Subgroups — #25079312

Order 11008.

596 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	#19145179	C₂	1	1	yes	maximal, characteristic, complemented
2	#19144378	C₂	1	2	yes	maximal
3	#19143560	C₂	1	1	yes	maximal, characteristic
4	#19134175	C₂	1	2	yes	maximal, complemented
5	#19134169	C₂	1	2	yes	maximal, complemented
6	#19134123	C₂	1	1	yes	maximal, characteristic, complemented
7	#19134119	C₂	1	1	yes	maximal, characteristic, complemented
8	#19134081	C₂	1	1	yes	maximal, characteristic, complemented
9	#334×D₄₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#19133251	C₂	1	2	yes	maximal, complemented
11	C₄₃×#1690	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583454	C₂²	1	2	yes	—
13	C₄×#10621902	C₂²	1	1	yes	characteristic
14	#14583304	C₂²	1	1	yes	characteristic
15	#14583249	C₂²	1	2	yes	—
16	#14583204	—	2	2	no	—
17	#14583198	—	2	2	no	—
18	#14583194	C₂²	1	2	yes	—
19	#14583190	C₂²	1	2	yes	—
20	#14583176	C₂²	1	2	yes	—
21	#14583137	—	2	1	no	—
22	#14583137	—	2	1	no	—
23	#14583030	C₂²	1	1	yes	characteristic
24	Q₁₆×Dic₄₃	—	2	1	no	—
25	Q₁₆×Dic₄₃	—	2	1	no	—
26	#14582978	C₂²	1	1	yes	characteristic
27	#14582976	C₂²	1	1	yes	characteristic
28	#14582972	—	2	1	no	—
29	#14582969	—	2	1	no	—
30	#14582951	—	2	1	no	—
31	#14582951	C₂²	1	1	yes	characteristic
32	#14582934	—	2	2	no	—
33	C₄⋊C₈×D₄₃	C₂²	1	2	yes	—
34	#14582894	—	2	2	no	—
35	#14582851	—	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	#14582647	C₂²	1	2	yes	—
40	#14582647	C₂²	1	2	yes	—
41	#14582645	C₂²	1	2	yes	—
42	C₄₃×#494	C₂²	1	1	yes	characteristic
43	C₄₃×#475	C₂²	1	2	yes	—
44	C₄₃×#446	C₂²	1	1	yes	characteristic
45	C₄₃×#334	C₂²	1	1	yes	characteristic
46	C₄₃×#332	C₂²	1	2	yes	—
47	C₂×#10621902	D₄	1	1	yes	characteristic
48	C₂×#10621872	—	2	1	no	—
49	C₂×#10621872	—	2	1	no	—
50	C₂×C₈×D₄₃	—	2	2	no	—
51	C₂×C₈×D₄₃	D₄	1	1	yes	characteristic
52	C₂×C₈×D₄₃	D₄	1	1	yes	characteristic
53	#13063325	C₂³	1	2	yes	—
54	#13063316	—	2	2	no	—
55	#13063305	—	2	2	no	—
56	#13063301	—	2	2	no	—
57	Q₈×Dic₄₃	—	2	2	no	—
58	Q₈×Dic₄₃	D₄	1	1	yes	characteristic
59	#13063290	—	2	2	no	—
60	C₄×D₁₇₂	—	2	1	no	—
61	C₄×D₁₇₂	C₂³	1	1	yes	characteristic
62	#13063287	—	2	1	no	—
63	#13063287	D₄	1	1	yes	characteristic
64	#13063283	—	2	1	no	—
65	#13063283	—	2	2	no	—
66	#13063283	D₄	1	1	yes	characteristic
67	C₄²×D₄₃	—	2	1	no	—
68	C₄²×D₄₃	C₂³	1	1	yes	characteristic
69	#13063259	—	2	1	no	—
70	#13063259	—	2	1	no	—
71	#13063258	—	2	2	no	—
72	#13063258	—	2	2	no	—
73	#13063257	—	2	2	no	—
74	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
75	C₄⋊(C₄₃⋊C₈)	C₂³	1	1	yes	characteristic
76	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
77	Dic₄₃⋊C₈	—	2	2	no	—
78	#13063249	—	2	2	no	—
79	#13063249	—	2	2	no	—
80	#13063249	—	2	2	no	—
81	#13063247	—	2	2	no	—
82	#13063245	—	2	2	no	—
83	(C₂×D₄₃)⋊C₈	—	2	1	no	—
84	(C₂×D₄₃)⋊C₈	—	2	1	no	—
85	(C₂×D₄₃)⋊C₈	—	2	1	no	—
86	#13063236	—	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	1	no	—
92	C₂×Q₁₆×C₄₃	—	2	1	no	—
93	C₄⋊Q₈×C₄₃	C₂³	1	2	yes	—
94	C₄×Q₈×C₄₃	C₂³	1	1	yes	characteristic
95	#106×C₄₃	—	2	2	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	#97×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	#10621902	—	2	4	no	—
105	C₂×D₁₇₂	—	2	1	no	—
106	C₂×D₁₇₂	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×D₄₃	—	2	1	no	—
108	C₂×C₄×D₄₃	—	2	1	no	—
109	C₂×C₄×D₄₃	—	2	1	no	—
110	C₂×C₄×D₄₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×D₄₃	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×D₄₃	#41	1	1	yes	characteristic
113	C₂×Dic₈₆	—	2	2	no	—
114	Q₁₆×C₄₃	—	4	2	no	—
115	Q₁₆×C₄₃	—	4	2	no	—
116	C₂×C₈×C₄₃	—	2	1	no	—
117	C₂×C₈×C₄₃	—	2	2	no	—
118	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
120	C₄⋊C₄×C₄₃	—	2	1	no	—
121	C₄⋊C₄×C₄₃	—	2	2	no	—
122	C₄⋊C₄×C₄₃	—	2	2	no	—
123	C₄⋊C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
124	C₄²×C₄₃	—	2	1	no	—
125	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
126	(C₂×D₄₃)⋊C₄	—	2	1	no	—
127	(C₂×D₄₃)⋊C₄	—	2	1	no	—
128	(C₂×D₄₃)⋊C₄	—	2	1	no	—
129	(C₂×D₄₃)⋊C₄	—	4	2	no	—
130	C₄⋊Dic₄₃	—	2	1	no	—
131	C₄⋊Dic₄₃	—	2	2	no	—
132	C₄⋊Dic₄₃	—	2	2	no	—
133	C₄⋊Dic₄₃	—	4	2	no	—
134	C₄⋊Dic₄₃	C₂×D₄	1	1	yes	characteristic
135	Dic₄₃⋊C₄	—	4	2	no	—
136	C₄×Dic₄₃	—	2	1	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	C₂×D₄	1	1	yes	characteristic
145	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
146	#10621872	—	4	2	no	—
147	#10621872	—	4	2	no	—
148	C₈×D₄₃	—	2	2	no	—
149	C₈×D₄₃	—	4	1	no	—
150	C₈×D₄₃	—	4	4	no	—
151	C₂²×D₄₃	—	2	1	no	—
152	C₂²×D₄₃	#121	1	1	yes	characteristic
153	Q₈×C₄₃	—	2	2	no	—
154	Q₈×C₄₃	—	4	2	no	—
155	Q₈×C₄₃	—	4	2	no	—
156	C₂×C₄×C₄₃	—	2	1	no	—
157	C₂×C₄×C₄₃	—	2	1	no	—
158	C₂×C₄×C₄₃	—	4	2	no	—
159	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
161	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
162	C₂×Dic₄₃	—	2	1	no	—
163	C₂×Dic₄₃	—	4	2	no	—
164	C₂×Dic₄₃	#121	1	1	yes	characteristic
165	C₂×Dic₄₃	#121	1	1	yes	characteristic
166	C₂×Dic₄₃	#121	1	1	yes	characteristic
167	D₁₇₂	—	2	2	no	—
168	D₁₇₂	—	4	2	no	—
169	C₄×D₄₃	—	2	1	no	—
170	C₄×D₄₃	—	2	1	no	—
171	C₄×D₄₃	—	4	1	no	—
172	C₄×D₄₃	—	4	2	no	—
173	C₄×D₄₃	—	4	1	no	—
174	C₄×D₄₃	—	4	1	no	—
175	C₄×D₄₃	—	4	2	no	—
176	C₄×D₄₃	#137	1	4	yes	—
177	Dic₈₆	—	8	2	no	—
178	C₈×C₄₃	—	2	1	no	—
179	C₈×C₄₃	—	4	2	no	—
180	C₈×C₄₃	—	4	1	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	#19158	—	43	1	no	maximal, Sylow
187	C₂²×C₄₃	C₂×#121	1	1	yes	characteristic
188	C₂×D₄₃	—	2	1	no	—
189	C₂×D₄₃	—	2	1	no	—
190	C₂×D₄₃	—	4	1	no	—
191	C₂×D₄₃	—	4	1	no	—
192	C₂×D₄₃	—	4	1	no	—
193	C₂×D₄₃	#470	1	2	yes	—
194	C₄×C₄₃	—	2	1	no	—
195	C₄×C₄₃	—	4	1	no	—
196	C₄×C₄₃	—	4	1	no	—
197	C₄×C₄₃	—	8	2	no	—
198	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
199	C₄×C₄₃	C₂×#137	1	2	yes	—
200	C₄×C₄₃	#575	1	1	yes	characteristic
201	Dic₄₃	—	2	1	no	—
202	Dic₄₃	—	2	1	no	—
203	Dic₄₃	—	4	1	no	—
204	Dic₄₃	—	8	2	no	—
205	Dic₄₃	#470	1	2	yes	—
206	#3170	—	43	1	no	—
207	#3076	—	43	2	no	—
208	#2965	—	43	1	no	—
209	#1704	—	43	2	no	—
210	#1703	—	43	2	no	—
211	#1696	—	43	1	no	—
212	#1695	—	43	1	no	—
213	#1690	D₄₃	1	1	yes	characteristic, complemented
214	#1689	—	43	1	no	—
215	C₂×#334	—	43	1	no	—
216	#1542	—	43	2	no	—
217	C₂×C₄₃	#3035	1	1	yes	characteristic
218	C₂×C₄₃	C₂×#470	1	1	yes	characteristic
219	C₂×C₄₃	#3067	1	1	yes	characteristic
220	D₄₃	—	2	2	no	—
221	D₄₃	—	8	1	no	—
222	#533	—	43	2	no	—
223	C₄×#41	—	43	1	no	—
224	#500	—	43	1	no	—
225	#494	C₂×D₄₃	1	1	yes	characteristic
226	#487	—	43	2	no	—
227	#479	—	86	2	no	—
228	#478	—	86	2	no	—
229	#477	—	43	2	no	—
230	#476	—	43	2	no	—
231	#475	C₂×D₄₃	1	2	yes	—
232	#474	—	43	2	no	—
233	#467	—	86	1	no	—
234	#467	—	86	1	no	—
235	#446	—	43	1	no	—
236	#446	C₂×D₄₃	1	1	yes	characteristic
237	C₄×Q₁₆	—	86	1	no	—
238	C₄×Q₁₆	—	86	1	no	—
239	#436	—	43	1	no	—
240	#436	—	43	1	no	—
241	#435	—	86	1	no	—
242	#435	—	86	1	no	—
243	#431	—	43	1	no	—
244	#431	—	86	1	no	—
245	#427	—	86	2	no	—
246	C₂×C₄⋊C₈	—	43	2	no	—
247	#416	—	86	2	no	—
248	C₄×M₁₆	—	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	#333	—	43	2	no	—
254	#333	—	43	2	no	—
255	#332	—	43	2	no	—
256	#332	C₂×D₄₃	1	2	yes	—
257	C₄₃	#19158	1	1	yes	characteristic, complemented, Sylow
258	C₂×#41	—	43	1	no	—
259	C₂×Q₁₆	—	2	1	no	—
260	C₂×Q₁₆	—	2	1	no	—
261	C₂×M₁₆	—	86	1	no	—
262	C₂×M₁₆	—	86	1	no	—
263	C₂²×C₈	—	43	1	no	—
264	C₂²×C₈	—	43	1	no	—
265	C₂²×C₈	—	86	2	no	—
266	C₄⋊Q₈	—	43	2	no	—
267	C₄⋊Q₈	C₂²×D₄₃	1	2	yes	—
268	#125	—	86	2	no	—
269	C₂²⋊Q₈	—	86	2	no	—
270	C₂²⋊Q₈	—	86	2	no	—
271	C₄×Q₈	—	43	1	no	—
272	C₄×Q₈	—	86	2	no	—
273	C₄×Q₈	—	86	2	no	—
274	C₄×Q₈	C₂²×D₄₃	1	1	yes	characteristic
275	C₄×D₄	—	43	1	no	—
276	C₄×D₄	—	43	1	no	—
277	C₄×D₄	—	86	1	no	—
278	C₄×D₄	—	86	1	no	—
279	#117	—	43	1	no	—
280	#117	—	86	1	no	—
281	#117	—	86	2	no	—
282	C₂×C₄²	—	43	1	no	—
283	C₂×C₄²	—	86	1	no	—
284	#107	—	86	2	no	—
285	#107	—	86	1	no	—
286	#107	—	86	1	no	—
287	#107	—	86	2	no	—
288	#106	—	2	2	no	—
289	#106	—	86	2	no	—
290	C₄⋊C₈	—	2	1	no	—
291	C₄⋊C₈	—	43	1	no	—
292	C₄⋊C₈	—	43	2	no	—
293	C₄⋊C₈	—	86	1	no	—
294	C₄⋊C₈	—	86	2	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	#102	—	86	2	no	—
302	#102	—	86	2	no	—
303	C₂²⋊C₈	—	86	1	no	—
304	C₂²⋊C₈	—	86	1	no	—
305	C₂²⋊C₈	—	86	1	no	—
306	#97	—	2	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	#41	—	86	4	no	—
314	C₂×Q₈	—	2	2	no	—
315	C₂×Q₈	—	86	2	no	—
316	C₂×Q₈	D₄×D₄₃	1	1	yes	characteristic
317	C₂×D₄	—	43	1	no	—
318	C₂×D₄	—	86	1	no	—
319	C₂²×C₄	—	43	1	no	—
320	C₂²×C₄	—	43	1	no	—
321	C₂²×C₄	—	43	1	no	—
322	C₂²×C₄	—	86	1	no	—
323	C₂²×C₄	—	86	1	no	—
324	C₂²×C₄	—	86	1	no	—
325	Q₁₆	—	4	2	no	—
326	Q₁₆	—	4	2	no	—
327	M₁₆	—	172	2	no	—
328	M₁₆	—	172	2	no	—
329	C₂×C₈	—	2	1	no	—
330	C₂×C₈	—	2	2	no	—
331	C₂×C₈	—	43	1	no	—
332	C₂×C₈	—	43	1	no	—
333	C₂×C₈	—	86	2	no	—
334	C₂×C₈	—	86	1	no	—
335	C₂×C₈	—	86	2	no	—
336	C₂×C₈	—	172	1	no	—
337	C₂×C₈	—	172	4	no	—
338	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
339	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
340	C₄⋊C₄	—	2	1	no	—
341	C₄⋊C₄	—	2	2	no	—
342	C₄⋊C₄	—	2	2	no	—
343	C₄⋊C₄	—	43	1	no	—
344	C₄⋊C₄	—	86	1	no	—
345	C₄⋊C₄	—	86	2	no	—
346	C₄⋊C₄	—	86	2	no	—
347	C₄⋊C₄	—	172	2	no	—
348	C₄⋊C₄	—	172	2	no	—
349	C₄⋊C₄	D₄×D₄₃	1	1	yes	characteristic
350	C₂²⋊C₄	—	86	1	no	—
351	C₂²⋊C₄	—	86	1	no	—
352	C₂²⋊C₄	—	86	1	no	—
353	C₂²⋊C₄	—	172	2	no	—
354	C₄²	—	2	1	no	—
355	C₄²	—	43	1	no	—
356	C₄²	—	43	1	no	—
357	C₄²	—	43	1	no	—
358	C₄²	—	86	1	no	—
359	C₄²	—	86	1	no	—
360	C₄²	—	172	2	no	—
361	C₄²	C₂³×D₄₃	1	1	yes	characteristic
362	C₂³	—	43	1	no	—
363	C₂³	—	86	1	no	—
364	Q₈	—	2	2	no	—
365	Q₈	—	4	2	no	—
366	Q₈	—	4	2	no	—
367	Q₈	—	344	2	no	—
368	D₄	—	86	2	no	—
369	D₄	—	172	2	no	—
370	C₂×C₄	—	2	1	no	—
371	C₂×C₄	—	2	1	no	—
372	C₂×C₄	—	4	2	no	—
373	C₂×C₄	—	43	1	no	—
374	C₂×C₄	—	43	4	no	—
375	C₂×C₄	—	43	1	no	—
376	C₂×C₄	—	43	1	no	—
377	C₂×C₄	—	86	1	no	—
378	C₂×C₄	—	86	1	no	—
379	C₂×C₄	—	86	1	no	—
380	C₂×C₄	—	172	1	no	—
381	C₂×C₄	—	172	2	no	—
382	C₂×C₄	—	172	2	no	—
383	C₂×C₄	—	172	2	no	—
384	C₂×C₄	—	172	1	no	—
385	C₂×C₄	—	172	1	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	2	no	—
391	C₈	—	4	1	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	—	43	2	no	—
398	C₂²	—	86	1	no	—
399	C₂²	—	86	1	no	—
400	C₂²	—	172	1	no	—
401	C₂²	—	172	1	no	—
402	C₂²	—	172	1	no	—
403	C₂²	#121×D₄₃	1	1	yes	characteristic
404	C₄	—	2	1	no	—
405	C₄	—	4	1	no	—
406	C₄	—	4	1	no	—
407	C₄	—	8	2	no	—
408	C₄	—	43	2	no	—
409	C₄	—	86	1	no	—
410	C₄	—	86	1	no	—
411	C₄	—	344	2	no	—
412	C₄	—	172	1	no	—
413	C₄	C₂×#13063346	1	1	yes	characteristic
414	C₄	#137×D₄₃	1	2	yes	—
415	C₄	#14583695	1	1	yes	characteristic
416	C₂	—	86	2	no	—
417	C₂	—	344	1	no	—
418	C₂	#19144045	1	1	yes	characteristic
419	C₂	#470×D₄₃	1	1	yes	characteristic
420	C₂	#19144317	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#121×D₄₃
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting	C₄₃×#1690	C₂
socle	C₂²×C₄₃	C₂×#121
prime radical	C₂×#10621902	D₄
hypercenter	#1690	D₄₃
hypocenter	C₄₃	#19158
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1690

Complements

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

N	H ≅ G/N	type
1. #19145179	C₂	semidirect
4. #19134175	C₂	semidirect
5. #19134169	C₂	semidirect
6. #19134123	C₂	semidirect
7. #19134119	C₂	semidirect
8. #19134081	C₂	semidirect
9. #334×D₄₃	C₂	semidirect
10. #19133251	C₂	semidirect
11. C₄₃×#1690	C₂	semidirect
213. #1690	D₄₃	semidirect
257. C₄₃	#19158	semidirect

Maximal subgroups

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

subgroup	index
#19145179	2
#19144378	2
#19143560	2
#19134175	2
#19134169	2
#19134123	2
#19134119	2
#19134081	2
#334×D₄₃	2
#19133251	2
C₄₃×#1690	2
#19158	43