Subgroups — #25080057

Order 11008.

584 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145928	C₂	1	2	yes	maximal
2	#19145182	C₂	1	1	yes	maximal, characteristic, complemented
3	#19143573	C₂	1	1	yes	maximal, characteristic
4	#19134110	C₂	1	1	yes	maximal, characteristic, complemented
5	#19134090	C₂	1	1	yes	maximal, characteristic, complemented
6	#19134084	C₂	1	1	yes	maximal, characteristic, complemented
7	#19134028	C₂	1	2	yes	maximal, complemented
8	#19134020	C₂	1	2	yes	maximal, complemented
9	#334×D₄₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#19133294	C₂	1	2	yes	maximal, complemented
11	C₄₃×#1693	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583454	C₂²	1	2	yes	—
13	#14583431	C₂²	1	1	yes	characteristic
14	#14583279	C₂²	1	1	yes	characteristic
15	#14583251	C₂²	1	2	yes	—
16	#14583204	—	2	2	no	—
17	#14583185	C₂²	1	2	yes	—
18	#14583179	C₂²	1	2	yes	—
19	#14583141	—	2	1	no	—
20	#14583141	—	2	1	no	—
21	#14583139	—	2	2	no	—
22	#14583125	—	2	2	no	—
23	#14583119	—	2	2	no	—
24	#14583106	C₂²	1	2	yes	—
25	#14583029	C₂²	1	1	yes	characteristic
26	#14583023	C₂²	1	1	yes	characteristic
27	#14582993	—	2	1	no	—
28	#14582993	—	2	1	no	—
29	#14582968	—	2	1	no	—
30	D₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
31	(C₂²⋊D₄₃)⋊C₈	—	2	1	no	—
32	#14582957	—	2	1	no	—
33	#14582955	C₂²	1	1	yes	characteristic
34	C₄⋊C₈×D₄₃	C₂²	1	2	yes	—
35	#14582857	—	2	1	no	—
36	C₄×C₈×D₄₃	C₂²	1	1	yes	characteristic
37	#14582650	C₂²	1	1	yes	characteristic
38	#14582649	C₂²	1	2	yes	—
39	#14582646	C₂²	1	2	yes	—
40	#14582645	C₂²	1	2	yes	—
41	#14582644	C₂²	1	2	yes	—
42	C₄₃×#500	C₂²	1	1	yes	characteristic
43	C₄₃×#461	C₂²	1	2	yes	—
44	C₄₃×#436	C₂²	1	1	yes	characteristic
45	C₄₃×#334	C₂²	1	1	yes	characteristic
46	C₄₃×#332	C₂²	1	2	yes	—
47	C₂×Dic₁₇₂	—	2	1	no	—
48	C₂×Dic₁₇₂	—	2	1	no	—
49	C₂×#10621885	—	2	2	no	—
50	C₂²×C₄₃⋊C₈	—	2	1	no	—
51	C₂×C₈×D₄₃	—	2	2	no	—
52	C₂×C₈×D₄₃	D₄	1	1	yes	characteristic
53	C₂×C₈×D₄₃	D₄	1	1	yes	characteristic
54	#13063325	C₂³	1	2	yes	—
55	#13063318	—	2	1	no	—
56	#13063316	—	2	2	no	—
57	#13063316	D₄	1	1	yes	characteristic
58	#13063312	D₄	1	1	yes	characteristic
59	#13063307	—	2	1	no	—
60	#13063305	—	2	2	no	—
61	#13063305	D₄	1	1	yes	characteristic
62	#13063301	—	2	2	no	—
63	#13063296	D₄	1	1	yes	characteristic
64	Q₈×Dic₄₃	—	2	2	no	—
65	C₄×Dic₈₆	C₂³	1	1	yes	characteristic
66	#13063290	—	2	2	no	—
67	C₄×C₂²⋊D₄₃	—	2	1	no	—
68	#13063284	—	2	1	no	—
69	#13063283	—	2	2	no	—
70	C₄²×D₄₃	C₂³	1	1	yes	characteristic
71	#13063258	—	2	2	no	—
72	#13063257	—	2	2	no	—
73	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
74	C₄⋊(C₄₃⋊C₈)	C₂³	1	1	yes	characteristic
75	Dic₄₃⋊C₈	—	2	1	no	—
76	Dic₄₃⋊C₈	—	2	2	no	—
77	Dic₄₃⋊C₈	—	2	1	no	—
78	#13063250	—	2	2	no	—
79	#13063249	—	2	2	no	—
80	#13063248	—	2	2	no	—
81	#13063248	—	2	2	no	—
82	#13063246	—	2	2	no	—
83	C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
84	(C₂×D₄₃)⋊C₈	—	2	1	no	—
85	#13063235	—	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×SD₁₆×C₄₃	—	2	2	no	—
92	C₂×M₁₆×C₄₃	—	2	1	no	—
93	C₄⋊Q₈×C₄₃	C₂³	1	2	yes	—
94	C₄×D₄×C₄₃	C₂³	1	1	yes	characteristic
95	#107×C₄₃	—	2	1	no	—
96	#107×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	C₂²⋊C₈×C₄₃	—	2	1	no	—
101	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
102	C₂×Q₈×C₄₃	—	2	2	no	—
103	C₂×D₄×C₄₃	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₄₃	—	2	1	no	—
105	C₂×C₂²⋊D₄₃	—	2	1	no	—
106	C₂×C₄×D₄₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×D₄₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×D₄₃	#41	1	1	yes	characteristic
109	C₂×Dic₈₆	—	2	2	no	—
110	C₂×Dic₈₆	C₂×D₄	1	1	yes	characteristic
111	SD₁₆×C₄₃	—	8	2	no	—
112	M₁₆×C₄₃	—	4	2	no	—
113	C₂×C₈×C₄₃	—	2	1	no	—
114	C₂×C₈×C₄₃	—	2	2	no	—
115	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
117	C₄⋊C₄×C₄₃	—	2	2	no	—
118	C₄⋊C₄×C₄₃	—	2	2	no	—
119	C₄⋊C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
120	C₂²⋊C₄×C₄₃	—	2	1	no	—
121	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
122	C₂²⋊Dic₄₃	—	2	1	no	—
123	C₂²⋊Dic₄₃	—	2	1	no	—
124	C₂²⋊Dic₄₃	—	2	1	no	—
125	#10621885	—	8	2	no	—
126	(C₂×D₄₃)⋊C₄	—	2	1	no	—
127	(C₂×D₄₃)⋊C₄	—	4	2	no	—
128	C₄⋊Dic₄₃	—	2	1	no	—
129	C₄⋊Dic₄₃	—	2	1	no	—
130	C₄⋊Dic₄₃	—	2	2	no	—
131	C₄⋊Dic₄₃	—	2	2	no	—
132	C₄⋊Dic₄₃	—	4	2	no	—
133	C₄⋊Dic₄₃	C₂×D₄	1	1	yes	characteristic
134	Dic₄₃⋊C₄	—	2	1	no	—
135	Dic₄₃⋊C₄	—	2	1	no	—
136	Dic₄₃⋊C₄	—	4	2	no	—
137	C₄×Dic₄₃	—	2	1	no	—
138	C₄×Dic₄₃	—	4	2	no	—
139	C₄×Dic₄₃	C₂×D₄	1	1	yes	characteristic
140	C₄×Dic₄₃	C₂×D₄	1	1	yes	characteristic
141	C₄×Dic₄₃	#41	1	1	yes	characteristic
142	C₂×C₄₃⋊C₈	—	2	1	no	—
143	C₂×C₄₃⋊C₈	—	2	2	no	—
144	C₂×C₄₃⋊C₈	—	4	2	no	—
145	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
146	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
147	Dic₁₇₂	—	4	2	no	—
148	Dic₁₇₂	—	4	2	no	—
149	C₈×D₄₃	—	2	2	no	—
150	C₈×D₄₃	—	4	1	no	—
151	C₈×D₄₃	—	8	2	no	—
152	C₂³×C₄₃	—	2	1	no	—
153	C₂²×D₄₃	#121	1	1	yes	characteristic
154	Q₈×C₄₃	—	4	2	no	—
155	Q₈×C₄₃	—	4	2	no	—
156	D₄×C₄₃	—	2	2	no	—
157	C₂×C₄×C₄₃	—	2	1	no	—
158	C₂×C₄×C₄₃	—	4	1	no	—
159	C₂×C₄×C₄₃	—	4	1	no	—
160	C₂×C₄×C₄₃	—	4	2	no	—
161	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
163	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
164	C₂²⋊D₄₃	—	4	2	no	—
165	C₂×Dic₄₃	—	2	1	no	—
166	C₂×Dic₄₃	—	2	1	no	—
167	C₂×Dic₄₃	—	4	2	no	—
168	C₂×Dic₄₃	#121	1	1	yes	characteristic
169	C₂×Dic₄₃	#121	1	1	yes	characteristic
170	C₂×Dic₄₃	#121	1	1	yes	characteristic
171	C₄×D₄₃	—	2	2	no	—
172	C₄×D₄₃	—	2	1	no	—
173	C₄×D₄₃	—	2	1	no	—
174	C₄×D₄₃	—	2	2	no	—
175	Dic₈₆	—	2	2	no	—
176	Dic₈₆	—	4	2	no	—
177	Dic₈₆	—	4	2	no	—
178	C₈×C₄₃	—	2	1	no	—
179	C₈×C₄₃	—	4	1	no	—
180	C₈×C₄₃	—	4	2	no	—
181	C₈×C₄₃	#135	1	2	yes	—
182	C₄₃⋊C₈	—	2	1	no	—
183	C₄₃⋊C₈	—	2	1	no	—
184	C₄₃⋊C₈	—	4	1	no	—
185	C₄₃⋊C₈	—	4	2	no	—
186	#19228	—	43	1	no	maximal, Sylow
187	C₂²×C₄₃	—	4	1	no	—
188	C₂²×C₄₃	—	4	1	no	—
189	C₂²×C₄₃	—	4	1	no	—
190	C₂²×C₄₃	C₂×#121	1	1	yes	characteristic
191	C₂×D₄₃	—	2	1	no	—
192	C₂×D₄₃	—	2	1	no	—
193	C₂×D₄₃	#473	1	2	yes	—
194	C₄×C₄₃	—	2	1	no	—
195	C₄×C₄₃	—	4	1	no	—
196	C₄×C₄₃	—	8	2	no	—
197	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
198	C₄×C₄₃	#575	1	1	yes	characteristic
199	C₄×C₄₃	#577	1	2	yes	—
200	Dic₄₃	—	2	1	no	—
201	Dic₄₃	—	2	1	no	—
202	Dic₄₃	—	2	1	no	—
203	Dic₄₃	—	4	1	no	—
204	Dic₄₃	—	4	1	no	—
205	Dic₄₃	—	8	2	no	—
206	#3248	—	43	2	no	—
207	#3170	—	43	1	no	—
208	#2966	—	43	1	no	—
209	#1694	—	43	1	no	—
210	#1693	D₄₃	1	1	yes	characteristic, complemented
211	#1690	—	43	1	no	—
212	#1689	—	43	1	no	—
213	#1680	—	43	2	no	—
214	#1679	—	43	2	no	—
215	C₂×#334	—	43	1	no	—
216	#1552	—	43	2	no	—
217	C₂×C₄₃	—	8	1	no	—
218	C₂×C₄₃	#3035	1	1	yes	characteristic
219	C₂×C₄₃	#3060	1	1	yes	characteristic
220	C₂×C₄₃	C₂×#473	1	1	yes	characteristic
221	D₄₃	—	2	2	no	—
222	#533	—	43	2	no	—
223	#529	—	43	1	no	—
224	#500	C₂×D₄₃	1	1	yes	characteristic
225	#494	—	43	1	no	—
226	#487	—	43	2	no	—
227	#479	—	86	2	no	—
228	#475	—	43	2	no	—
229	#474	—	43	2	no	—
230	#467	—	86	2	no	—
231	#467	—	86	1	no	—
232	#467	—	86	1	no	—
233	#465	—	86	2	no	—
234	#464	—	86	2	no	—
235	#462	—	43	2	no	—
236	#461	C₂×D₄₃	1	2	yes	—
237	#446	—	43	1	no	—
238	C₈×Q₈	—	43	1	no	—
239	C₄×Q₁₆	—	86	1	no	—
240	C₄×Q₁₆	—	86	1	no	—
241	#436	C₂×D₄₃	1	1	yes	characteristic
242	#435	—	86	1	no	—
243	C₈×D₄	—	43	1	no	—
244	C₈×D₄	—	86	1	no	—
245	#433	—	86	1	no	—
246	#432	—	43	1	no	—
247	C₂×C₄⋊C₈	—	43	2	no	—
248	#405	—	86	1	no	—
249	C₂×C₄×C₈	—	43	1	no	—
250	#334	—	43	2	no	—
251	#334	—	43	1	no	—
252	#334	C₂×D₄₃	1	1	yes	characteristic
253	#332	—	43	2	no	—
254	#332	—	43	2	no	—
255	#332	—	43	2	no	—
256	#332	C₂×D₄₃	1	2	yes	—
257	C₄₃	#19228	1	1	yes	characteristic, complemented, Sylow
258	C₂×Q₁₆	—	86	2	no	—
259	C₂×Q₁₆	—	86	1	no	—
260	C₂×Q₁₆	—	86	1	no	—
261	C₂×SD₁₆	—	2	2	no	—
262	C₂×M₁₆	—	2	1	no	—
263	C₂²×C₈	—	43	1	no	—
264	C₂²×C₈	—	43	1	no	—
265	C₂²×C₈	—	86	2	no	—
266	C₂²×C₈	—	86	1	no	—
267	C₄⋊Q₈	—	43	2	no	—
268	C₄⋊Q₈	C₂²×D₄₃	1	2	yes	—
269	#126	—	86	1	no	—
270	#125	—	43	1	no	—
271	#125	—	86	2	no	—
272	#124	—	43	1	no	—
273	#123	—	86	1	no	—
274	C₂²⋊Q₈	—	43	1	no	—
275	C₂²⋊Q₈	—	86	2	no	—
276	C₂²⋊Q₈	—	86	2	no	—
277	#121	—	43	1	no	—
278	C₄×Q₈	—	43	1	no	—
279	C₄×Q₈	—	86	2	no	—
280	C₄×Q₈	—	86	2	no	—
281	C₄×D₄	—	86	1	no	—
282	C₄×D₄	C₂²×D₄₃	1	1	yes	characteristic
283	#117	—	86	1	no	—
284	#117	—	86	2	no	—
285	C₂×C₄²	—	43	1	no	—
286	#107	—	2	1	no	—
287	#107	—	2	1	no	—
288	#107	—	86	2	no	—
289	#106	—	86	2	no	—
290	C₄⋊C₈	—	43	1	no	—
291	C₄⋊C₈	—	43	2	no	—
292	C₄⋊C₈	—	86	2	no	—
293	C₄⋊C₈	—	86	1	no	—
294	C₄⋊C₈	—	86	1	no	—
295	C₄⋊C₈	C₂²×D₄₃	1	1	yes	characteristic
296	C₄⋊C₈	C₂²×D₄₃	1	2	yes	—
297	#103	—	2	2	no	—
298	#103	—	86	2	no	—
299	#103	—	86	2	no	—
300	#103	—	86	2	no	—
301	#103	—	86	2	no	—
302	#102	—	86	2	no	—
303	C₂²⋊C₈	—	2	1	no	—
304	C₂²⋊C₈	—	86	1	no	—
305	C₂²⋊C₈	—	86	1	no	—
306	#97	—	86	1	no	—
307	#97	—	86	1	no	—
308	C₄×C₈	—	43	1	no	—
309	C₄×C₈	—	43	1	no	—
310	C₄×C₈	—	43	1	no	—
311	C₄×C₈	—	86	1	no	—
312	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
313	C₂×Q₈	—	2	2	no	—
314	C₂×Q₈	—	43	1	no	—
315	C₂×Q₈	—	86	2	no	—
316	C₂×D₄	—	86	1	no	—
317	C₂×D₄	D₄×D₄₃	1	1	yes	characteristic
318	C₂²×C₄	—	2	1	no	—
319	C₂²×C₄	—	43	1	no	—
320	C₂²×C₄	—	43	1	no	—
321	C₂²×C₄	—	43	1	no	—
322	Q₁₆	—	344	2	no	—
323	Q₁₆	—	172	2	no	—
324	Q₁₆	—	172	2	no	—
325	SD₁₆	—	8	2	no	—
326	M₁₆	—	4	2	no	—
327	C₂×C₈	—	2	1	no	—
328	C₂×C₈	—	2	2	no	—
329	C₂×C₈	—	43	1	no	—
330	C₂×C₈	—	43	1	no	—
331	C₂×C₈	—	86	2	no	—
332	C₂×C₈	—	86	1	no	—
333	C₂×C₈	—	86	2	no	—
334	C₂×C₈	—	344	2	no	—
335	C₂×C₈	—	172	1	no	—
336	C₂×C₈	—	172	2	no	—
337	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
338	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
339	C₄⋊C₄	—	2	2	no	—
340	C₄⋊C₄	—	2	2	no	—
341	C₄⋊C₄	—	43	1	no	—
342	C₄⋊C₄	—	86	1	no	—
343	C₄⋊C₄	—	86	1	no	—
344	C₄⋊C₄	—	86	1	no	—
345	C₄⋊C₄	—	86	1	no	—
346	C₄⋊C₄	—	86	2	no	—
347	C₄⋊C₄	—	86	2	no	—
348	C₄⋊C₄	—	172	2	no	—
349	C₄⋊C₄	—	172	2	no	—
350	C₄⋊C₄	D₄×D₄₃	1	1	yes	characteristic
351	C₂²⋊C₄	—	2	1	no	—
352	C₂²⋊C₄	—	86	1	no	—
353	C₂²⋊C₄	—	86	1	no	—
354	C₂²⋊C₄	—	86	1	no	—
355	C₂²⋊C₄	—	86	1	no	—
356	C₂²⋊C₄	—	172	2	no	—
357	C₄²	—	43	1	no	—
358	C₄²	—	43	1	no	—
359	C₄²	—	43	1	no	—
360	C₄²	—	86	1	no	—
361	C₄²	—	172	2	no	—
362	C₄²	C₂³×D₄₃	1	1	yes	characteristic
363	C₂³	—	2	1	no	—
364	C₂³	—	43	1	no	—
365	Q₈	—	4	2	no	—
366	Q₈	—	4	2	no	—
367	Q₈	—	86	2	no	—
368	Q₈	—	172	2	no	—
369	Q₈	—	172	2	no	—
370	D₄	—	2	2	no	—
371	D₄	—	172	2	no	—
372	C₂×C₄	—	2	1	no	—
373	C₂×C₄	—	4	1	no	—
374	C₂×C₄	—	4	1	no	—
375	C₂×C₄	—	4	2	no	—
376	C₂×C₄	—	43	1	no	—
377	C₂×C₄	—	43	1	no	—
378	C₂×C₄	—	43	1	no	—
379	C₂×C₄	—	86	2	no	—
380	C₂×C₄	—	86	1	no	—
381	C₂×C₄	—	86	1	no	—
382	C₂×C₄	—	86	2	no	—
383	C₂×C₄	—	86	1	no	—
384	C₂×C₄	—	86	1	no	—
385	C₂×C₄	—	172	2	no	—
386	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
387	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
388	C₂×C₄	#41×D₄₃	1	1	yes	characteristic
389	C₈	—	2	1	no	—
390	C₈	—	4	1	no	—
391	C₈	—	4	2	no	—
392	C₈	—	86	1	no	—
393	C₈	—	86	1	no	—
394	C₈	—	172	1	no	—
395	C₈	—	172	2	no	—
396	C₈	#13063346	1	2	yes	—
397	C₂²	—	4	1	no	—
398	C₂²	—	4	1	no	—
399	C₂²	—	4	1	no	—
400	C₂²	—	43	2	no	—
401	C₂²	—	86	1	no	—
402	C₂²	—	86	1	no	—
403	C₂²	#121×D₄₃	1	1	yes	characteristic
404	C₄	—	2	1	no	—
405	C₄	—	4	1	no	—
406	C₄	—	8	2	no	—
407	C₄	—	86	1	no	—
408	C₄	—	86	1	no	—
409	C₄	—	86	1	no	—
410	C₄	—	344	2	no	—
411	C₄	—	172	1	no	—
412	C₄	—	172	1	no	—
413	C₄	C₂×#13063346	1	1	yes	characteristic
414	C₄	#14583698	1	1	yes	characteristic
415	C₄	#14583709	1	2	yes	—
416	C₂	—	8	1	no	—
417	C₂	—	86	2	no	—
418	C₂	#19144051	1	1	yes	characteristic
419	C₂	#19144255	1	1	yes	characteristic
420	C₂	#473×D₄₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#121×D₄₃
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting	C₄₃×#1693	C₂
socle	C₂²×C₄₃	C₂×#121
prime radical	#13063296	D₄
hypercenter	#1693	D₄₃
hypocenter	C₄₃	#19228
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1693

Complements

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

N	H ≅ G/N	type
2. #19145182	C₂	semidirect
4. #19134110	C₂	semidirect
5. #19134090	C₂	semidirect
6. #19134084	C₂	semidirect
7. #19134028	C₂	semidirect
8. #19134020	C₂	semidirect
9. #334×D₄₃	C₂	semidirect
10. #19133294	C₂	semidirect
11. C₄₃×#1693	C₂	semidirect
210. #1693	D₄₃	semidirect
257. C₄₃	#19228	semidirect

Maximal subgroups

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

subgroup	index
#19145928	2
#19145182	2
#19143573	2
#19134110	2
#19134090	2
#19134084	2
#19134028	2
#19134020	2
#334×D₄₃	2
#19133294	2
C₄₃×#1693	2
#19228	43