Subgroups — #25084556

Order 11008.

710 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145544	C₂	1	2	yes	maximal
2	#19145179	C₂	1	1	yes	maximal, characteristic
3	#19145136	C₂	1	2	yes	maximal
4	#500×D₄₃	C₂	1	1	yes	maximal, characteristic
5	#19134274	C₂	1	2	yes	maximal, complemented
6	#19134269	C₂	1	2	yes	maximal, complemented
7	#19134213	C₂	1	1	yes	maximal, characteristic, complemented
8	#19134212	C₂	1	1	yes	maximal, characteristic, complemented
9	#19134189	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×#10621872	C₂	1	1	yes	maximal, characteristic
11	C₄₃×#1707	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583454	C₂²	1	2	yes	—
13	C₄⋊Q₈×D₄₃	C₂²	1	2	yes	—
14	#14583306	—	2	4	no	—
15	#14583304	C₂²	1	1	yes	characteristic
16	#14583304	C₂²	1	2	yes	—
17	#14583304	C₂²	1	1	yes	characteristic
18	#14583303	—	2	2	no	—
19	#14583302	C₂²	1	2	yes	—
20	#14583302	C₂²	1	2	yes	—
21	#14583295	C₂²	1	2	yes	—
22	#14583279	C₂²	1	2	yes	—
23	#14583278	C₂²	1	2	yes	—
24	#14583251	C₂²	1	2	yes	—
25	#14583249	C₂²	1	2	yes	—
26	#14583137	—	2	2	no	—
27	#14583108	—	2	4	no	—
28	#14583056	—	2	4	no	—
29	Q₁₆×Dic₄₃	—	2	2	no	—
30	#14582950	C₂²	1	1	yes	characteristic
31	#14582938	—	2	4	no	—
32	#107×D₄₃	—	2	2	no	—
33	C₄×#10621872	C₂²	1	1	yes	characteristic
34	C₄×C₈×D₄₃	C₂²	1	1	yes	characteristic
35	#14582621	C₂²	1	1	yes	characteristic
36	#14582620	C₂²	1	1	yes	characteristic
37	C₈×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
38	C₄₃×#500	C₂²	1	1	yes	characteristic
39	C₄₃×#498	C₂²	1	2	yes	—
40	C₄₃×#494	C₂²	1	1	yes	characteristic
41	C₄₃×#487	C₂²	1	2	yes	—
42	C₈²×C₄₃	C₂²	1	1	yes	characteristic
43	C₂×Q₈×D₄₃	—	2	2	no	—
44	C₂×Dic₁₇₂	—	2	4	no	—
45	C₂×#10621885	—	2	4	no	—
46	C₂×#10621872	D₄	1	2	yes	—
47	C₂×C₈×D₄₃	D₄	1	2	yes	—
48	#13063325	C₂³	1	2	yes	—
49	#13063325	C₂³	1	2	yes	—
50	#13063324	—	2	2	no	—
51	#13063324	—	2	2	no	—
52	#13063323	—	2	2	no	—
53	#13063316	—	2	2	no	—
54	#13063305	—	2	2	no	—
55	#13063301	—	2	2	no	—
56	Q₈×Dic₄₃	—	2	2	no	—
57	#13063290	—	2	2	no	—
58	#13063283	—	2	2	no	—
59	C₄⋊C₄×D₄₃	—	2	2	no	—
60	C₄⋊C₄×D₄₃	—	2	2	no	—
61	C₄²×D₄₃	C₂³	1	1	yes	characteristic
62	#13063259	—	2	2	no	—
63	#13063259	—	2	4	no	—
64	#13063259	—	2	2	no	—
65	#13063258	—	2	4	no	—
66	#13063258	—	2	4	no	—
67	#13063256	—	2	4	no	—
68	Dic₄₃⋊C₈	D₄	1	2	yes	—
69	#13063249	—	2	4	no	—
70	#13063248	—	2	4	no	—
71	(C₂×D₄₃)⋊C₈	D₄	1	2	yes	—
72	#13063235	D₄	1	2	yes	—
73	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
74	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
75	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
76	C₈×Dic₄₃	D₄	1	2	yes	—
77	C₂×Q₁₆×C₄₃	—	2	2	no	—
78	C₄⋊Q₈×C₄₃	C₂³	1	2	yes	—
79	C₄⋊Q₈×C₄₃	C₂³	1	2	yes	—
80	#107×C₄₃	—	2	2	no	—
81	#106×C₄₃	—	2	4	no	—
82	#103×C₄₃	—	2	4	no	—
83	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
84	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
86	C₂×Q₈×C₄₃	—	2	2	no	—
87	C₂×Q₈×C₄₃	—	2	2	no	—
88	Q₈×D₄₃	—	8	4	no	—
89	C₂×C₄×D₄₃	—	4	2	no	—
90	C₂×C₄×D₄₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×D₄₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×D₄₃	C₂×D₄	1	1	yes	characteristic
93	C₂×Dic₈₆	—	2	2	no	—
94	C₂×Dic₈₆	—	2	2	no	—
95	C₂×Dic₈₆	—	4	2	no	—
96	Q₁₆×C₄₃	—	4	4	no	—
97	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
98	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
99	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
100	C₄⋊C₄×C₄₃	—	2	2	no	—
101	C₄⋊C₄×C₄₃	—	2	2	no	—
102	C₄⋊C₄×C₄₃	—	2	2	no	—
103	C₄⋊C₄×C₄₃	—	2	2	no	—
104	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
105	#10621885	—	8	4	no	—
106	(C₂×D₄₃)⋊C₄	—	4	2	no	—
107	C₄⋊Dic₄₃	—	2	2	no	—
108	C₄⋊Dic₄₃	—	2	2	no	—
109	C₄⋊Dic₄₃	—	2	2	no	—
110	C₄⋊Dic₄₃	—	2	2	no	—
111	C₄⋊Dic₄₃	—	4	2	no	—
112	Dic₄₃⋊C₄	—	4	2	no	—
113	Dic₄₃⋊C₄	—	4	2	no	—
114	Dic₄₃⋊C₄	—	4	2	no	—
115	C₄×Dic₄₃	—	4	2	no	—
116	C₄×Dic₄₃	C₂×D₄	1	1	yes	characteristic
117	C₄×Dic₄₃	C₂×D₄	1	1	yes	characteristic
118	C₄×Dic₄₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
120	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
121	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
122	Dic₁₇₂	—	8	4	no	—
123	#10621872	Q₁₆	1	8	yes	complemented
124	C₈×D₄₃	—	2	4	no	—
125	C₂²×D₄₃	#127	1	1	yes	characteristic
126	Q₈×C₄₃	—	4	4	no	—
127	Q₈×C₄₃	—	4	4	no	—
128	C₂×C₄×C₄₃	—	4	2	no	—
129	C₂×C₄×C₄₃	—	4	2	no	—
130	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
133	C₂×Dic₄₃	—	4	2	no	—
134	C₂×Dic₄₃	—	4	2	no	—
135	C₂×Dic₄₃	#127	1	1	yes	characteristic
136	C₂×Dic₄₃	#127	1	1	yes	characteristic
137	C₂×Dic₄₃	#127	1	1	yes	characteristic
138	C₄×D₄₃	—	2	2	no	—
139	C₄×D₄₃	—	2	2	no	—
140	C₄×D₄₃	—	8	4	no	—
141	C₄×D₄₃	C₂×Q₁₆	1	4	yes	—
142	Dic₈₆	—	4	4	no	—
143	Dic₈₆	—	4	4	no	—
144	Dic₈₆	—	8	4	no	—
145	C₈×C₄₃	—	2	2	no	—
146	C₈×C₄₃	C₂×Q₁₆	1	4	yes	—
147	C₈×C₄₃	#137	1	4	yes	—
148	C₄₃⋊C₈	—	2	2	no	—
149	C₄₃⋊C₈	—	2	2	no	—
150	C₄₃⋊C₈	C₂×Q₁₆	1	4	yes	—
151	#19633	—	43	1	no	maximal, Sylow
152	C₂²×C₄₃	C₂×#127	1	1	yes	characteristic
153	C₂×D₄₃	—	2	1	no	—
154	C₂×D₄₃	—	2	1	no	—
155	C₂×D₄₃	#494	1	2	yes	—
156	C₄×C₄₃	—	8	2	no	—
157	C₄×C₄₃	—	8	2	no	—
158	C₄×C₄₃	C₂²×Q₁₆	1	2	yes	—
159	C₄×C₄₃	C₂×#137	1	2	yes	—
160	C₄×C₄₃	#576	1	2	yes	—
161	Dic₄₃	—	2	1	no	—
162	Dic₄₃	—	2	1	no	—
163	Dic₄₃	—	8	2	no	—
164	Dic₄₃	—	8	2	no	—
165	Dic₄₃	#494	1	2	yes	—
166	#3209	—	43	2	no	—
167	#3170	—	43	1	no	—
168	#3165	—	43	2	no	—
169	C₂×#500	—	43	1	no	—
170	#1723	—	43	2	no	—
171	#1723	—	43	2	no	—
172	#1713	—	43	1	no	—
173	#1713	—	43	1	no	—
174	#1707	—	43	1	no	—
175	#1707	D₄₃	1	1	yes	characteristic, complemented
176	C₈×M₁₆	—	43	1	no	—
177	C₂×C₄₃	C₂×#494	1	1	yes	characteristic
178	C₂×C₄₃	#3152	1	1	yes	characteristic
179	C₂×C₄₃	#3155	1	1	yes	characteristic
180	D₄₃	—	2	2	no	—
181	#533	—	43	2	no	—
182	C₂×C₄⋊Q₈	—	43	2	no	—
183	#501	—	86	4	no	—
184	#500	—	43	2	no	—
185	#500	—	43	2	no	—
186	#500	—	43	1	no	—
187	#500	—	43	2	no	—
188	#500	—	43	1	no	—
189	#500	—	86	2	no	—
190	#500	C₂×D₄₃	1	1	yes	characteristic
191	#498	—	43	2	no	—
192	#498	C₂×D₄₃	1	2	yes	—
193	#494	—	43	2	no	—
194	#494	—	43	2	no	—
195	#494	C₂×D₄₃	1	1	yes	characteristic
196	#487	—	43	2	no	—
197	#487	—	43	2	no	—
198	#487	C₂×D₄₃	1	2	yes	—
199	#467	—	86	2	no	—
200	#462	—	86	4	no	—
201	#451	—	86	4	no	—
202	C₄×Q₁₆	—	86	2	no	—
203	#431	—	43	1	no	—
204	#428	—	86	4	no	—
205	C₂×#107	—	86	2	no	—
206	C₄×M₁₆	—	43	1	no	—
207	C₂×C₄×C₈	—	43	1	no	—
208	#322	—	43	1	no	—
209	#322	—	43	1	no	—
210	C₈²	—	43	1	no	—
211	C₈²	C₂×D₄₃	1	1	yes	characteristic
212	C₄₃	#19633	1	1	yes	characteristic, complemented, Sylow
213	C₂²×Q₈	—	86	2	no	—
214	C₂×Q₁₆	—	2	2	no	—
215	C₂×Q₁₆	—	86	4	no	—
216	C₂×Q₁₆	—	86	4	no	—
217	C₂×M₁₆	—	43	2	no	—
218	C₂²×C₈	—	43	2	no	—
219	C₄⋊Q₈	—	43	2	no	—
220	C₄⋊Q₈	—	43	2	no	—
221	C₄⋊Q₈	—	86	2	no	—
222	C₄⋊Q₈	—	86	2	no	—
223	C₄⋊Q₈	—	86	2	no	—
224	C₄⋊Q₈	C₂²×D₄₃	1	2	yes	—
225	C₄⋊Q₈	C₂²×D₄₃	1	2	yes	—
226	#125	—	86	2	no	—
227	C₂²⋊Q₈	—	86	2	no	—
228	C₂²⋊Q₈	—	86	2	no	—
229	C₄×Q₈	—	86	2	no	—
230	C₄×Q₈	—	86	2	no	—
231	#117	—	86	2	no	—
232	C₂×C₄⋊C₄	—	86	2	no	—
233	C₂×C₄⋊C₄	—	86	2	no	—
234	C₂×C₄²	—	43	1	no	—
235	#107	—	2	2	no	—
236	#107	—	86	4	no	—
237	#107	—	86	4	no	—
238	#107	—	86	2	no	—
239	#107	—	86	4	no	—
240	#107	—	86	2	no	—
241	#106	—	2	4	no	—
242	#106	—	86	4	no	—
243	C₄⋊C₈	—	43	2	no	—
244	#103	—	2	4	no	—
245	#103	—	86	4	no	—
246	#103	—	86	4	no	—
247	C₂²⋊C₈	—	43	2	no	—
248	#97	—	43	2	no	—
249	C₄×C₈	—	43	2	no	—
250	C₄×C₈	—	43	1	no	—
251	C₄×C₈	—	43	1	no	—
252	C₄×C₈	—	43	1	no	—
253	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
254	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
255	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
256	C₂×Q₈	—	2	2	no	—
257	C₂×Q₈	—	2	2	no	—
258	C₂×Q₈	—	86	2	no	—
259	C₂×Q₈	—	86	2	no	—
260	C₂×Q₈	—	344	4	no	—
261	C₂×Q₈	—	172	2	no	—
262	C₂²×C₄	—	43	1	no	—
263	C₂²×C₄	—	43	1	no	—
264	C₂²×C₄	—	43	1	no	—
265	C₂²×C₄	—	172	2	no	—
266	Q₁₆	—	4	4	no	—
267	Q₁₆	—	344	4	no	—
268	Q₁₆	—	344	4	no	—
269	M₁₆	—	43	8	no	—
270	C₂×C₈	—	43	2	no	—
271	C₂×C₈	—	43	2	no	—
272	C₂×C₈	—	43	2	no	—
273	C₂×C₈	—	86	4	no	—
274	C₂×C₈	D₄×D₄₃	1	2	yes	—
275	C₂×C₈	D₄×D₄₃	1	2	yes	—
276	C₂×C₈	D₄×D₄₃	1	2	yes	—
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	86	2	no	—
282	C₄⋊C₄	—	86	2	no	—
283	C₄⋊C₄	—	86	2	no	—
284	C₄⋊C₄	—	86	2	no	—
285	C₄⋊C₄	—	172	2	no	—
286	C₄⋊C₄	—	172	2	no	—
287	C₄⋊C₄	—	172	2	no	—
288	C₄⋊C₄	—	172	2	no	—
289	C₂²⋊C₄	—	172	2	no	—
290	C₄²	—	43	1	no	—
291	C₄²	—	43	1	no	—
292	C₄²	—	43	1	no	—
293	C₄²	—	172	2	no	—
294	C₄²	C₂³×D₄₃	1	1	yes	characteristic
295	C₂³	—	43	1	no	—
296	Q₈	—	4	4	no	—
297	Q₈	—	4	4	no	—
298	Q₈	—	344	4	no	—
299	Q₈	—	172	4	no	—
300	Q₈	—	172	4	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	43	1	no	—
304	C₂×C₄	—	43	4	no	—
305	C₂×C₄	—	43	1	no	—
306	C₂×C₄	—	43	1	no	—
307	C₂×C₄	—	86	2	no	—
308	C₂×C₄	—	86	2	no	—
309	C₂×C₄	—	344	4	no	—
310	C₂×C₄	—	172	2	no	—
311	C₂×C₄	—	172	2	no	—
312	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
313	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
314	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
315	C₈	—	2	2	no	—
316	C₈	—	43	4	no	—
317	C₈	—	86	2	no	—
318	C₈	—	86	2	no	—
319	C₈	Q₁₆×D₄₃	1	4	yes	—
320	C₈	#13063354	1	4	yes	—
321	C₂²	—	43	2	no	—
322	C₂²	—	86	1	no	—
323	C₂²	—	86	1	no	—
324	C₂²	#127×D₄₃	1	1	yes	characteristic
325	C₄	—	8	2	no	—
326	C₄	—	8	2	no	—
327	C₄	—	43	2	no	—
328	C₄	—	86	1	no	—
329	C₄	—	86	1	no	—
330	C₄	—	344	2	no	—
331	C₄	—	344	2	no	—
332	C₄	C₂×Q₁₆×D₄₃	1	2	yes	—
333	C₄	C₂×#13063354	1	2	yes	—
334	C₄	#14583701	1	2	yes	—
335	C₂	—	86	2	no	—
336	C₂	#494×D₄₃	1	1	yes	characteristic
337	C₂	#19145049	1	1	yes	characteristic
338	C₂	#19145077	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#127×D₄₃
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting	C₄₃×#1707	C₂
socle	C₂²×C₄₃	C₂×#127
prime radical	C₂²×D₄₃	#127
hypercenter	#1707	D₄₃
hypocenter	C₄₃	#19633
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1707

Complements

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

N	H ≅ G/N	type
5. #19134274	C₂	semidirect
6. #19134269	C₂	semidirect
7. #19134213	C₂	semidirect
8. #19134212	C₂	semidirect
9. #19134189	C₂	semidirect
11. C₄₃×#1707	C₂	semidirect
123. #10621872	Q₁₆	semidirect
175. #1707	D₄₃	semidirect
212. C₄₃	#19633	semidirect

Maximal subgroups

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

subgroup	index
#19145544	2
#19145179	2
#19145136	2
#500×D₄₃	2
#19134274	2
#19134269	2
#19134213	2
#19134212	2
#19134189	2
C₈×#10621872	2
C₄₃×#1707	2
#19633	43