Subgroups — #544×D₄₃ #19147991

Order 5504.

1080 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583530	C₂	1	1	yes	maximal, characteristic, complemented
2	#14583529	C₂	1	4	yes	maximal, complemented
3	#14583528	C₂	1	4	yes	maximal, complemented
4	#14583527	C₂	1	2	yes	maximal, complemented
5	#14583526	C₂	1	4	yes	maximal, complemented
6	C₄⋊Q₈×D₄₃	C₂	1	4	yes	maximal, complemented
7	#125×D₄₃	C₂	1	4	yes	maximal, complemented
8	C₂²⋊Q₈×D₄₃	C₂	1	4	yes	maximal
9	#117×D₄₃	C₂	1	2	yes	maximal
10	C₂×C₄⋊C₄×D₄₃	C₂	1	1	yes	maximal, characteristic
11	C₄₃×#544	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₄₃	C₂²	1	4	yes	—
13	C₂²×C₄×D₄₃	C₂²	1	2	yes	—
14	C₂²×C₄×D₄₃	C₂²	1	1	yes	characteristic
15	#13063325	C₂²	1	4	yes	complemented
16	#13063324	C₂²	1	4	yes	complemented
17	#13063324	C₂²	1	8	yes	complemented
18	#13063324	C₂²	1	4	yes	complemented
19	#13063323	C₂²	1	8	yes	complemented
20	#13063317	C₂²	1	4	yes	complemented
21	#13063316	C₂²	1	4	yes	complemented
22	#13063316	C₂²	1	4	yes	complemented
23	#13063315	C₂²	1	8	yes	complemented
24	#13063315	C₂²	1	8	yes	complemented
25	#13063305	C₂²	1	4	yes	—
26	#13063304	C₂²	1	4	yes	—
27	#13063303	C₂²	1	8	yes	—
28	#13063302	C₂²	1	8	yes	—
29	#13063301	C₂²	1	4	yes	—
30	#13063284	C₂²	1	2	yes	—
31	#13063283	C₂²	1	2	yes	—
32	#13063283	C₂²	1	2	yes	—
33	#13063282	C₂²	1	4	yes	—
34	#13063281	C₂²	1	4	yes	—
35	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
36	C₂×Dic₄₃⋊C₄	C₂²	1	2	yes	—
37	C₄⋊C₄×D₄₃	C₂²	1	4	yes	—
38	C₄⋊C₄×D₄₃	C₂²	1	4	yes	—
39	C₄⋊C₄×D₄₃	C₂²	1	2	yes	—
40	C₄⋊C₄×D₄₃	C₂²	1	2	yes	—
41	C₄⋊C₄×D₄₃	C₂²	1	4	yes	—
42	C₄⋊C₄×D₄₃	C₂²	1	8	yes	—
43	C₂²⋊C₄×D₄₃	C₂²	1	4	yes	—
44	C₄²×D₄₃	C₂²	1	4	yes	—
45	C₄⋊Q₈×C₄₃	C₂²	1	4	yes	complemented
46	#125×C₄₃	C₂²	1	4	yes	complemented
47	C₂²⋊Q₈×C₄₃	C₂²	1	4	yes	—
48	#117×C₄₃	C₂²	1	2	yes	—
49	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
50	C₂³×D₄₃	C₂³	1	1	yes	characteristic
51	C₂×Q₈×C₄₃	C₂³	1	4	yes	—
52	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
53	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
54	C₂²×Dic₄₃	C₂³	1	2	yes	—
55	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
56	Q₈×D₄₃	—	4	8	no	—
57	C₂×C₄×D₄₃	—	4	4	no	—
58	C₂×C₄×D₄₃	Q₈	1	8	yes	complemented
59	C₂×C₄×D₄₃	C₂³	1	4	yes	—
60	C₂×C₄×D₄₃	C₂³	1	4	yes	—
61	C₂×C₄×D₄₃	C₂³	1	4	yes	—
62	C₂×C₄×D₄₃	C₂³	1	4	yes	—
63	C₂×C₄×D₄₃	C₂³	1	2	yes	—
64	C₂×C₄×D₄₃	C₂³	1	2	yes	—
65	C₂×Dic₈₆	C₂³	1	8	yes	—
66	C₂×Dic₈₆	C₂³	1	4	yes	—
67	C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
68	C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
69	C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
70	C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
71	C₄⋊C₄×C₄₃	C₂³	1	8	yes	—
72	C₂²⋊C₄×C₄₃	C₂³	1	4	yes	—
73	C₄²×C₄₃	C₂³	1	4	yes	—
74	C₂²⋊Dic₄₃	C₂³	1	4	yes	—
75	(C₂×D₄₃)⋊C₄	C₂³	1	4	yes	—
76	(C₂×D₄₃)⋊C₄	C₂³	1	4	yes	—
77	C₄⋊Dic₄₃	C₂³	1	4	yes	—
78	C₄⋊Dic₄₃	C₂³	1	2	yes	—
79	C₄⋊Dic₄₃	C₂³	1	2	yes	—
80	C₄⋊Dic₄₃	C₂³	1	4	yes	—
81	C₄⋊Dic₄₃	C₂³	1	8	yes	—
82	Dic₄₃⋊C₄	C₂³	1	8	yes	—
83	Dic₄₃⋊C₄	C₂³	1	4	yes	—
84	Dic₄₃⋊C₄	C₂³	1	4	yes	—
85	Dic₄₃⋊C₄	C₂³	1	8	yes	—
86	Dic₄₃⋊C₄	C₂³	1	8	yes	—
87	Dic₄₃⋊C₄	C₂³	1	8	yes	—
88	C₄×Dic₄₃	C₂³	1	4	yes	—
89	C₄×Dic₄₃	C₂³	1	4	yes	—
90	C₄×Dic₄₃	C₂³	1	4	yes	—
91	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
92	C₂²×D₄₃	—	2	2	no	—
93	C₂²×D₄₃	—	2	2	no	—
94	C₂²×D₄₃	C₂×Q₈	1	4	yes	—
95	C₂²×D₄₃	C₂⁴	1	1	yes	characteristic
96	C₂²×D₄₃	C₂⁴	1	1	yes	characteristic
97	Q₈×C₄₃	—	4	4	no	—
98	C₂×C₄×C₄₃	—	4	2	no	—
99	C₂×C₄×C₄₃	C₂×Q₈	1	4	yes	complemented
100	C₂×C₄×C₄₃	C₂⁴	1	4	yes	—
101	C₂×C₄×C₄₃	C₂⁴	1	4	yes	—
102	C₂×C₄×C₄₃	C₂⁴	1	4	yes	—
103	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
104	C₂×Dic₄₃	—	4	2	no	—
105	C₂×Dic₄₃	C₂×Q₈	1	4	yes	complemented
106	C₂×Dic₄₃	C₂⁴	1	4	yes	—
107	C₂×Dic₄₃	C₂⁴	1	4	yes	—
108	C₂×Dic₄₃	C₂⁴	1	4	yes	—
109	C₂×Dic₄₃	C₂⁴	1	2	yes	—
110	C₄×D₄₃	—	2	8	no	—
111	C₄×D₄₃	—	2	8	no	—
112	C₄×D₄₃	—	2	8	no	—
113	C₄×D₄₃	—	4	4	no	—
114	C₄×D₄₃	C₂×Q₈	1	8	yes	—
115	C₄×D₄₃	C₂×Q₈	1	8	yes	complemented
116	Dic₈₆	—	4	4	no	—
117	Dic₈₆	—	4	8	no	—
118	C₂²×C₄₃	—	2	1	no	—
119	C₂²×C₄₃	—	2	1	no	—
120	C₂²×C₄₃	C₂²×Q₈	1	2	yes	—
121	C₂²×C₄₃	C₂⁵	1	1	yes	characteristic
122	C₂×D₄₃	—	2	1	no	—
123	C₂×D₄₃	—	2	1	no	—
124	C₂×D₄₃	—	2	8	no	—
125	C₂×D₄₃	C₂²×Q₈	1	2	yes	—
126	C₂×D₄₃	C₂²×Q₈	1	2	yes	—
127	C₂×D₄₃	H(5,𝔽₂)	1	2	yes	—
128	C₂×D₄₃	2₋¹⁺⁴	1	2	yes	—
129	C₄×C₄₃	—	2	4	no	—
130	C₄×C₄₃	—	2	4	no	—
131	C₄×C₄₃	—	2	4	no	—
132	C₄×C₄₃	C₂²×Q₈	1	4	yes	—
133	Dic₄₃	—	2	4	no	—
134	Dic₄₃	—	2	4	no	—
135	Dic₄₃	—	2	4	no	—
136	Dic₄₃	C₂²×Q₈	1	4	yes	—
137	C₂×#544	—	43	1	no	maximal, Sylow
138	C₂×C₄₃	—	2	2	no	—
139	C₂×C₄₃	C₂³×Q₈	1	1	yes	characteristic
140	C₂×C₄₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
141	C₂×C₄₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
142	D₄₃	—	2	2	no	—
143	D₄₃	#544	1	4	yes	direct product factor
144	#544	—	43	4	no	—
145	#544	—	43	4	no	—
146	#544	—	43	4	no	—
147	#544	—	43	2	no	—
148	#544	—	43	1	no	—
149	#544	D₄₃	1	1	yes	characteristic, direct product factor
150	C₂×C₄⋊Q₈	—	43	4	no	—
151	C₂×#125	—	43	4	no	—
152	C₂×C₂²⋊Q₈	—	43	4	no	—
153	C₂×#117	—	43	2	no	—
154	C₂²×C₄⋊C₄	—	43	1	no	—
155	C₄₃	C₂×#544	1	1	yes	characteristic, complemented, Sylow
156	C₂²×Q₈	—	43	4	no	—
157	C₂³×C₄	—	43	1	no	—
158	C₂³×C₄	—	43	2	no	—
159	C₄⋊Q₈	—	43	8	no	—
160	C₄⋊Q₈	—	43	8	no	—
161	C₄⋊Q₈	—	43	4	no	—
162	C₄⋊Q₈	—	43	4	no	—
163	C₄⋊Q₈	—	43	4	no	—
164	C₄⋊Q₈	C₂×D₄₃	1	4	yes	complemented
165	#125	—	43	8	no	—
166	#125	—	43	8	no	—
167	#125	—	43	4	no	—
168	#125	—	43	4	no	—
169	#125	—	43	4	no	—
170	#125	C₂×D₄₃	1	4	yes	complemented
171	C₂²⋊Q₈	—	43	4	no	—
172	C₂²⋊Q₈	—	43	4	no	—
173	C₂²⋊Q₈	—	43	8	no	—
174	C₂²⋊Q₈	—	43	4	no	—
175	C₂²⋊Q₈	—	43	8	no	—
176	C₂²⋊Q₈	C₂×D₄₃	1	4	yes	—
177	#117	—	43	2	no	—
178	#117	—	43	2	no	—
179	#117	—	43	2	no	—
180	#117	—	43	4	no	—
181	#117	—	43	4	no	—
182	#117	C₂×D₄₃	1	2	yes	—
183	C₂×C₄⋊C₄	—	43	2	no	—
184	C₂×C₄⋊C₄	—	43	1	no	—
185	C₂×C₄⋊C₄	—	43	4	no	—
186	C₂×C₄⋊C₄	—	43	4	no	—
187	C₂×C₄⋊C₄	—	43	2	no	—
188	C₂×C₄⋊C₄	—	43	2	no	—
189	C₂×C₄⋊C₄	—	43	4	no	—
190	C₂×C₄⋊C₄	—	43	8	no	—
191	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
192	C₂×C₂²⋊C₄	—	43	4	no	—
193	C₂×C₄²	—	43	4	no	—
194	C₂⁴	—	43	1	no	—
195	C₂×Q₈	—	43	4	no	—
196	C₂×Q₈	—	43	8	no	—
197	C₂×Q₈	—	172	8	no	—
198	C₂×Q₈	C₂²×D₄₃	1	4	yes	—
199	C₂²×C₄	—	43	1	no	—
200	C₂²×C₄	—	43	8	no	—
201	C₂²×C₄	—	43	2	no	—
202	C₂²×C₄	—	43	2	no	—
203	C₂²×C₄	—	43	2	no	—
204	C₂²×C₄	—	43	4	no	—
205	C₂²×C₄	—	43	4	no	—
206	C₂²×C₄	—	43	4	no	—
207	C₂²×C₄	—	43	4	no	—
208	C₂²×C₄	—	172	4	no	—
209	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
210	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
211	C₄⋊C₄	—	43	8	no	—
212	C₄⋊C₄	—	43	4	no	—
213	C₄⋊C₄	—	43	4	no	—
214	C₄⋊C₄	—	43	2	no	—
215	C₄⋊C₄	—	43	4	no	—
216	C₄⋊C₄	—	43	2	no	—
217	C₄⋊C₄	—	43	8	no	—
218	C₄⋊C₄	—	43	4	no	—
219	C₄⋊C₄	—	43	8	no	—
220	C₄⋊C₄	—	43	8	no	—
221	C₄⋊C₄	—	43	8	no	—
222	C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
223	C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
224	C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
225	C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
226	C₄⋊C₄	C₂²×D₄₃	1	8	yes	—
227	C₂²⋊C₄	—	43	4	no	—
228	C₂²⋊C₄	—	43	4	no	—
229	C₂²⋊C₄	—	43	4	no	—
230	C₂²⋊C₄	C₂²×D₄₃	1	4	yes	—
231	C₄²	—	43	4	no	—
232	C₄²	—	43	4	no	—
233	C₄²	—	43	4	no	—
234	C₄²	C₂²×D₄₃	1	4	yes	—
235	C₂³	—	43	1	no	—
236	C₂³	—	43	4	no	—
237	C₂³	—	43	1	no	—
238	C₂³	—	86	2	no	—
239	C₂³	—	86	2	no	—
240	C₂³	C₂³×D₄₃	1	1	yes	characteristic
241	Q₈	—	4	4	no	—
242	Q₈	—	172	4	no	—
243	Q₈	—	172	8	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	43	4	no	—
246	C₂×C₄	—	43	2	no	—
247	C₂×C₄	—	43	8	no	—
248	C₂×C₄	—	43	8	no	—
249	C₂×C₄	—	43	4	no	—
250	C₂×C₄	—	43	4	no	—
251	C₂×C₄	—	43	4	no	—
252	C₂×C₄	—	86	8	no	—
253	C₂×C₄	—	86	8	no	—
254	C₂×C₄	—	86	8	no	—
255	C₂×C₄	—	172	4	no	—
256	C₂×C₄	—	172	2	no	—
257	C₂×C₄	Q₈×D₄₃	1	4	yes	complemented
258	C₂×C₄	C₂³×D₄₃	1	4	yes	—
259	C₂×C₄	C₂³×D₄₃	1	4	yes	—
260	C₂×C₄	C₂³×D₄₃	1	4	yes	—
261	C₂×C₄	C₂³×D₄₃	1	2	yes	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	43	2	no	—
265	C₂²	—	43	2	no	—
266	C₂²	—	43	2	no	—
267	C₂²	—	43	2	no	—
268	C₂²	—	86	8	no	—
269	C₂²	—	86	1	no	—
270	C₂²	—	86	1	no	—
271	C₂²	C₂×Q₈×D₄₃	1	2	yes	—
272	C₂²	C₂⁴×D₄₃	1	1	yes	characteristic
273	C₄	—	2	4	no	—
274	C₄	—	2	4	no	—
275	C₄	—	2	4	no	—
276	C₄	—	43	4	no	—
277	C₄	—	86	4	no	—
278	C₄	—	86	4	no	—
279	C₄	—	86	4	no	—
280	C₄	C₂×Q₈×D₄₃	1	4	yes	—
281	C₂	—	2	2	no	—
282	C₂	—	43	4	no	—
283	C₂	—	86	2	no	—
284	C₂	C₂²×Q₈×D₄₃	1	1	yes	characteristic
285	C₂	H(5,𝔽₂)×D₄₃	1	1	yes	characteristic
286	C₂	2₋¹⁺⁴×D₄₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₄₃
commutator, socle	C₂²×C₄₃	C₂⁵
Fitting	C₄₃×#544	C₂
hypercenter	#544	D₄₃
hypocenter	C₄₃	C₂×#544
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ #544

Complements

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

N	H ≅ G/N	type
1. #14583530	C₂	semidirect
2. #14583529	C₂	semidirect
3. #14583528	C₂	semidirect
4. #14583527	C₂	semidirect
5. #14583526	C₂	semidirect
6. C₄⋊Q₈×D₄₃	C₂	semidirect
7. #125×D₄₃	C₂	semidirect
11. C₄₃×#544	C₂	semidirect
15. #13063325	C₂²	semidirect
16. #13063324	C₂²	semidirect
17. #13063324	C₂²	semidirect
18. #13063324	C₂²	semidirect
19. #13063323	C₂²	semidirect
20. #13063317	C₂²	semidirect
21. #13063316	C₂²	semidirect
22. #13063316	C₂²	semidirect
23. #13063315	C₂²	semidirect
24. #13063315	C₂²	semidirect
45. C₄⋊Q₈×C₄₃	C₂²	semidirect
46. #125×C₄₃	C₂²	semidirect
58. C₂×C₄×D₄₃	Q₈	semidirect
99. C₂×C₄×C₄₃	C₂×Q₈	semidirect
105. C₂×Dic₄₃	C₂×Q₈	semidirect
115. C₄×D₄₃	C₂×Q₈	semidirect
143. D₄₃	#544	direct (N × H)
149. #544	D₄₃	direct (N × H)
155. C₄₃	C₂×#544	semidirect
164. C₄⋊Q₈	C₂×D₄₃	semidirect
170. #125	C₂×D₄₃	semidirect
257. C₂×C₄	Q₈×D₄₃	semidirect

Subgroups — #544×D43 #19147991

Conjugacy classes of subgroups

Complements

Subgroups — #544×D₄₃ #19147991