Subgroups — #19145310

Order 5504.

402 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583453	C₂	1	1	yes	maximal, characteristic, complemented
2	#14583446	C₂	1	1	yes	maximal, characteristic
3	#14583303	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583270	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583248	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583244	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583148	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583146	C₂	1	1	yes	maximal, characteristic, complemented
9	#14583144	C₂	1	1	yes	maximal, characteristic, complemented
10	#14583142	C₂	1	1	yes	maximal, characteristic, complemented
11	#14583001	C₂	1	1	yes	maximal, characteristic, complemented
12	#14582999	C₂	1	1	yes	maximal, characteristic, complemented
13	#14582997	C₂	1	1	yes	maximal, characteristic, complemented
14	#14582851	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄₃×#440	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#10621898	—	2	1	no	—
17	C₂×#10621873	C₂²	1	1	yes	characteristic
18	C₂×#10621883	C₂²	1	1	yes	characteristic
19	C₂×#10621884	C₂²	1	1	yes	characteristic
20	C₂×#10621872	C₂²	1	1	yes	characteristic
21	C₂×#10621872	C₂²	1	1	yes	characteristic
22	#13063324	C₂²	1	1	yes	characteristic
23	#13063324	C₂²	1	1	yes	characteristic
24	#13063323	C₂²	1	1	yes	characteristic
25	#13063321	C₂²	1	1	yes	characteristic
26	#13063315	—	2	1	no	—
27	#13063311	—	2	1	no	—
28	#13063305	C₂²	1	1	yes	characteristic
29	#13063303	—	2	1	no	—
30	#13063301	C₂²	1	1	yes	characteristic
31	#13063300	C₂²	1	1	yes	characteristic
32	#13063298	—	2	1	no	—
33	#13063296	C₂²	1	1	yes	characteristic
34	C₄×Dic₈₆	—	2	1	no	—
35	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
36	C₄×D₁₇₂	C₂²	1	1	yes	characteristic
37	C₄×C₂²⋊D₄₃	—	2	1	no	—
38	#13063282	—	2	1	no	—
39	C₄²×D₄₃	C₂²	1	1	yes	characteristic
40	#13063259	C₂²	1	1	yes	characteristic
41	#13063258	C₂²	1	1	yes	characteristic
42	#13063258	C₂²	1	1	yes	characteristic
43	#13063250	C₂²	1	1	yes	characteristic
44	#13063249	C₂²	1	1	yes	characteristic
45	#13063248	C₂²	1	1	yes	characteristic
46	#13063247	C₂²	1	1	yes	characteristic
47	#13063246	C₂²	1	1	yes	characteristic
48	#13063245	C₂²	1	1	yes	characteristic
49	#13063236	C₂²	1	1	yes	characteristic
50	C₈×Dic₄₃	C₂²	1	1	yes	characteristic
51	C₈×Dic₄₃	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₄₃	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₄₃	C₂²	1	1	yes	characteristic
55	#107×C₄₃	C₂²	1	1	yes	characteristic
56	#103×C₄₃	C₂²	1	1	yes	characteristic
57	#102×C₄₃	C₂²	1	1	yes	characteristic
58	#97×C₄₃	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₄₃	C₂³	1	1	yes	characteristic
60	C₂×D₄×C₄₃	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	—	2	1	no	—
62	C₂×C₂²⋊D₄₃	—	2	1	no	—
63	C₂×C₂²⋊D₄₃	—	2	1	no	—
64	C₂×C₂²⋊D₄₃	—	2	1	no	—
65	#10621898	—	4	4	no	—
66	C₂×D₁₇₂	—	2	1	no	—
67	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
68	C₂×C₄×D₄₃	—	2	1	no	—
69	C₂×C₄×D₄₃	D₄	1	1	yes	characteristic
70	C₂×C₄×D₄₃	D₄	1	1	yes	characteristic
71	C₂×C₄×D₄₃	C₂³	1	1	yes	characteristic
72	C₂×Dic₈₆	—	2	1	no	—
73	C₂×Dic₈₆	—	2	1	no	—
74	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
75	SD₁₆×C₄₃	—	2	2	no	—
76	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₄₃	—	2	1	no	—
79	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
80	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
81	C₂²⋊C₄×C₄₃	—	2	1	no	—
82	C₄²×C₄₃	—	2	1	no	—
83	C₄²×C₄₃	C₂³	1	1	yes	characteristic
84	C₂²⋊Dic₄₃	—	2	1	no	—
85	#10621884	—	2	2	no	—
86	#10621883	—	2	2	no	—
87	(C₂×D₄₃)⋊C₄	—	2	1	no	—
88	(C₂×D₄₃)⋊C₄	—	2	1	no	—
89	(C₂×D₄₃)⋊C₄	—	2	1	no	—
90	(C₂×D₄₃)⋊C₄	—	2	1	no	—
91	C₄⋊Dic₄₃	—	2	1	no	—
92	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
93	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
94	Dic₄₃⋊C₄	—	2	1	no	—
95	Dic₄₃⋊C₄	—	2	1	no	—
96	Dic₄₃⋊C₄	—	2	1	no	—
97	Dic₄₃⋊C₄	—	2	1	no	—
98	Dic₄₃⋊C₄	—	2	1	no	—
99	Dic₄₃⋊C₄	—	2	1	no	—
100	C₄×Dic₄₃	—	2	1	no	—
101	C₄×Dic₄₃	D₄	1	1	yes	characteristic
102	C₄×Dic₄₃	D₄	1	1	yes	characteristic
103	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
104	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
105	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
106	#10621873	—	2	2	no	—
107	#10621872	—	2	2	no	—
108	#10621872	—	2	2	no	—
109	C₂³×C₄₃	—	2	1	no	—
110	C₂²×D₄₃	—	2	1	no	—
111	C₂²×D₄₃	C₂×D₄	1	1	yes	characteristic
112	Q₈×C₄₃	—	2	1	no	—
113	Q₈×C₄₃	—	2	1	no	—
114	D₄×C₄₃	—	2	1	no	—
115	D₄×C₄₃	—	2	1	no	—
116	C₂×C₄×C₄₃	—	2	1	no	—
117	C₂×C₄×C₄₃	—	2	1	no	—
118	C₂×C₄×C₄₃	—	2	1	no	—
119	C₂×C₄×C₄₃	—	4	2	no	—
120	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
123	C₂²⋊D₄₃	—	4	2	no	—
124	C₂²⋊D₄₃	—	4	2	no	—
125	C₂²⋊D₄₃	—	4	2	no	—
126	C₂×Dic₄₃	—	2	1	no	—
127	C₂×Dic₄₃	—	2	1	no	—
128	C₂×Dic₄₃	—	2	1	no	—
129	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
131	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
132	D₁₇₂	—	2	1	no	—
133	D₁₇₂	—	2	1	no	—
134	D₁₇₂	—	4	2	no	—
135	C₄×D₄₃	—	2	1	no	—
136	C₄×D₄₃	—	2	1	no	—
137	C₄×D₄₃	—	2	2	no	—
138	C₄×D₄₃	—	2	2	no	—
139	C₄×D₄₃	—	4	2	no	—
140	Dic₈₆	—	2	1	no	—
141	Dic₈₆	—	2	1	no	—
142	Dic₈₆	—	4	2	no	—
143	Dic₈₆	—	4	2	no	—
144	C₈×C₄₃	—	2	1	no	—
145	C₈×C₄₃	#41	1	2	yes	complemented
146	C₄₃⋊C₈	—	2	1	no	—
147	C₄₃⋊C₈	#41	1	2	yes	complemented
148	C₂²×C₄₃	—	4	1	no	—
149	C₂²×C₄₃	—	4	1	no	—
150	C₂²×C₄₃	—	4	1	no	—
151	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
152	C₂×D₄₃	—	2	1	no	—
153	C₂×D₄₃	—	2	1	no	—
154	C₂×D₄₃	—	2	1	no	—
155	C₂×D₄₃	—	4	1	no	—
156	C₂×D₄₃	—	4	1	no	—
157	C₂×D₄₃	—	4	1	no	—
158	C₄×C₄₃	—	2	1	no	—
159	C₄×C₄₃	—	2	1	no	—
160	C₄×C₄₃	—	4	1	no	—
161	C₄×C₄₃	—	4	1	no	—
162	C₄×C₄₃	—	4	1	no	—
163	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
164	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
165	Dic₄₃	—	2	1	no	—
166	Dic₄₃	—	4	1	no	—
167	Dic₄₃	—	4	1	no	—
168	Dic₄₃	—	4	1	no	—
169	Dic₄₃	AGL(1,ℤ/8)	1	2	yes	—
170	Dic₄₃	#137	1	2	yes	—
171	#3182	—	43	1	no	maximal, Sylow
172	C₂×C₄₃	—	8	1	no	—
173	C₂×C₄₃	#532	1	1	yes	characteristic
174	C₂×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
175	C₂×C₄₃	C₂×#137	1	1	yes	characteristic
176	D₄₃	—	4	1	no	—
177	D₄₃	—	8	1	no	—
178	#533	—	43	1	no	—
179	#532	—	43	1	no	—
180	#500	—	43	1	no	—
181	#492	—	43	1	no	—
182	#487	—	43	1	no	—
183	#486	—	43	1	no	—
184	#468	—	43	1	no	—
185	#468	—	43	1	no	—
186	#468	—	43	1	no	—
187	#468	—	43	1	no	—
188	#440	—	43	1	no	—
189	#440	—	43	1	no	—
190	#440	—	43	1	no	—
191	#440	D₄₃	1	1	yes	characteristic, complemented
192	C₄×M₁₆	—	43	1	no	—
193	C₄₃	#3182	1	1	yes	characteristic, complemented, Sylow
194	C₂×#41	—	86	1	no	—
195	C₂×SD₁₆	—	43	1	no	—
196	C₂×SD₁₆	—	43	1	no	—
197	C₂×SD₁₆	—	43	1	no	—
198	C₂×SD₁₆	C₂×D₄₃	1	1	yes	characteristic
199	C₂×M₁₆	—	43	1	no	—
200	C₂×M₁₆	—	43	1	no	—
201	C₄⋊Q₈	—	43	1	no	—
202	C₄⋊Q₈	—	43	1	no	—
203	C₄⋊Q₈	—	43	1	no	—
204	#127	—	43	1	no	—
205	#125	—	86	1	no	—
206	#124	—	86	1	no	—
207	C₂²⋊Q₈	—	43	1	no	—
208	C₂²⋊Q₈	—	43	1	no	—
209	C₂²⋊Q₈	—	86	1	no	—
210	#121	—	43	1	no	—
211	#121	—	43	1	no	—
212	#121	—	86	1	no	—
213	C₄×Q₈	—	43	1	no	—
214	C₄×Q₈	—	86	1	no	—
215	C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
216	C₄×D₄	—	43	1	no	—
217	C₄×D₄	—	86	1	no	—
218	C₄×D₄	C₂×D₄₃	1	1	yes	characteristic
219	#117	—	86	1	no	—
220	C₂×C₄²	—	43	1	no	—
221	#107	—	43	1	no	—
222	#107	—	43	1	no	—
223	#107	—	43	1	no	—
224	#107	C₂×D₄₃	1	1	yes	characteristic
225	#103	—	43	1	no	—
226	#103	—	43	1	no	—
227	#103	—	43	1	no	—
228	#103	C₂×D₄₃	1	1	yes	characteristic
229	#102	—	43	1	no	—
230	#102	—	43	1	no	—
231	#102	—	43	1	no	—
232	#102	C₂×D₄₃	1	1	yes	characteristic
233	#97	—	43	1	no	—
234	#97	C₂×D₄₃	1	1	yes	characteristic
235	C₄×C₈	—	43	1	no	—
236	C₄×C₈	—	43	1	no	—
237	#41	—	172	4	no	—
238	C₂×Q₈	—	43	1	no	—
239	C₂×Q₈	—	86	1	no	—
240	C₂×Q₈	—	86	1	no	—
241	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
242	C₂×D₄	—	43	1	no	—
243	C₂×D₄	—	86	1	no	—
244	C₂×D₄	—	86	1	no	—
245	C₂×D₄	—	86	1	no	—
246	C₂×D₄	—	86	1	no	—
247	C₂×D₄	C₂²×D₄₃	1	1	yes	characteristic
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	—	43	1	no	—
250	C₂²×C₄	—	43	1	no	—
251	C₂²×C₄	—	43	1	no	—
252	C₂²×C₄	—	86	1	no	—
253	SD₁₆	—	2	2	no	—
254	SD₁₆	—	86	2	no	—
255	SD₁₆	—	86	2	no	—
256	SD₁₆	—	86	2	no	—
257	M₁₆	—	86	2	no	—
258	M₁₆	—	86	2	no	—
259	C₂×C₈	—	43	1	no	—
260	C₂×C₈	—	43	1	no	—
261	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
262	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
263	C₄⋊C₄	—	2	1	no	—
264	C₄⋊C₄	—	43	1	no	—
265	C₄⋊C₄	—	43	1	no	—
266	C₄⋊C₄	—	86	1	no	—
267	C₄⋊C₄	—	86	1	no	—
268	C₄⋊C₄	—	86	1	no	—
269	C₄⋊C₄	—	86	1	no	—
270	C₄⋊C₄	—	86	1	no	—
271	C₄⋊C₄	—	86	1	no	—
272	C₄⋊C₄	—	86	1	no	—
273	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
274	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
275	C₂²⋊C₄	—	2	1	no	—
276	C₂²⋊C₄	—	86	1	no	—
277	C₂²⋊C₄	—	86	1	no	—
278	C₂²⋊C₄	—	86	1	no	—
279	C₂²⋊C₄	—	86	1	no	—
280	C₂²⋊C₄	—	86	1	no	—
281	C₄²	—	2	1	no	—
282	C₄²	—	43	1	no	—
283	C₄²	—	43	1	no	—
284	C₄²	—	43	1	no	—
285	C₄²	—	86	1	no	—
286	C₄²	C₂²×D₄₃	1	1	yes	characteristic
287	C₂³	—	2	1	no	—
288	C₂³	—	43	1	no	—
289	C₂³	—	86	1	no	—
290	Q₈	—	2	1	no	—
291	Q₈	—	2	1	no	—
292	Q₈	—	86	1	no	—
293	Q₈	—	86	1	no	—
294	Q₈	—	172	2	no	—
295	Q₈	—	172	2	no	—
296	D₄	—	2	1	no	—
297	D₄	—	2	1	no	—
298	D₄	—	86	1	no	—
299	D₄	—	86	1	no	—
300	D₄	—	172	2	no	—
301	D₄	—	172	2	no	—
302	D₄	—	172	2	no	—
303	D₄	—	172	2	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	43	1	no	—
309	C₂×C₄	—	43	1	no	—
310	C₂×C₄	—	43	1	no	—
311	C₂×C₄	—	86	2	no	—
312	C₂×C₄	—	86	2	no	—
313	C₂×C₄	—	86	1	no	—
314	C₂×C₄	—	86	1	no	—
315	C₂×C₄	—	86	1	no	—
316	C₂×C₄	—	86	1	no	—
317	C₂×C₄	—	86	1	no	—
318	C₂×C₄	—	172	2	no	—
319	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
320	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
321	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
322	C₈	—	2	1	no	—
323	C₈	—	43	2	no	—
324	C₈	—	86	1	no	—
325	C₈	#10621898	1	2	yes	complemented
326	C₂²	—	4	1	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	86	1	no	—
330	C₂²	—	86	1	no	—
331	C₂²	—	86	1	no	—
332	C₂²	—	172	1	no	—
333	C₂²	—	172	1	no	—
334	C₂²	—	172	1	no	—
335	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	4	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	4	1	no	—
341	C₄	—	43	2	no	—
342	C₄	—	43	2	no	—
343	C₄	—	86	1	no	—
344	C₄	—	172	1	no	—
345	C₄	—	172	1	no	—
346	C₄	—	172	1	no	—
347	C₄	#41×D₄₃	1	1	yes	characteristic
348	C₄	C₂×#10621898	1	1	yes	characteristic
349	C₂	—	8	1	no	—
350	C₂	—	344	1	no	—
351	C₂	—	172	1	no	—
352	C₂	#14583446	1	1	yes	characteristic
353	C₂	AGL(1,ℤ/8)×D₄₃	1	1	yes	characteristic
354	C₂	#137×D₄₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#440	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#440	D₄₃
hypocenter	C₄₃	#3182
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #440

Complements

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

N	H ≅ G/N	type
1. #14583453	C₂	semidirect
3. #14583303	C₂	semidirect
4. #14583270	C₂	semidirect
5. #14583248	C₂	semidirect
6. #14583244	C₂	semidirect
7. #14583148	C₂	semidirect
8. #14583146	C₂	semidirect
9. #14583144	C₂	semidirect
10. #14583142	C₂	semidirect
11. #14583001	C₂	semidirect
12. #14582999	C₂	semidirect
13. #14582997	C₂	semidirect
14. #14582851	C₂	semidirect
15. C₄₃×#440	C₂	semidirect
145. C₈×C₄₃	#41	semidirect
147. C₄₃⋊C₈	#41	semidirect
191. #440	D₄₃	semidirect
193. C₄₃	#3182	semidirect
325. C₈	#10621898	semidirect