Subgroups — #25079860

Order 11008.

570 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145938	C₂	1	2	yes	maximal
2	#19145106	C₂	1	1	yes	maximal, characteristic
3	#19143524	C₂	1	1	yes	maximal, characteristic
4	#19134100	C₂	1	2	yes	maximal, complemented
5	#19134099	C₂	1	2	yes	maximal, complemented
6	#19134000	C₂	1	1	yes	maximal, characteristic, complemented
7	#19133999	C₂	1	1	yes	maximal, characteristic, complemented
8	#19133986	C₂	1	1	yes	maximal, characteristic, complemented
9	#335×D₄₃	C₂	1	1	yes	maximal, characteristic
10	#19133284	C₂	1	2	yes	maximal
11	C₄₃×#1674	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583454	C₂²	1	2	yes	—
13	#14583453	C₂²	1	1	yes	characteristic
14	#14583304	C₂²	1	1	yes	characteristic
15	#14583300	—	2	1	no	—
16	#14583300	—	2	1	no	—
17	#14583295	C₂²	1	2	yes	—
18	#14583195	C₂²	1	2	yes	—
19	#14583194	C₂²	1	2	yes	—
20	#14583192	—	2	2	no	—
21	#14583139	—	2	2	no	—
22	#14583137	—	2	2	no	—
23	#14583107	—	2	2	no	—
24	#14583106	C₂²	1	2	yes	—
25	#14583030	C₂²	1	1	yes	characteristic
26	#14583028	—	2	1	no	—
27	#14583026	C₂²	1	1	yes	characteristic
28	C₈×Dic₈₆	—	2	1	no	—
29	C₈×Dic₈₆	C₂²	1	1	yes	characteristic
30	#14582936	—	2	1	no	—
31	#14582936	—	2	1	no	—
32	#14582924	—	2	1	no	—
33	#14582921	C₂²	1	1	yes	characteristic
34	C₄⋊C₈×D₄₃	C₂²	1	2	yes	—
35	#14582853	—	2	1	no	—
36	C₄×C₈×D₄₃	C₂²	1	1	yes	characteristic
37	#14582652	C₂²	1	1	yes	characteristic
38	#14582651	C₂²	1	2	yes	—
39	#14582647	C₂²	1	2	yes	—
40	#14582647	C₂²	1	2	yes	—
41	#14582645	C₂²	1	2	yes	—
42	C₄₃×#500	C₂²	1	1	yes	characteristic
43	C₄₃×#462	C₂²	1	2	yes	—
44	C₈×Q₈×C₄₃	C₂²	1	1	yes	characteristic
45	C₄₃×#335	C₂²	1	1	yes	characteristic
46	C₄₃×#332	C₂²	1	2	yes	—
47	C₂×#10621885	—	2	2	no	—
48	C₂×#10621872	—	2	1	no	—
49	C₂×C₈×D₄₃	—	2	2	no	—
50	C₂×C₈×D₄₃	D₄	1	1	yes	characteristic
51	C₂×C₈×D₄₃	D₄	1	1	yes	characteristic
52	#13063325	C₂³	1	2	yes	—
53	#13063324	D₄	1	1	yes	characteristic
54	#13063323	D₄	1	1	yes	characteristic
55	#13063316	—	2	2	no	—
56	#13063315	—	2	1	no	—
57	#13063305	—	2	2	no	—
58	#13063305	D₄	1	1	yes	characteristic
59	#13063303	—	2	1	no	—
60	#13063301	—	2	2	no	—
61	#13063301	D₄	1	1	yes	characteristic
62	Q₈×Dic₄₃	—	2	2	no	—
63	C₄×Dic₈₆	—	2	1	no	—
64	C₄×Dic₈₆	C₂³	1	1	yes	characteristic
65	#13063290	—	2	2	no	—
66	#13063283	—	2	2	no	—
67	#13063282	—	2	1	no	—
68	C₄²×D₄₃	C₂³	1	1	yes	characteristic
69	#13063259	—	2	1	no	—
70	#13063259	—	2	1	no	—
71	#13063259	—	2	2	no	—
72	#13063258	—	2	2	no	—
73	#13063256	—	2	2	no	—
74	#13063256	—	2	2	no	—
75	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
76	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
77	C₄⋊(C₄₃⋊C₈)	C₂³	1	1	yes	characteristic
78	Dic₄₃⋊C₈	—	2	1	no	—
79	Dic₄₃⋊C₈	—	2	2	no	—
80	Dic₄₃⋊C₈	—	2	1	no	—
81	Dic₄₃⋊C₈	—	2	1	no	—
82	#13063250	—	2	2	no	—
83	#13063249	—	2	2	no	—
84	#13063249	—	2	2	no	—
85	#13063236	—	2	1	no	—
86	#13063235	—	2	1	no	—
87	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
88	C₈×Dic₄₃	—	2	1	no	—
89	C₈×Dic₄₃	D₄	1	1	yes	characteristic
90	C₈×Dic₄₃	D₄	1	1	yes	characteristic
91	C₂×Q₁₆×C₄₃	—	2	2	no	—
92	C₄⋊Q₈×C₄₃	C₂³	1	2	yes	—
93	C₄×Q₈×C₄₃	C₂³	1	1	yes	characteristic
94	#107×C₄₃	—	2	1	no	—
95	#107×C₄₃	—	2	1	no	—
96	C₄⋊C₈×C₄₃	—	2	1	no	—
97	C₄⋊C₈×C₄₃	C₂³	1	1	yes	characteristic
98	C₄⋊C₈×C₄₃	C₂³	1	2	yes	—
99	#103×C₄₃	—	2	2	no	—
100	C₄×C₈×C₄₃	—	2	1	no	—
101	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
102	C₂×Q₈×C₄₃	—	2	2	no	—
103	C₂×Q₈×C₄₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×D₄₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×D₄₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×D₄₃	#41	1	1	yes	characteristic
107	C₂×Dic₈₆	—	2	1	no	—
108	C₂×Dic₈₆	—	2	2	no	—
109	C₂×Dic₈₆	C₂×D₄	1	1	yes	characteristic
110	Q₁₆×C₄₃	—	8	2	no	—
111	C₂×C₈×C₄₃	—	2	1	no	—
112	C₂×C₈×C₄₃	—	2	2	no	—
113	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
115	C₄⋊C₄×C₄₃	—	2	1	no	—
116	C₄⋊C₄×C₄₃	—	2	2	no	—
117	C₄⋊C₄×C₄₃	—	2	2	no	—
118	C₄⋊C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₄₃	—	2	1	no	—
120	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
121	#10621885	—	8	2	no	—
122	(C₂×D₄₃)⋊C₄	—	2	1	no	—
123	(C₂×D₄₃)⋊C₄	—	2	1	no	—
124	(C₂×D₄₃)⋊C₄	—	4	2	no	—
125	C₄⋊Dic₄₃	—	2	1	no	—
126	C₄⋊Dic₄₃	—	2	2	no	—
127	C₄⋊Dic₄₃	—	2	2	no	—
128	C₄⋊Dic₄₃	—	4	2	no	—
129	C₄⋊Dic₄₃	C₂×D₄	1	1	yes	characteristic
130	Dic₄₃⋊C₄	—	2	1	no	—
131	Dic₄₃⋊C₄	—	2	1	no	—
132	Dic₄₃⋊C₄	—	2	1	no	—
133	Dic₄₃⋊C₄	—	2	1	no	—
134	Dic₄₃⋊C₄	—	2	1	no	—
135	Dic₄₃⋊C₄	—	4	2	no	—
136	C₄×Dic₄₃	—	2	1	no	—
137	C₄×Dic₄₃	—	4	2	no	—
138	C₄×Dic₄₃	C₂×D₄	1	1	yes	characteristic
139	C₄×Dic₄₃	C₂×D₄	1	1	yes	characteristic
140	C₄×Dic₄₃	#41	1	1	yes	characteristic
141	C₂×C₄₃⋊C₈	—	2	1	no	—
142	C₂×C₄₃⋊C₈	—	2	2	no	—
143	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
144	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
145	#10621872	—	4	2	no	—
146	C₈×D₄₃	—	2	2	no	—
147	C₈×D₄₃	—	8	2	no	—
148	C₈×D₄₃	Q₁₆	1	4	yes	complemented
149	C₂²×D₄₃	#121	1	1	yes	characteristic
150	Q₈×C₄₃	—	2	1	no	—
151	Q₈×C₄₃	—	2	1	no	—
152	Q₈×C₄₃	—	4	2	no	—
153	Q₈×C₄₃	—	4	2	no	—
154	C₂×C₄×C₄₃	—	2	1	no	—
155	C₂×C₄×C₄₃	—	2	1	no	—
156	C₂×C₄×C₄₃	—	4	2	no	—
157	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
160	C₂×Dic₄₃	—	2	1	no	—
161	C₂×Dic₄₃	—	2	1	no	—
162	C₂×Dic₄₃	—	4	2	no	—
163	C₂×Dic₄₃	#121	1	1	yes	characteristic
164	C₂×Dic₄₃	#121	1	1	yes	characteristic
165	C₂×Dic₄₃	#121	1	1	yes	characteristic
166	C₄×D₄₃	—	2	1	no	—
167	C₄×D₄₃	—	2	1	no	—
168	C₄×D₄₃	—	4	1	no	—
169	C₄×D₄₃	C₂×Q₁₆	1	2	yes	—
170	C₄×D₄₃	#137	1	2	yes	—
171	Dic₈₆	—	2	1	no	—
172	Dic₈₆	—	2	1	no	—
173	Dic₈₆	—	4	2	no	—
174	Dic₈₆	—	8	2	no	—
175	C₈×C₄₃	—	2	1	no	—
176	C₈×C₄₃	—	4	1	no	—
177	C₈×C₄₃	—	4	2	no	—
178	C₈×C₄₃	C₂×Q₁₆	1	2	yes	—
179	C₄₃⋊C₈	—	2	1	no	—
180	C₄₃⋊C₈	—	4	1	no	—
181	C₄₃⋊C₈	—	4	2	no	—
182	C₄₃⋊C₈	C₂×Q₁₆	1	2	yes	—
183	#19211	—	43	1	no	maximal, Sylow
184	C₂²×C₄₃	C₂×#121	1	1	yes	characteristic
185	C₂×D₄₃	—	2	1	no	—
186	C₂×D₄₃	—	2	1	no	—
187	C₂×D₄₃	#462	1	2	yes	—
188	C₄×C₄₃	—	2	1	no	—
189	C₄×C₄₃	—	4	1	no	—
190	C₄×C₄₃	—	4	1	no	—
191	C₄×C₄₃	—	8	2	no	—
192	C₄×C₄₃	C₂²×Q₁₆	1	1	yes	characteristic
193	C₄×C₄₃	C₂×#137	1	1	yes	characteristic
194	C₄×C₄₃	#576	1	1	yes	characteristic
195	C₄×C₄₃	#578	1	1	yes	characteristic
196	Dic₄₃	—	2	1	no	—
197	Dic₄₃	—	2	1	no	—
198	Dic₄₃	—	4	1	no	—
199	Dic₄₃	—	4	1	no	—
200	Dic₄₃	—	8	2	no	—
201	Dic₄₃	#462	1	2	yes	—
202	#3249	—	43	2	no	—
203	#3160	—	43	1	no	—
204	#2960	—	43	1	no	—
205	#1692	—	43	2	no	—
206	#1692	—	43	2	no	—
207	#1676	—	43	1	no	—
208	#1676	—	43	1	no	—
209	#1674	—	43	1	no	—
210	#1674	D₄₃	1	1	yes	characteristic, complemented
211	C₂×#335	—	43	1	no	—
212	#1550	—	43	2	no	—
213	C₂×C₄₃	C₂×#462	1	1	yes	characteristic
214	C₂×C₄₃	#3061	1	1	yes	characteristic
215	C₂×C₄₃	#3067	1	1	yes	characteristic
216	D₄₃	—	2	2	no	—
217	#533	—	43	1	no	—
218	#533	—	43	2	no	—
219	#500	—	43	1	no	—
220	#500	C₂×D₄₃	1	1	yes	characteristic
221	#499	—	86	1	no	—
222	#499	—	86	1	no	—
223	#498	—	43	2	no	—
224	#477	—	43	2	no	—
225	#477	—	43	2	no	—
226	#477	—	86	2	no	—
227	#467	—	86	2	no	—
228	#467	—	86	2	no	—
229	#462	—	43	2	no	—
230	#462	—	86	2	no	—
231	#462	C₂×D₄₃	1	2	yes	—
232	#446	—	43	1	no	—
233	#446	—	43	1	no	—
234	#446	—	86	1	no	—
235	C₈×Q₈	—	43	1	no	—
236	C₈×Q₈	—	86	1	no	—
237	C₈×Q₈	C₂×D₄₃	1	1	yes	characteristic
238	#427	—	86	1	no	—
239	#427	—	86	1	no	—
240	#424	—	86	1	no	—
241	#423	—	43	1	no	—
242	C₂×C₄⋊C₈	—	43	2	no	—
243	#405	—	86	1	no	—
244	C₂×C₄×C₈	—	43	1	no	—
245	#335	—	43	2	no	—
246	#335	—	43	1	no	—
247	#335	C₂×D₄₃	1	1	yes	characteristic
248	#333	—	43	2	no	—
249	#333	—	43	2	no	—
250	#332	—	43	2	no	—
251	#332	C₂×D₄₃	1	2	yes	—
252	C₄₃	#19211	1	1	yes	characteristic, complemented, Sylow
253	C₂×Q₁₆	—	2	2	no	—
254	C₂×Q₁₆	—	86	2	no	—
255	C₂×M₁₆	—	86	1	no	—
256	C₂²×C₈	—	43	1	no	—
257	C₂²×C₈	—	43	1	no	—
258	C₂²×C₈	—	86	2	no	—
259	C₄⋊Q₈	—	43	1	no	—
260	C₄⋊Q₈	—	43	1	no	—
261	C₄⋊Q₈	—	43	2	no	—
262	C₄⋊Q₈	C₂²×D₄₃	1	2	yes	—
263	#125	—	86	1	no	—
264	#125	—	86	2	no	—
265	C₂²⋊Q₈	—	43	1	no	—
266	C₂²⋊Q₈	—	43	1	no	—
267	C₂²⋊Q₈	—	86	1	no	—
268	C₂²⋊Q₈	—	86	2	no	—
269	C₂²⋊Q₈	—	86	2	no	—
270	C₄×Q₈	—	43	1	no	—
271	C₄×Q₈	—	86	1	no	—
272	C₄×Q₈	—	86	2	no	—
273	C₄×Q₈	—	86	2	no	—
274	C₄×Q₈	C₂²×D₄₃	1	1	yes	characteristic
275	#117	—	86	1	no	—
276	#117	—	86	2	no	—
277	C₂×C₄²	—	43	1	no	—
278	#107	—	2	1	no	—
279	#107	—	2	1	no	—
280	#107	—	86	2	no	—
281	#107	—	86	1	no	—
282	#107	—	86	1	no	—
283	#107	—	86	2	no	—
284	#106	—	86	2	no	—
285	#106	—	86	2	no	—
286	C₄⋊C₈	—	2	1	no	—
287	C₄⋊C₈	—	43	1	no	—
288	C₄⋊C₈	—	43	2	no	—
289	C₄⋊C₈	—	86	1	no	—
290	C₄⋊C₈	—	86	2	no	—
291	C₄⋊C₈	—	86	1	no	—
292	C₄⋊C₈	—	86	1	no	—
293	C₄⋊C₈	—	86	1	no	—
294	C₄⋊C₈	C₂²×D₄₃	1	1	yes	characteristic
295	C₄⋊C₈	C₂²×D₄₃	1	2	yes	—
296	#103	—	2	2	no	—
297	#103	—	86	2	no	—
298	#103	—	86	2	no	—
299	#103	—	86	2	no	—
300	#97	—	86	1	no	—
301	#97	—	86	1	no	—
302	C₄×C₈	—	2	1	no	—
303	C₄×C₈	—	43	1	no	—
304	C₄×C₈	—	43	1	no	—
305	C₄×C₈	—	43	1	no	—
306	C₄×C₈	—	86	1	no	—
307	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
308	C₂×Q₈	—	2	2	no	—
309	C₂×Q₈	—	43	1	no	—
310	C₂×Q₈	—	86	1	no	—
311	C₂×Q₈	—	86	2	no	—
312	C₂×Q₈	D₄×D₄₃	1	1	yes	characteristic
313	C₂²×C₄	—	43	1	no	—
314	C₂²×C₄	—	43	1	no	—
315	C₂²×C₄	—	43	1	no	—
316	Q₁₆	—	8	2	no	—
317	Q₁₆	—	344	2	no	—
318	M₁₆	—	172	2	no	—
319	C₂×C₈	—	2	1	no	—
320	C₂×C₈	—	2	2	no	—
321	C₂×C₈	—	43	1	no	—
322	C₂×C₈	—	43	4	no	—
323	C₂×C₈	—	43	1	no	—
324	C₂×C₈	—	86	2	no	—
325	C₂×C₈	—	86	1	no	—
326	C₂×C₈	—	86	2	no	—
327	C₂×C₈	—	344	2	no	—
328	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
329	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
330	C₄⋊C₄	—	2	1	no	—
331	C₄⋊C₄	—	2	2	no	—
332	C₄⋊C₄	—	2	2	no	—
333	C₄⋊C₄	—	43	1	no	—
334	C₄⋊C₄	—	86	1	no	—
335	C₄⋊C₄	—	86	1	no	—
336	C₄⋊C₄	—	86	1	no	—
337	C₄⋊C₄	—	86	1	no	—
338	C₄⋊C₄	—	86	1	no	—
339	C₄⋊C₄	—	86	1	no	—
340	C₄⋊C₄	—	86	2	no	—
341	C₄⋊C₄	—	86	2	no	—
342	C₄⋊C₄	—	172	2	no	—
343	C₄⋊C₄	—	172	2	no	—
344	C₄⋊C₄	D₄×D₄₃	1	1	yes	characteristic
345	C₂²⋊C₄	—	86	1	no	—
346	C₂²⋊C₄	—	86	1	no	—
347	C₂²⋊C₄	—	172	2	no	—
348	C₄²	—	2	1	no	—
349	C₄²	—	43	1	no	—
350	C₄²	—	43	1	no	—
351	C₄²	—	43	1	no	—
352	C₄²	—	86	1	no	—
353	C₄²	—	172	2	no	—
354	C₄²	C₂³×D₄₃	1	1	yes	characteristic
355	C₂³	—	43	1	no	—
356	Q₈	—	2	1	no	—
357	Q₈	—	2	1	no	—
358	Q₈	—	4	2	no	—
359	Q₈	—	4	2	no	—
360	Q₈	—	86	1	no	—
361	Q₈	—	86	1	no	—
362	Q₈	—	344	2	no	—
363	Q₈	—	172	2	no	—
364	C₂×C₄	—	2	1	no	—
365	C₂×C₄	—	2	1	no	—
366	C₂×C₄	—	4	2	no	—
367	C₂×C₄	—	43	1	no	—
368	C₂×C₄	—	43	2	no	—
369	C₂×C₄	—	43	2	no	—
370	C₂×C₄	—	43	1	no	—
371	C₂×C₄	—	43	1	no	—
372	C₂×C₄	—	86	1	no	—
373	C₂×C₄	—	86	1	no	—
374	C₂×C₄	—	86	1	no	—
375	C₂×C₄	—	86	1	no	—
376	C₂×C₄	—	172	1	no	—
377	C₂×C₄	—	172	2	no	—
378	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
379	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
380	C₂×C₄	#41×D₄₃	1	1	yes	characteristic
381	C₈	—	2	1	no	—
382	C₈	—	4	1	no	—
383	C₈	—	4	2	no	—
384	C₈	—	43	2	no	—
385	C₈	—	86	1	no	—
386	C₈	—	172	1	no	—
387	C₈	—	172	2	no	—
388	C₈	Q₁₆×D₄₃	1	2	yes	—
389	C₂²	—	43	2	no	—
390	C₂²	—	86	1	no	—
391	C₂²	—	86	1	no	—
392	C₂²	#121×D₄₃	1	1	yes	characteristic
393	C₄	—	2	1	no	—
394	C₄	—	4	1	no	—
395	C₄	—	4	1	no	—
396	C₄	—	8	2	no	—
397	C₄	—	43	2	no	—
398	C₄	—	86	1	no	—
399	C₄	—	86	1	no	—
400	C₄	—	344	2	no	—
401	C₄	—	172	1	no	—
402	C₄	—	172	1	no	—
403	C₄	C₂×Q₁₆×D₄₃	1	1	yes	characteristic
404	C₄	#137×D₄₃	1	1	yes	characteristic
405	C₄	#14583700	1	1	yes	characteristic
406	C₄	#14583713	1	1	yes	characteristic
407	C₂	—	86	2	no	—
408	C₂	#462×D₄₃	1	1	yes	characteristic
409	C₂	#19144260	1	1	yes	characteristic
410	C₂	#19144317	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#121×D₄₃
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting	C₄₃×#1674	C₂
socle	C₂²×C₄₃	C₂×#121
prime radical	C₂²×D₄₃	#121
hypercenter	#1674	D₄₃
hypocenter	C₄₃	#19211
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1674

Complements

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

N	H ≅ G/N	type
4. #19134100	C₂	semidirect
5. #19134099	C₂	semidirect
6. #19134000	C₂	semidirect
7. #19133999	C₂	semidirect
8. #19133986	C₂	semidirect
11. C₄₃×#1674	C₂	semidirect
148. C₈×D₄₃	Q₁₆	semidirect
210. #1674	D₄₃	semidirect
252. C₄₃	#19211	semidirect

Maximal subgroups

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

subgroup	index
#19145938	2
#19145106	2
#19143524	2
#19134100	2
#19134099	2
#19134000	2
#19133999	2
#19133986	2
#335×D₄₃	2
#19133284	2
C₄₃×#1674	2
#19211	43