Subgroups — #25062306

Order 11008.

900 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	#19143226	C₂	1	1	yes	maximal, characteristic
2	C₄×M₁₆×D₄₃	C₂	1	2	yes	maximal
3	#19132898	C₂	1	2	yes	maximal, complemented
4	#19132898	C₂	1	1	yes	maximal, characteristic, complemented
5	#19132883	C₂	1	2	yes	maximal, complemented
6	#19132881	C₂	1	2	yes	maximal, complemented
7	#19132874	C₂	1	2	yes	maximal, complemented
8	C₈×#10621872	C₂	1	2	yes	maximal, complemented
9	C₄₃×#1455	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×M₁₆×D₄₃	C₄	1	4	yes	—
11	C₂×C₄²×D₄₃	C₂²	1	1	yes	characteristic
12	#14582954	C₄	1	4	yes	—
13	#14582950	C₂²	1	2	yes	—
14	#14582950	C₂²	1	2	yes	—
15	#14582922	C₄	1	2	yes	—
16	#14582862	C₄	1	2	yes	—
17	C₄×#10621877	C₂²	1	1	yes	characteristic
18	C₄×#10621877	C₂²	1	2	yes	—
19	#14582851	C₄	1	4	yes	—
20	M₁₆×Dic₄₃	C₄	1	4	yes	—
21	C₄×#10621872	C₂²	1	2	yes	—
22	C₄×#10621872	C₂²	1	2	yes	—
23	#97×D₄₃	C₄	1	4	yes	—
24	C₄×C₈×D₄₃	C₂²	1	2	yes	—
25	C₄×C₈×D₄₃	C₂²	1	2	yes	—
26	#14582621	C₂²	1	2	yes	complemented
27	#14582621	C₂²	1	2	yes	complemented
28	#14582621	C₂²	1	2	yes	complemented
29	#14582620	C₂²	1	2	yes	complemented
30	#14582620	C₂²	1	1	yes	characteristic, complemented
31	C₈×C₄₃⋊C₈	C₂²	1	2	yes	complemented
32	C₈×C₄₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
33	C₄×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
34	C₄×M₁₆×C₄₃	C₂²	1	2	yes	—
35	C₄₃×#322	C₂²	1	2	yes	complemented
36	C₄₃×#322	C₂²	1	1	yes	characteristic, complemented
37	C₈²×C₄₃	C₂²	1	1	yes	characteristic, complemented
38	C₂²×C₄×D₄₃	C₂×C₄	1	2	yes	—
39	C₂²×C₄×D₄₃	C₂×C₄	1	1	yes	characteristic
40	C₂×#10621877	C₂×C₄	1	4	yes	—
41	C₂×#10621877	C₂×C₄	1	2	yes	—
42	C₂×#10621872	C₂×C₄	1	4	yes	—
43	C₂×#10621872	C₂×C₄	1	4	yes	—
44	M₁₆×D₄₃	—	2	16	no	—
45	C₂×C₈×D₄₃	C₂×C₄	1	4	yes	—
46	C₂×C₈×D₄₃	C₂×C₄	1	4	yes	—
47	C₂×C₄×Dic₄₃	C₂×C₄	1	2	yes	—
48	C₂×C₄×Dic₄₃	C₂×C₄	1	1	yes	characteristic
49	C₄²×D₄₃	C₂×C₄	1	2	yes	—
50	C₄²×D₄₃	C₂×C₄	1	2	yes	—
51	C₄²×D₄₃	C₂×C₄	1	2	yes	—
52	C₄²×D₄₃	C₂³	1	2	yes	—
53	Dic₄₃⋊C₈	C₂×C₄	1	4	yes	—
54	Dic₄₃⋊C₈	C₂×C₄	1	4	yes	—
55	(C₂×D₄₃)⋊C₈	C₂×C₄	1	4	yes	—
56	(C₂×D₄₃)⋊C₈	C₂×C₄	1	4	yes	—
57	#13063236	C₂×C₄	1	4	yes	—
58	#13063236	C₂×C₄	1	2	yes	—
59	#13063235	C₂×C₄	1	4	yes	—
60	#13063235	C₂×C₄	1	4	yes	—
61	C₄×C₄₃⋊C₈	C₂³	1	2	yes	—
62	C₄×C₄₃⋊C₈	C₂³	1	2	yes	—
63	C₄×C₄₃⋊C₈	C₂³	1	2	yes	—
64	C₈×Dic₄₃	C₂×C₄	1	4	yes	—
65	C₈×Dic₄₃	C₂×C₄	1	4	yes	—
66	C₂×M₁₆×C₄₃	C₂×C₄	1	4	yes	—
67	C₂×M₁₆×C₄₃	C₂×C₄	1	2	yes	—
68	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
69	#97×C₄₃	C₂×C₄	1	4	yes	—
70	#97×C₄₃	C₂×C₄	1	2	yes	—
71	C₄×C₈×C₄₃	C₂³	1	2	yes	—
72	C₄×C₈×C₄₃	C₂³	1	2	yes	—
73	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
75	C₂³×D₄₃	C₄²	1	1	yes	characteristic
76	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
78	C₂²×Dic₄₃	C₄²	1	2	yes	—
79	C₂²×Dic₄₃	C₄²	1	1	yes	characteristic
80	C₂×C₄×D₄₃	—	2	8	no	—
81	C₂×C₄×D₄₃	—	4	2	no	—
82	C₂×C₄×D₄₃	C₄²	1	2	yes	—
83	C₂×C₄×D₄₃	C₄²	1	2	yes	—
84	C₂×C₄×D₄₃	C₄²	1	2	yes	—
85	C₂×C₄×D₄₃	C₂²×C₄	1	2	yes	—
86	C₂×C₄×D₄₃	C₂²×C₄	1	2	yes	—
87	C₂×C₄×D₄₃	C₂²×C₄	1	2	yes	—
88	M₁₆×C₄₃	—	2	8	no	—
89	M₁₆×C₄₃	—	4	2	no	—
90	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
91	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
92	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
93	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
94	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
95	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
97	C₄×Dic₄₃	C₄²	1	1	yes	characteristic
98	C₄×Dic₄₃	C₄²	1	2	yes	—
99	C₄×Dic₄₃	C₄²	1	2	yes	—
100	C₄×Dic₄₃	C₄²	1	1	yes	characteristic
101	C₄×Dic₄₃	C₂²×C₄	1	2	yes	—
102	C₄×Dic₄₃	C₂²×C₄	1	2	yes	—
103	C₄×Dic₄₃	C₂²×C₄	1	2	yes	—
104	#10621877	—	2	8	no	—
105	#10621877	—	4	2	no	—
106	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
107	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
108	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
109	#10621872	—	2	8	no	—
110	#10621872	M₁₆	1	16	yes	complemented
111	C₈×D₄₃	—	2	8	no	—
112	C₈×D₄₃	—	2	8	no	—
113	C₂³×C₄₃	C₂×C₄²	1	1	yes	characteristic
114	C₂²×D₄₃	—	2	4	no	—
115	C₂²×D₄₃	—	4	1	no	—
116	C₂²×D₄₃	C₂×C₄²	1	2	yes	—
117	C₂×C₄×C₄₃	—	2	4	no	—
118	C₂×C₄×C₄₃	—	2	2	no	—
119	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
120	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
121	C₂×C₄×C₄₃	C₂³×C₄	1	2	yes	—
122	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
123	C₂×Dic₄₃	—	2	4	no	—
124	C₂×Dic₄₃	—	4	1	no	—
125	C₂×Dic₄₃	C₂×C₄²	1	2	yes	—
126	C₂×Dic₄₃	C₂×C₄²	1	2	yes	—
127	C₂×Dic₄₃	C₂×C₄²	1	2	yes	—
128	C₄×D₄₃	—	2	4	no	—
129	C₄×D₄₃	—	2	4	no	—
130	C₄×D₄₃	—	2	4	no	—
131	C₄×D₄₃	—	2	4	no	—
132	C₄×D₄₃	—	4	2	no	—
133	C₄×D₄₃	C₂×M₁₆	1	8	yes	—
134	C₈×C₄₃	—	2	4	no	—
135	C₈×C₄₃	—	2	2	no	—
136	C₈×C₄₃	—	2	2	no	—
137	C₈×C₄₃	C₂×M₁₆	1	8	yes	complemented
138	C₄₃⋊C₈	—	2	4	no	—
139	C₄₃⋊C₈	—	2	4	no	—
140	C₄₃⋊C₈	C₂×M₁₆	1	8	yes	complemented
141	M₁₆²	—	43	1	no	maximal, Sylow
142	C₂²×C₄₃	—	2	2	no	—
143	C₂²×C₄₃	—	2	1	no	—
144	C₂²×C₄₃	C₂²×C₄²	1	1	yes	characteristic
145	C₂×D₄₃	—	2	2	no	—
146	C₂×D₄₃	—	2	2	no	—
147	C₂×D₄₃	—	4	4	no	—
148	C₂×D₄₃	C₄×M₁₆	1	4	yes	—
149	C₄×C₄₃	—	2	2	no	—
150	C₄×C₄₃	—	2	1	no	—
151	C₄×C₄₃	C₂²×M₁₆	1	4	yes	—
152	C₄×C₄₃	#568	1	2	yes	—
153	Dic₄₃	—	2	2	no	—
154	Dic₄₃	—	2	2	no	—
155	Dic₄₃	C₄×M₁₆	1	4	yes	—
156	#2920	—	43	1	no	—
157	C₂×C₄×M₁₆	—	43	2	no	—
158	#1455	—	43	2	no	—
159	#1455	—	43	1	no	—
160	#1455	D₄₃	1	1	yes	characteristic, complemented
161	#1451	—	43	2	no	—
162	#1451	—	43	2	no	—
163	C₈×M₁₆	—	43	2	no	—
164	C₈×M₁₆	—	43	2	no	—
165	C₂×C₄₃	—	4	1	no	—
166	C₂×C₄₃	C₂×C₄×M₁₆	1	2	yes	—
167	C₂×C₄₃	#2873	1	1	yes	characteristic
168	D₄₃	—	2	4	no	—
169	C₂²×M₁₆	—	43	4	no	—
170	C₂²×C₄²	—	43	1	no	—
171	#432	—	43	4	no	—
172	#431	—	43	2	no	—
173	#431	—	43	2	no	—
174	#423	—	43	2	no	—
175	#407	—	43	2	no	—
176	C₄×M₁₆	—	43	4	no	—
177	C₄×M₁₆	—	43	2	no	—
178	C₄×M₁₆	—	43	1	no	—
179	C₄×M₁₆	—	43	4	no	—
180	C₄×M₁₆	—	43	2	no	—
181	C₄×M₁₆	—	43	2	no	—
182	C₄×M₁₆	C₂×D₄₃	1	2	yes	—
183	C₄×M₁₆	C₂×D₄₃	1	1	yes	characteristic
184	C₂×#97	—	43	4	no	—
185	C₂×C₄×C₈	—	43	2	no	—
186	C₂×C₄×C₈	—	43	2	no	—
187	#322	—	43	2	no	—
188	#322	—	43	1	no	—
189	#322	—	43	2	no	—
190	#322	—	43	2	no	—
191	#322	—	43	2	no	—
192	#322	C₂×D₄₃	1	2	yes	complemented
193	#322	C₂×D₄₃	1	1	yes	characteristic, complemented
194	C₈²	—	43	1	no	—
195	C₈²	—	43	2	no	—
196	C₈²	C₂×D₄₃	1	1	yes	characteristic, complemented
197	C₄₃	M₁₆²	1	1	yes	characteristic, complemented, Sylow
198	C₂³×C₄	—	43	2	no	—
199	C₂³×C₄	—	43	1	no	—
200	C₂×M₁₆	—	43	4	no	—
201	C₂×M₁₆	—	43	2	no	—
202	C₂×M₁₆	—	43	4	no	—
203	C₂×M₁₆	—	43	4	no	—
204	C₂×M₁₆	—	86	16	no	—
205	C₂×M₁₆	C₄×D₄₃	1	4	yes	—
206	C₂×M₁₆	C₄×D₄₃	1	2	yes	—
207	C₂²×C₈	—	43	4	no	—
208	C₂²×C₈	—	43	4	no	—
209	C₂×C₄²	—	43	2	no	—
210	C₂×C₄²	—	43	1	no	—
211	C₂×C₄²	—	43	2	no	—
212	C₂×C₄²	—	43	2	no	—
213	C₂×C₄²	—	43	2	no	—
214	C₂×C₄²	—	43	2	no	—
215	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
216	C₄⋊C₈	—	43	4	no	—
217	C₄⋊C₈	—	43	4	no	—
218	C₂²⋊C₈	—	43	4	no	—
219	C₂²⋊C₈	—	43	4	no	—
220	#97	—	43	4	no	—
221	#97	—	43	4	no	—
222	#97	—	43	4	no	—
223	#97	—	43	2	no	—
224	#97	C₄×D₄₃	1	4	yes	—
225	#97	C₄×D₄₃	1	2	yes	—
226	C₄×C₈	—	43	4	no	—
227	C₄×C₈	—	43	4	no	—
228	C₄×C₈	—	43	2	no	—
229	C₄×C₈	—	43	2	no	—
230	C₄×C₈	—	43	2	no	—
231	C₄×C₈	C₂²×D₄₃	1	2	yes	—
232	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
233	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
234	C₄×C₈	C₂²×D₄₃	1	2	yes	—
235	C₂⁴	—	43	1	no	—
236	C₂²×C₄	—	43	2	no	—
237	C₂²×C₄	—	43	2	no	—
238	C₂²×C₄	—	43	1	no	—
239	C₂²×C₄	—	43	2	no	—
240	C₂²×C₄	—	43	2	no	—
241	C₂²×C₄	—	43	2	no	—
242	C₂²×C₄	—	43	2	no	—
243	C₂²×C₄	—	43	2	no	—
244	C₂²×C₄	—	86	8	no	—
245	C₂²×C₄	—	172	2	no	—
246	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
247	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
248	M₁₆	—	2	8	no	—
249	M₁₆	—	4	2	no	—
250	M₁₆	—	43	16	no	—
251	M₁₆	—	86	8	no	—
252	M₁₆	—	86	8	no	—
253	M₁₆	—	172	2	no	—
254	C₂×C₈	—	43	4	no	—
255	C₂×C₈	—	43	4	no	—
256	C₂×C₈	—	43	4	no	—
257	C₂×C₈	—	86	8	no	—
258	C₂×C₈	—	86	8	no	—
259	C₂×C₈	C₂×C₄×D₄₃	1	4	yes	—
260	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
261	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
262	C₂×C₈	C₂×C₄×D₄₃	1	4	yes	—
263	C₄²	—	43	1	no	—
264	C₄²	—	43	2	no	—
265	C₄²	—	43	2	no	—
266	C₄²	—	43	1	no	—
267	C₄²	—	43	2	no	—
268	C₄²	—	43	2	no	—
269	C₄²	—	43	2	no	—
270	C₄²	C₂×C₄×D₄₃	1	2	yes	—
271	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
272	C₄²	C₂³×D₄₃	1	1	yes	characteristic
273	C₂³	—	43	2	no	—
274	C₂³	—	86	4	no	—
275	C₂³	—	172	1	no	—
276	C₂³	C₄²×D₄₃	1	1	yes	characteristic
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	43	2	no	—
280	C₂×C₄	—	43	8	no	—
281	C₂×C₄	—	43	2	no	—
282	C₂×C₄	—	43	2	no	—
283	C₂×C₄	—	86	4	no	—
284	C₂×C₄	—	86	4	no	—
285	C₂×C₄	—	86	4	no	—
286	C₂×C₄	—	86	4	no	—
287	C₂×C₄	—	86	4	no	—
288	C₂×C₄	—	172	2	no	—
289	C₂×C₄	—	172	1	no	—
290	C₂×C₄	C₄²×D₄₃	1	2	yes	—
291	C₂×C₄	C₄²×D₄₃	1	1	yes	characteristic
292	C₂×C₄	C₂²×C₄×D₄₃	1	2	yes	—
293	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
294	C₈	—	2	4	no	—
295	C₈	—	2	2	no	—
296	C₈	—	2	2	no	—
297	C₈	—	43	8	no	—
298	C₈	—	86	4	no	—
299	C₈	—	86	4	no	—
300	C₈	M₁₆×D₄₃	1	8	yes	complemented
301	C₂²	—	2	2	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	43	4	no	—
304	C₂²	—	86	2	no	—
305	C₂²	—	86	2	no	—
306	C₂²	—	172	4	no	—
307	C₂²	C₂×C₄²×D₄₃	1	1	yes	characteristic
308	C₄	—	2	2	no	—
309	C₄	—	2	1	no	—
310	C₄	—	43	4	no	—
311	C₄	—	86	2	no	—
312	C₄	—	86	2	no	—
313	C₄	C₂×M₁₆×D₄₃	1	4	yes	—
314	C₄	#14583665	1	2	yes	—
315	C₂	—	4	1	no	—
316	C₂	—	86	4	no	—
317	C₂	C₄×M₁₆×D₄₃	1	2	yes	—
318	C₂	#19143012	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₄₃
commutator, socle	C₂²×C₄₃	C₂²×C₄²
Fitting	C₄₃×#1455	C₂
prime radical	C₂³×D₄₃	C₄²
cosocle	C₄²×C₄₃	C₂⁴
hypercenter	#1455	D₄₃
hypocenter	C₄₃	M₁₆²
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₄² ◁ #1455

Complements

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

N	H ≅ G/N	type
3. #19132898	C₂	semidirect
4. #19132898	C₂	semidirect
5. #19132883	C₂	semidirect
6. #19132881	C₂	semidirect
7. #19132874	C₂	semidirect
8. C₈×#10621872	C₂	semidirect
9. C₄₃×#1455	C₂	semidirect
26. #14582621	C₂²	semidirect
27. #14582621	C₂²	semidirect
28. #14582621	C₂²	semidirect
29. #14582620	C₂²	semidirect
30. #14582620	C₂²	semidirect
31. C₈×C₄₃⋊C₈	C₂²	semidirect
32. C₈×C₄₃⋊C₈	C₂²	semidirect
35. C₄₃×#322	C₂²	semidirect
36. C₄₃×#322	C₂²	semidirect
37. C₈²×C₄₃	C₂²	semidirect
110. #10621872	M₁₆	semidirect
137. C₈×C₄₃	C₂×M₁₆	semidirect
140. C₄₃⋊C₈	C₂×M₁₆	semidirect
160. #1455	D₄₃	semidirect
192. #322	C₂×D₄₃	semidirect
193. #322	C₂×D₄₃	semidirect
196. C₈²	C₂×D₄₃	semidirect
197. C₄₃	M₁₆²	semidirect
300. C₈	M₁₆×D₄₃	semidirect

Maximal subgroups

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

subgroup	index
#19143226	2
C₄×M₁₆×D₄₃	2
#19132898	2
#19132883	2
#19132881	2
#19132874	2
C₈×#10621872	2
C₄₃×#1455	2
M₁₆²	43