Subgroups — #25011500

Order 11008.

570 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₃₂×D₄₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#108×D₄₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#14582934	C₂	1	1	yes	maximal, characteristic
4	#19132695	C₂	1	1	yes	maximal, characteristic, complemented
5	#19132694	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132693	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1383	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×M₁₆×D₄₃	—	2	1	no	—
9	C₂²×C₈×D₄₃	C₂²	1	1	yes	characteristic
10	C₂×#13063283	—	2	1	no	—
11	C₂×#13063265	C₂²	1	1	yes	characteristic
12	C₂×#13063264	C₄	1	2	yes	complemented
13	M₃₂×D₄₃	—	2	2	no	—
14	M₃₂×D₄₃	—	2	2	no	—
15	C₂×C₁₆×D₄₃	C₄	1	2	yes	complemented
16	C₂×#13063261	C₂²	1	1	yes	characteristic
17	C₂×#13063260	—	2	1	no	—
18	#108×D₄₃	—	2	2	no	—
19	#108×D₄₃	C₄	1	4	yes	complemented
20	#14582934	—	2	2	no	—
21	#14582934	C₄	1	4	yes	—
22	C₂×#13063258	—	2	1	no	—
23	C₂×#13063257	C₂²	1	1	yes	characteristic
24	C₂×M₃₂×C₄₃	C₂²	1	1	yes	characteristic
25	C₂×#108×C₄₃	C₂²	1	1	yes	characteristic
26	C₂×#106×C₄₃	C₂²	1	1	yes	characteristic
27	C₂²×C₄×D₄₃	D₄	1	1	yes	characteristic
28	C₂×#10621877	—	2	1	no	—
29	C₂×#10621872	—	4	1	no	—
30	M₁₆×D₄₃	—	2	4	no	—
31	M₁₆×D₄₃	—	4	2	no	—
32	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
33	C₂×C₈×D₄₃	—	4	1	no	—
34	C₂×C₈×D₄₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₈×D₄₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×D₄₃	C₂×C₄	1	2	yes	—
37	C₂×C₈×D₄₃	C₂×C₄	1	2	yes	—
38	C₂×C₈×D₄₃	D₄	1	2	yes	—
39	C₂×C₈×D₄₃	D₄	1	2	yes	—
40	C₂×C₈×D₄₃	Q₈	1	1	yes	characteristic
41	C₂×C₈×D₄₃	Q₈	1	1	yes	characteristic
42	#13063283	—	2	4	no	—
43	#13063283	—	4	2	no	—
44	C₂×C₄⋊Dic₄₃	—	2	1	no	—
45	C₂×(C₂×D₄₃)⋊C₄	—	4	1	no	—
46	C₂×C₄×Dic₄₃	—	4	1	no	—
47	#13063265	—	2	1	no	—
48	#13063265	—	2	1	no	—
49	#13063264	—	2	4	no	—
50	C₂×C₄₃⋊C₁₆	C₂×C₄	1	2	yes	—
51	C₁₆×D₄₃	—	2	4	no	—
52	#13063261	—	2	1	no	—
53	#13063261	C₂×C₄	1	2	yes	—
54	#13063260	—	2	4	no	—
55	#13063258	—	2	4	no	—
56	#13063257	—	2	1	no	—
57	#13063257	C₂×C₄	1	2	yes	—
58	C₂×M₁₆×C₄₃	—	2	1	no	—
59	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
60	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
61	M₃₂×C₄₃	—	2	1	no	—
62	M₃₂×C₄₃	—	2	1	no	—
63	C₂×C₁₆×C₄₃	C₂×C₄	1	2	yes	—
64	#108×C₄₃	—	2	1	no	—
65	#108×C₄₃	C₂×C₄	1	2	yes	—
66	#106×C₄₃	—	2	1	no	—
67	#106×C₄₃	C₂×C₄	1	2	yes	—
68	C₂³×D₄₃	SD₁₆	1	1	yes	characteristic
69	C₂²×C₄×C₄₃	—	4	1	no	—
70	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×Dic₄₃	—	4	1	no	—
72	C₂²×Dic₄₃	SD₁₆	1	1	yes	characteristic
73	C₂×C₄×D₄₃	C₂²⋊C₄	1	2	yes	—
74	C₂×C₄×D₄₃	C₂²⋊C₄	1	2	yes	—
75	C₂×C₄×D₄₃	C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×D₄₃	C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×D₄₃	D₈	1	2	yes	—
78	C₂×C₄×D₄₃	SD₁₆	1	1	yes	characteristic
79	C₂×C₄×D₄₃	SD₁₆	1	1	yes	characteristic
80	C₂×C₄×D₄₃	Q₁₆	1	2	yes	—
81	M₁₆×C₄₃	—	2	2	no	—
82	M₁₆×C₄₃	—	4	1	no	—
83	C₂×C₈×C₄₃	—	4	1	no	—
84	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
90	C₄⋊C₄×C₄₃	—	2	2	no	—
91	C₄⋊C₄×C₄₃	—	4	1	no	—
92	(C₂×D₄₃)⋊C₄	—	4	4	no	—
93	C₄⋊Dic₄₃	—	2	2	no	—
94	C₄⋊Dic₄₃	—	4	1	no	—
95	C₄×Dic₄₃	—	4	4	no	—
96	#10621877	—	2	2	no	—
97	#10621877	—	4	1	no	—
98	C₂×C₄₃⋊C₈	—	4	1	no	—
99	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	2	yes	—
100	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	2	yes	—
101	C₂×C₄₃⋊C₈	C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄₃⋊C₈	C₄⋊C₄	1	1	yes	characteristic
103	#10621872	—	4	4	no	—
104	C₈×D₄₃	—	4	4	no	—
105	C₈×D₄₃	C₄²	1	2	yes	—
106	C₈×D₄₃	C₄²	1	2	yes	—
107	C₈×D₄₃	C₂²⋊C₄	1	2	yes	—
108	C₈×D₄₃	C₂²⋊C₄	1	2	yes	—
109	C₈×D₄₃	C₄⋊C₄	1	2	yes	—
110	C₈×D₄₃	C₄⋊C₄	1	2	yes	—
111	C₈×D₄₃	C₄⋊C₄	1	2	yes	—
112	C₈×D₄₃	C₄⋊C₄	1	2	yes	—
113	C₁₆×C₄₃	—	2	2	no	—
114	C₄₃⋊C₁₆	—	2	2	no	—
115	C₂³×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
116	C₂²×D₄₃	—	4	2	no	—
117	C₂²×D₄₃	#102	1	2	yes	—
118	C₂²×D₄₃	#103	1	2	yes	—
119	C₂²×D₄₃	#106	1	1	yes	characteristic
120	C₂²×D₄₃	#106	1	1	yes	characteristic
121	C₂×C₄×C₄₃	—	4	2	no	—
122	C₂×C₄×C₄₃	—	8	2	no	—
123	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
128	C₂×C₄×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
129	C₂×Dic₄₃	—	4	2	no	—
130	C₂×Dic₄₃	—	8	2	no	—
131	C₂×Dic₄₃	#102	1	2	yes	—
132	C₂×Dic₄₃	#103	1	2	yes	—
133	C₂×Dic₄₃	#106	1	1	yes	characteristic
134	C₂×Dic₄₃	#106	1	1	yes	characteristic
135	C₄×D₄₃	#95	1	2	yes	—
136	C₄×D₄₃	#95	1	2	yes	—
137	C₄×D₄₃	#102	1	2	yes	—
138	C₄×D₄₃	#102	1	2	yes	—
139	C₄×D₄₃	#103	1	2	yes	—
140	C₄×D₄₃	#103	1	2	yes	—
141	C₄×D₄₃	#107	1	2	yes	—
142	C₄×D₄₃	#107	1	2	yes	—
143	C₈×C₄₃	—	4	2	no	—
144	C₈×C₄₃	C₂×C₄²	1	1	yes	characteristic
145	C₈×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
146	C₈×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₈×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₄₃⋊C₈	—	4	2	no	—
149	C₄₃⋊C₈	#95	1	2	yes	—
150	C₄₃⋊C₈	#95	1	2	yes	—
151	#12563	—	43	1	no	maximal, Sylow
152	C₂²×C₄₃	—	2	2	no	—
153	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
154	C₂²×C₄₃	C₂×#103	1	1	yes	characteristic
155	C₂²×C₄₃	C₂×#106	1	1	yes	characteristic
156	C₂×D₄₃	—	2	2	no	—
157	C₂×D₄₃	—	2	2	no	—
158	C₂×D₄₃	—	4	2	no	—
159	C₂×D₄₃	—	4	2	no	—
160	C₂×D₄₃	#340	1	2	yes	—
161	C₂×D₄₃	#340	1	2	yes	—
162	C₄×C₄₃	—	8	2	no	—
163	C₄×C₄₃	C₂×#95	1	1	yes	characteristic
164	C₄×C₄₃	C₂×#102	1	1	yes	characteristic
165	C₄×C₄₃	C₂×#103	1	1	yes	characteristic
166	C₄×C₄₃	C₂×#107	1	1	yes	characteristic
167	Dic₄₃	—	8	2	no	—
168	Dic₄₃	#340	1	2	yes	—
169	Dic₄₃	#340	1	2	yes	—
170	C₂²×M₃₂	—	43	1	no	—
171	C₂²×#108	—	43	1	no	—
172	C₂×#427	—	43	1	no	—
173	#1383	—	43	1	no	—
174	#1383	—	43	1	no	—
175	#1383	—	43	1	no	—
176	#1383	D₄₃	1	1	yes	characteristic, complemented
177	C₂×C₄₃	—	2	1	no	—
178	C₂×C₄₃	—	2	1	no	—
179	C₂×C₄₃	C₂×#340	1	1	yes	characteristic
180	C₂×C₄₃	#2155	1	2	yes	—
181	D₄₃	—	2	2	no	—
182	D₄₃	—	2	2	no	—
183	C₂²×M₁₆	—	86	1	no	—
184	C₂³×C₈	—	43	1	no	—
185	C₂×#117	—	86	1	no	—
186	C₂×M₃₂	—	43	1	no	—
187	C₂×M₃₂	—	43	2	no	—
188	C₂×M₃₂	—	86	2	no	—
189	C₂×M₃₂	—	86	2	no	—
190	C₂×M₃₂	C₂×D₄₃	1	1	yes	characteristic
191	C₂²×C₁₆	—	43	2	no	—
192	C₂×#108	—	43	1	no	—
193	C₂×#108	—	43	4	no	—
194	C₂×#108	—	86	2	no	—
195	C₂×#108	—	86	1	no	—
196	C₂×#108	C₂×D₄₃	1	1	yes	characteristic
197	#427	—	43	4	no	—
198	#427	—	86	2	no	—
199	C₂×#107	—	86	1	no	—
200	C₂×#106	—	43	1	no	—
201	C₂×#106	C₂×D₄₃	1	1	yes	characteristic
202	C₄₃	#12563	1	1	yes	characteristic, complemented, Sylow
203	C₂³×C₄	—	43	1	no	—
204	C₂×M₁₆	—	2	1	no	—
205	C₂×M₁₆	—	86	4	no	—
206	C₂×M₁₆	—	86	1	no	—
207	C₂×M₁₆	—	172	1	no	—
208	C₂×M₁₆	—	172	2	no	—
209	C₂²×C₈	—	43	1	no	—
210	C₂²×C₈	—	43	2	no	—
211	C₂²×C₈	—	43	2	no	—
212	C₂²×C₈	—	43	1	no	—
213	C₂²×C₈	—	43	1	no	—
214	C₂²×C₈	—	43	2	no	—
215	C₂²×C₈	—	43	2	no	—
216	C₂²×C₈	—	43	1	no	—
217	C₂²×C₈	—	43	1	no	—
218	C₂²×C₈	—	172	1	no	—
219	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
220	#117	—	86	4	no	—
221	#117	—	172	2	no	—
222	C₂×C₄⋊C₄	—	2	1	no	—
223	C₂×C₄⋊C₄	—	86	1	no	—
224	C₂×C₂²⋊C₄	—	172	1	no	—
225	C₂×C₄²	—	172	1	no	—
226	M₃₂	—	2	1	no	—
227	M₃₂	—	2	1	no	—
228	M₃₂	—	86	4	no	—
229	M₃₂	—	86	1	no	—
230	M₃₂	—	86	1	no	—
231	C₂×C₁₆	—	43	2	no	—
232	C₂×C₁₆	—	86	4	no	—
233	C₂×C₁₆	C₄×D₄₃	1	2	yes	—
234	#108	—	2	1	no	—
235	#108	—	43	2	no	—
236	#108	—	86	4	no	—
237	#108	—	86	1	no	—
238	#108	C₄×D₄₃	1	2	yes	—
239	#107	—	86	4	no	—
240	#106	—	2	1	no	—
241	#106	—	43	2	no	—
242	#106	—	86	1	no	—
243	#106	C₄×D₄₃	1	2	yes	—
244	C₂⁴	—	43	1	no	—
245	C₂²×C₄	—	4	1	no	—
246	C₂²×C₄	—	43	1	no	—
247	C₂²×C₄	—	43	2	no	—
248	C₂²×C₄	—	43	2	no	—
249	C₂²×C₄	—	43	1	no	—
250	C₂²×C₄	—	43	1	no	—
251	C₂²×C₄	—	43	2	no	—
252	C₂²×C₄	—	43	1	no	—
253	C₂²×C₄	—	43	1	no	—
254	C₂²×C₄	—	43	2	no	—
255	C₂²×C₄	—	172	1	no	—
256	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
257	M₁₆	—	2	2	no	—
258	M₁₆	—	4	1	no	—
259	M₁₆	—	86	2	no	—
260	M₁₆	—	172	4	no	—
261	M₁₆	—	172	1	no	—
262	C₂×C₈	—	4	1	no	—
263	C₂×C₈	—	43	2	no	—
264	C₂×C₈	—	43	2	no	—
265	C₂×C₈	—	43	1	no	—
266	C₂×C₈	—	43	1	no	—
267	C₂×C₈	—	43	2	no	—
268	C₂×C₈	—	43	2	no	—
269	C₂×C₈	—	43	2	no	—
270	C₂×C₈	—	43	2	no	—
271	C₂×C₈	—	43	2	no	—
272	C₂×C₈	—	43	2	no	—
273	C₂×C₈	—	43	2	no	—
274	C₂×C₈	—	43	2	no	—
275	C₂×C₈	—	172	4	no	—
276	C₂×C₈	—	172	1	no	—
277	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
278	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
279	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
280	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
281	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
282	C₂×C₈	Q₈×D₄₃	1	1	yes	characteristic
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	4	1	no	—
285	C₄⋊C₄	—	86	2	no	—
286	C₄⋊C₄	—	172	1	no	—
287	C₂²⋊C₄	—	172	4	no	—
288	C₄²	—	172	4	no	—
289	C₁₆	—	2	2	no	—
290	C₁₆	—	86	2	no	—
291	C₂³	—	43	1	no	—
292	C₂³	—	43	1	no	—
293	C₂³	—	43	2	no	—
294	C₂³	—	43	2	no	—
295	C₂³	—	172	2	no	—
296	C₂³	SD₁₆×D₄₃	1	1	yes	characteristic
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	8	2	no	—
299	C₂×C₄	—	43	2	no	—
300	C₂×C₄	—	43	2	no	—
301	C₂×C₄	—	43	1	no	—
302	C₂×C₄	—	43	1	no	—
303	C₂×C₄	—	43	2	no	—
304	C₂×C₄	—	43	2	no	—
305	C₂×C₄	—	43	2	no	—
306	C₂×C₄	—	43	2	no	—
307	C₂×C₄	—	43	2	no	—
308	C₂×C₄	—	43	2	no	—
309	C₂×C₄	—	43	2	no	—
310	C₂×C₄	—	43	2	no	—
311	C₂×C₄	—	344	2	no	—
312	C₂×C₄	—	172	2	no	—
313	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
314	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
315	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
316	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
317	C₂×C₄	SD₁₆×D₄₃	1	1	yes	characteristic
318	C₂×C₄	Q₁₆×D₄₃	1	1	yes	characteristic
319	C₈	—	4	2	no	—
320	C₈	—	43	2	no	—
321	C₈	—	43	2	no	—
322	C₈	—	172	2	no	—
323	C₈	C₄²×D₄₃	1	1	yes	characteristic
324	C₈	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
325	C₈	C₄⋊C₄×D₄₃	1	1	yes	characteristic
326	C₈	C₄⋊C₄×D₄₃	1	1	yes	characteristic
327	C₂²	—	2	2	no	—
328	C₂²	—	43	2	no	—
329	C₂²	—	43	2	no	—
330	C₂²	—	86	2	no	—
331	C₂²	—	86	2	no	—
332	C₂²	—	172	2	no	—
333	C₂²	—	172	2	no	—
334	C₂²	#102×D₄₃	1	1	yes	characteristic
335	C₂²	#103×D₄₃	1	1	yes	characteristic
336	C₂²	#106×D₄₃	1	1	yes	characteristic
337	C₄	—	8	2	no	—
338	C₄	—	43	2	no	—
339	C₄	—	43	2	no	—
340	C₄	—	344	2	no	—
341	C₄	#95×D₄₃	1	1	yes	characteristic
342	C₄	#102×D₄₃	1	1	yes	characteristic
343	C₄	#103×D₄₃	1	1	yes	characteristic
344	C₄	#107×D₄₃	1	1	yes	characteristic
345	C₂	—	2	1	no	—
346	C₂	—	2	1	no	—
347	C₂	—	86	2	no	—
348	C₂	—	86	2	no	—
349	C₂	#340×D₄₃	1	1	yes	characteristic
350	C₂	#19136366	1	2	yes	—

	subgroup	G/H
center	C₂²	#106×D₄₃
commutator	C₈×C₄₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄₃×#1383	C₂
socle	C₂²×C₄₃	C₂×#106
prime radical	C₂³×D₄₃	SD₁₆
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	#1383	D₄₃
hypocenter	C₄₃	#12563
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1383

Complements

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

N	H ≅ G/N	type
4. #19132695	C₂	semidirect
5. #19132694	C₂	semidirect
6. #19132693	C₂	semidirect
7. C₄₃×#1383	C₂	semidirect
12. C₂×#13063264	C₄	semidirect
15. C₂×C₁₆×D₄₃	C₄	semidirect
19. #108×D₄₃	C₄	semidirect
176. #1383	D₄₃	semidirect
202. C₄₃	#12563	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₃₂×D₄₃	2
C₂×#108×D₄₃	2
C₂×#14582934	2
#19132695	2
#19132694	2
#19132693	2
C₄₃×#1383	2
#12563	43