Subgroups — #25251251

Order 11008.

598 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145107	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14582981	C₂	1	1	yes	maximal, characteristic
3	#19136755	C₂	1	2	yes	maximal, complemented
4	#19136751	C₂	1	2	yes	maximal, complemented
5	#19136741	C₂	1	2	yes	maximal, complemented
6	#19136731	C₂	1	2	yes	maximal, complemented
7	#19136381	C₂	1	1	yes	maximal, characteristic, complemented
8	#19136350	C₂	1	1	yes	maximal, characteristic, complemented
9	#19136312	C₂	1	1	yes	maximal, characteristic
10	#19136310	C₂	1	1	yes	maximal, characteristic
11	C₄₃×#2159	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₃₄₄	C₂²	1	1	yes	characteristic
13	#14583452	—	2	1	no	—
14	C₂×#13063287	—	2	1	no	—
15	C₂×#13063265	C₂²	1	1	yes	characteristic
16	#14583303	C₂²	1	1	yes	characteristic, complemented
17	#14583303	C₂²	1	1	yes	characteristic, complemented
18	#14583300	C₂²	1	1	yes	characteristic, complemented
19	#14583300	C₂²	1	1	yes	characteristic, complemented
20	#14583299	—	2	1	no	—
21	#14583296	—	2	1	no	—
22	#14582981	—	2	1	no	—
23	#14582981	—	2	1	no	—
24	#14582981	C₂²	1	2	yes	—
25	#14582981	C₂²	1	2	yes	—
26	#14582937	C₂²	1	1	yes	characteristic
27	#14582935	—	2	1	no	—
28	C₂×#13063245	—	2	1	no	—
29	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
30	#14582726	C₂²	1	2	yes	complemented
31	#14582724	C₂²	1	2	yes	complemented
32	Dic₄₃⋊C₁₆	C₂²	1	2	yes	complemented
33	Dic₄₃⋊C₁₆	C₂²	1	2	yes	complemented
34	#14582701	C₂²	1	2	yes	—
35	#14582701	C₂²	1	2	yes	—
36	#14582699	C₂²	1	2	yes	—
37	#14582699	C₂²	1	2	yes	—
38	C₂×M₃₂×C₄₃	C₂²	1	1	yes	characteristic
39	C₄₃×#427	C₂²	1	1	yes	characteristic
40	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
41	C₄₃×#367	C₂²	1	2	yes	complemented
42	C₄₃×#366	C₂²	1	2	yes	complemented
43	C₂²×D₁₇₂	—	2	1	no	—
44	C₂²×C₄×D₄₃	—	4	1	no	—
45	C₂×D₃₄₄	—	2	1	no	—
46	C₂×D₃₄₄	—	2	2	no	—
47	C₂×D₃₄₄	Q₈	1	4	yes	complemented
48	C₂×D₃₄₄	C₂³	1	2	yes	—
49	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
50	#13063324	—	2	1	no	—
51	#13063324	—	2	1	no	—
52	#13063316	—	2	1	no	—
53	#13063316	—	2	1	no	—
54	#13063302	—	4	1	no	—
55	#13063302	—	4	1	no	—
56	C₄×Dic₈₆	—	4	1	no	—
57	C₄×Dic₈₆	—	4	1	no	—
58	#13063287	—	2	2	no	—
59	#13063287	—	2	2	no	—
60	#13063287	—	4	1	no	—
61	#13063287	—	4	1	no	—
62	#13063284	—	2	1	no	—
63	C₂×(C₂×D₄₃)⋊C₄	—	4	1	no	—
64	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
65	#13063265	—	2	2	no	—
66	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
67	#13063259	C₂³	1	1	yes	characteristic
68	#13063259	C₂³	1	1	yes	characteristic
69	#13063258	—	2	1	no	—
70	#13063258	—	2	1	no	—
71	#13063257	—	2	1	no	—
72	#13063256	—	2	1	no	—
73	#13063256	—	2	1	no	—
74	#13063245	—	2	2	no	—
75	#13063245	—	2	2	no	—
76	C₈×Dic₄₃	D₄	1	1	yes	characteristic
77	C₈×Dic₄₃	D₄	1	1	yes	characteristic
78	C₈×Dic₄₃	C₂³	1	1	yes	characteristic
79	C₈×Dic₄₃	C₂³	1	1	yes	characteristic
80	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
81	#117×C₄₃	—	2	1	no	—
82	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
83	M₃₂×C₄₃	—	2	2	no	—
84	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
85	#107×C₄₃	—	2	1	no	—
86	#107×C₄₃	C₂³	1	1	yes	characteristic
87	#107×C₄₃	C₂³	1	1	yes	characteristic
88	#107×C₄₃	C₂³	1	2	yes	—
89	#106×C₄₃	—	2	1	no	—
90	C₂³×D₄₃	—	4	1	no	—
91	C₂²×C₄×C₄₃	—	4	1	no	—
92	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
93	C₂²×Dic₄₃	D₈	1	1	yes	characteristic
94	C₂²×Dic₄₃	D₈	1	1	yes	characteristic
95	C₂×D₁₇₂	—	2	4	no	—
96	C₂×D₁₇₂	—	2	2	no	—
97	C₂×D₁₇₂	—	4	1	no	—
98	C₂×D₁₇₂	—	4	2	no	—
99	C₂×C₄×D₄₃	—	4	2	no	—
100	C₂×C₄×D₄₃	—	4	2	no	—
101	C₂×C₄×D₄₃	—	8	4	no	—
102	C₂×Dic₈₆	—	4	1	no	—
103	C₂×Dic₈₆	—	4	1	no	—
104	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
106	C₂×C₈×C₄₃	#41	1	2	yes	—
107	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
108	C₄⋊C₄×C₄₃	—	2	1	no	—
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	—	4	1	no	—
114	C₂²⋊Dic₄₃	—	4	1	no	—
115	(C₂×D₄₃)⋊C₄	—	4	2	no	—
116	(C₂×D₄₃)⋊C₄	—	4	2	no	—
117	C₄⋊Dic₄₃	—	2	1	no	—
118	C₄⋊Dic₄₃	—	2	1	no	—
119	C₄⋊Dic₄₃	—	4	1	no	—
120	C₄⋊Dic₄₃	—	4	1	no	—
121	Dic₄₃⋊C₄	—	4	1	no	—
122	Dic₄₃⋊C₄	—	4	1	no	—
123	Dic₄₃⋊C₄	—	4	1	no	—
124	Dic₄₃⋊C₄	—	4	1	no	—
125	C₄×Dic₄₃	—	4	1	no	—
126	C₄×Dic₄₃	D₈	1	1	yes	characteristic
127	C₄×Dic₄₃	D₈	1	1	yes	characteristic
128	C₄×Dic₄₃	C₂×D₄	1	1	yes	characteristic
129	C₄×Dic₄₃	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄₃⋊C₈	—	2	2	no	—
131	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
132	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
133	D₃₄₄	—	2	4	no	—
134	D₃₄₄	—	2	2	no	—
135	D₃₄₄	C₂×Q₈	1	4	yes	—
136	C₁₆×C₄₃	—	2	2	no	—
137	C₄₃⋊C₁₆	—	2	2	no	—
138	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
139	C₂²×D₄₃	—	4	2	no	—
140	C₂²×D₄₃	—	4	4	no	—
141	C₂²×D₄₃	—	8	2	no	—
142	C₂²×D₄₃	—	8	2	no	—
143	C₂×C₄×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	C₂²⋊Q₈	1	2	yes	—
148	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
149	C₂×C₄×C₄₃	#137	1	2	yes	—
150	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
151	C₂×Dic₄₃	—	2	2	no	—
152	C₂×Dic₄₃	—	4	1	no	—
153	C₂×Dic₄₃	—	4	1	no	—
154	C₂×Dic₄₃	—	4	1	no	—
155	C₂×Dic₄₃	C₂×D₈	1	1	yes	characteristic
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	D₁₇₂	—	2	4	no	—
160	D₁₇₂	—	4	2	no	—
161	D₁₇₂	—	4	4	no	—
162	C₄×D₄₃	—	8	4	no	—
163	C₄×D₄₃	—	8	4	no	—
164	Dic₈₆	—	8	2	no	—
165	Dic₈₆	—	8	2	no	—
166	C₈×C₄₃	C₂²⋊Q₈	1	2	yes	—
167	C₈×C₄₃	C₂²×Q₈	1	1	yes	characteristic
168	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
169	C₄₃⋊C₈	—	2	1	no	—
170	C₄₃⋊C₈	—	2	1	no	—
171	#33627	—	43	1	no	maximal, Sylow
172	C₂²×C₄₃	—	2	1	no	—
173	C₂²×C₄₃	—	2	1	no	—
174	C₂²×C₄₃	#474	1	2	yes	—
175	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
176	C₂×D₄₃	—	4	4	no	—
177	C₂×D₄₃	—	8	8	no	—
178	C₂×D₄₃	—	8	2	no	—
179	C₂×D₄₃	—	8	2	no	—
180	C₄×C₄₃	—	8	1	no	—
181	C₄×C₄₃	—	8	2	no	—
182	C₄×C₄₃	—	8	1	no	—
183	C₄×C₄₃	#474	1	2	yes	—
184	C₄×C₄₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
185	C₄×C₄₃	C₂×#137	1	1	yes	characteristic
186	Dic₄₃	—	2	1	no	—
187	Dic₄₃	—	2	1	no	—
188	Dic₄₃	—	2	1	no	—
189	Dic₄₃	—	8	1	no	—
190	Dic₄₃	—	8	1	no	—
191	Dic₄₃	#509	1	2	yes	—
192	#3160	—	43	1	no	—
193	C₂×C₄×D₈	—	43	1	no	—
194	#2228	—	43	2	no	—
195	#2228	—	43	2	no	—
196	#2226	—	43	2	no	—
197	#2224	—	43	2	no	—
198	#2159	—	43	1	no	—
199	#2159	D₄₃	1	1	yes	characteristic, complemented
200	#2150	—	43	1	no	—
201	#2140	—	43	1	no	—
202	#2140	—	43	1	no	—
203	C₂×C₄₃	—	2	2	no	—
204	C₂×C₄₃	C₂×#474	1	1	yes	characteristic
205	C₂×C₄₃	C₂×#509	1	1	yes	characteristic
206	C₂×C₄₃	#3414	1	1	yes	characteristic
207	D₄₃	—	8	4	no	—
208	C₂²×D₈	—	43	1	no	—
209	#533	—	86	1	no	—
210	C₂×C₄×D₄	—	86	1	no	—
211	C₂×M₃₂	—	43	1	no	—
212	C₂×M₃₂	C₂×D₄₃	1	1	yes	characteristic
213	#500	—	43	1	no	—
214	#500	—	43	1	no	—
215	#499	—	43	1	no	—
216	#499	—	43	1	no	—
217	#499	—	86	1	no	—
218	#498	—	86	1	no	—
219	C₄×D₈	—	43	2	no	—
220	C₄×D₈	—	43	2	no	—
221	C₄×D₈	—	86	1	no	—
222	C₄×D₈	—	86	1	no	—
223	#427	—	43	1	no	—
224	#427	—	86	1	no	—
225	#427	C₂×D₄₃	1	1	yes	characteristic
226	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
227	C₂×#102	—	86	1	no	—
228	C₂×C₄×C₈	—	43	1	no	—
229	#367	—	43	2	no	—
230	#367	C₂×D₄₃	1	2	yes	complemented
231	#366	—	43	2	no	—
232	#366	C₂×D₄₃	1	2	yes	complemented
233	C₄⋊C₁₆	—	43	2	no	—
234	C₄⋊C₁₆	—	43	2	no	—
235	#357	—	43	2	no	—
236	#357	—	43	2	no	—
237	#357	—	43	2	no	—
238	#357	—	43	2	no	—
239	C₄₃	#33627	1	1	yes	characteristic, complemented, Sylow
240	C₂²×D₄	—	86	1	no	—
241	C₂³×C₄	—	172	1	no	—
242	C₂×D₈	—	43	4	no	—
243	C₂×D₈	—	43	2	no	—
244	C₂×D₈	—	86	2	no	—
245	C₂×D₈	—	86	1	no	—
246	C₂²×C₈	—	43	1	no	—
247	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
248	C₄⋊Q₈	—	86	1	no	—
249	C₄⋊Q₈	—	86	1	no	—
250	#125	—	86	1	no	—
251	#125	—	86	1	no	—
252	C₂²⋊Q₈	—	172	1	no	—
253	C₂²⋊Q₈	—	172	1	no	—
254	C₄×Q₈	—	172	1	no	—
255	C₄×Q₈	—	172	1	no	—
256	C₄×D₄	—	86	2	no	—
257	C₄×D₄	—	86	2	no	—
258	C₄×D₄	—	172	1	no	—
259	C₄×D₄	—	172	1	no	—
260	#117	—	2	1	no	—
261	#117	—	86	1	no	—
262	C₂×C₄⋊C₄	—	2	1	no	—
263	C₂×C₂²⋊C₄	—	172	1	no	—
264	C₂×C₄²	—	43	1	no	—
265	M₃₂	—	2	2	no	—
266	M₃₂	—	86	2	no	—
267	C₂×C₁₆	—	43	2	no	—
268	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
269	#107	—	2	1	no	—
270	#107	—	43	1	no	—
271	#107	—	43	1	no	—
272	#107	—	86	1	no	—
273	#107	—	86	1	no	—
274	#107	C₂²×D₄₃	1	1	yes	characteristic
275	#107	C₂²×D₄₃	1	1	yes	characteristic
276	#107	C₂²×D₄₃	1	2	yes	—
277	#106	—	2	1	no	—
278	#106	—	86	1	no	—
279	#106	—	86	1	no	—
280	#106	—	86	1	no	—
281	#102	—	86	2	no	—
282	#102	—	86	2	no	—
283	C₄×C₈	—	43	1	no	—
284	C₄×C₈	—	43	1	no	—
285	C₄×C₈	—	43	1	no	—
286	C₄×C₈	—	43	1	no	—
287	C₂⁴	—	172	1	no	—
288	C₂×Q₈	—	172	1	no	—
289	C₂×Q₈	—	172	1	no	—
290	C₂×D₄	—	86	4	no	—
291	C₂×D₄	—	86	2	no	—
292	C₂×D₄	—	172	1	no	—
293	C₂×D₄	—	172	2	no	—
294	C₂²×C₄	—	4	1	no	—
295	C₂²×C₄	—	43	1	no	—
296	C₂²×C₄	—	43	1	no	—
297	C₂²×C₄	—	344	4	no	—
298	C₂²×C₄	—	172	2	no	—
299	C₂²×C₄	—	172	2	no	—
300	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
301	D₈	—	43	4	no	—
302	D₈	—	86	4	no	—
303	D₈	—	86	2	no	—
304	C₂×C₈	—	43	1	no	—
305	C₂×C₈	—	43	1	no	—
306	C₂×C₈	—	86	2	no	—
307	C₂×C₈	#10621898	1	2	yes	—
308	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
309	C₂×C₈	Q₈×D₄₃	1	2	yes	—
310	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
311	C₄⋊C₄	—	2	2	no	—
312	C₄⋊C₄	—	2	1	no	—
313	C₄⋊C₄	—	2	1	no	—
314	C₄⋊C₄	—	4	1	no	—
315	C₄⋊C₄	—	86	1	no	—
316	C₄⋊C₄	—	86	1	no	—
317	C₄⋊C₄	—	172	1	no	—
318	C₄⋊C₄	—	172	1	no	—
319	C₄⋊C₄	—	172	1	no	—
320	C₄⋊C₄	—	172	1	no	—
321	C₄⋊C₄	—	172	1	no	—
322	C₄⋊C₄	—	172	1	no	—
323	C₂²⋊C₄	—	4	1	no	—
324	C₂²⋊C₄	—	172	2	no	—
325	C₂²⋊C₄	—	172	2	no	—
326	C₂²⋊C₄	—	172	1	no	—
327	C₄²	—	4	1	no	—
328	C₄²	—	43	1	no	—
329	C₄²	—	43	1	no	—
330	C₄²	—	43	1	no	—
331	C₄²	—	43	1	no	—
332	C₄²	—	172	1	no	—
333	C₁₆	—	2	2	no	—
334	C₁₆	—	86	2	no	—
335	C₂³	—	344	2	no	—
336	C₂³	—	344	2	no	—
337	C₂³	—	172	2	no	—
338	C₂³	—	172	4	no	—
339	C₂³	D₈×D₄₃	1	1	yes	characteristic
340	Q₈	—	344	2	no	—
341	Q₈	—	344	2	no	—
342	D₄	—	86	4	no	—
343	D₄	—	172	4	no	—
344	D₄	—	172	2	no	—
345	C₂×C₄	—	4	1	no	—
346	C₂×C₄	—	4	1	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	—	8	2	no	—
349	C₂×C₄	—	43	1	no	—
350	C₂×C₄	—	43	1	no	—
351	C₂×C₄	—	43	1	no	—
352	C₂×C₄	—	43	1	no	—
353	C₂×C₄	—	86	2	no	—
354	C₂×C₄	—	344	4	no	—
355	C₂×C₄	—	344	4	no	—
356	C₂×C₄	—	172	1	no	—
357	C₂×C₄	—	172	1	no	—
358	C₂×C₄	—	172	1	no	—
359	C₂×C₄	#13063303	1	2	yes	—
360	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
361	C₂×C₄	#13063357	1	2	yes	—
362	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
363	C₈	—	86	1	no	—
364	C₈	—	86	1	no	—
365	C₈	#13063303	1	2	yes	—
366	C₈	C₂×Q₈×D₄₃	1	1	yes	characteristic
367	C₈	C₂×#10621898	1	1	yes	characteristic
368	C₂²	—	2	1	no	—
369	C₂²	—	2	1	no	—
370	C₂²	—	344	8	no	—
371	C₂²	—	344	2	no	—
372	C₂²	—	344	2	no	—
373	C₂²	—	172	4	no	—
374	C₂²	#14583178	1	2	yes	—
375	C₂²	C₂×D₈×D₄₃	1	1	yes	characteristic
376	C₄	—	8	2	no	—
377	C₄	—	8	1	no	—
378	C₄	—	8	1	no	—
379	C₄	—	43	2	no	—
380	C₄	—	86	1	no	—
381	C₄	—	86	1	no	—
382	C₄	—	86	1	no	—
383	C₄	—	344	1	no	—
384	C₄	—	344	1	no	—
385	C₄	#14583178	1	2	yes	—
386	C₄	C₂×#13063303	1	1	yes	characteristic
387	C₄	C₂×#13063357	1	1	yes	characteristic
388	C₂	—	2	2	no	—
389	C₂	—	344	4	no	—
390	C₂	C₂×#14583178	1	1	yes	characteristic
391	C₂	#509×D₄₃	1	1	yes	characteristic
392	C₂	#19147730	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×#2159	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	C₂²×D₃₄₄	C₂²
hypercenter	#2159	D₄₃
hypocenter	C₄₃	#33627
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2159

Complements

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

N	H ≅ G/N	type
1. #19145107	C₂	semidirect
3. #19136755	C₂	semidirect
4. #19136751	C₂	semidirect
5. #19136741	C₂	semidirect
6. #19136731	C₂	semidirect
7. #19136381	C₂	semidirect
8. #19136350	C₂	semidirect
11. C₄₃×#2159	C₂	semidirect
16. #14583303	C₂²	semidirect
17. #14583303	C₂²	semidirect
18. #14583300	C₂²	semidirect
19. #14583300	C₂²	semidirect
30. #14582726	C₂²	semidirect
31. #14582724	C₂²	semidirect
32. Dic₄₃⋊C₁₆	C₂²	semidirect
33. Dic₄₃⋊C₁₆	C₂²	semidirect
41. C₄₃×#367	C₂²	semidirect
42. C₄₃×#366	C₂²	semidirect
47. C₂×D₃₄₄	Q₈	semidirect
199. #2159	D₄₃	semidirect
230. #367	C₂×D₄₃	semidirect
232. #366	C₂×D₄₃	semidirect
239. C₄₃	#33627	semidirect

Maximal subgroups

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

subgroup	index
#19145107	2
C₂×#14582981	2
#19136755	2
#19136751	2
#19136741	2
#19136731	2
#19136381	2
#19136350	2
#19136312	2
#19136310	2
C₄₃×#2159	2
#33627	43