Subgroups — #25013963

Order 11008.

526 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₆×D₄₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#14583141	C₂	1	1	yes	maximal, characteristic
3	C₂×#14582704	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14582702	C₂	1	1	yes	maximal, characteristic, complemented
5	#19134621	C₂	1	1	yes	maximal, characteristic
6	#19132686	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1380	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Dic₁₇₂	C₂²	1	1	yes	characteristic
9	C₂²×C₈×D₄₃	C₂²	1	1	yes	characteristic
10	C₂×#13063305	—	2	1	no	—
11	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×D₄₃	—	2	2	no	—
13	C₂×C₁₆×D₄₃	—	2	2	no	—
14	C₂×C₁₆×D₄₃	—	2	2	no	—
15	#14583141	—	2	2	no	—
16	#14583141	C₄	1	4	yes	complemented
17	C₂×#13063259	C₂²	1	1	yes	characteristic
18	C₂×#13063248	—	2	1	no	—
19	#14582756	—	2	1	no	—
20	#14582704	—	2	2	no	—
21	#14582702	—	2	2	no	—
22	#14582661	—	2	1	no	—
23	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
24	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
25	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
26	C₂²×Dic₈₆	—	2	1	no	—
27	C₂²×C₄×D₄₃	D₄	1	1	yes	characteristic
28	C₂×Dic₁₇₂	—	2	1	no	—
29	C₂×Dic₁₇₂	—	2	1	no	—
30	C₂×Dic₁₇₂	—	2	1	no	—
31	C₂×Dic₁₇₂	—	2	1	no	—
32	C₂×Dic₁₇₂	—	2	1	no	—
33	C₂×Dic₁₇₂	C₂×C₄	1	2	yes	—
34	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
35	C₂×C₈×D₄₃	—	2	1	no	—
36	C₂×C₈×D₄₃	—	2	1	no	—
37	C₂×C₈×D₄₃	—	2	1	no	—
38	C₂×C₈×D₄₃	—	2	1	no	—
39	C₂×C₈×D₄₃	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₄₃	D₄	1	2	yes	—
41	#13063305	—	2	4	no	—
42	#13063305	—	4	2	no	—
43	C₂×C₄⋊Dic₄₃	—	2	1	no	—
44	C₂×C₄⋊Dic₄₃	—	4	1	no	—
45	C₂×(C₂×D₄₃)⋊C₄	—	4	1	no	—
46	C₂×(C₂×D₄₃)⋊C₄	—	4	1	no	—
47	C₂×C₄₃⋊C₁₆	—	2	1	no	—
48	C₂×C₄₃⋊C₁₆	—	2	1	no	—
49	C₂×C₄₃⋊C₁₆	D₄	1	2	yes	—
50	C₁₆×D₄₃	—	2	8	no	—
51	#13063259	—	2	1	no	—
52	#13063259	—	2	1	no	—
53	#13063248	—	2	4	no	—
54	C₄₃⋊#95	—	4	1	no	—
55	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
56	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
57	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
58	C₂×C₁₆×C₄₃	—	2	1	no	—
59	C₂×C₁₆×C₄₃	—	2	1	no	—
60	C₂×C₁₆×C₄₃	D₄	1	2	yes	—
61	#107×C₄₃	—	2	1	no	—
62	#107×C₄₃	—	2	1	no	—
63	#107×C₄₃	—	2	1	no	—
64	#107×C₄₃	C₂×C₄	1	2	yes	—
65	C₂³×D₄₃	D₈	1	1	yes	characteristic
66	C₂²×C₄×C₄₃	—	4	1	no	—
67	C₂²×C₄×C₄₃	—	4	1	no	—
68	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×Dic₄₃	—	4	1	no	—
70	C₂²×Dic₄₃	—	4	1	no	—
71	C₂²×Dic₄₃	D₈	1	1	yes	characteristic
72	C₂×C₄×D₄₃	—	2	1	no	—
73	C₂×C₄×D₄₃	—	2	1	no	—
74	C₂×C₄×D₄₃	—	2	1	no	—
75	C₂×C₄×D₄₃	—	2	1	no	—
76	C₂×C₄×D₄₃	C₂²⋊C₄	1	2	yes	—
77	C₂×C₄×D₄₃	SD₁₆	1	2	yes	—
78	C₂×Dic₈₆	—	2	2	no	—
79	C₂×Dic₈₆	—	4	1	no	—
80	C₂×Dic₈₆	—	4	1	no	—
81	C₂×Dic₈₆	—	4	1	no	—
82	C₂×Dic₈₆	—	4	1	no	—
83	C₂×Dic₈₆	—	4	1	no	—
84	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
89	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
90	C₄⋊C₄×C₄₃	—	2	2	no	—
91	C₄⋊C₄×C₄₃	—	4	1	no	—
92	C₄⋊C₄×C₄₃	—	4	1	no	—
93	C₄⋊C₄×C₄₃	—	4	1	no	—
94	(C₂×D₄₃)⋊C₄	—	4	4	no	—
95	(C₂×D₄₃)⋊C₄	—	8	2	no	—
96	C₄⋊Dic₄₃	—	4	4	no	—
97	C₄⋊Dic₄₃	—	4	1	no	—
98	C₄⋊Dic₄₃	—	4	1	no	—
99	C₂×C₄₃⋊C₈	—	2	1	no	—
100	C₂×C₄₃⋊C₈	—	2	1	no	—
101	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	2	yes	—
102	Dic₁₇₂	—	2	2	no	—
103	Dic₁₇₂	—	2	2	no	—
104	Dic₁₇₂	—	4	1	no	—
105	Dic₁₇₂	—	4	1	no	—
106	C₈×D₄₃	—	2	2	no	—
107	C₈×D₄₃	—	2	2	no	—
108	C₈×D₄₃	—	2	2	no	—
109	C₈×D₄₃	—	2	2	no	—
110	C₁₆×C₄₃	—	2	2	no	—
111	C₄₃⋊C₁₆	—	2	2	no	—
112	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
113	C₂²×D₄₃	—	2	1	no	—
114	C₂²×D₄₃	—	2	1	no	—
115	C₂²×D₄₃	—	2	1	no	—
116	C₂²×D₄₃	—	2	1	no	—
117	C₂²×D₄₃	#102	1	2	yes	—
118	C₂²×D₄₃	SD₃₂	1	2	yes	—
119	C₂²×D₄₃	Q₃₂	1	2	yes	—
120	C₂×C₄×C₄₃	—	4	2	no	—
121	C₂×C₄×C₄₃	—	8	1	no	—
122	C₂×C₄×C₄₃	—	8	1	no	—
123	C₂×C₄×C₄₃	—	8	1	no	—
124	C₂×C₄×C₄₃	—	8	1	no	—
125	C₂×C₄×C₄₃	—	8	1	no	—
126	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
128	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
129	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
130	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
131	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
132	C₂×Dic₄₃	—	2	1	no	—
133	C₂×Dic₄₃	—	2	1	no	—
134	C₂×Dic₄₃	—	4	2	no	—
135	C₂×Dic₄₃	—	8	1	no	—
136	C₂×Dic₄₃	—	8	1	no	—
137	C₂×Dic₄₃	—	8	1	no	—
138	C₂×Dic₄₃	—	8	1	no	—
139	C₂×Dic₄₃	—	8	1	no	—
140	C₂×Dic₄₃	#102	1	2	yes	—
141	C₄×D₄₃	—	2	2	no	—
142	C₄×D₄₃	—	2	2	no	—
143	C₄×D₄₃	—	2	2	no	—
144	C₄×D₄₃	—	2	2	no	—
145	Dic₈₆	—	4	2	no	—
146	Dic₈₆	—	4	2	no	—
147	Dic₈₆	—	8	1	no	—
148	Dic₈₆	—	8	1	no	—
149	C₈×C₄₃	C₄×D₄	1	1	yes	characteristic
150	C₈×C₄₃	C₄×D₄	1	1	yes	characteristic
151	C₈×C₄₃	#120	1	1	yes	characteristic
152	C₈×C₄₃	#123	1	1	yes	characteristic
153	C₄₃⋊C₈	—	2	2	no	—
154	#13060	—	43	1	no	maximal, Sylow
155	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
156	C₂²×C₄₃	#466	1	1	yes	characteristic
157	C₂²×C₄₃	#466	1	1	yes	characteristic
158	C₂²×C₄₃	C₂×SD₃₂	1	1	yes	characteristic
159	C₂²×C₄₃	C₂×Q₃₂	1	1	yes	characteristic
160	C₂²×C₄₃	#508	1	1	yes	characteristic
161	C₂²×C₄₃	#508	1	1	yes	characteristic
162	C₂×D₄₃	—	2	2	no	—
163	C₂×D₄₃	—	2	2	no	—
164	C₂×D₄₃	—	2	2	no	—
165	C₂×D₄₃	—	2	2	no	—
166	C₂×D₄₃	—	2	2	no	—
167	C₂×D₄₃	—	2	2	no	—
168	C₂×D₄₃	#358	1	4	yes	complemented
169	C₄×C₄₃	—	8	2	no	—
170	C₄×C₄₃	—	16	1	no	—
171	C₄×C₄₃	#386	1	1	yes	characteristic
172	C₄×C₄₃	#416	1	1	yes	characteristic
173	C₄×C₄₃	#465	1	1	yes	characteristic
174	C₄×C₄₃	#465	1	1	yes	characteristic
175	Dic₄₃	—	2	2	no	—
176	Dic₄₃	—	8	2	no	—
177	Dic₄₃	—	16	1	no	—
178	C₂³×C₁₆	—	43	1	no	—
179	C₂×#467	—	43	1	no	—
180	C₂×#358	—	43	1	no	—
181	C₂×#358	—	43	1	no	—
182	#1808	—	43	1	no	—
183	#1380	—	43	1	no	—
184	#1380	D₄₃	1	1	yes	characteristic, complemented
185	C₂×C₄₃	#1893	1	1	yes	characteristic
186	C₂×C₄₃	C₂×#358	1	1	yes	characteristic
187	C₂×C₄₃	#2138	1	1	yes	characteristic
188	C₂×C₄₃	#2216	1	1	yes	characteristic
189	C₂×C₄₃	#2216	1	1	yes	characteristic
190	C₂×C₄₃	#2217	1	1	yes	characteristic
191	C₂×C₄₃	#2217	1	1	yes	characteristic
192	D₄₃	—	2	4	no	—
193	C₂²×Q₁₆	—	43	1	no	—
194	C₂³×C₈	—	43	1	no	—
195	C₂×C₂²⋊Q₈	—	86	1	no	—
196	C₂²×C₁₆	—	43	1	no	—
197	C₂²×C₁₆	—	86	2	no	—
198	C₂²×C₁₆	—	86	2	no	—
199	C₂²×C₁₆	—	86	2	no	—
200	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
201	#467	—	43	4	no	—
202	#467	—	86	2	no	—
203	C₂×#107	—	43	1	no	—
204	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
205	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
206	C₂×#103	—	86	1	no	—
207	#380	—	86	1	no	—
208	#358	—	86	2	no	—
209	#358	—	86	2	no	—
210	#340	—	86	1	no	—
211	C₄₃	#13060	1	1	yes	characteristic, complemented, Sylow
212	C₂²×Q₈	—	86	1	no	—
213	C₂³×C₄	—	43	1	no	—
214	C₂×Q₁₆	—	43	2	no	—
215	C₂×Q₁₆	—	86	1	no	—
216	C₂×Q₁₆	—	86	1	no	—
217	C₂×Q₁₆	—	86	1	no	—
218	C₂×Q₁₆	—	86	1	no	—
219	C₂×Q₁₆	—	86	1	no	—
220	C₂²×C₈	—	43	1	no	—
221	C₂²×C₈	—	43	2	no	—
222	C₂²×C₈	—	43	2	no	—
223	C₂²×C₈	—	86	1	no	—
224	C₂²×C₈	—	86	1	no	—
225	C₂²×C₈	—	86	1	no	—
226	C₂²×C₈	—	86	1	no	—
227	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
228	C₂²⋊Q₈	—	86	4	no	—
229	C₂²⋊Q₈	—	172	2	no	—
230	C₂×C₄⋊C₄	—	2	1	no	—
231	C₂×C₄⋊C₄	—	2	1	no	—
232	C₂×C₄⋊C₄	—	86	1	no	—
233	C₂×C₄⋊C₄	—	172	1	no	—
234	C₂×C₂²⋊C₄	—	172	1	no	—
235	C₂×C₂²⋊C₄	—	172	1	no	—
236	C₂×C₁₆	—	2	1	no	—
237	C₂×C₁₆	—	2	1	no	—
238	C₂×C₁₆	—	43	2	no	—
239	C₂×C₁₆	—	86	8	no	—
240	C₂×C₁₆	—	86	1	no	—
241	C₂×C₁₆	—	86	1	no	—
242	C₂×C₁₆	D₁₇₂	1	2	yes	—
243	#107	—	2	1	no	—
244	#107	—	2	1	no	—
245	#107	—	2	1	no	—
246	#107	—	86	1	no	—
247	#107	—	86	1	no	—
248	#107	C₄×D₄₃	1	2	yes	—
249	#103	—	86	4	no	—
250	#95	—	172	1	no	—
251	C₂⁴	—	43	1	no	—
252	C₂×Q₈	—	86	2	no	—
253	C₂×Q₈	—	172	1	no	—
254	C₂×Q₈	—	172	1	no	—
255	C₂×Q₈	—	172	1	no	—
256	C₂×Q₈	—	172	1	no	—
257	C₂×Q₈	—	172	1	no	—
258	C₂²×C₄	—	4	1	no	—
259	C₂²×C₄	—	4	1	no	—
260	C₂²×C₄	—	43	1	no	—
261	C₂²×C₄	—	43	2	no	—
262	C₂²×C₄	—	43	2	no	—
263	C₂²×C₄	—	86	1	no	—
264	C₂²×C₄	—	86	1	no	—
265	C₂²×C₄	—	86	1	no	—
266	C₂²×C₄	—	86	1	no	—
267	C₂²×C₄	—	172	1	no	—
268	C₂²×C₄	—	172	1	no	—
269	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
270	Q₁₆	—	86	2	no	—
271	Q₁₆	—	86	2	no	—
272	Q₁₆	—	172	1	no	—
273	Q₁₆	—	172	1	no	—
274	C₂×C₈	—	43	2	no	—
275	C₂×C₈	—	86	1	no	—
276	C₂×C₈	—	86	1	no	—
277	C₂×C₈	—	86	2	no	—
278	C₂×C₈	—	86	2	no	—
279	C₂×C₈	—	86	2	no	—
280	C₂×C₈	—	86	2	no	—
281	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
282	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
283	C₂×C₈	#10621898	1	1	yes	characteristic
284	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
285	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
286	C₂×C₈	#10621902	1	1	yes	characteristic
287	C₄⋊C₄	—	2	2	no	—
288	C₄⋊C₄	—	4	1	no	—
289	C₄⋊C₄	—	4	1	no	—
290	C₄⋊C₄	—	4	1	no	—
291	C₄⋊C₄	—	172	4	no	—
292	C₄⋊C₄	—	172	1	no	—
293	C₄⋊C₄	—	172	1	no	—
294	C₂²⋊C₄	—	344	2	no	—
295	C₂²⋊C₄	—	172	4	no	—
296	C₁₆	—	2	2	no	—
297	C₁₆	—	86	2	no	—
298	C₂³	—	43	2	no	—
299	C₂³	—	43	2	no	—
300	C₂³	—	43	2	no	—
301	C₂³	—	86	1	no	—
302	C₂³	—	86	1	no	—
303	C₂³	—	86	1	no	—
304	C₂³	—	86	1	no	—
305	C₂³	D₈×D₄₃	1	1	yes	characteristic
306	Q₈	—	344	1	no	—
307	Q₈	—	344	1	no	—
308	Q₈	—	172	2	no	—
309	Q₈	—	172	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	8	1	no	—
312	C₂×C₄	—	8	1	no	—
313	C₂×C₄	—	8	1	no	—
314	C₂×C₄	—	8	1	no	—
315	C₂×C₄	—	8	1	no	—
316	C₂×C₄	—	43	2	no	—
317	C₂×C₄	—	86	1	no	—
318	C₂×C₄	—	86	1	no	—
319	C₂×C₄	—	86	2	no	—
320	C₂×C₄	—	86	2	no	—
321	C₂×C₄	—	86	2	no	—
322	C₂×C₄	—	86	2	no	—
323	C₂×C₄	—	344	1	no	—
324	C₂×C₄	—	344	1	no	—
325	C₂×C₄	—	344	1	no	—
326	C₂×C₄	—	344	1	no	—
327	C₂×C₄	—	344	1	no	—
328	C₂×C₄	—	172	2	no	—
329	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
330	C₂×C₄	#13063298	1	1	yes	characteristic
331	C₂×C₄	#13063300	1	1	yes	characteristic
332	C₂×C₄	SD₁₆×D₄₃	1	1	yes	characteristic
333	C₂×C₄	#13063346	1	1	yes	characteristic
334	C₂×C₄	#13063348	1	1	yes	characteristic
335	C₈	—	86	2	no	—
336	C₈	#13063287	1	1	yes	characteristic
337	C₈	C₄×D₁₇₂	1	1	yes	characteristic
338	C₈	#13063295	1	1	yes	characteristic
339	C₈	#13063310	1	1	yes	characteristic
340	C₂²	—	43	4	no	—
341	C₂²	—	86	2	no	—
342	C₂²	—	86	2	no	—
343	C₂²	—	86	2	no	—
344	C₂²	—	86	2	no	—
345	C₂²	—	86	2	no	—
346	C₂²	—	86	2	no	—
347	C₂²	#102×D₄₃	1	1	yes	characteristic
348	C₂²	#14583134	1	1	yes	characteristic
349	C₂²	#14583136	1	1	yes	characteristic
350	C₂²	SD₃₂×D₄₃	1	1	yes	characteristic
351	C₂²	Q₃₂×D₄₃	1	1	yes	characteristic
352	C₂²	#14583328	1	1	yes	characteristic
353	C₂²	#14583330	1	1	yes	characteristic
354	C₄	—	8	2	no	—
355	C₄	—	16	1	no	—
356	C₄	—	86	2	no	—
357	C₄	—	344	2	no	—
358	C₄	—	688	1	no	—
359	C₄	#14582777	1	1	yes	characteristic
360	C₄	#14582894	1	1	yes	characteristic
361	C₄	#14583129	1	1	yes	characteristic
362	C₄	#14583131	1	1	yes	characteristic
363	C₂	—	86	4	no	—
364	C₂	#19135022	1	1	yes	characteristic
365	C₂	#358×D₄₃	1	1	yes	characteristic
366	C₂	#19136299	1	1	yes	characteristic
367	C₂	#19136685	1	1	yes	characteristic
368	C₂	#19136687	1	1	yes	characteristic
369	C₂	#19136690	1	1	yes	characteristic
370	C₂	#19136692	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₄₃
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄₃×#1380	C₂
socle	C₂³×C₄₃	C₂×D₈
prime radical	C₂³×D₄₃	D₈
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	#1380	D₄₃
hypocenter	C₄₃	#13060
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1380

Complements

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

N	H ≅ G/N	type
3. C₂×#14582704	C₂	semidirect
4. C₂×#14582702	C₂	semidirect
6. #19132686	C₂	semidirect
7. C₄₃×#1380	C₂	semidirect
16. #14583141	C₄	semidirect
168. C₂×D₄₃	#358	semidirect
184. #1380	D₄₃	semidirect
211. C₄₃	#13060	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₆×D₄₃	2
C₂×#14583141	2
C₂×#14582704	2
C₂×#14582702	2
#19134621	2
#19132686	2
C₄₃×#1380	2
#13060	43