Subgroups — #576×D₄₃ #19149343

Order 5504.

1512 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	H(5,𝔽₂)×D₄₃	C₂	1	2	yes	maximal, complemented
2	#14583704	C₂	1	1	yes	maximal, characteristic, complemented
3	#14583703	C₂	1	6	yes	maximal, complemented
4	#14583702	C₂	1	2	yes	maximal, complemented
5	#14583701	C₂	1	3	yes	maximal, complemented
6	#14583700	C₂	1	3	yes	maximal, complemented
7	AGL(1,ℤ/8)×D₄₃	C₂	1	6	yes	maximal, complemented
8	#135×D₄₃	C₂	1	3	yes	maximal, complemented
9	C₂×D₈×D₄₃	C₂	1	3	yes	maximal, complemented
10	#131×D₄₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#576	C₂	1	1	yes	maximal, characteristic, complemented
12	#13063371	—	2	2	no	—
13	#13063371	C₂²	1	6	yes	complemented
14	#13063371	C₂²	1	2	yes	complemented
15	#13063370	—	2	6	no	—
16	#13063370	C₂²	1	6	yes	complemented
17	#41×D₄₃	—	2	2	no	—
18	#41×D₄₃	C₂²	1	1	yes	characteristic
19	#41×D₄₃	C₂²	1	6	yes	complemented
20	C₂×D₄×D₄₃	—	2	6	no	—
21	C₂×D₄×D₄₃	C₂²	1	6	yes	complemented
22	#13063353	C₂²	1	6	yes	complemented
23	#13063352	C₂²	1	12	yes	complemented
24	#13063351	C₂²	1	6	yes	complemented
25	#13063350	C₂²	1	6	yes	complemented
26	#13063349	C₂²	1	12	yes	complemented
27	#13063348	C₂²	1	3	yes	complemented
28	#13063347	C₂²	1	3	yes	complemented
29	#13063346	C₂²	1	6	yes	complemented
30	#13063345	C₂²	1	6	yes	complemented
31	#13063344	C₂²	1	3	yes	complemented
32	Q₁₆×D₄₃	C₂²	1	1	yes	characteristic, complemented
33	SD₁₆×D₄₃	C₂²	1	6	yes	complemented
34	C₂×D₃₄₄	C₂²	1	3	yes	complemented
35	C₂×#10621882	C₂²	1	6	yes	complemented
36	D₈×D₄₃	C₂²	1	3	yes	complemented
37	D₈×D₄₃	C₂²	1	6	yes	complemented
38	D₈×D₄₃	C₂²	1	6	yes	complemented
39	D₈×D₄₃	C₂²	1	6	yes	complemented
40	#13063333	C₂²	1	3	yes	complemented
41	#13063332	C₂²	1	1	yes	characteristic, complemented
42	#13063331	C₂²	1	3	yes	complemented
43	M₁₆×D₄₃	C₂²	1	3	yes	complemented
44	C₂×C₈×D₄₃	C₂²	1	3	yes	complemented
45	H(5,𝔽₂)×C₄₃	C₂²	1	2	yes	complemented
46	AGL(1,ℤ/8)×C₄₃	C₂²	1	6	yes	complemented
47	#135×C₄₃	C₂²	1	3	yes	complemented
48	C₂×D₈×C₄₃	C₂²	1	3	yes	complemented
49	#131×C₄₃	C₂²	1	1	yes	characteristic, complemented
50	C₂³×D₄₃	—	2	6	no	—
51	#41×C₄₃	—	2	2	no	—
52	#41×C₄₃	C₂³	1	1	yes	characteristic
53	#41×C₄₃	C₂³	1	6	yes	complemented
54	C₂×D₄×C₄₃	—	2	6	no	—
55	C₂×D₄×C₄₃	C₂³	1	6	yes	complemented
56	C₂×C₂²⋊D₄₃	—	2	12	no	—
57	C₂×C₂²⋊D₄₃	—	2	6	no	—
58	#10621902	—	2	2	no	—
59	#10621902	D₄	1	1	yes	characteristic
60	#10621902	C₂³	1	6	yes	complemented
61	Q₈×D₄₃	D₄	1	1	yes	characteristic
62	Q₈×D₄₃	C₂³	1	2	yes	—
63	#10621900	—	2	6	no	—
64	#10621900	—	2	6	no	—
65	#10621900	D₄	1	3	yes	—
66	#10621900	C₂³	1	6	yes	complemented
67	D₄×D₄₃	—	2	6	no	—
68	D₄×D₄₃	—	2	12	no	—
69	D₄×D₄₃	—	2	6	no	—
70	D₄×D₄₃	—	2	6	no	—
71	D₄×D₄₃	—	2	6	no	—
72	D₄×D₄₃	—	2	6	no	—
73	D₄×D₄₃	D₄	1	3	yes	—
74	D₄×D₄₃	C₂³	1	6	yes	—
75	D₄×D₄₃	C₂³	1	12	yes	complemented
76	#10621898	—	2	6	no	—
77	#10621898	—	2	6	no	—
78	#10621898	C₂³	1	3	yes	—
79	#10621898	C₂³	1	6	yes	complemented
80	C₂×D₁₇₂	—	2	6	no	—
81	C₂×D₁₇₂	C₂³	1	6	yes	complemented
82	C₂×C₄×D₄₃	—	2	6	no	—
83	C₂×C₄×D₄₃	C₂³	1	3	yes	—
84	Q₁₆×C₄₃	C₂³	1	1	yes	characteristic, complemented
85	SD₁₆×C₄₃	C₂³	1	6	yes	complemented
86	D₈×C₄₃	C₂³	1	3	yes	complemented
87	D₈×C₄₃	C₂³	1	6	yes	complemented
88	M₁₆×C₄₃	C₂³	1	3	yes	complemented
89	C₂×C₈×C₄₃	C₂³	1	3	yes	complemented
90	#10621885	C₂³	1	2	yes	complemented
91	#10621884	C₂³	1	6	yes	complemented
92	#10621883	C₂³	1	6	yes	complemented
93	#10621882	C₂³	1	6	yes	complemented
94	#10621882	C₂³	1	12	yes	complemented
95	#10621877	C₂³	1	3	yes	complemented
96	C₂×C₄₃⋊C₈	C₂³	1	3	yes	complemented
97	Dic₁₇₂	C₂³	1	1	yes	characteristic, complemented
98	D₃₄₄	C₂³	1	3	yes	complemented
99	D₃₄₄	C₂³	1	6	yes	complemented
100	#10621873	C₂³	1	6	yes	complemented
101	#10621872	C₂³	1	6	yes	complemented
102	C₈×D₄₃	C₂³	1	1	yes	characteristic
103	C₈×D₄₃	C₂³	1	3	yes	—
104	C₈×D₄₃	C₂³	1	3	yes	complemented
105	C₈×D₄₃	C₂³	1	3	yes	complemented
106	C₂³×C₄₃	—	2	6	no	—
107	C₂²×D₄₃	—	2	6	no	—
108	C₂²×D₄₃	—	2	6	no	—
109	C₂²×D₄₃	—	2	12	no	—
110	C₂²×D₄₃	—	8	12	no	—
111	C₂²×D₄₃	C₂×D₄	1	3	yes	—
112	Q₈×C₄₃	C₂×D₄	1	1	yes	characteristic
113	Q₈×C₄₃	C₂⁴	1	2	yes	—
114	D₄×C₄₃	—	2	6	no	—
115	D₄×C₄₃	—	2	6	no	—
116	D₄×C₄₃	C₂×D₄	1	3	yes	—
117	D₄×C₄₃	C₂⁴	1	6	yes	—
118	C₂×C₄×C₄₃	—	2	6	no	—
119	C₂×C₄×C₄₃	C₂⁴	1	3	yes	—
120	C₂²⋊D₄₃	—	2	6	no	—
121	C₂²⋊D₄₃	—	2	6	no	—
122	C₂²⋊D₄₃	—	2	6	no	—
123	C₂²⋊D₄₃	—	2	12	no	—
124	C₂²⋊D₄₃	C₂×D₄	1	6	yes	—
125	C₂×Dic₄₃	—	2	6	no	—
126	C₂×Dic₄₃	C₂×D₄	1	3	yes	—
127	D₁₇₂	—	2	6	no	—
128	D₁₇₂	—	2	6	no	—
129	D₁₇₂	C₂×D₄	1	3	yes	—
130	D₁₇₂	C₂⁴	1	6	yes	—
131	C₄×D₄₃	—	2	2	no	—
132	C₄×D₄₃	—	2	6	no	—
133	C₄×D₄₃	—	2	6	no	—
134	C₄×D₄₃	C₂×D₄	1	3	yes	—
135	C₄×D₄₃	C₂×D₄	1	3	yes	—
136	C₄×D₄₃	C₂⁴	1	1	yes	characteristic
137	C₄×D₄₃	C₂⁴	1	3	yes	—
138	Dic₈₆	—	2	2	no	—
139	Dic₈₆	C₂×D₄	1	3	yes	—
140	Dic₈₆	C₂⁴	1	2	yes	—
141	C₈×C₄₃	C₂⁴	1	1	yes	characteristic
142	C₈×C₄₃	C₂⁴	1	3	yes	—
143	C₄₃⋊C₈	C₂⁴	1	1	yes	characteristic
144	C₄₃⋊C₈	C₂⁴	1	3	yes	—
145	C₂²×C₄₃	—	2	6	no	—
146	C₂²×C₄₃	—	8	6	no	—
147	C₂²×C₄₃	C₂²×D₄	1	3	yes	—
148	C₂×D₄₃	—	2	6	no	—
149	C₂×D₄₃	—	2	6	no	—
150	C₂×D₄₃	—	4	12	no	—
151	C₂×D₄₃	—	8	12	no	—
152	C₂×D₄₃	—	8	6	no	—
153	C₂×D₄₃	C₂²×D₄	1	3	yes	—
154	C₂×D₄₃	C₂²×D₄	1	1	yes	characteristic
155	C₄×C₄₃	—	2	2	no	—
156	C₄×C₄₃	C₂²×D₄	1	3	yes	—
157	C₄×C₄₃	C₂⁵	1	1	yes	characteristic
158	Dic₄₃	—	2	2	no	—
159	Dic₄₃	C₂²×D₄	1	1	yes	characteristic
160	Dic₄₃	C₂²×D₄	1	3	yes	—
161	C₂×#576	—	43	1	no	maximal, Sylow
162	C₂×C₄₃	—	2	3	no	—
163	C₂×C₄₃	—	4	6	no	—
164	C₂×C₄₃	C₂³×D₄	1	1	yes	characteristic
165	D₄₃	—	2	3	no	—
166	D₄₃	—	4	6	no	—
167	D₄₃	#576	1	2	yes	direct product factor
168	C₂×H(5,𝔽₂)	—	43	2	no	—
169	#576	—	43	2	no	—
170	#576	—	43	6	no	—
171	#576	—	43	3	no	—
172	#576	—	43	3	no	—
173	#576	—	43	1	no	—
174	#576	D₄₃	1	1	yes	characteristic, direct product factor
175	C₂×AGL(1,ℤ/8)	—	43	6	no	—
176	C₂×#135	—	43	3	no	—
177	C₂²×D₈	—	43	3	no	—
178	C₂×#131	—	43	1	no	—
179	C₄₃	C₂×#576	1	1	yes	characteristic, complemented, Sylow
180	H(5,𝔽₂)	—	43	2	no	—
181	H(5,𝔽₂)	—	43	6	no	—
182	H(5,𝔽₂)	—	43	6	no	—
183	H(5,𝔽₂)	—	86	6	no	—
184	H(5,𝔽₂)	—	86	2	no	—
185	H(5,𝔽₂)	C₂×D₄₃	1	2	yes	complemented
186	C₂×#41	—	43	1	no	—
187	C₂×#41	—	43	6	no	—
188	C₂×#41	—	86	2	no	—
189	C₂²×D₄	—	43	6	no	—
190	C₂²×D₄	—	86	6	no	—
191	AGL(1,ℤ/8)	—	43	6	no	—
192	AGL(1,ℤ/8)	—	43	6	no	—
193	AGL(1,ℤ/8)	—	43	12	no	—
194	AGL(1,ℤ/8)	—	43	12	no	—
195	AGL(1,ℤ/8)	—	43	6	no	—
196	AGL(1,ℤ/8)	C₂×D₄₃	1	6	yes	complemented
197	#135	—	43	6	no	—
198	#135	—	43	3	no	—
199	#135	—	43	6	no	—
200	#135	—	43	3	no	—
201	#135	—	43	3	no	—
202	#135	C₂×D₄₃	1	3	yes	complemented
203	C₂×Q₁₆	—	43	1	no	—
204	C₂×SD₁₆	—	43	6	no	—
205	C₂×D₈	—	43	6	no	—
206	C₂×D₈	—	43	3	no	—
207	C₂×D₈	—	43	6	no	—
208	C₂×D₈	—	43	6	no	—
209	C₂×D₈	—	43	6	no	—
210	C₂×D₈	—	43	3	no	—
211	C₂×D₈	C₂×D₄₃	1	3	yes	complemented
212	#131	—	43	1	no	—
213	#131	—	43	3	no	—
214	#131	—	43	3	no	—
215	#131	C₂×D₄₃	1	1	yes	characteristic, complemented
216	C₂×M₁₆	—	43	3	no	—
217	C₂²×C₈	—	43	3	no	—
218	C₂⁴	—	86	6	no	—
219	#41	—	2	2	no	—
220	#41	—	43	1	no	—
221	#41	—	43	3	no	—
222	#41	—	43	3	no	—
223	#41	—	43	6	no	—
224	#41	—	43	6	no	—
225	#41	—	43	6	no	—
226	#41	—	86	6	no	—
227	#41	—	86	6	no	—
228	#41	—	86	6	no	—
229	#41	—	86	6	no	—
230	#41	—	86	2	no	—
231	#41	C₂²×D₄₃	1	1	yes	characteristic
232	#41	C₂²×D₄₃	1	6	yes	complemented
233	C₂×Q₈	—	43	1	no	—
234	C₂×Q₈	—	43	2	no	—
235	C₂×D₄	—	2	6	no	—
236	C₂×D₄	—	43	3	no	—
237	C₂×D₄	—	43	6	no	—
238	C₂×D₄	—	43	12	no	—
239	C₂×D₄	—	43	6	no	—
240	C₂×D₄	—	86	6	no	—
241	C₂×D₄	—	86	6	no	—
242	C₂×D₄	—	86	6	no	—
243	C₂×D₄	—	86	6	no	—
244	C₂×D₄	—	86	12	no	—
245	C₂×D₄	—	86	6	no	—
246	C₂×D₄	—	86	6	no	—
247	C₂×D₄	—	86	12	no	—
248	C₂×D₄	—	86	6	no	—
249	C₂×D₄	C₂²×D₄₃	1	6	yes	complemented
250	C₂²×C₄	—	43	3	no	—
251	C₂²×C₄	—	86	6	no	—
252	Q₁₆	—	43	2	no	—
253	Q₁₆	—	43	1	no	—
254	Q₁₆	C₂²×D₄₃	1	1	yes	characteristic, complemented
255	SD₁₆	—	43	6	no	—
256	SD₁₆	—	43	6	no	—
257	SD₁₆	—	43	6	no	—
258	SD₁₆	C₂²×D₄₃	1	6	yes	complemented
259	D₈	—	43	6	no	—
260	D₈	—	43	12	no	—
261	D₈	—	43	3	no	—
262	D₈	—	43	6	no	—
263	D₈	C₂²×D₄₃	1	3	yes	complemented
264	D₈	C₂²×D₄₃	1	6	yes	complemented
265	M₁₆	—	43	3	no	—
266	M₁₆	—	43	6	no	—
267	M₁₆	C₂²×D₄₃	1	3	yes	complemented
268	C₂×C₈	—	43	3	no	—
269	C₂×C₈	—	43	1	no	—
270	C₂×C₈	—	43	3	no	—
271	C₂×C₈	—	43	3	no	—
272	C₂×C₈	—	43	3	no	—
273	C₂×C₈	C₂²×D₄₃	1	3	yes	complemented
274	C₂³	—	2	6	no	—
275	C₂³	—	43	3	no	—
276	C₂³	—	86	12	no	—
277	C₂³	—	86	6	no	—
278	C₂³	—	86	6	no	—
279	C₂³	—	344	12	no	—
280	Q₈	—	43	3	no	—
281	Q₈	—	43	2	no	—
282	Q₈	—	86	2	no	—
283	Q₈	D₄×D₄₃	1	1	yes	characteristic
284	Q₈	C₂³×D₄₃	1	2	yes	—
285	D₄	—	2	6	no	—
286	D₄	—	2	6	no	—
287	D₄	—	43	6	no	—
288	D₄	—	43	3	no	—
289	D₄	—	43	6	no	—
290	D₄	—	86	6	no	—
291	D₄	—	86	6	no	—
292	D₄	—	86	12	no	—
293	D₄	—	86	6	no	—
294	D₄	—	86	6	no	—
295	D₄	—	86	6	no	—
296	D₄	D₄×D₄₃	1	3	yes	—
297	D₄	C₂³×D₄₃	1	6	yes	—
298	C₂×C₄	—	2	6	no	—
299	C₂×C₄	—	43	3	no	—
300	C₂×C₄	—	43	3	no	—
301	C₂×C₄	—	43	3	no	—
302	C₂×C₄	—	43	1	no	—
303	C₂×C₄	—	43	3	no	—
304	C₂×C₄	—	86	6	no	—
305	C₂×C₄	—	86	2	no	—
306	C₂×C₄	—	86	6	no	—
307	C₂×C₄	—	86	6	no	—
308	C₂×C₄	C₂³×D₄₃	1	3	yes	—
309	C₈	—	43	1	no	—
310	C₈	—	43	3	no	—
311	C₈	C₂³×D₄₃	1	1	yes	characteristic
312	C₈	C₂³×D₄₃	1	3	yes	—
313	C₂²	—	2	6	no	—
314	C₂²	—	8	6	no	—
315	C₂²	—	43	1	no	—
316	C₂²	—	43	3	no	—
317	C₂²	—	86	6	no	—
318	C₂²	—	86	6	no	—
319	C₂²	—	344	6	no	—
320	C₂²	—	344	12	no	—
321	C₂²	—	172	12	no	—
322	C₂²	C₂×D₄×D₄₃	1	3	yes	—
323	C₄	—	2	2	no	—
324	C₄	—	43	1	no	—
325	C₄	—	43	3	no	—
326	C₄	—	86	2	no	—
327	C₄	C₂×D₄×D₄₃	1	3	yes	—
328	C₄	C₂⁴×D₄₃	1	1	yes	characteristic
329	C₂	—	2	3	no	—
330	C₂	—	4	6	no	—
331	C₂	—	43	2	no	—
332	C₂	—	86	3	no	—
333	C₂	—	172	6	no	—
334	C₂	C₂²×D₄×D₄₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₄×D₄₃
commutator	C₄×C₄₃	C₂⁵
Frattini	C₄	C₂⁴×D₄₃
Fitting	C₄₃×#576	C₂
socle	C₂×C₄₃	C₂³×D₄
hypercenter	#576	D₄₃
hypocenter	C₄₃	C₂×#576
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂ ◁ #41 ◁ #576

Complements

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

N	H ≅ G/N	type
1. H(5,𝔽₂)×D₄₃	C₂	semidirect
2. #14583704	C₂	semidirect
3. #14583703	C₂	semidirect
4. #14583702	C₂	semidirect
5. #14583701	C₂	semidirect
6. #14583700	C₂	semidirect
7. AGL(1,ℤ/8)×D₄₃	C₂	semidirect
8. #135×D₄₃	C₂	semidirect
9. C₂×D₈×D₄₃	C₂	semidirect
10. #131×D₄₃	C₂	semidirect
11. C₄₃×#576	C₂	semidirect
13. #13063371	C₂²	semidirect
14. #13063371	C₂²	semidirect
16. #13063370	C₂²	semidirect
19. #41×D₄₃	C₂²	semidirect
21. C₂×D₄×D₄₃	C₂²	semidirect
22. #13063353	C₂²	semidirect
23. #13063352	C₂²	semidirect
24. #13063351	C₂²	semidirect
25. #13063350	C₂²	semidirect
26. #13063349	C₂²	semidirect
27. #13063348	C₂²	semidirect
28. #13063347	C₂²	semidirect
29. #13063346	C₂²	semidirect
30. #13063345	C₂²	semidirect
31. #13063344	C₂²	semidirect
32. Q₁₆×D₄₃	C₂²	semidirect
33. SD₁₆×D₄₃	C₂²	semidirect
34. C₂×D₃₄₄	C₂²	semidirect
35. C₂×#10621882	C₂²	semidirect
36. D₈×D₄₃	C₂²	semidirect
37. D₈×D₄₃	C₂²	semidirect
38. D₈×D₄₃	C₂²	semidirect
39. D₈×D₄₃	C₂²	semidirect
40. #13063333	C₂²	semidirect
41. #13063332	C₂²	semidirect
42. #13063331	C₂²	semidirect
43. M₁₆×D₄₃	C₂²	semidirect
44. C₂×C₈×D₄₃	C₂²	semidirect
45. H(5,𝔽₂)×C₄₃	C₂²	semidirect
46. AGL(1,ℤ/8)×C₄₃	C₂²	semidirect
47. #135×C₄₃	C₂²	semidirect
48. C₂×D₈×C₄₃	C₂²	semidirect
49. #131×C₄₃	C₂²	semidirect
53. #41×C₄₃	C₂³	semidirect
55. C₂×D₄×C₄₃	C₂³	semidirect
60. #10621902	C₂³	semidirect
66. #10621900	C₂³	semidirect
75. D₄×D₄₃	C₂³	semidirect
79. #10621898	C₂³	semidirect
81. C₂×D₁₇₂	C₂³	semidirect
84. Q₁₆×C₄₃	C₂³	semidirect
85. SD₁₆×C₄₃	C₂³	semidirect
86. D₈×C₄₃	C₂³	semidirect
87. D₈×C₄₃	C₂³	semidirect
88. M₁₆×C₄₃	C₂³	semidirect
89. C₂×C₈×C₄₃	C₂³	semidirect
90. #10621885	C₂³	semidirect
91. #10621884	C₂³	semidirect
92. #10621883	C₂³	semidirect
93. #10621882	C₂³	semidirect
94. #10621882	C₂³	semidirect
95. #10621877	C₂³	semidirect
96. C₂×C₄₃⋊C₈	C₂³	semidirect
97. Dic₁₇₂	C₂³	semidirect
98. D₃₄₄	C₂³	semidirect
99. D₃₄₄	C₂³	semidirect
100. #10621873	C₂³	semidirect
101. #10621872	C₂³	semidirect
104. C₈×D₄₃	C₂³	semidirect
105. C₈×D₄₃	C₂³	semidirect
167. D₄₃	#576	direct (N × H)
174. #576	D₄₃	direct (N × H)
179. C₄₃	C₂×#576	semidirect
185. H(5,𝔽₂)	C₂×D₄₃	semidirect
196. AGL(1,ℤ/8)	C₂×D₄₃	semidirect
202. #135	C₂×D₄₃	semidirect
211. C₂×D₈	C₂×D₄₃	semidirect
215. #131	C₂×D₄₃	semidirect
232. #41	C₂²×D₄₃	semidirect
249. C₂×D₄	C₂²×D₄₃	semidirect
254. Q₁₆	C₂²×D₄₃	semidirect
258. SD₁₆	C₂²×D₄₃	semidirect
263. D₈	C₂²×D₄₃	semidirect
264. D₈	C₂²×D₄₃	semidirect
267. M₁₆	C₂²×D₄₃	semidirect
273. C₂×C₈	C₂²×D₄₃	semidirect

Subgroups — #576×D43 #19149343

Conjugacy classes of subgroups

Complements

Subgroups — #576×D₄₃ #19149343