Subgroups — #25013993

Order 11008.

484 conjugacy classes of subgroups.

The subgroup lattice has 400 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×#14583141	C₂	1	1	yes	maximal, characteristic
3	C₂×#14582716	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14582714	C₂	1	1	yes	maximal, characteristic, complemented
5	#19134630	C₂	1	1	yes	maximal, characteristic
6	#19132703	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1387	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	C₂×#13063265	C₂²	1	1	yes	characteristic
12	C₂×#13063264	—	2	1	no	—
13	M₃₂×D₄₃	—	2	2	no	—
14	M₃₂×D₄₃	—	2	2	no	—
15	C₂×C₁₆×D₄₃	—	2	1	no	—
16	#14583141	—	2	1	no	—
17	#14583141	—	2	1	no	—
18	#14583141	C₄	1	2	yes	—
19	#14583141	C₄	1	2	yes	—
20	C₂×#13063261	C₂²	1	1	yes	characteristic
21	C₂×#13063248	—	2	1	no	—
22	#14582862	—	2	1	no	—
23	#14582716	—	2	2	no	—
24	#14582714	—	2	2	no	—
25	#14582663	—	2	1	no	—
26	C₂×M₃₂×C₄₃	C₂²	1	1	yes	characteristic
27	C₂×#108×C₄₃	C₂²	1	1	yes	characteristic
28	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
29	C₂²×Dic₈₆	—	2	1	no	—
30	C₂²×C₄×D₄₃	D₄	1	1	yes	characteristic
31	C₂×Dic₁₇₂	—	2	1	no	—
32	C₂×Dic₁₇₂	—	2	1	no	—
33	C₂×Dic₁₇₂	—	2	1	no	—
34	C₂×Dic₁₇₂	—	2	1	no	—
35	C₂×Dic₁₇₂	—	2	1	no	—
36	C₂×Dic₁₇₂	C₂×C₄	1	2	yes	—
37	C₂×#10621877	—	2	1	no	—
38	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
39	C₂×C₈×D₄₃	—	2	1	no	—
40	C₂×C₈×D₄₃	—	2	1	no	—
41	C₂×C₈×D₄₃	—	2	1	no	—
42	C₂×C₈×D₄₃	—	2	1	no	—
43	C₂×C₈×D₄₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₈×D₄₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₈×D₄₃	D₄	1	1	yes	characteristic
46	C₂×C₈×D₄₃	D₄	1	1	yes	characteristic
47	#13063305	—	2	2	no	—
48	#13063305	—	2	2	no	—
49	#13063305	—	4	1	no	—
50	#13063305	—	4	1	no	—
51	C₂×C₄⋊Dic₄₃	—	4	1	no	—
52	C₂×(C₂×D₄₃)⋊C₄	—	4	1	no	—
53	#13063265	—	2	1	no	—
54	#13063265	D₄	1	2	yes	—
55	#13063264	—	2	4	no	—
56	C₂×C₄₃⋊C₁₆	—	2	1	no	—
57	C₁₆×D₄₃	—	2	4	no	—
58	#13063261	—	2	1	no	—
59	#13063261	—	2	1	no	—
60	#13063248	—	2	2	no	—
61	#13063248	—	2	2	no	—
62	(C₂×D₄₃)⋊C₈	—	4	1	no	—
63	(C₂×D₄₃)⋊C₈	—	4	1	no	—
64	C₂×M₁₆×C₄₃	—	2	1	no	—
65	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
67	M₃₂×C₄₃	—	2	1	no	—
68	M₃₂×C₄₃	D₄	1	2	yes	—
69	C₂×C₁₆×C₄₃	—	2	1	no	—
70	#108×C₄₃	—	2	1	no	—
71	#108×C₄₃	—	2	1	no	—
72	#107×C₄₃	—	2	1	no	—
73	#107×C₄₃	C₂×C₄	1	2	yes	—
74	C₂³×D₄₃	SD₁₆	1	1	yes	characteristic
75	C₂²×C₄×C₄₃	—	4	1	no	—
76	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×Dic₄₃	—	4	1	no	—
78	C₂²×Dic₄₃	SD₁₆	1	1	yes	characteristic
79	C₂×C₄×D₄₃	—	2	1	no	—
80	C₂×C₄×D₄₃	—	2	1	no	—
81	C₂×C₄×D₄₃	—	2	1	no	—
82	C₂×C₄×D₄₃	—	2	1	no	—
83	C₂×C₄×D₄₃	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×D₄₃	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×D₄₃	D₈	1	1	yes	characteristic
86	C₂×C₄×D₄₃	D₈	1	1	yes	characteristic
87	C₂×Dic₈₆	—	2	2	no	—
88	C₂×Dic₈₆	—	4	1	no	—
89	C₂×Dic₈₆	—	4	1	no	—
90	C₂×Dic₈₆	—	4	1	no	—
91	C₂×Dic₈₆	—	4	1	no	—
92	C₂×Dic₈₆	—	4	1	no	—
93	M₁₆×C₄₃	—	4	1	no	—
94	M₁₆×C₄₃	—	4	1	no	—
95	C₂×C₈×C₄₃	—	4	1	no	—
96	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
101	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
102	C₄⋊C₄×C₄₃	—	2	2	no	—
103	C₄⋊C₄×C₄₃	—	4	1	no	—
104	(C₂×D₄₃)⋊C₄	—	4	2	no	—
105	(C₂×D₄₃)⋊C₄	—	4	2	no	—
106	C₄⋊Dic₄₃	—	4	2	no	—
107	C₄⋊Dic₄₃	—	4	2	no	—
108	#10621877	—	4	1	no	—
109	#10621877	—	4	1	no	—
110	C₂×C₄₃⋊C₈	—	2	1	no	—
111	C₂×C₄₃⋊C₈	—	2	1	no	—
112	C₂×C₄₃⋊C₈	—	4	1	no	—
113	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
115	Dic₁₇₂	—	2	2	no	—
116	Dic₁₇₂	—	2	2	no	—
117	Dic₁₇₂	—	4	1	no	—
118	Dic₁₇₂	—	4	1	no	—
119	C₈×D₄₃	—	2	1	no	—
120	C₈×D₄₃	—	2	1	no	—
121	C₈×D₄₃	—	2	1	no	—
122	C₈×D₄₃	—	2	1	no	—
123	C₈×D₄₃	—	2	1	no	—
124	C₈×D₄₃	—	2	1	no	—
125	C₈×D₄₃	—	2	1	no	—
126	C₈×D₄₃	—	2	1	no	—
127	C₁₆×C₄₃	—	2	2	no	—
128	C₄₃⋊C₁₆	—	2	2	no	—
129	C₂³×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
130	C₂²×D₄₃	—	2	1	no	—
131	C₂²×D₄₃	—	2	1	no	—
132	C₂²×D₄₃	—	2	2	no	—
133	C₂²×D₄₃	—	4	1	no	—
134	C₂²×D₄₃	#102	1	1	yes	characteristic
135	C₂²×D₄₃	#102	1	1	yes	characteristic
136	C₂×C₄×C₄₃	—	4	2	no	—
137	C₂×C₄×C₄₃	—	8	2	no	—
138	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
140	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
141	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
143	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
144	C₂×Dic₄₃	—	2	1	no	—
145	C₂×Dic₄₃	—	2	1	no	—
146	C₂×Dic₄₃	—	4	2	no	—
147	C₂×Dic₄₃	—	8	1	no	—
148	C₂×Dic₄₃	—	8	1	no	—
149	C₂×Dic₄₃	#102	1	1	yes	characteristic
150	C₂×Dic₄₃	#102	1	1	yes	characteristic
151	C₄×D₄₃	—	2	1	no	—
152	C₄×D₄₃	—	2	1	no	—
153	C₄×D₄₃	—	2	1	no	—
154	C₄×D₄₃	—	2	1	no	—
155	C₄×D₄₃	—	2	1	no	—
156	C₄×D₄₃	—	2	1	no	—
157	C₄×D₄₃	—	2	1	no	—
158	C₄×D₄₃	—	2	1	no	—
159	Dic₈₆	—	4	2	no	—
160	Dic₈₆	—	4	2	no	—
161	Dic₈₆	—	8	1	no	—
162	Dic₈₆	—	8	1	no	—
163	C₈×C₄₃	—	8	1	no	—
164	C₈×C₄₃	C₄×D₄	1	1	yes	characteristic
165	C₈×C₄₃	C₄×D₄	1	1	yes	characteristic
166	C₈×C₄₃	#120	1	1	yes	characteristic
167	C₈×C₄₃	#123	1	1	yes	characteristic
168	C₄₃⋊C₈	—	2	1	no	—
169	C₄₃⋊C₈	—	2	1	no	—
170	C₄₃⋊C₈	—	8	1	no	—
171	#13066	—	43	1	no	maximal, Sylow
172	C₂²×C₄₃	—	2	1	no	—
173	C₂²×C₄₃	—	2	1	no	—
174	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
175	C₂²×C₄₃	#465	1	1	yes	characteristic
176	C₂²×C₄₃	#465	1	1	yes	characteristic
177	C₂×D₄₃	—	2	1	no	—
178	C₂×D₄₃	—	2	1	no	—
179	C₂×D₄₃	—	2	1	no	—
180	C₂×D₄₃	—	2	1	no	—
181	C₂×D₄₃	—	2	1	no	—
182	C₂×D₄₃	—	4	2	no	—
183	C₂×D₄₃	—	4	2	no	—
184	C₂×D₄₃	#362	1	2	yes	complemented
185	C₄×C₄₃	—	8	2	no	—
186	C₄×C₄₃	#386	1	1	yes	characteristic
187	C₄×C₄₃	#416	1	1	yes	characteristic
188	C₄×C₄₃	#466	1	1	yes	characteristic
189	C₄×C₄₃	#466	1	1	yes	characteristic
190	Dic₄₃	—	2	1	no	—
191	Dic₄₃	—	2	1	no	—
192	Dic₄₃	—	8	2	no	—
193	C₂²×M₃₂	—	43	1	no	—
194	C₂×#467	—	43	1	no	—
195	C₂×#362	—	43	1	no	—
196	C₂×#362	—	43	1	no	—
197	#1810	—	43	1	no	—
198	#1387	—	43	1	no	—
199	#1387	D₄₃	1	1	yes	characteristic, complemented
200	C₂×C₄₃	—	2	1	no	—
201	C₂×C₄₃	—	2	1	no	—
202	C₂×C₄₃	#1893	1	1	yes	characteristic
203	C₂×C₄₃	C₂×#362	1	1	yes	characteristic
204	C₂×C₄₃	#2148	1	1	yes	characteristic
205	D₄₃	—	2	2	no	—
206	D₄₃	—	4	1	no	—
207	C₂²×Q₁₆	—	43	1	no	—
208	C₂³×C₈	—	43	1	no	—
209	C₂×C₂²⋊Q₈	—	86	1	no	—
210	C₂×M₃₂	—	43	1	no	—
211	C₂×M₃₂	—	86	1	no	—
212	C₂×M₃₂	—	86	2	no	—
213	C₂×M₃₂	—	86	2	no	—
214	C₂×M₃₂	C₂×D₄₃	1	1	yes	characteristic
215	C₂²×C₁₆	—	86	1	no	—
216	#467	—	43	2	no	—
217	#467	—	43	2	no	—
218	#467	—	86	1	no	—
219	#467	—	86	1	no	—
220	C₂×#108	—	43	1	no	—
221	C₂×#108	C₂×D₄₃	1	1	yes	characteristic
222	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
223	C₂×#103	—	86	1	no	—
224	#407	—	86	1	no	—
225	#362	—	86	2	no	—
226	#362	—	86	2	no	—
227	#341	—	86	1	no	—
228	C₄₃	#13066	1	1	yes	characteristic, complemented, Sylow
229	C₂²×Q₈	—	86	1	no	—
230	C₂³×C₄	—	43	1	no	—
231	C₂×Q₁₆	—	43	2	no	—
232	C₂×Q₁₆	—	86	1	no	—
233	C₂×Q₁₆	—	86	1	no	—
234	C₂×Q₁₆	—	86	1	no	—
235	C₂×Q₁₆	—	86	1	no	—
236	C₂×Q₁₆	—	86	1	no	—
237	C₂×M₁₆	—	2	1	no	—
238	C₂×M₁₆	—	86	1	no	—
239	C₂²×C₈	—	43	1	no	—
240	C₂²×C₈	—	43	1	no	—
241	C₂²×C₈	—	43	1	no	—
242	C₂²×C₈	—	43	1	no	—
243	C₂²×C₈	—	43	1	no	—
244	C₂²×C₈	—	86	1	no	—
245	C₂²×C₈	—	86	1	no	—
246	C₂²×C₈	—	86	1	no	—
247	C₂²×C₈	—	86	1	no	—
248	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
249	C₂²⋊Q₈	—	86	2	no	—
250	C₂²⋊Q₈	—	86	2	no	—
251	C₂²⋊Q₈	—	172	1	no	—
252	C₂²⋊Q₈	—	172	1	no	—
253	C₂×C₄⋊C₄	—	2	1	no	—
254	C₂×C₄⋊C₄	—	172	1	no	—
255	C₂×C₂²⋊C₄	—	172	1	no	—
256	M₃₂	—	2	1	no	—
257	M₃₂	—	43	2	no	—
258	M₃₂	—	86	4	no	—
259	M₃₂	—	86	1	no	—
260	M₃₂	D₁₇₂	1	2	yes	—
261	C₂×C₁₆	—	2	1	no	—
262	C₂×C₁₆	—	86	4	no	—
263	C₂×C₁₆	—	86	1	no	—
264	#108	—	2	1	no	—
265	#108	—	2	1	no	—
266	#108	—	86	1	no	—
267	#108	—	86	1	no	—
268	#107	—	2	1	no	—
269	#107	C₄×D₄₃	1	2	yes	—
270	#103	—	86	2	no	—
271	#103	—	86	2	no	—
272	C₂²⋊C₈	—	172	1	no	—
273	C₂²⋊C₈	—	172	1	no	—
274	C₂⁴	—	43	1	no	—
275	C₂×Q₈	—	86	2	no	—
276	C₂×Q₈	—	172	1	no	—
277	C₂×Q₈	—	172	1	no	—
278	C₂×Q₈	—	172	1	no	—
279	C₂×Q₈	—	172	1	no	—
280	C₂×Q₈	—	172	1	no	—
281	C₂²×C₄	—	4	1	no	—
282	C₂²×C₄	—	43	1	no	—
283	C₂²×C₄	—	43	1	no	—
284	C₂²×C₄	—	43	1	no	—
285	C₂²×C₄	—	43	1	no	—
286	C₂²×C₄	—	43	1	no	—
287	C₂²×C₄	—	86	1	no	—
288	C₂²×C₄	—	86	1	no	—
289	C₂²×C₄	—	86	1	no	—
290	C₂²×C₄	—	86	1	no	—
291	C₂²×C₄	—	172	1	no	—
292	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
293	Q₁₆	—	86	2	no	—
294	Q₁₆	—	86	2	no	—
295	Q₁₆	—	172	1	no	—
296	Q₁₆	—	172	1	no	—
297	M₁₆	—	4	1	no	—
298	M₁₆	—	4	1	no	—
299	M₁₆	—	172	1	no	—
300	M₁₆	—	172	1	no	—
301	C₂×C₈	—	4	1	no	—
302	C₂×C₈	—	43	1	no	—
303	C₂×C₈	—	43	1	no	—
304	C₂×C₈	—	86	1	no	—
305	C₂×C₈	—	86	1	no	—
306	C₂×C₈	—	86	1	no	—
307	C₂×C₈	—	86	1	no	—
308	C₂×C₈	—	86	1	no	—
309	C₂×C₈	—	86	1	no	—
310	C₂×C₈	—	86	1	no	—
311	C₂×C₈	—	86	1	no	—
312	C₂×C₈	—	86	1	no	—
313	C₂×C₈	—	86	1	no	—
314	C₂×C₈	—	172	1	no	—
315	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
316	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
317	C₂×C₈	#10621898	1	1	yes	characteristic
318	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
319	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
320	C₂×C₈	#10621902	1	1	yes	characteristic
321	C₄⋊C₄	—	2	2	no	—
322	C₄⋊C₄	—	4	1	no	—
323	C₄⋊C₄	—	172	2	no	—
324	C₄⋊C₄	—	172	2	no	—
325	C₂²⋊C₄	—	172	2	no	—
326	C₂²⋊C₄	—	172	2	no	—
327	C₁₆	—	2	2	no	—
328	C₁₆	—	86	2	no	—
329	C₂³	—	43	1	no	—
330	C₂³	—	43	1	no	—
331	C₂³	—	86	2	no	—
332	C₂³	—	86	1	no	—
333	C₂³	—	86	1	no	—
334	C₂³	—	172	1	no	—
335	C₂³	SD₁₆×D₄₃	1	1	yes	characteristic
336	Q₈	—	344	1	no	—
337	Q₈	—	344	1	no	—
338	Q₈	—	172	2	no	—
339	Q₈	—	172	2	no	—
340	C₂×C₄	—	4	2	no	—
341	C₂×C₄	—	8	2	no	—
342	C₂×C₄	—	43	1	no	—
343	C₂×C₄	—	43	1	no	—
344	C₂×C₄	—	86	1	no	—
345	C₂×C₄	—	86	1	no	—
346	C₂×C₄	—	86	1	no	—
347	C₂×C₄	—	86	1	no	—
348	C₂×C₄	—	86	1	no	—
349	C₂×C₄	—	86	1	no	—
350	C₂×C₄	—	86	1	no	—
351	C₂×C₄	—	86	1	no	—
352	C₂×C₄	—	86	1	no	—
353	C₂×C₄	—	86	1	no	—
354	C₂×C₄	—	344	1	no	—
355	C₂×C₄	—	344	1	no	—
356	C₂×C₄	—	172	2	no	—
357	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
358	C₂×C₄	#13063298	1	1	yes	characteristic
359	C₂×C₄	#13063300	1	1	yes	characteristic
360	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
361	C₂×C₄	#13063346	1	1	yes	characteristic
362	C₂×C₄	#13063348	1	1	yes	characteristic
363	C₈	—	8	1	no	—
364	C₈	—	86	1	no	—
365	C₈	—	86	1	no	—
366	C₈	—	344	1	no	—
367	C₈	#13063287	1	1	yes	characteristic
368	C₈	C₄×D₁₇₂	1	1	yes	characteristic
369	C₈	#13063295	1	1	yes	characteristic
370	C₈	#13063310	1	1	yes	characteristic
371	C₂²	—	2	1	no	—
372	C₂²	—	2	1	no	—
373	C₂²	—	43	2	no	—
374	C₂²	—	86	1	no	—
375	C₂²	—	86	1	no	—
376	C₂²	—	86	1	no	—
377	C₂²	—	86	1	no	—
378	C₂²	—	86	1	no	—
379	C₂²	—	172	2	no	—
380	C₂²	—	172	2	no	—
381	C₂²	#102×D₄₃	1	1	yes	characteristic
382	C₂²	#14583129	1	1	yes	characteristic
383	C₂²	#14583131	1	1	yes	characteristic
384	C₄	—	8	2	no	—
385	C₄	—	86	1	no	—
386	C₄	—	86	1	no	—
387	C₄	—	344	2	no	—
388	C₄	#14582777	1	1	yes	characteristic
389	C₄	#14582894	1	1	yes	characteristic
390	C₄	#14583134	1	1	yes	characteristic
391	C₄	#14583136	1	1	yes	characteristic
392	C₂	—	2	1	no	—
393	C₂	—	2	1	no	—
394	C₂	—	86	2	no	—
395	C₂	—	172	1	no	—
396	C₂	#19135022	1	1	yes	characteristic
397	C₂	#362×D₄₃	1	1	yes	characteristic
398	C₂	#19136341	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#102×D₄₃
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄₃×#1387	C₂
socle	C₂²×C₄₃	C₂×#102
prime radical	C₂³×D₄₃	SD₁₆
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	#1387	D₄₃
hypocenter	C₄₃	#13066
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1387

Complements

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

N	H ≅ G/N	type
3. C₂×#14582716	C₂	semidirect
4. C₂×#14582714	C₂	semidirect
6. #19132703	C₂	semidirect
7. C₄₃×#1387	C₂	semidirect
184. C₂×D₄₃	#362	semidirect
199. #1387	D₄₃	semidirect
228. C₄₃	#13066	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×#14583141	2
C₂×#14582716	2
C₂×#14582714	2
#19134630	2
#19132703	2
C₄₃×#1387	2
#13066	43