Subgroups — #25252120

Order 11008.

514 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145108	C₂	1	1	yes	maximal, characteristic
2	#500×D₄₃	C₂	1	1	yes	maximal, characteristic
3	#19136882	C₂	1	1	yes	maximal, characteristic, complemented
4	#19136881	C₂	1	1	yes	maximal, characteristic, complemented
5	#19136878	C₂	1	1	yes	maximal, characteristic, complemented
6	#19136876	C₂	1	2	yes	maximal, complemented
7	#19136869	C₂	1	2	yes	maximal, complemented
8	#19136382	C₂	1	2	yes	maximal
9	#19136380	C₂	1	2	yes	maximal
10	#19136188	C₂	1	1	yes	maximal, characteristic
11	C₄₃×#2255	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583455	—	2	1	no	—
13	C₄⋊Q₈×D₄₃	—	2	1	no	—
14	C₂×#13063264	C₂²	1	2	yes	—
15	#14583304	C₂²	1	1	yes	characteristic
16	#14583303	—	2	1	no	—
17	#14583303	C₂²	1	1	yes	characteristic
18	#14583303	C₂²	1	2	yes	—
19	#14583303	C₂²	1	1	yes	characteristic
20	#14583302	—	2	1	no	—
21	#14583301	C₂²	1	1	yes	characteristic
22	#14583298	—	2	1	no	—
23	#14583296	—	2	1	no	—
24	#14582936	C₂²	1	1	yes	characteristic
25	#14582936	C₂²	1	1	yes	characteristic
26	#14582934	—	2	1	no	—
27	#107×D₄₃	—	2	1	no	—
28	#107×D₄₃	C₂²	1	2	yes	—
29	C₄×C₈×D₄₃	C₂²	1	1	yes	characteristic
30	#14582727	C₂²	1	2	yes	—
31	#14582726	C₂²	1	2	yes	—
32	#14582726	C₂²	1	2	yes	—
33	#14582725	C₂²	1	2	yes	—
34	#14582724	C₂²	1	2	yes	—
35	#14582724	C₂²	1	2	yes	—
36	#14582675	C₂²	1	1	yes	characteristic
37	C₁₆×Dic₄₃	C₂²	1	2	yes	—
38	C₄₃×#500	C₂²	1	1	yes	characteristic
39	C₄₃×#499	C₂²	1	1	yes	characteristic
40	C₄₃×#367	C₂²	1	2	yes	—
41	C₄₃×#366	C₂²	1	2	yes	—
42	C₄₃×#346	C₂²	1	1	yes	characteristic
43	C₂×Q₈×D₄₃	—	4	1	no	—
44	C₂×C₈×D₄₃	D₄	1	1	yes	characteristic
45	C₂×C₈×D₄₃	C₂³	1	1	yes	characteristic
46	#13063325	—	2	1	no	—
47	#13063324	—	2	1	no	—
48	#13063324	—	2	1	no	—
49	#13063324	—	2	2	no	—
50	#13063324	—	4	1	no	—
51	#13063323	—	4	1	no	—
52	#13063317	—	2	1	no	—
53	#13063303	—	4	1	no	—
54	#13063303	—	4	1	no	—
55	#13063290	—	4	1	no	—
56	#13063290	—	4	1	no	—
57	#13063283	—	2	1	no	—
58	#13063283	—	2	1	no	—
59	C₄⋊C₄×D₄₃	—	2	2	no	—
60	C₄⋊C₄×D₄₃	—	4	1	no	—
61	C₄²×D₄₃	D₄	1	1	yes	characteristic
62	#13063264	Q₈	1	8	yes	complemented
63	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
64	#13063259	—	2	1	no	—
65	#13063259	C₂³	1	1	yes	characteristic
66	#13063259	C₂³	1	1	yes	characteristic
67	#13063259	C₂³	1	2	yes	—
68	#13063258	—	2	1	no	—
69	#13063258	—	2	1	no	—
70	#13063258	—	2	2	no	—
71	#13063258	—	2	2	no	—
72	#13063257	—	2	1	no	—
73	#13063256	—	2	1	no	—
74	#13063256	—	2	1	no	—
75	C₄×C₄₃⋊C₈	D₄	1	1	yes	characteristic
76	C₈×Dic₄₃	D₄	1	1	yes	characteristic
77	C₈×Dic₄₃	C₂³	1	1	yes	characteristic
78	C₄⋊Q₈×C₄₃	—	2	1	no	—
79	#125×C₄₃	—	2	1	no	—
80	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
81	#107×C₄₃	—	2	1	no	—
82	#107×C₄₃	C₂³	1	1	yes	characteristic
83	#107×C₄₃	C₂³	1	1	yes	characteristic
84	#107×C₄₃	C₂³	1	2	yes	—
85	#106×C₄₃	—	2	1	no	—
86	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
87	C₂×Q₈×C₄₃	—	4	1	no	—
88	Q₈×D₄₃	—	8	4	no	—
89	C₂×C₄×D₄₃	—	4	2	no	—
90	C₂×C₄×D₄₃	D₈	1	1	yes	characteristic
91	C₂×C₄×D₄₃	D₈	1	1	yes	characteristic
92	C₂×C₄×D₄₃	C₂×D₄	1	1	yes	characteristic
93	C₂×Dic₈₆	—	4	1	no	—
94	C₂×Dic₈₆	—	4	1	no	—
95	C₂×Dic₈₆	—	4	1	no	—
96	C₂×Dic₈₆	—	4	2	no	—
97	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
99	C₄⋊C₄×C₄₃	—	2	1	no	—
100	C₄⋊C₄×C₄₃	—	2	1	no	—
101	C₄⋊C₄×C₄₃	—	2	2	no	—
102	C₄⋊C₄×C₄₃	—	4	1	no	—
103	C₄⋊C₄×C₄₃	—	4	1	no	—
104	C₄⋊C₄×C₄₃	—	4	1	no	—
105	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
106	(C₂×D₄₃)⋊C₄	—	4	1	no	—
107	(C₂×D₄₃)⋊C₄	—	4	1	no	—
108	C₄⋊Dic₄₃	—	2	1	no	—
109	C₄⋊Dic₄₃	—	2	1	no	—
110	C₄⋊Dic₄₃	—	2	2	no	—
111	C₄⋊Dic₄₃	—	4	1	no	—
112	Dic₄₃⋊C₄	—	4	1	no	—
113	Dic₄₃⋊C₄	—	4	1	no	—
114	Dic₄₃⋊C₄	—	4	2	no	—
115	Dic₄₃⋊C₄	—	4	2	no	—
116	Dic₄₃⋊C₄	—	4	1	no	—
117	Dic₄₃⋊C₄	—	4	1	no	—
118	C₄×Dic₄₃	—	4	1	no	—
119	C₄×Dic₄₃	—	4	1	no	—
120	C₄×Dic₄₃	D₈	1	1	yes	characteristic
121	C₄×Dic₄₃	D₈	1	1	yes	characteristic
122	C₄×Dic₄₃	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
125	C₈×D₄₃	Q₁₆	1	4	yes	—
126	C₈×D₄₃	C₂×Q₈	1	4	yes	—
127	C₁₆×C₄₃	C₂×Q₈	1	4	yes	complemented
128	C₄₃⋊C₁₆	C₂×Q₈	1	4	yes	complemented
129	C₂²×D₄₃	C₂×D₈	1	1	yes	characteristic
130	Q₈×C₄₃	—	8	2	no	—
131	C₂×C₄×C₄₃	—	4	1	no	—
132	C₂×C₄×C₄₃	—	4	1	no	—
133	C₂×C₄×C₄₃	—	4	2	no	—
134	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
137	C₂×Dic₄₃	—	4	1	no	—
138	C₂×Dic₄₃	—	4	1	no	—
139	C₂×Dic₄₃	—	4	2	no	—
140	C₂×Dic₄₃	C₂×D₈	1	1	yes	characteristic
141	C₂×Dic₄₃	C₂×D₈	1	1	yes	characteristic
142	C₂×Dic₄₃	C₂×D₈	1	1	yes	characteristic
143	C₄×D₄₃	—	2	2	no	—
144	C₄×D₄₃	—	2	2	no	—
145	C₄×D₄₃	—	8	4	no	—
146	C₄×D₄₃	C₄⋊Q₈	1	2	yes	—
147	C₄×D₄₃	C₂×Q₁₆	1	2	yes	—
148	Dic₈₆	—	8	2	no	—
149	Dic₈₆	—	8	2	no	—
150	Dic₈₆	—	8	4	no	—
151	Dic₈₆	—	8	2	no	—
152	C₈×C₄₃	C₂×Q₁₆	1	2	yes	—
153	C₈×C₄₃	C₂²×Q₈	1	2	yes	—
154	C₄₃⋊C₈	C₄⋊Q₈	1	2	yes	—
155	C₄₃⋊C₈	C₂×Q₁₆	1	2	yes	—
156	#33731	—	43	1	no	maximal, Sylow
157	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
158	C₂×D₄₃	—	2	1	no	—
159	C₂×D₄₃	—	2	1	no	—
160	C₂×D₄₃	#500	1	2	yes	—
161	C₄×C₄₃	—	2	1	no	—
162	C₄×C₄₃	—	2	1	no	—
163	C₄×C₄₃	—	8	2	no	—
164	C₄×C₄₃	—	8	1	no	—
165	C₄×C₄₃	—	8	1	no	—
166	C₄×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
167	C₄×C₄₃	C₂²×Q₁₆	1	1	yes	characteristic
168	Dic₄₃	—	8	2	no	—
169	Dic₄₃	—	8	1	no	—
170	Dic₄₃	—	8	1	no	—
171	Dic₄₃	#500	1	2	yes	—
172	Dic₄₃	#509	1	2	yes	—
173	Dic₄₃	#510	1	2	yes	—
174	#3160	—	43	1	no	—
175	C₂×#500	—	43	1	no	—
176	#2255	—	43	1	no	—
177	#2255	—	43	1	no	—
178	#2255	—	43	1	no	—
179	#2255	D₄₃	1	1	yes	characteristic, complemented
180	#2254	—	43	2	no	—
181	#2252	—	43	2	no	—
182	#2159	—	43	2	no	—
183	#2159	—	43	2	no	—
184	C₄×M₃₂	—	43	1	no	—
185	C₂×C₄₃	C₂×#500	1	1	yes	characteristic
186	C₂×C₄₃	C₂×#509	1	1	yes	characteristic
187	C₂×C₄₃	C₂×#510	1	1	yes	characteristic
188	D₄₃	—	2	2	no	—
189	#533	—	86	1	no	—
190	C₂×C₄⋊Q₈	—	86	1	no	—
191	C₂×M₃₂	—	43	2	no	—
192	#500	—	43	2	no	—
193	#500	—	43	1	no	—
194	#500	—	43	1	no	—
195	#500	—	43	1	no	—
196	#500	—	86	1	no	—
197	#500	—	86	1	no	—
198	#500	C₂×D₄₃	1	1	yes	characteristic
199	#499	—	43	1	no	—
200	#499	—	86	1	no	—
201	#499	C₂×D₄₃	1	1	yes	characteristic
202	#498	—	86	1	no	—
203	#427	—	43	1	no	—
204	#427	—	43	1	no	—
205	#427	—	86	1	no	—
206	C₂×#107	—	43	2	no	—
207	C₂×#107	—	86	1	no	—
208	C₂×C₄×C₈	—	43	1	no	—
209	#367	—	43	2	no	—
210	#367	—	43	2	no	—
211	#367	—	43	2	no	—
212	#367	C₂×D₄₃	1	2	yes	—
213	#366	—	43	2	no	—
214	#366	—	43	2	no	—
215	#366	—	43	2	no	—
216	#366	C₂×D₄₃	1	2	yes	—
217	#346	—	43	1	no	—
218	#346	C₂×D₄₃	1	1	yes	characteristic
219	C₄×C₁₆	—	43	2	no	—
220	C₄₃	#33731	1	1	yes	characteristic, complemented, Sylow
221	C₂²×Q₈	—	172	1	no	—
222	C₂²×C₈	—	43	1	no	—
223	C₂²×C₈	—	43	1	no	—
224	C₄⋊Q₈	—	2	1	no	—
225	C₄⋊Q₈	—	86	2	no	—
226	C₄⋊Q₈	—	86	1	no	—
227	C₄⋊Q₈	—	86	1	no	—
228	C₄⋊Q₈	—	86	1	no	—
229	C₄⋊Q₈	—	172	1	no	—
230	C₄⋊Q₈	—	172	1	no	—
231	#125	—	2	1	no	—
232	#125	—	86	1	no	—
233	C₂²⋊Q₈	—	172	1	no	—
234	C₂²⋊Q₈	—	172	1	no	—
235	C₄×Q₈	—	172	1	no	—
236	C₄×Q₈	—	172	1	no	—
237	#117	—	86	1	no	—
238	#117	—	86	1	no	—
239	C₂×C₄⋊C₄	—	86	2	no	—
240	C₂×C₄⋊C₄	—	172	1	no	—
241	C₂×C₄²	—	43	1	no	—
242	M₃₂	—	43	8	no	—
243	C₂×C₁₆	—	43	2	no	—
244	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
245	#107	—	2	1	no	—
246	#107	—	43	1	no	—
247	#107	—	43	2	no	—
248	#107	—	43	1	no	—
249	#107	—	86	2	no	—
250	#107	—	86	2	no	—
251	#107	—	86	1	no	—
252	#107	—	86	1	no	—
253	#107	—	86	1	no	—
254	#107	C₂²×D₄₃	1	1	yes	characteristic
255	#107	C₂²×D₄₃	1	2	yes	—
256	#107	C₂²×D₄₃	1	1	yes	characteristic
257	#106	—	2	1	no	—
258	#106	—	86	1	no	—
259	#106	—	86	1	no	—
260	#106	—	86	1	no	—
261	C₄×C₈	—	43	1	no	—
262	C₄×C₈	—	43	1	no	—
263	C₄×C₈	—	43	1	no	—
264	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
265	C₂×Q₈	—	4	1	no	—
266	C₂×Q₈	—	344	4	no	—
267	C₂×Q₈	—	172	1	no	—
268	C₂×Q₈	—	172	1	no	—
269	C₂×Q₈	—	172	2	no	—
270	C₂×Q₈	—	172	1	no	—
271	C₂²×C₄	—	43	1	no	—
272	C₂²×C₄	—	43	1	no	—
273	C₂²×C₄	—	43	1	no	—
274	C₂²×C₄	—	172	2	no	—
275	C₂×C₈	—	43	1	no	—
276	C₂×C₈	—	43	1	no	—
277	C₂×C₈	—	43	4	no	—
278	C₂×C₈	—	43	4	no	—
279	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
280	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
281	C₄⋊C₄	—	2	1	no	—
282	C₄⋊C₄	—	2	1	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	4	1	no	—
285	C₄⋊C₄	—	4	1	no	—
286	C₄⋊C₄	—	4	1	no	—
287	C₄⋊C₄	—	86	1	no	—
288	C₄⋊C₄	—	86	1	no	—
289	C₄⋊C₄	—	86	2	no	—
290	C₄⋊C₄	—	172	2	no	—
291	C₄⋊C₄	—	172	2	no	—
292	C₄⋊C₄	—	172	1	no	—
293	C₄⋊C₄	—	172	1	no	—
294	C₄⋊C₄	—	172	1	no	—
295	C₄⋊C₄	—	172	1	no	—
296	C₄⋊C₄	—	172	1	no	—
297	C₂²⋊C₄	—	172	1	no	—
298	C₂²⋊C₄	—	172	1	no	—
299	C₄²	—	43	1	no	—
300	C₄²	—	43	1	no	—
301	C₄²	—	43	1	no	—
302	C₄²	—	172	1	no	—
303	C₄²	—	172	1	no	—
304	C₄²	D₄×D₄₃	1	1	yes	characteristic
305	C₁₆	—	43	4	no	—
306	C₁₆	Q₈×D₄₃	1	4	yes	complemented
307	C₂³	—	43	1	no	—
308	Q₈	—	8	2	no	—
309	Q₈	—	344	2	no	—
310	Q₈	—	344	4	no	—
311	Q₈	—	344	2	no	—
312	Q₈	—	344	2	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	43	1	no	—
317	C₂×C₄	—	43	1	no	—
318	C₂×C₄	—	43	1	no	—
319	C₂×C₄	—	43	2	no	—
320	C₂×C₄	—	43	2	no	—
321	C₂×C₄	—	86	2	no	—
322	C₂×C₄	—	86	2	no	—
323	C₂×C₄	—	344	4	no	—
324	C₂×C₄	—	172	1	no	—
325	C₂×C₄	—	172	1	no	—
326	C₂×C₄	—	172	2	no	—
327	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
328	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
329	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
330	C₈	—	43	2	no	—
331	C₈	—	43	2	no	—
332	C₈	Q₁₆×D₄₃	1	2	yes	—
333	C₈	C₂×Q₈×D₄₃	1	2	yes	—
334	C₂²	—	43	2	no	—
335	C₂²	—	86	1	no	—
336	C₂²	—	86	1	no	—
337	C₂²	C₂×D₈×D₄₃	1	1	yes	characteristic
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	8	2	no	—
341	C₄	—	8	1	no	—
342	C₄	—	8	1	no	—
343	C₄	—	43	2	no	—
344	C₄	—	43	2	no	—
345	C₄	—	43	2	no	—
346	C₄	—	344	2	no	—
347	C₄	—	344	1	no	—
348	C₄	—	344	1	no	—
349	C₄	C₄⋊Q₈×D₄₃	1	1	yes	characteristic
350	C₄	C₂×Q₁₆×D₄₃	1	1	yes	characteristic
351	C₂	—	86	2	no	—
352	C₂	#500×D₄₃	1	1	yes	characteristic
353	C₂	#509×D₄₃	1	1	yes	characteristic
354	C₂	#510×D₄₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₃
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄₃×#2255	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	C₂²×D₄₃	C₂×D₈
hypercenter	#2255	D₄₃
hypocenter	C₄₃	#33731
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2255

Complements

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

N	H ≅ G/N	type
3. #19136882	C₂	semidirect
4. #19136881	C₂	semidirect
5. #19136878	C₂	semidirect
6. #19136876	C₂	semidirect
7. #19136869	C₂	semidirect
11. C₄₃×#2255	C₂	semidirect
62. #13063264	Q₈	semidirect
127. C₁₆×C₄₃	C₂×Q₈	semidirect
128. C₄₃⋊C₁₆	C₂×Q₈	semidirect
179. #2255	D₄₃	semidirect
220. C₄₃	#33731	semidirect
306. C₁₆	Q₈×D₄₃	semidirect

Maximal subgroups

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

subgroup	index
#19145108	2
#500×D₄₃	2
#19136882	2
#19136881	2
#19136878	2
#19136876	2
#19136869	2
#19136382	2
#19136380	2
#19136188	2
C₄₃×#2255	2
#33731	43