Subgroups — #25251096

Order 11008.

650 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₃₂×D₄₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14583315	C₂	1	1	yes	maximal, characteristic
3	#19144245	C₂	1	2	yes	maximal
4	#19136702	C₂	1	2	yes	maximal, complemented
5	#19136701	C₂	1	2	yes	maximal, complemented
6	#19136687	C₂	1	1	yes	maximal, characteristic, complemented
7	#19136686	C₂	1	1	yes	maximal, characteristic, complemented
8	#19136683	C₂	1	1	yes	maximal, characteristic, complemented
9	#19136376	C₂	1	1	yes	maximal, characteristic, complemented
10	#19136313	C₂	1	2	yes	maximal, complemented
11	C₄₃×#2216	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₄₃	C₂²	1	2	yes	—
13	C₂×#13063331	C₂²	1	1	yes	characteristic
14	#14583472	—	2	2	no	—
15	C₂×Dic₃₄₄	C₂²	1	1	yes	characteristic
16	C₂×#13063271	C₂²	1	2	yes	—
17	Q₃₂×D₄₃	—	2	4	no	—
18	#14583315	—	2	4	no	—
19	C₂×#13063265	C₂²	1	1	yes	characteristic
20	C₂×#13063264	C₂²	1	1	yes	characteristic
21	C₂×#13063264	C₂²	1	1	yes	characteristic
22	C₂×C₁₆×D₄₃	C₂²	1	1	yes	characteristic
23	C₂×C₁₆×D₄₃	C₂²	1	1	yes	characteristic
24	#14583156	—	2	2	no	—
25	#14583155	—	2	2	no	—
26	#14583141	C₂²	1	2	yes	—
27	#14583140	C₂²	1	2	yes	—
28	#14583137	C₂²	1	2	yes	—
29	#14582936	C₂²	1	2	yes	—
30	#14582906	—	2	2	no	—
31	#14582726	C₂²	1	2	yes	—
32	#14582725	C₂²	1	1	yes	characteristic
33	#14582704	C₂²	1	2	yes	—
34	#14582703	C₂²	1	2	yes	—
35	#14582702	C₂²	1	2	yes	—
36	C₂×Q₃₂×C₄₃	C₂²	1	1	yes	characteristic
37	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
38	C₄₃×#467	C₂²	1	2	yes	—
39	C₄₃×#366	C₂²	1	1	yes	characteristic
40	C₄₃×#358	C₂²	1	2	yes	—
41	C₂×#10621898	D₄	1	1	yes	characteristic
42	C₂×Q₈×D₄₃	—	2	2	no	—
43	C₂×Dic₁₇₂	C₂³	1	2	yes	—
44	C₂×#10621885	—	2	2	no	—
45	Q₁₆×D₄₃	—	2	4	no	—
46	Q₁₆×D₄₃	—	2	4	no	—
47	#13063331	—	2	2	no	—
48	#13063331	—	2	2	no	—
49	C₂×#10621877	D₄	1	1	yes	characteristic
50	C₂×#10621872	D₄	1	1	yes	characteristic
51	C₂×#10621872	D₄	1	1	yes	characteristic
52	C₂×C₈×D₄₃	C₂³	1	1	yes	characteristic
53	C₂×C₈×D₄₃	C₂³	1	1	yes	characteristic
54	#13063324	—	4	2	no	—
55	#13063311	—	4	2	no	—
56	#13063310	—	4	2	no	—
57	#13063307	—	4	2	no	—
58	#13063305	—	2	2	no	—
59	#13063304	—	2	2	no	—
60	#13063301	—	2	2	no	—
61	#13063283	—	2	2	no	—
62	Dic₃₄₄	—	2	2	no	—
63	#13063271	—	2	4	no	—
64	#13063265	—	2	2	no	—
65	#13063264	—	2	2	no	—
66	#13063264	—	2	2	no	—
67	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
68	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
69	C₁₆×D₄₃	—	2	2	no	—
70	C₁₆×D₄₃	D₄	1	4	yes	—
71	#13063259	C₂³	1	2	yes	—
72	#13063256	—	2	2	no	—
73	#13063250	—	2	2	no	—
74	#13063249	—	2	2	no	—
75	#13063248	—	2	2	no	—
76	C₂×Q₁₆×C₄₃	C₂³	1	2	yes	—
77	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
78	C₂²⋊Q₈×C₄₃	—	2	2	no	—
79	Q₃₂×C₄₃	—	2	2	no	—
80	C₂×C₁₆×C₄₃	—	2	2	no	—
81	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
82	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
83	#107×C₄₃	C₂³	1	2	yes	—
84	#103×C₄₃	—	2	2	no	—
85	C₂×Q₈×C₄₃	—	2	2	no	—
86	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₂²⋊D₄₃	D₈	1	1	yes	characteristic
88	C₂×C₂²⋊D₄₃	D₈	1	1	yes	characteristic
89	Q₈×D₄₃	—	4	4	no	—
90	Q₈×D₄₃	—	4	4	no	—
91	#10621898	—	2	2	no	—
92	#10621898	—	2	2	no	—
93	C₂×D₁₇₂	D₈	1	1	yes	characteristic
94	C₂×C₄×D₄₃	—	4	2	no	—
95	C₂×C₄×D₄₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×D₄₃	C₂×D₄	1	1	yes	characteristic
97	C₂×Dic₈₆	—	2	2	no	—
98	C₂×Dic₈₆	—	4	2	no	—
99	C₂×Dic₈₆	D₈	1	1	yes	characteristic
100	Q₁₆×C₄₃	—	2	2	no	—
101	Q₁₆×C₄₃	—	2	2	no	—
102	C₂×C₈×C₄₃	—	2	1	no	—
103	C₂×C₈×C₄₃	—	2	1	no	—
104	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
106	C₄⋊C₄×C₄₃	—	2	2	no	—
107	C₄⋊C₄×C₄₃	—	4	2	no	—
108	C₂²⋊C₄×C₄₃	—	4	2	no	—
109	C₂²⋊Dic₄₃	—	4	2	no	—
110	#10621885	—	4	4	no	—
111	(C₂×D₄₃)⋊C₄	—	4	2	no	—
112	(C₂×D₄₃)⋊C₄	—	4	2	no	—
113	(C₂×D₄₃)⋊C₄	—	4	2	no	—
114	C₄⋊Dic₄₃	—	2	2	no	—
115	C₄⋊Dic₄₃	—	4	2	no	—
116	Dic₄₃⋊C₄	—	4	2	no	—
117	Dic₄₃⋊C₄	—	4	2	no	—
118	C₄×Dic₄₃	—	4	2	no	—
119	#10621877	—	2	2	no	—
120	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
122	Dic₁₇₂	—	2	2	no	—
123	Dic₁₇₂	—	2	2	no	—
124	#10621872	—	2	2	no	—
125	#10621872	—	2	2	no	—
126	C₈×D₄₃	—	2	1	no	—
127	C₈×D₄₃	—	2	1	no	—
128	C₈×D₄₃	C₂×D₄	1	2	yes	—
129	C₈×D₄₃	#41	1	2	yes	—
130	C₁₆×C₄₃	—	2	1	no	—
131	C₁₆×C₄₃	C₂×D₄	1	2	yes	—
132	C₄₃⋊C₁₆	—	2	1	no	—
133	C₄₃⋊C₁₆	C₂×D₄	1	2	yes	—
134	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
135	C₂²×D₄₃	C₂×D₈	1	1	yes	characteristic
136	C₂²×D₄₃	C₂×D₈	1	1	yes	characteristic
137	Q₈×C₄₃	—	4	2	no	—
138	Q₈×C₄₃	—	4	2	no	—
139	C₂×C₄×C₄₃	—	2	1	no	—
140	C₂×C₄×C₄₃	—	2	1	no	—
141	C₂×C₄×C₄₃	—	4	2	no	—
142	C₂×C₄×C₄₃	—	4	2	no	—
143	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
145	C₂²⋊D₄₃	—	2	2	no	—
146	C₂²⋊D₄₃	—	2	2	no	—
147	C₂×Dic₄₃	—	4	2	no	—
148	C₂×Dic₄₃	—	4	2	no	—
149	C₂×Dic₄₃	C₂×D₈	1	1	yes	characteristic
150	C₂×Dic₄₃	C₂×D₈	1	1	yes	characteristic
151	D₁₇₂	—	2	2	no	—
152	C₄×D₄₃	—	2	1	no	—
153	C₄×D₄₃	—	2	1	no	—
154	C₄×D₄₃	—	8	4	no	—
155	C₄×D₄₃	#121	1	2	yes	—
156	C₄×D₄₃	#135	1	2	yes	—
157	Dic₈₆	—	2	2	no	—
158	Dic₈₆	—	4	2	no	—
159	Dic₈₆	—	4	2	no	—
160	Dic₈₆	—	8	4	no	—
161	C₈×C₄₃	—	2	1	no	—
162	C₈×C₄₃	C₂²×D₄	1	1	yes	characteristic
163	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
164	C₄₃⋊C₈	—	2	1	no	—
165	C₄₃⋊C₈	#121	1	2	yes	—
166	#33608	—	43	1	no	maximal, Sylow
167	C₂²×C₄₃	—	2	1	no	—
168	C₂²×C₄₃	—	2	1	no	—
169	C₂²×C₄₃	—	2	1	no	—
170	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
171	C₂×D₄₃	—	2	1	no	—
172	C₂×D₄₃	—	2	1	no	—
173	C₂×D₄₃	—	2	1	no	—
174	C₂×D₄₃	—	2	1	no	—
175	C₂×D₄₃	—	2	1	no	—
176	C₂×D₄₃	#466	1	2	yes	—
177	C₄×C₄₃	—	2	1	no	—
178	C₄×C₄₃	—	8	2	no	—
179	C₄×C₄₃	—	8	2	no	—
180	C₄×C₄₃	C₂×#121	1	1	yes	characteristic
181	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
182	Dic₄₃	—	2	1	no	—
183	Dic₄₃	—	8	2	no	—
184	Dic₄₃	—	8	2	no	—
185	Dic₄₃	#466	1	2	yes	—
186	C₂²×Q₃₂	—	43	1	no	—
187	C₂×#504	—	43	1	no	—
188	#3059	—	43	2	no	—
189	#2219	—	43	2	no	—
190	#2219	—	43	2	no	—
191	#2216	—	43	1	no	—
192	#2216	—	43	1	no	—
193	#2216	—	43	1	no	—
194	#2216	D₄₃	1	1	yes	characteristic, complemented
195	#2158	—	43	1	no	—
196	#2141	—	43	2	no	—
197	C₂×C₄₃	—	4	1	no	—
198	C₂×C₄₃	C₂×#466	1	1	yes	characteristic
199	C₂×C₄₃	#3417	1	1	yes	characteristic
200	C₂×C₄₃	#3417	1	1	yes	characteristic
201	D₄₃	—	2	2	no	—
202	D₄₃	—	4	1	no	—
203	C₂²×Q₁₆	—	43	2	no	—
204	C₂×#131	—	43	1	no	—
205	#536	—	86	2	no	—
206	C₂×Q₃₂	—	43	2	no	—
207	C₂×Q₃₂	—	43	1	no	—
208	C₂×Q₃₂	—	86	4	no	—
209	C₂×Q₃₂	C₂×D₄₃	1	1	yes	characteristic
210	#504	—	86	4	no	—
211	C₂×M₃₂	—	43	1	no	—
212	C₂×M₃₂	—	43	1	no	—
213	C₂×M₃₂	—	43	1	no	—
214	C₂²×C₁₆	—	43	1	no	—
215	C₂²×C₁₆	—	43	1	no	—
216	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
217	#470	—	86	2	no	—
218	#470	—	86	2	no	—
219	#467	—	43	2	no	—
220	#467	—	43	2	no	—
221	#467	—	43	2	no	—
222	#467	C₂×D₄₃	1	2	yes	—
223	#427	—	43	2	no	—
224	#419	—	86	2	no	—
225	#367	—	43	2	no	—
226	#366	—	43	1	no	—
227	#366	C₂×D₄₃	1	1	yes	characteristic
228	#358	—	43	2	no	—
229	#358	—	43	2	no	—
230	#358	—	43	2	no	—
231	#358	C₂×D₄₃	1	2	yes	—
232	C₄₃	#33608	1	1	yes	characteristic, complemented, Sylow
233	C₂×#41	—	43	1	no	—
234	C₂²×Q₈	—	86	2	no	—
235	C₂×Q₁₆	—	43	2	no	—
236	C₂×Q₁₆	—	86	4	no	—
237	C₂×Q₁₆	—	86	4	no	—
238	C₂×Q₁₆	—	86	2	no	—
239	C₂×Q₁₆	C₂²×D₄₃	1	2	yes	—
240	#131	—	86	2	no	—
241	#131	—	86	2	no	—
242	C₂×M₁₆	—	43	1	no	—
243	C₂×M₁₆	—	43	1	no	—
244	C₂×M₁₆	—	43	1	no	—
245	C₂²×C₈	—	43	1	no	—
246	C₂²×C₈	—	43	1	no	—
247	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
248	C₄⋊Q₈	—	172	2	no	—
249	#124	—	172	2	no	—
250	#123	—	172	2	no	—
251	#123	—	172	2	no	—
252	C₂²⋊Q₈	—	2	2	no	—
253	C₂²⋊Q₈	—	86	2	no	—
254	C₂²⋊Q₈	—	86	2	no	—
255	C₂²⋊Q₈	—	86	2	no	—
256	#117	—	86	2	no	—
257	Q₃₂	—	2	2	no	—
258	Q₃₂	—	86	4	no	—
259	Q₃₂	—	86	2	no	—
260	M₃₂	—	86	2	no	—
261	M₃₂	—	86	2	no	—
262	M₃₂	—	86	2	no	—
263	C₂×C₁₆	—	2	2	no	—
264	C₂×C₁₆	—	43	1	no	—
265	C₂×C₁₆	—	43	4	no	—
266	C₂×C₁₆	—	43	1	no	—
267	C₂×C₁₆	—	86	2	no	—
268	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
269	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
270	#107	—	43	2	no	—
271	#107	C₂²×D₄₃	1	2	yes	—
272	#106	—	86	2	no	—
273	#103	—	2	2	no	—
274	#103	—	86	2	no	—
275	#103	—	86	2	no	—
276	#103	—	86	2	no	—
277	#41	—	86	2	no	—
278	#41	—	86	2	no	—
279	C₂×Q₈	—	2	2	no	—
280	C₂×Q₈	—	43	1	no	—
281	C₂×Q₈	—	86	2	no	—
282	C₂×Q₈	—	172	2	no	—
283	C₂×Q₈	—	172	4	no	—
284	C₂×Q₈	—	172	4	no	—
285	C₂×D₄	—	43	1	no	—
286	C₂×D₄	—	43	1	no	—
287	C₂×D₄	—	43	1	no	—
288	C₂²×C₄	—	43	1	no	—
289	C₂²×C₄	—	43	1	no	—
290	C₂²×C₄	—	172	2	no	—
291	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
292	Q₁₆	—	2	2	no	—
293	Q₁₆	—	2	2	no	—
294	Q₁₆	—	86	2	no	—
295	Q₁₆	—	86	2	no	—
296	Q₁₆	—	172	4	no	—
297	M₁₆	—	86	2	no	—
298	M₁₆	—	86	2	no	—
299	M₁₆	—	86	2	no	—
300	C₂×C₈	—	2	1	no	—
301	C₂×C₈	—	2	1	no	—
302	C₂×C₈	—	43	1	no	—
303	C₂×C₈	—	43	2	no	—
304	C₂×C₈	—	43	2	no	—
305	C₂×C₈	—	43	1	no	—
306	C₂×C₈	—	86	1	no	—
307	C₂×C₈	—	86	1	no	—
308	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
309	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
310	C₄⋊C₄	—	2	2	no	—
311	C₄⋊C₄	—	4	2	no	—
312	C₄⋊C₄	—	86	2	no	—
313	C₄⋊C₄	—	172	2	no	—
314	C₄⋊C₄	—	172	2	no	—
315	C₄⋊C₄	—	172	2	no	—
316	C₂²⋊C₄	—	4	2	no	—
317	C₂²⋊C₄	—	172	2	no	—
318	C₂²⋊C₄	—	172	2	no	—
319	C₂²⋊C₄	—	172	2	no	—
320	C₂²⋊C₄	—	172	2	no	—
321	C₄²	—	172	2	no	—
322	C₁₆	—	2	1	no	—
323	C₁₆	—	43	2	no	—
324	C₁₆	—	86	1	no	—
325	C₁₆	D₄×D₄₃	1	2	yes	—
326	C₂³	—	43	1	no	—
327	C₂³	—	43	1	no	—
328	C₂³	D₈×D₄₃	1	1	yes	characteristic
329	Q₈	—	4	2	no	—
330	Q₈	—	4	2	no	—
331	Q₈	—	86	2	no	—
332	Q₈	—	344	4	no	—
333	Q₈	—	172	2	no	—
334	Q₈	—	172	2	no	—
335	D₄	—	86	2	no	—
336	D₄	—	86	2	no	—
337	D₄	—	86	2	no	—
338	C₂×C₄	—	2	1	no	—
339	C₂×C₄	—	2	1	no	—
340	C₂×C₄	—	4	2	no	—
341	C₂×C₄	—	4	2	no	—
342	C₂×C₄	—	43	1	no	—
343	C₂×C₄	—	43	2	no	—
344	C₂×C₄	—	43	2	no	—
345	C₂×C₄	—	43	1	no	—
346	C₂×C₄	—	86	1	no	—
347	C₂×C₄	—	86	1	no	—
348	C₂×C₄	—	344	4	no	—
349	C₂×C₄	—	172	2	no	—
350	C₂×C₄	—	172	2	no	—
351	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
352	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
353	C₈	—	2	1	no	—
354	C₈	—	43	2	no	—
355	C₈	—	86	1	no	—
356	C₈	C₂×D₄×D₄₃	1	1	yes	characteristic
357	C₈	#41×D₄₃	1	1	yes	characteristic
358	C₂²	—	2	1	no	—
359	C₂²	—	2	1	no	—
360	C₂²	—	2	1	no	—
361	C₂²	—	43	2	no	—
362	C₂²	—	86	1	no	—
363	C₂²	—	86	1	no	—
364	C₂²	—	86	1	no	—
365	C₂²	—	86	1	no	—
366	C₂²	—	86	1	no	—
367	C₂²	C₂×D₈×D₄₃	1	1	yes	characteristic
368	C₄	—	2	1	no	—
369	C₄	—	8	2	no	—
370	C₄	—	8	2	no	—
371	C₄	—	43	2	no	—
372	C₄	—	86	1	no	—
373	C₄	—	344	2	no	—
374	C₄	—	344	2	no	—
375	C₄	#121×D₄₃	1	1	yes	characteristic
376	C₄	#135×D₄₃	1	1	yes	characteristic
377	C₂	—	4	1	no	—
378	C₂	—	86	2	no	—
379	C₂	—	172	1	no	—
380	C₂	#466×D₄₃	1	1	yes	characteristic
381	C₂	#19147744	1	1	yes	characteristic
382	C₂	#19147745	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×#2216	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	C₂×#10621898	D₄
hypercenter	#2216	D₄₃
hypocenter	C₄₃	#33608
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2216

Complements

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

N	H ≅ G/N	type
1. C₂×Q₃₂×D₄₃	C₂	semidirect
4. #19136702	C₂	semidirect
5. #19136701	C₂	semidirect
6. #19136687	C₂	semidirect
7. #19136686	C₂	semidirect
8. #19136683	C₂	semidirect
9. #19136376	C₂	semidirect
10. #19136313	C₂	semidirect
11. C₄₃×#2216	C₂	semidirect
194. #2216	D₄₃	semidirect
232. C₄₃	#33608	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₃₂×D₄₃	2
C₂×#14583315	2
#19144245	2
#19136702	2
#19136701	2
#19136687	2
#19136686	2
#19136683	2
#19136376	2
#19136313	2
C₄₃×#2216	2
#33608	43