	2	no	—
56	C₄×C₁₉⋊C₈	—	2	1	no	—
57	C₄×C₁₉⋊C₈	—	2	1	no	—
58	C₄×C₁₉⋊C₈	—	2	1	no	—
59	C₄×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
60	C₄×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
61	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
62	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
63	#41×C₁₉	—	2	2	no	—
64	#41×C₁₉	—	2	2	no	—
65	C₂×Q₈×C₁₉	C₄²	1	1	yes	characteristic
66	C₂×D₄×C₁₉	—	2	1	no	—
67	C₂×D₄×C₁₉	C₄²	1	1	yes	characteristic
68	C₂²×C₄×C₁₉	—	2	2	no	—
69	C₂²×C₄×C₁₉	—	2	1	no	—
70	C₂²×C₄×C₁₉	—	2	1	no	—
71	C₂²×C₄×C₁₉	—	2	1	no	—
72	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₉	#41	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	C₄⋊C₄×C₁₉	—	2	1	no	—
79	C₄⋊C₄×C₁₉	—	2	1	no	—
80	C₄⋊C₄×C₁₉	C₄²	1	1	yes	characteristic
81	C₄⋊C₄×C₁₉	C₄²	1	1	yes	characteristic
82	C₂²⋊C₄×C₁₉	—	2	1	no	—
83	C₂²⋊C₄×C₁₉	—	2	1	no	—
84	C₂²⋊C₄×C₁₉	—	2	1	no	—
85	C₄²×C₁₉	—	2	4	no	—
86	C₄²×C₁₉	—	2	1	no	—
87	C₄²×C₁₉	—	2	1	no	—
88	C₄²×C₁₉	—	2	1	no	—
89	C₄²×C₁₉	—	2	2	no	—
90	C₄²×C₁₉	—	2	2	no	—
91	C₄²×C₁₉	—	4	2	no	—
92	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₁₉	#41	1	1	yes	characteristic
96	#64520	—	4	4	no	—
97	C₂×C₁₉⋊C₈	—	2	4	no	—
98	C₂×C₁₉⋊C₈	—	2	2	no	—
99	C₂×C₁₉⋊C₈	—	2	4	no	—
100	C₂×C₁₉⋊C₈	—	2	2	no	—
101	C₂×C₁₉⋊C₈	—	2	2	no	—
102	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	2	yes	—
103	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	2	yes	—
104	#8196	—	19	1	no	maximal, Sylow
105	C₂³×C₁₉	—	2	1	no	—
106	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
107	Q₈×C₁₉	—	2	1	no	—
108	Q₈×C₁₉	—	2	1	no	—
109	D₄×C₁₉	—	2	1	no	—
110	D₄×C₁₉	—	2	1	no	—
111	D₄×C₁₉	—	4	2	no	—
112	C₂×C₄×C₁₉	—	2	2	no	—
113	C₂×C₄×C₁₉	—	2	2	no	—
114	C₂×C₄×C₁₉	—	2	4	no	—
115	C₂×C₄×C₁₉	—	2	1	no	—
116	C₂×C₄×C₁₉	—	2	1	no	—
117	C₂×C₄×C₁₉	—	2	1	no	—
118	C₂×C₄×C₁₉	—	4	2	no	—
119	C₂×C₄×C₁₉	—	4	2	no	—
120	C₂×C₄×C₁₉	—	4	2	no	—
121	C₂×C₄×C₁₉	—	4	2	no	—
122	C₂×C₄×C₁₉	—	4	2	no	—
123	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
124	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
126	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
130	C₂×C₄×C₁₉	#131	1	2	yes	—
131	C₈×C₁₉	—	2	4	no	—
132	C₁₉⋊C₈	—	2	4	no	—
133	C₁₉⋊C₈	—	2	2	no	—
134	C₁₉⋊C₈	—	4	2	no	—
135	C₁₉⋊C₈	—	4	2	no	—
136	C₁₉⋊C₈	#104	1	4	yes	complemented
137	#2965	—	19	1	no	—
138	C₄×#104	D₁₉	1	1	yes	characteristic
139	C₄×#97	—	19	1	no	—
140	#1476	—	19	1	no	—
141	C₈×M₁₆	—	19	1	no	—
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	—	4	1	no	—
147	C₂²×C₁₉	—	4	1	no	—
148	C₂²×C₁₉	—	4	1	no	—
149	C₂²×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
150	C₂²×C₁₉	#436	1	2	yes	—
151	C₄×C₁₉	—	2	1	no	—
152	C₄×C₁₉	—	2	1	no	—
153	C₄×C₁₉	—	4	4	no	—
154	C₄×C₁₉	—	4	1	no	—
155	C₄×C₁₉	—	4	1	no	—
156	C₄×C₁₉	—	4	1	no	—
157	C₄×C₁₉	C₄×M₁₆	1	1	yes	characteristic
158	C₄×C₁₉	#405	1	1	yes	characteristic
159	C₄×C₁₉	C₂×#104	1	2	yes	—
160	C₄×C₁₉	#435	1	2	yes	—
161	C₄×C₁₉	#444	1	2	yes	—
162	C₄×#41	C₂×D₁₉	1	1	yes	characteristic
163	#446	—	19	2	no	—
164	#436	—	19	2	no	—
165	#435	—	38	2	no	—
166	#431	—	19	1	no	—
167	#431	—	19	1	no	—
168	#431	—	38	1	no	—
169	C₂×#104	Dic₁₉	1	2	yes	—
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	—	19	1	no	—
175	C₄³	C₂×D₁₉	1	1	yes	characteristic
176	#339	Dic₁₉	1	2	yes	—
177	Q₈⋊C₈	—	19	2	no	—
178	D₄⋊C₈	—	19	2	no	—
179	#322	—	38	1	no	—
180	C₈²	—	38	1	no	—
181	C₂×C₁₉	—	2	2	no	—
182	C₂×C₁₉	—	8	1	no	—
183	C₂×C₁₉	#1753	1	1	yes	characteristic
184	C₂×C₁₉	C₄×#104	1	1	yes	characteristic
185	C₂×C₁₉	#1765	1	1	yes	characteristic
186	C₂×#41	C₂×Dic₁₉	1	1	yes	characteristic
187	C₂×M₁₆	—	38	2	no	—
188	C₂×M₁₆	C₂×Dic₁₉	1	2	yes	—
189	C₂²×C₈	—	19	2	no	—
190	C₂²×C₈	—	38	1	no	—
191	C₄×Q₈	C₄×D₁₉	1	1	yes	characteristic
192	C₄×Q₈	C₄×D₁₉	1	1	yes	characteristic
193	C₄×D₄	—	2	1	no	—
194	C₄×D₄	—	2	1	no	—
195	C₄×D₄	C₄×D₁₉	1	1	yes	characteristic
196	C₄×D₄	C₄×D₁₉	1	1	yes	characteristic
197	#117	—	2	1	no	—
198	#117	C₂×Dic₁₉	1	1	yes	characteristic
199	C₂×C₄²	—	2	1	no	—
200	C₂×C₄²	—	2	1	no	—
201	C₂×C₄²	—	2	1	no	—
202	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
203	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
204	C₄⋊C₈	—	19	2	no	—
205	C₄⋊C₈	—	19	2	no	—
206	C₄⋊C₈	—	38	2	no	—
207	#104	—	2	4	no	—
208	C₂²⋊C₈	—	38	2	no	—
209	C₂²⋊C₈	—	38	1	no	—
210	C₂²⋊C₈	—	38	1	no	—
211	#97	—	2	1	no	—
212	#97	—	38	2	no	—
213	#97	—	38	8	no	—
214	C₄×C₈	—	2	1	no	—
215	C₄×C₈	—	19	2	no	—
216	C₄×C₈	—	19	2	no	—
217	C₄×C₈	—	19	1	no	—
218	C₄×C₈	—	19	1	no	—
219	C₄×C₈	—	38	1	no	—
220	C₄×C₈	—	38	1	no	—
221	C₄×C₈	—	38	1	no	—
222	C₁₉	#8196	1	1	yes	characteristic, complemented, Sylow
223	#41	—	2	2	no	—
224	#41	—	2	2	no	—
225	C₂×Q₈	C₄×Dic₁₉	1	1	yes	characteristic
226	C₂×D₄	—	2	1	no	—
227	C₂×D₄	C₄×Dic₁₉	1	1	yes	characteristic
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	—	2	1	no	—
232	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
233	C₂²×C₄	#64543	1	1	yes	characteristic
234	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
235	M₁₆	—	2	2	no	—
236	M₁₆	—	76	4	no	—
237	M₁₆	#64520	1	4	yes	complemented
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	19	2	no	—
240	C₂×C₈	—	19	2	no	—
241	C₂×C₈	—	38	4	no	—
242	C₂×C₈	—	38	4	no	—
243	C₂×C₈	—	38	2	no	—
244	C₂×C₈	—	38	2	no	—
245	C₂×C₈	—	38	2	no	—
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	C₄×Dic₁₉	1	1	yes	characteristic
249	C₄⋊C₄	C₄×Dic₁₉	1	1	yes	characteristic
250	C₂²⋊C₄	—	2	1	no	—
251	C₂²⋊C₄	—	2	1	no	—
252	C₂²⋊C₄	—	2	1	no	—
253	C₄²	—	2	4	no	—
254	C₄²	—	2	1	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	4	2	no	—
260	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
261	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
262	C₄²	D₄×D₁₉	1	1	yes	characteristic
263	C₄²	#64543	1	1	yes	characteristic
264	C₂³	—	2	1	no	—
265	C₂³	D₄×Dic₁₉	1	1	yes	characteristic
266	Q₈	—	2	1	no	—
267	Q₈	—	2	1	no	—
268	D₄	—	2	1	no	—
269	D₄	—	2	1	no	—
270	D₄	—	4	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	#10597323	1	1	yes	characteristic
283	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
284	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
285	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
286	C₂×C₄	C₂×#64520	1	2	yes	—
287	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
288	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
289	C₂×C₄	#10597394	1	2	yes	—
290	C₈	—	2	4	no	—
291	C₈	—	19	4	no	—
292	C₈	—	38	4	no	—
293	C₈	—	38	2	no	—
294	C₈	—	76	2	no	—
295	C₈	—	76	2	no	—
296	C₂²	—	2	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	4	1	no	—
299	C₂²	—	4	1	no	—
300	C₂²	—	4	1	no	—
301	C₂²	C₂²⋊C₄×Dic₁₉	1	1	yes	characteristic
302	C₂²	#11926622	1	2	yes	—
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	4	4	no	—
306	C₄	—	4	1	no	—
307	C₄	—	4	1	no	—
308	C₄	—	4	1	no	—
309	C₄	C₄×#64520	1	1	yes	characteristic
310	C₄	#11926500	1	1	yes	characteristic
311	C₄	#104×D₁₉	1	2	yes	—
312	C₄	#11926616	1	2	yes	—
313	C₄	#11926663	1	2	yes	—
314	C₂	—	2	2	no	—
315	C₂	—	8	1	no	—
316	C₂	#14563419	1	1	yes	characteristic
317	C₂	#104×Dic₁₉	1	1	yes	characteristic
318	C₂	#14563480	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₁₉
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₄×C₁₉×#104	C₂
socle	C₂²×C₁₉	C₄×C₂²⋊C₄
prime radical	C₂×#41×C₁₉	C₂×C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₄×#104	D₁₉
hypocenter	C₁₉	#8196
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#104

Complements

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

N	H ≅ G/N	type
2. C₄×#10597260	C₂	semidirect
4. M₁₆×C₁₉⋊C₈	C₂	semidirect
5. #14561473	C₂	semidirect
6. #14561468	C₂	semidirect
8. #11926675	C₄	semidirect
9. #11926622	C₄	semidirect
18. Q₈⋊(C₁₉⋊C₈)	C₄	semidirect
19. D₄⋊(C₁₉⋊C₈)	C₄	semidirect
76. M₁₆×C₁₉	M₁₆	semidirect
136. C₁₉⋊C₈	#104	semidirect
222. C₁₉	#8196	semidirect
237. M₁₆	#64520	semidirect

Maximal subgroups

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

subgroup	index
#14572602	2
C₄×#10597260	2
#14562017	2
M₁₆×C₁₉⋊C₈	2
#14561473	2
#14561468	2
C₄×C₁₉×#104	2
#8196	19