Subgroups — #25084284

Order 11008.

750 conjugacy classes of subgroups.

The subgroup lattice has 352 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	#19145175	C₂	1	1	yes	maximal, characteristic
3	#19145156	C₂	1	2	yes	maximal, complemented
4	#19144919	C₂	1	2	yes	maximal, complemented
5	#19134208	C₂	1	2	yes	maximal, complemented
6	#19134204	C₂	1	1	yes	maximal, characteristic, complemented
7	#19134202	C₂	1	1	yes	maximal, characteristic, complemented
8	#19134197	C₂	1	2	yes	maximal, complemented
9	#19134187	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈²×D₄₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#1707	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583454	C₂²	1	2	yes	—
13	#14583450	C₂²	1	2	yes	—
14	#14583304	C₂²	1	1	yes	characteristic
15	#14583276	C₂²	1	1	yes	characteristic
16	#14583272	C₂²	1	2	yes	—
17	#14583255	—	2	4	no	—
18	#14583251	C₂²	1	2	yes	—
19	#14583249	C₂²	1	2	yes	—
20	#14583249	C₂²	1	2	yes	—
21	#14583247	C₂²	1	2	yes	—
22	#14583247	C₂²	1	2	yes	—
23	#14583247	C₂²	1	2	yes	—
24	#14583246	C₂²	1	2	yes	—
25	#14583137	—	2	2	no	—
26	#14583136	—	2	2	no	—
27	#14583129	—	2	4	no	—
28	Q₁₆×Dic₄₃	—	2	2	no	—
29	#14582986	—	2	4	no	—
30	#14582981	—	2	2	no	—
31	#14582894	—	2	4	no	—
32	C₄×C₈×D₄₃	C₂²	1	1	yes	characteristic
33	C₄×C₈×D₄₃	C₂²	1	1	yes	characteristic
34	C₄×C₈×D₄₃	C₂²	1	1	yes	characteristic
35	C₈×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
36	C₈×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
37	C₈×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
38	C₄₃×#500	C₂²	1	1	yes	characteristic
39	C₄₃×#498	C₂²	1	2	yes	—
40	C₄₃×#494	C₂²	1	1	yes	characteristic
41	C₄₃×#487	C₂²	1	2	yes	—
42	C₈²×C₄₃	C₂²	1	1	yes	characteristic
43	C₂×#10621902	—	2	2	no	—
44	C₂×#10621873	—	2	4	no	—
45	C₂×D₃₄₄	—	2	2	no	—
46	C₂×C₈×D₄₃	D₄	1	2	yes	—
47	C₂×C₈×D₄₃	D₄	1	2	yes	—
48	C₂×C₈×D₄₃	D₄	1	2	yes	—
49	#13063325	C₂³	1	2	yes	—
50	#13063322	C₂³	1	2	yes	—
51	#13063316	—	2	2	no	—
52	#13063314	—	2	2	no	—
53	#13063305	—	2	2	no	—
54	#13063301	—	2	2	no	—
55	#13063300	—	2	2	no	—
56	#13063300	—	2	2	no	—
57	Q₈×Dic₄₃	—	2	2	no	—
58	#13063290	—	2	2	no	—
59	#13063287	—	2	2	no	—
60	#13063287	—	2	2	no	—
61	#13063283	—	2	2	no	—
62	C₄²×D₄₃	C₂³	1	1	yes	characteristic
63	#13063259	—	2	2	no	—
64	#13063249	—	2	4	no	—
65	#13063249	—	2	4	no	—
66	#13063248	—	2	4	no	—
67	#13063247	—	2	4	no	—
68	#13063245	—	2	4	no	—
69	#13063245	—	2	4	no	—
70	#13063245	—	2	4	no	—
71	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
72	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
73	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
74	C₈×Dic₄₃	D₄	1	2	yes	—
75	C₈×Dic₄₃	D₄	1	2	yes	—
76	C₈×Dic₄₃	D₄	1	2	yes	—
77	C₂×Q₁₆×C₄₃	—	2	2	no	—
78	C₄⋊Q₈×C₄₃	C₂³	1	2	yes	—
79	C₄⋊Q₈×C₄₃	C₂³	1	2	yes	—
80	#107×C₄₃	—	2	2	no	—
81	#106×C₄₃	—	2	4	no	—
82	#103×C₄₃	—	2	4	no	—
83	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
84	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
86	C₂×Q₈×C₄₃	—	2	2	no	—
87	C₂×Q₈×C₄₃	—	2	2	no	—
88	#10621902	—	4	8	no	—
89	C₂×D₁₇₂	—	2	2	no	—
90	C₂×D₁₇₂	—	2	2	no	—
91	C₂×D₁₇₂	—	2	2	no	—
92	C₂×D₁₇₂	—	4	2	no	—
93	C₂×C₄×D₄₃	—	4	2	no	—
94	C₂×C₄×D₄₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×D₄₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×D₄₃	C₂×D₄	1	1	yes	characteristic
97	C₂×Dic₈₆	—	2	2	no	—
98	Q₁₆×C₄₃	—	4	4	no	—
99	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
100	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
101	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
102	C₄⋊C₄×C₄₃	—	2	2	no	—
103	C₄⋊C₄×C₄₃	—	2	2	no	—
104	C₄⋊C₄×C₄₃	—	2	2	no	—
105	C₄⋊C₄×C₄₃	—	2	2	no	—
106	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
107	(C₂×D₄₃)⋊C₄	—	4	2	no	—
108	(C₂×D₄₃)⋊C₄	—	4	2	no	—
109	(C₂×D₄₃)⋊C₄	—	4	2	no	—
110	C₄⋊Dic₄₃	—	2	2	no	—
111	C₄⋊Dic₄₃	—	2	2	no	—
112	C₄⋊Dic₄₃	—	4	2	no	—
113	Dic₄₃⋊C₄	—	4	2	no	—
114	C₄×Dic₄₃	—	4	2	no	—
115	C₄×Dic₄₃	C₂×D₄	1	1	yes	characteristic
116	C₄×Dic₄₃	C₂×D₄	1	1	yes	characteristic
117	C₄×Dic₄₃	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
119	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
120	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
121	D₃₄₄	—	4	4	no	—
122	#10621873	—	4	8	no	—
123	C₈×D₄₃	—	2	4	no	—
124	C₈×D₄₃	—	2	4	no	—
125	C₈×D₄₃	—	2	4	no	—
126	C₂²×D₄₃	—	4	2	no	—
127	C₂²×D₄₃	#127	1	1	yes	characteristic
128	Q₈×C₄₃	—	4	4	no	—
129	Q₈×C₄₃	—	4	4	no	—
130	C₂×C₄×C₄₃	—	4	2	no	—
131	C₂×C₄×C₄₃	—	4	2	no	—
132	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
135	C₂×Dic₄₃	—	4	2	no	—
136	C₂×Dic₄₃	#127	1	1	yes	characteristic
137	C₂×Dic₄₃	#127	1	1	yes	characteristic
138	C₂×Dic₄₃	#127	1	1	yes	characteristic
139	D₁₇₂	—	4	4	no	—
140	D₁₇₂	—	4	4	no	—
141	D₁₇₂	—	4	4	no	—
142	D₁₇₂	—	8	4	no	—
143	C₄×D₄₃	—	2	2	no	—
144	C₄×D₄₃	—	2	2	no	—
145	C₄×D₄₃	—	8	4	no	—
146	C₄×D₄₃	#135	1	4	yes	—
147	Dic₈₆	—	4	4	no	—
148	C₈×C₄₃	—	2	2	no	—
149	C₈×C₄₃	#135	1	4	yes	—
150	C₈×C₄₃	#135	1	4	yes	—
151	C₄₃⋊C₈	—	2	2	no	—
152	C₄₃⋊C₈	—	2	2	no	—
153	C₄₃⋊C₈	—	2	2	no	—
154	#19600	—	43	1	no	maximal, Sylow
155	C₂²×C₄₃	C₂×#127	1	1	yes	characteristic
156	C₂×D₄₃	—	2	1	no	—
157	C₂×D₄₃	—	2	1	no	—
158	C₂×D₄₃	—	8	2	no	—
159	C₂×D₄₃	—	8	2	no	—
160	C₂×D₄₃	—	8	2	no	—
161	C₂×D₄₃	#495	1	2	yes	—
162	C₄×C₄₃	—	8	2	no	—
163	C₄×C₄₃	—	8	2	no	—
164	C₄×C₄₃	C₂×#135	1	2	yes	—
165	C₄×C₄₃	C₂×#135	1	2	yes	—
166	C₄×C₄₃	C₂×#135	1	2	yes	—
167	Dic₄₃	—	2	1	no	—
168	Dic₄₃	—	2	1	no	—
169	Dic₄₃	—	8	2	no	—
170	Dic₄₃	#495	1	2	yes	—
171	#3170	—	43	1	no	—
172	#3169	—	43	1	no	—
173	#3167	—	43	2	no	—
174	#3131	—	43	2	no	—
175	#1711	—	43	2	no	—
176	#1710	—	43	1	no	—
177	#1709	—	43	1	no	—
178	#1708	—	43	2	no	—
179	#1707	D₄₃	1	1	yes	characteristic, complemented
180	#1706	—	43	1	no	—
181	C₂×C₈²	—	43	1	no	—
182	C₂×C₄₃	C₂×#495	1	1	yes	characteristic
183	C₂×C₄₃	#3147	1	1	yes	characteristic
184	C₂×C₄₃	#3147	1	1	yes	characteristic
185	D₄₃	—	2	2	no	—
186	D₄₃	—	16	2	no	—
187	#533	—	43	2	no	—
188	#532	—	43	2	no	—
189	#500	—	43	1	no	—
190	#500	C₂×D₄₃	1	1	yes	characteristic
191	#498	C₂×D₄₃	1	2	yes	—
192	#494	C₂×D₄₃	1	1	yes	characteristic
193	#493	—	43	1	no	—
194	#492	—	43	2	no	—
195	#488	—	86	4	no	—
196	#487	—	43	2	no	—
197	#487	—	43	2	no	—
198	#487	—	43	2	no	—
199	#487	C₂×D₄₃	1	2	yes	—
200	#486	—	43	2	no	—
201	#486	—	43	2	no	—
202	#486	—	43	2	no	—
203	#486	—	43	2	no	—
204	#467	—	86	2	no	—
205	#466	—	86	2	no	—
206	#465	—	86	4	no	—
207	C₄×Q₁₆	—	86	2	no	—
208	C₄×SD₁₆	—	86	4	no	—
209	C₄×D₈	—	86	2	no	—
210	#416	—	86	4	no	—
211	C₂×C₄×C₈	—	43	1	no	—
212	C₂×C₄×C₈	—	43	1	no	—
213	C₂×C₄×C₈	—	43	1	no	—
214	C₈²	—	43	1	no	—
215	C₈²	—	43	1	no	—
216	C₈²	—	43	1	no	—
217	C₈²	C₂×D₄₃	1	1	yes	characteristic
218	C₄₃	#19600	1	1	yes	characteristic, complemented, Sylow
219	C₂×#41	—	86	2	no	—
220	C₂×Q₁₆	—	2	2	no	—
221	C₂×SD₁₆	—	86	4	no	—
222	C₂×D₈	—	86	2	no	—
223	C₂²×C₈	—	43	2	no	—
224	C₂²×C₈	—	43	2	no	—
225	C₂²×C₈	—	43	2	no	—
226	C₄⋊Q₈	—	43	2	no	—
227	C₄⋊Q₈	C₂²×D₄₃	1	2	yes	—
228	C₄⋊Q₈	C₂²×D₄₃	1	2	yes	—
229	#127	—	43	2	no	—
230	#125	—	86	2	no	—
231	#124	—	86	2	no	—
232	C₂²⋊Q₈	—	86	2	no	—
233	C₂²⋊Q₈	—	86	2	no	—
234	#121	—	86	2	no	—
235	#121	—	86	2	no	—
236	C₄×Q₈	—	86	2	no	—
237	C₄×Q₈	—	86	2	no	—
238	C₄×D₄	—	86	2	no	—
239	C₄×D₄	—	86	2	no	—
240	#117	—	86	2	no	—
241	C₂×C₄²	—	43	1	no	—
242	#107	—	2	2	no	—
243	#107	—	86	2	no	—
244	#106	—	2	4	no	—
245	#103	—	2	4	no	—
246	#103	—	86	4	no	—
247	#103	—	86	4	no	—
248	#103	—	86	4	no	—
249	#102	—	86	4	no	—
250	#102	—	86	4	no	—
251	#102	—	86	4	no	—
252	#102	—	86	4	no	—
253	C₄×C₈	—	43	2	no	—
254	C₄×C₈	—	43	2	no	—
255	C₄×C₈	—	43	2	no	—
256	C₄×C₈	—	43	1	no	—
257	C₄×C₈	—	43	1	no	—
258	C₄×C₈	—	43	1	no	—
259	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
260	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
261	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
262	#41	—	172	8	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×Q₈	—	2	2	no	—
265	C₂×Q₈	—	86	2	no	—
266	C₂×D₄	—	86	2	no	—
267	C₂×D₄	—	86	2	no	—
268	C₂×D₄	—	86	2	no	—
269	C₂×D₄	—	172	2	no	—
270	C₂²×C₄	—	43	1	no	—
271	C₂²×C₄	—	43	1	no	—
272	C₂²×C₄	—	43	1	no	—
273	C₂²×C₄	—	172	2	no	—
274	Q₁₆	—	4	4	no	—
275	SD₁₆	—	172	8	no	—
276	D₈	—	172	4	no	—
277	C₂×C₈	—	43	2	no	—
278	C₂×C₈	—	43	2	no	—
279	C₂×C₈	—	43	2	no	—
280	C₂×C₈	—	86	4	no	—
281	C₂×C₈	—	86	4	no	—
282	C₂×C₈	—	86	4	no	—
283	C₂×C₈	D₄×D₄₃	1	2	yes	—
284	C₂×C₈	D₄×D₄₃	1	2	yes	—
285	C₂×C₈	D₄×D₄₃	1	2	yes	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	2	2	no	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	86	2	no	—
291	C₄⋊C₄	—	86	2	no	—
292	C₄⋊C₄	—	172	2	no	—
293	C₄⋊C₄	—	172	2	no	—
294	C₂²⋊C₄	—	172	2	no	—
295	C₂²⋊C₄	—	172	2	no	—
296	C₂²⋊C₄	—	172	2	no	—
297	C₄²	—	43	1	no	—
298	C₄²	—	43	1	no	—
299	C₄²	—	43	1	no	—
300	C₄²	—	172	2	no	—
301	C₄²	C₂³×D₄₃	1	1	yes	characteristic
302	C₂³	—	43	1	no	—
303	C₂³	—	172	2	no	—
304	Q₈	—	4	4	no	—
305	Q₈	—	4	4	no	—
306	Q₈	—	172	4	no	—
307	D₄	—	344	4	no	—
308	D₄	—	172	4	no	—
309	D₄	—	172	4	no	—
310	D₄	—	172	4	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	43	1	no	—
314	C₂×C₄	—	43	4	no	—
315	C₂×C₄	—	43	1	no	—
316	C₂×C₄	—	43	1	no	—
317	C₂×C₄	—	86	2	no	—
318	C₂×C₄	—	86	2	no	—
319	C₂×C₄	—	344	4	no	—
320	C₂×C₄	—	172	2	no	—
321	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
322	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
323	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
324	C₈	—	2	2	no	—
325	C₈	—	86	2	no	—
326	C₈	—	86	2	no	—
327	C₈	—	86	2	no	—
328	C₈	#13063346	1	4	yes	—
329	C₈	#13063348	1	4	yes	—
330	C₂²	—	43	2	no	—
331	C₂²	—	86	1	no	—
332	C₂²	—	86	1	no	—
333	C₂²	—	344	2	no	—
334	C₂²	—	344	2	no	—
335	C₂²	—	344	2	no	—
336	C₂²	#127×D₄₃	1	1	yes	characteristic
337	C₄	—	8	2	no	—
338	C₄	—	8	2	no	—
339	C₄	—	43	2	no	—
340	C₄	—	86	1	no	—
341	C₄	—	86	1	no	—
342	C₄	—	344	2	no	—
343	C₄	#135×D₄₃	1	2	yes	—
344	C₄	C₂×#13063346	1	2	yes	—
345	C₄	C₂×#13063348	1	2	yes	—
346	C₂	—	86	2	no	—
347	C₂	—	688	2	no	—
348	C₂	#495×D₄₃	1	1	yes	characteristic
349	C₂	#19145018	1	1	yes	characteristic
350	C₂	#19145024	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#127×D₄₃
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting	C₄₃×#1707	C₂
socle	C₂²×C₄₃	C₂×#127
prime radical	#19145175	C₂
hypercenter	#1707	D₄₃
hypocenter	C₄₃	#19600
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1707

Complements

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

N	H ≅ G/N	type
1. #19145179	C₂	semidirect
3. #19145156	C₂	semidirect
4. #19144919	C₂	semidirect
5. #19134208	C₂	semidirect
6. #19134204	C₂	semidirect
7. #19134202	C₂	semidirect
8. #19134197	C₂	semidirect
9. #19134187	C₂	semidirect
10. C₈²×D₄₃	C₂	semidirect
11. C₄₃×#1707	C₂	semidirect
179. #1707	D₄₃	semidirect
218. C₄₃	#19600	semidirect

Maximal subgroups

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

subgroup	index
#19145179	2
#19145175	2
#19145156	2
#19144919	2
#19134208	2
#19134204	2
#19134202	2
#19134197	2
#19134187	2
C₈²×D₄₃	2
C₄₃×#1707	2
#19600	43