Subgroups — #25251267

Order 11008.

574 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#500×D₄₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#13063344	C₂	1	1	yes	maximal, characteristic
3	#19136748	C₂	1	1	yes	maximal, characteristic, complemented
4	#19136744	C₂	1	1	yes	maximal, characteristic, complemented
5	#19136742	C₂	1	1	yes	maximal, characteristic, complemented
6	#19136736	C₂	1	2	yes	maximal, complemented
7	#19136730	C₂	1	2	yes	maximal, complemented
8	#19136382	C₂	1	2	yes	maximal, complemented
9	#19136349	C₂	1	1	yes	maximal, characteristic, complemented
10	#19136304	C₂	1	2	yes	maximal
11	C₄₃×#2226	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13063344	C₂²	1	1	yes	characteristic
13	C₄⋊Q₈×D₄₃	—	2	1	no	—
14	C₄×#10621900	—	2	1	no	—
15	C₂×#13063264	C₂²	1	2	yes	—
16	#14583304	C₂²	1	1	yes	characteristic
17	#14583303	—	2	1	no	—
18	#14583303	C₂²	1	2	yes	—
19	#14583302	—	2	1	no	—
20	C₄×Dic₁₇₂	C₂²	1	1	yes	characteristic
21	#14582993	C₂²	1	1	yes	characteristic
22	#14582990	—	2	1	no	—
23	C₄×#10621883	—	2	1	no	—
24	D₈×Dic₄₃	C₂²	1	1	yes	characteristic
25	#14582936	C₂²	1	1	yes	characteristic
26	#107×D₄₃	—	2	1	no	—
27	#107×D₄₃	C₂²	1	2	yes	—
28	#14582892	—	2	1	no	—
29	C₄×C₈×D₄₃	C₂²	1	1	yes	characteristic
30	#14582726	C₂²	1	2	yes	—
31	#14582725	C₂²	1	2	yes	—
32	#14582724	C₂²	1	2	yes	—
33	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	1	yes	characteristic
34	Dic₄₃⋊C₁₆	C₂²	1	2	yes	—
35	#14582704	C₂²	1	2	yes	—
36	#14582702	C₂²	1	2	yes	—
37	#14582700	C₂²	1	2	yes	—
38	C₄₃×#500	C₂²	1	1	yes	characteristic
39	C₄×D₈×C₄₃	C₂²	1	1	yes	characteristic
40	C₄₃×#367	C₂²	1	2	yes	—
41	C₄₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
42	C₄₃×#357	C₂²	1	2	yes	—
43	C₂×#10621900	—	2	1	no	—
44	C₂×Q₈×D₄₃	—	4	1	no	—
45	#13063344	—	2	2	no	—
46	#13063344	Q₈	1	4	yes	complemented
47	C₂×Dic₁₇₂	C₂³	1	1	yes	characteristic
48	C₂×#10621883	—	2	1	no	—
49	C₂×C₈×D₄₃	D₄	1	1	yes	characteristic
50	C₂×C₈×D₄₃	C₂³	1	1	yes	characteristic
51	#13063325	—	2	1	no	—
52	#13063324	—	2	2	no	—
53	#13063324	—	4	1	no	—
54	#13063323	—	4	1	no	—
55	C₄×Dic₈₆	—	2	1	no	—
56	#13063290	—	2	1	no	—
57	D₄×Dic₄₃	—	2	1	no	—
58	C₄×C₂²⋊D₄₃	—	4	1	no	—
59	(C₂²⋊D₄₃)⋊C₄	—	4	1	no	—
60	#13063283	—	2	1	no	—
61	#13063281	—	4	1	no	—
62	C₄⋊C₄×D₄₃	—	2	2	no	—
63	C₄⋊C₄×D₄₃	—	4	1	no	—
64	C₂×C₄×Dic₄₃	—	4	1	no	—
65	C₄²×D₄₃	D₄	1	1	yes	characteristic
66	#13063264	—	2	4	no	—
67	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
68	#13063259	—	2	1	no	—
69	#13063259	C₂³	1	2	yes	—
70	#13063259	C₂³	1	1	yes	characteristic
71	#13063258	—	2	2	no	—
72	#13063258	—	2	2	no	—
73	#13063256	—	2	1	no	—
74	#13063250	—	2	1	no	—
75	#13063248	—	2	1	no	—
76	#13063246	—	2	1	no	—
77	C₄×C₄₃⋊C₈	D₄	1	1	yes	characteristic
78	C₈×Dic₄₃	D₄	1	1	yes	characteristic
79	C₈×Dic₄₃	C₂³	1	1	yes	characteristic
80	C₂×D₈×C₄₃	C₂³	1	1	yes	characteristic
81	C₄⋊Q₈×C₄₃	—	2	1	no	—
82	C₄×D₄×C₄₃	—	2	1	no	—
83	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
84	#107×C₄₃	—	2	1	no	—
85	#107×C₄₃	C₂³	1	2	yes	—
86	#107×C₄₃	C₂³	1	1	yes	characteristic
87	#102×C₄₃	—	2	1	no	—
88	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
89	C₂×Q₈×C₄₃	—	4	1	no	—
90	C₂×D₄×C₄₃	—	2	1	no	—
91	C₂²×C₄×C₄₃	—	4	1	no	—
92	C₂×C₂²⋊D₄₃	—	4	1	no	—
93	C₂²×Dic₄₃	—	4	1	no	—
94	C₂²×Dic₄₃	—	4	1	no	—
95	Q₈×D₄₃	—	8	4	no	—
96	#10621900	—	2	4	no	—
97	#10621900	—	4	2	no	—
98	C₂×C₄×D₄₃	—	4	2	no	—
99	C₂×C₄×D₄₃	D₈	1	1	yes	characteristic
100	C₂×C₄×D₄₃	D₈	1	1	yes	characteristic
101	C₂×C₄×D₄₃	C₂×D₄	1	1	yes	characteristic
102	C₂×Dic₈₆	—	2	1	no	—
103	C₂×Dic₈₆	—	4	1	no	—
104	C₂×Dic₈₆	—	4	2	no	—
105	D₈×C₄₃	—	2	1	no	—
106	D₈×C₄₃	C₂×Q₈	1	2	yes	complemented
107	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
109	C₄⋊C₄×C₄₃	—	2	1	no	—
110	C₄⋊C₄×C₄₃	—	2	2	no	—
111	C₄⋊C₄×C₄₃	—	4	1	no	—
112	C₂²⋊C₄×C₄₃	—	4	1	no	—
113	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
114	C₂²⋊Dic₄₃	—	4	1	no	—
115	#10621883	—	2	4	no	—
116	(C₂×D₄₃)⋊C₄	—	4	1	no	—
117	C₄⋊Dic₄₃	—	2	1	no	—
118	C₄⋊Dic₄₃	—	2	2	no	—
119	C₄⋊Dic₄₃	—	4	1	no	—
120	Dic₄₃⋊C₄	—	4	1	no	—
121	Dic₄₃⋊C₄	—	4	2	no	—
122	Dic₄₃⋊C₄	—	4	2	no	—
123	Dic₄₃⋊C₄	—	4	1	no	—
124	C₄×Dic₄₃	—	4	1	no	—
125	C₄×Dic₄₃	—	4	1	no	—
126	C₄×Dic₄₃	—	4	1	no	—
127	C₄×Dic₄₃	D₈	1	1	yes	characteristic
128	C₄×Dic₄₃	D₈	1	1	yes	characteristic
129	C₄×Dic₄₃	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
131	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
132	Dic₁₇₂	—	2	1	no	—
133	Dic₁₇₂	C₂×Q₈	1	2	yes	complemented
134	C₈×D₄₃	—	2	2	no	—
135	C₈×D₄₃	C₂×Q₈	1	2	yes	—
136	C₈×D₄₃	#41	1	2	yes	—
137	C₁₆×C₄₃	—	2	2	no	—
138	C₄₃⋊C₁₆	—	2	2	no	—
139	C₂³×C₄₃	—	4	1	no	—
140	C₂²×D₄₃	C₂×D₈	1	1	yes	characteristic
141	Q₈×C₄₃	—	8	2	no	—
142	D₄×C₄₃	—	2	2	no	—
143	D₄×C₄₃	—	4	1	no	—
144	C₂×C₄×C₄₃	—	4	1	no	—
145	C₂×C₄×C₄₃	—	4	2	no	—
146	C₂×C₄×C₄₃	—	8	2	no	—
147	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
148	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
149	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
150	C₂²⋊D₄₃	—	4	4	no	—
151	C₂×Dic₄₃	—	4	1	no	—
152	C₂×Dic₄₃	—	4	4	no	—
153	C₂×Dic₄₃	—	4	1	no	—
154	C₂×Dic₄₃	—	4	2	no	—
155	C₂×Dic₄₃	—	8	2	no	—
156	C₂×Dic₄₃	C₂×D₈	1	1	yes	characteristic
157	C₂×Dic₄₃	C₂×D₈	1	1	yes	characteristic
158	C₂×Dic₄₃	C₂×D₈	1	1	yes	characteristic
159	C₄×D₄₃	—	2	2	no	—
160	C₄×D₄₃	—	2	2	no	—
161	C₄×D₄₃	—	8	4	no	—
162	C₄×D₄₃	C₂²⋊Q₈	1	2	yes	—
163	C₄×D₄₃	#137	1	2	yes	—
164	Dic₈₆	—	2	2	no	—
165	Dic₈₆	—	4	1	no	—
166	Dic₈₆	—	8	2	no	—
167	Dic₈₆	—	8	4	no	—
168	C₈×C₄₃	—	2	1	no	—
169	C₈×C₄₃	C₂²×Q₈	1	1	yes	characteristic
170	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
171	C₄₃⋊C₈	—	2	1	no	—
172	C₄₃⋊C₈	C₂²⋊Q₈	1	2	yes	—
173	#33629	—	43	1	no	maximal, Sylow
174	C₂²×C₄₃	—	4	2	no	—
175	C₂²×C₄₃	—	8	1	no	—
176	C₂²×C₄₃	—	8	1	no	—
177	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
178	C₂×D₄₃	—	2	1	no	—
179	C₂×D₄₃	—	2	1	no	—
180	C₂×D₄₃	#474	1	2	yes	—
181	C₄×C₄₃	—	2	1	no	—
182	C₄×C₄₃	—	8	2	no	—
183	C₄×C₄₃	—	8	1	no	—
184	C₄×C₄₃	#510	1	2	yes	—
185	C₄×C₄₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
186	C₄×C₄₃	C₂×#137	1	1	yes	characteristic
187	Dic₄₃	—	2	1	no	—
188	Dic₄₃	—	4	2	no	—
189	Dic₄₃	—	8	2	no	—
190	Dic₄₃	—	8	1	no	—
191	Dic₄₃	#474	1	2	yes	—
192	Dic₄₃	#509	1	2	yes	—
193	C₂×#500	—	43	1	no	—
194	C₄×#135	—	43	1	no	—
195	#2227	—	43	1	no	—
196	#2227	—	43	1	no	—
197	#2226	—	43	1	no	—
198	#2226	D₄₃	1	1	yes	characteristic, complemented
199	#2225	—	43	2	no	—
200	#2224	—	43	2	no	—
201	#2159	—	43	2	no	—
202	#2150	—	43	1	no	—
203	#2139	—	43	2	no	—
204	C₂×C₄₃	—	8	2	no	—
205	C₂×C₄₃	C₂×#474	1	1	yes	characteristic
206	C₂×C₄₃	C₂×#509	1	1	yes	characteristic
207	C₂×C₄₃	C₂×#510	1	1	yes	characteristic
208	D₄₃	—	2	2	no	—
209	C₂×#135	—	43	1	no	—
210	C₂×C₄⋊Q₈	—	86	1	no	—
211	C₄×#41	—	86	1	no	—
212	C₂×M₃₂	—	43	2	no	—
213	#500	—	43	2	no	—
214	#500	—	43	1	no	—
215	#500	—	86	1	no	—
216	#500	—	86	1	no	—
217	#500	C₂×D₄₃	1	1	yes	characteristic
218	C₄×Q₁₆	—	43	1	no	—
219	C₄×Q₁₆	—	43	1	no	—
220	C₄×SD₁₆	—	86	1	no	—
221	C₄×SD₁₆	—	86	1	no	—
222	C₄×D₈	—	43	1	no	—
223	C₄×D₈	C₂×D₄₃	1	1	yes	characteristic
224	#427	—	43	1	no	—
225	C₂×#107	—	43	2	no	—
226	C₂×#107	—	86	1	no	—
227	#416	—	86	1	no	—
228	C₂×C₄×C₈	—	43	1	no	—
229	#367	—	43	2	no	—
230	#367	C₂×D₄₃	1	2	yes	—
231	#366	—	43	2	no	—
232	#366	—	43	2	no	—
233	C₄⋊C₁₆	—	43	2	no	—
234	C₄⋊C₁₆	—	43	1	no	—
235	C₄⋊C₁₆	C₂×D₄₃	1	1	yes	characteristic
236	#358	—	43	2	no	—
237	#358	—	43	2	no	—
238	#357	—	43	2	no	—
239	#357	C₂×D₄₃	1	2	yes	—
240	C₄₃	#33629	1	1	yes	characteristic, complemented, Sylow
241	C₂×#41	—	86	1	no	—
242	C₂²×Q₈	—	172	1	no	—
243	#135	—	43	4	no	—
244	#135	—	86	2	no	—
245	C₂×Q₁₆	—	43	1	no	—
246	C₂×SD₁₆	—	86	1	no	—
247	C₂×D₈	C₂²×D₄₃	1	1	yes	characteristic
248	C₂²×C₈	—	43	1	no	—
249	C₂²×C₈	—	43	1	no	—
250	C₄⋊Q₈	—	2	1	no	—
251	C₄⋊Q₈	—	86	2	no	—
252	C₄⋊Q₈	—	86	1	no	—
253	C₄⋊Q₈	—	172	1	no	—
254	C₄⋊Q₈	—	172	1	no	—
255	C₄×Q₈	—	86	1	no	—
256	C₄×Q₈	—	86	1	no	—
257	C₄×D₄	—	2	1	no	—
258	C₄×D₄	—	86	1	no	—
259	C₄×D₄	—	172	1	no	—
260	C₄×D₄	—	172	1	no	—
261	#117	—	86	1	no	—
262	#117	—	172	1	no	—
263	C₂×C₄⋊C₄	—	86	2	no	—
264	C₂×C₄⋊C₄	—	172	1	no	—
265	C₂×C₄²	—	43	1	no	—
266	C₂×C₄²	—	172	1	no	—
267	M₃₂	—	86	4	no	—
268	C₂×C₁₆	—	43	2	no	—
269	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
270	#107	—	2	1	no	—
271	#107	—	43	2	no	—
272	#107	—	43	1	no	—
273	#107	—	86	2	no	—
274	#107	—	86	2	no	—
275	#107	—	86	1	no	—
276	#107	C₂²×D₄₃	1	2	yes	—
277	#107	C₂²×D₄₃	1	1	yes	characteristic
278	#106	—	86	1	no	—
279	#103	—	86	1	no	—
280	#103	—	86	1	no	—
281	#102	—	2	1	no	—
282	#102	—	86	1	no	—
283	C₄×C₈	—	43	1	no	—
284	C₄×C₈	—	43	1	no	—
285	C₄×C₈	—	43	1	no	—
286	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
287	#41	—	86	4	no	—
288	#41	—	172	2	no	—
289	C₂×Q₈	—	4	1	no	—
290	C₂×Q₈	—	86	1	no	—
291	C₂×Q₈	—	344	4	no	—
292	C₂×Q₈	—	172	2	no	—
293	C₂×Q₈	—	172	1	no	—
294	C₂×D₄	—	2	1	no	—
295	C₂×D₄	—	172	1	no	—
296	C₂²×C₄	—	4	1	no	—
297	C₂²×C₄	—	43	1	no	—
298	C₂²×C₄	—	43	1	no	—
299	C₂²×C₄	—	43	1	no	—
300	C₂²×C₄	—	172	2	no	—
301	C₂²×C₄	—	172	1	no	—
302	C₂²×C₄	—	172	1	no	—
303	Q₁₆	—	43	2	no	—
304	Q₁₆	—	86	1	no	—
305	SD₁₆	—	86	4	no	—
306	D₈	—	2	1	no	—
307	D₈	Q₈×D₄₃	1	2	yes	complemented
308	C₂×C₈	—	43	1	no	—
309	C₂×C₈	—	43	2	no	—
310	C₂×C₈	—	43	2	no	—
311	C₂×C₈	—	43	1	no	—
312	C₂×C₈	—	86	2	no	—
313	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
314	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
315	C₄⋊C₄	—	2	1	no	—
316	C₄⋊C₄	—	2	2	no	—
317	C₄⋊C₄	—	4	1	no	—
318	C₄⋊C₄	—	86	1	no	—
319	C₄⋊C₄	—	86	2	no	—
320	C₄⋊C₄	—	172	2	no	—
321	C₄⋊C₄	—	172	2	no	—
322	C₄⋊C₄	—	172	1	no	—
323	C₄⋊C₄	—	172	1	no	—
324	C₄⋊C₄	—	172	1	no	—
325	C₂²⋊C₄	—	4	1	no	—
326	C₂²⋊C₄	—	172	1	no	—
327	C₂²⋊C₄	—	172	1	no	—
328	C₄²	—	43	1	no	—
329	C₄²	—	43	1	no	—
330	C₄²	—	43	1	no	—
331	C₄²	—	172	1	no	—
332	C₄²	—	172	1	no	—
333	C₄²	—	172	1	no	—
334	C₄²	D₄×D₄₃	1	1	yes	characteristic
335	C₁₆	—	2	2	no	—
336	C₁₆	—	86	2	no	—
337	C₂³	—	4	1	no	—
338	C₂³	—	43	1	no	—
339	Q₈	—	8	2	no	—
340	Q₈	—	86	2	no	—
341	Q₈	—	344	2	no	—
342	Q₈	—	344	4	no	—
343	Q₈	—	172	1	no	—
344	D₄	—	2	2	no	—
345	D₄	—	4	1	no	—
346	D₄	—	172	4	no	—
347	C₂×C₄	—	4	1	no	—
348	C₂×C₄	—	4	2	no	—
349	C₂×C₄	—	8	2	no	—
350	C₂×C₄	—	43	1	no	—
351	C₂×C₄	—	43	1	no	—
352	C₂×C₄	—	43	2	no	—
353	C₂×C₄	—	43	2	no	—
354	C₂×C₄	—	43	1	no	—
355	C₂×C₄	—	86	2	no	—
356	C₂×C₄	—	86	2	no	—
357	C₂×C₄	—	344	4	no	—
358	C₂×C₄	—	344	2	no	—
359	C₂×C₄	—	172	4	no	—
360	C₂×C₄	—	172	1	no	—
361	C₂×C₄	—	172	2	no	—
362	C₂×C₄	—	172	1	no	—
363	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
364	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
365	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
366	C₈	—	2	1	no	—
367	C₈	—	43	2	no	—
368	C₈	—	86	1	no	—
369	C₈	C₂×Q₈×D₄₃	1	1	yes	characteristic
370	C₈	#41×D₄₃	1	1	yes	characteristic
371	C₂²	—	4	2	no	—
372	C₂²	—	8	1	no	—
373	C₂²	—	8	1	no	—
374	C₂²	—	43	2	no	—
375	C₂²	—	86	1	no	—
376	C₂²	—	86	1	no	—
377	C₂²	C₂×D₈×D₄₃	1	1	yes	characteristic
378	C₄	—	2	1	no	—
379	C₄	—	8	2	no	—
380	C₄	—	8	1	no	—
381	C₄	—	43	2	no	—
382	C₄	—	43	2	no	—
383	C₄	—	86	1	no	—
384	C₄	—	344	2	no	—
385	C₄	—	344	1	no	—
386	C₄	—	172	2	no	—
387	C₄	#14583338	1	2	yes	—
388	C₄	C₂²⋊Q₈×D₄₃	1	1	yes	characteristic
389	C₄	#137×D₄₃	1	1	yes	characteristic
390	C₂	—	8	2	no	—
391	C₂	—	86	2	no	—
392	C₂	#474×D₄₃	1	1	yes	characteristic
393	C₂	#509×D₄₃	1	1	yes	characteristic
394	C₂	C₂×#14583338	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×#2226	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	C₂×#13063344	C₂²
hypercenter	#2226	D₄₃
hypocenter	C₄₃	#33629
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2226

Complements

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

N	H ≅ G/N	type
1. #500×D₄₃	C₂	semidirect
3. #19136748	C₂	semidirect
4. #19136744	C₂	semidirect
5. #19136742	C₂	semidirect
6. #19136736	C₂	semidirect
7. #19136730	C₂	semidirect
8. #19136382	C₂	semidirect
9. #19136349	C₂	semidirect
11. C₄₃×#2226	C₂	semidirect
46. #13063344	Q₈	semidirect
106. D₈×C₄₃	C₂×Q₈	semidirect
133. Dic₁₇₂	C₂×Q₈	semidirect
198. #2226	D₄₃	semidirect
240. C₄₃	#33629	semidirect
307. D₈	Q₈×D₄₃	semidirect

Maximal subgroups

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

subgroup	index
#500×D₄₃	2
C₄×#13063344	2
#19136748	2
#19136744	2
#19136742	2
#19136736	2
#19136730	2
#19136382	2
#19136349	2
#19136304	2
C₄₃×#2226	2
#33629	43