Subgroups — #19149363

Order 5504.

1424 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583734	C₂	1	2	yes	maximal, complemented
2	#14583716	C₂	1	1	yes	maximal, characteristic, complemented
3	#14583713	C₂	1	3	yes	maximal, complemented
4	#14583710	C₂	1	2	yes	maximal, complemented
5	#14583709	C₂	1	6	yes	maximal, complemented
6	#14583700	C₂	1	3	yes	maximal, complemented
7	#14583695	C₂	1	6	yes	maximal, complemented
8	C₂×#13063344	C₂	1	3	yes	maximal, complemented
9	#135×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	#13063372	—	2	2	no	—
13	#13063372	C₂²	1	6	yes	complemented
14	#13063372	C₂²	1	2	yes	complemented
15	#13063370	—	2	6	no	—
16	#13063370	C₂²	1	6	yes	complemented
17	C₂×#10621900	—	2	6	no	—
18	C₂×#10621900	C₂²	1	6	yes	complemented
19	#41×D₄₃	—	2	2	no	—
20	#41×D₄₃	C₂²	1	1	yes	characteristic
21	#41×D₄₃	C₂²	1	6	yes	complemented
22	#13063358	C₂²	1	6	yes	complemented
23	#13063355	C₂²	1	6	yes	complemented
24	#13063354	C₂²	1	12	yes	complemented
25	#13063351	C₂²	1	6	yes	complemented
26	#13063349	C₂²	1	12	yes	complemented
27	#13063348	C₂²	1	1	yes	characteristic, 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	#13063344	C₂²	1	6	yes	complemented
33	#13063344	C₂²	1	6	yes	complemented
34	#13063344	C₂²	1	6	yes	complemented
35	C₂×Dic₁₇₂	C₂²	1	3	yes	complemented
36	Q₁₆×D₄₃	C₂²	1	3	yes	complemented
37	C₂×#10621883	C₂²	1	6	yes	complemented
38	SD₁₆×D₄₃	C₂²	1	6	yes	complemented
39	D₈×D₄₃	C₂²	1	3	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	#41×C₄₃	—	2	2	no	—
51	#41×C₄₃	C₂³	1	6	yes	—
52	#41×C₄₃	C₂³	1	1	yes	characteristic
53	C₂×D₄×C₄₃	—	2	6	no	—
54	C₂×D₄×C₄₃	C₂³	1	6	yes	—
55	C₂×C₂²⋊D₄₃	—	2	12	no	—
56	C₂×C₂²⋊D₄₃	—	2	6	no	—
57	C₂²×Dic₄₃	—	2	6	no	—
58	#10621902	D₄	1	1	yes	characteristic
59	#10621902	C₂³	1	2	yes	—
60	Q₈×D₄₃	—	2	2	no	—
61	Q₈×D₄₃	D₄	1	1	yes	characteristic
62	Q₈×D₄₃	C₂³	1	6	yes	—
63	#10621900	—	2	6	no	—
64	#10621900	—	2	12	no	—
65	#10621900	—	2	6	no	—
66	#10621900	—	2	6	no	—
67	#10621900	—	2	6	no	—
68	#10621900	—	2	6	no	—
69	#10621900	D₄	1	3	yes	—
70	#10621900	C₂³	1	6	yes	—
71	#10621900	C₂³	1	12	yes	—
72	D₄×D₄₃	—	2	6	no	—
73	D₄×D₄₃	—	2	6	no	—
74	D₄×D₄₃	D₄	1	3	yes	—
75	D₄×D₄₃	C₂³	1	6	yes	—
76	#10621898	—	2	6	no	—
77	#10621898	—	2	6	no	—
78	#10621898	C₂³	1	3	yes	—
79	#10621898	C₂³	1	6	yes	—
80	C₂×C₄×D₄₃	—	2	6	no	—
81	C₂×C₄×D₄₃	C₂³	1	3	yes	—
82	C₂×Dic₈₆	—	2	6	no	—
83	C₂×Dic₈₆	C₂³	1	6	yes	—
84	Q₁₆×C₄₃	C₂³	1	1	yes	characteristic
85	SD₁₆×C₄₃	C₂³	1	6	yes	—
86	D₈×C₄₃	C₂³	1	3	yes	—
87	D₈×C₄₃	C₂³	1	6	yes	—
88	M₁₆×C₄₃	C₂³	1	3	yes	—
89	C₂×C₈×C₄₃	C₂³	1	3	yes	—
90	#10621885	C₂³	1	6	yes	—
91	#10621884	C₂³	1	2	yes	—
92	#10621883	C₂³	1	6	yes	—
93	#10621883	C₂³	1	12	yes	—
94	#10621882	C₂³	1	6	yes	—
95	#10621877	C₂³	1	3	yes	—
96	C₂×C₄₃⋊C₈	C₂³	1	3	yes	—
97	Dic₁₇₂	C₂³	1	3	yes	—
98	Dic₁₇₂	C₂³	1	6	yes	—
99	D₃₄₄	C₂³	1	1	yes	characteristic
100	#10621873	C₂³	1	6	yes	—
101	#10621872	C₂³	1	6	yes	—
102	C₈×D₄₃	C₂³	1	1	yes	characteristic
103	C₈×D₄₃	C₂³	1	3	yes	—
104	C₈×D₄₃	C₂³	1	3	yes	—
105	C₈×D₄₃	C₂³	1	3	yes	—
106	C₂³×C₄₃	—	2	6	no	—
107	C₂²×D₄₃	—	2	6	no	—
108	C₂²×D₄₃	C₂×D₄	1	3	yes	—
109	Q₈×C₄₃	C₂×D₄	1	1	yes	characteristic
110	Q₈×C₄₃	C₂⁴	1	2	yes	—
111	D₄×C₄₃	—	2	6	no	—
112	D₄×C₄₃	—	2	6	no	—
113	D₄×C₄₃	C₂×D₄	1	3	yes	—
114	D₄×C₄₃	C₂⁴	1	6	yes	—
115	C₂×C₄×C₄₃	—	2	6	no	—
116	C₂×C₄×C₄₃	C₂⁴	1	3	yes	—
117	C₂²⋊D₄₃	—	2	6	no	—
118	C₂²⋊D₄₃	—	2	12	no	—
119	C₂²⋊D₄₃	—	2	6	no	—
120	C₂²⋊D₄₃	—	2	6	no	—
121	C₂²⋊D₄₃	C₂×D₄	1	6	yes	—
122	C₂×Dic₄₃	—	2	6	no	—
123	C₂×Dic₄₃	—	2	12	no	—
124	C₂×Dic₄₃	—	2	6	no	—
125	C₂×Dic₄₃	C₂×D₄	1	3	yes	—
126	D₁₇₂	—	2	2	no	—
127	D₁₇₂	C₂×D₄	1	3	yes	—
128	D₁₇₂	C₂⁴	1	2	yes	—
129	C₄×D₄₃	—	2	6	no	—
130	C₄×D₄₃	—	2	6	no	—
131	C₄×D₄₃	—	2	2	no	—
132	C₄×D₄₃	C₂×D₄	1	3	yes	—
133	C₄×D₄₃	C₂×D₄	1	3	yes	—
134	C₄×D₄₃	C₂⁴	1	1	yes	characteristic
135	C₄×D₄₃	C₂⁴	1	3	yes	—
136	Dic₈₆	—	2	6	no	—
137	Dic₈₆	—	2	6	no	—
138	Dic₈₆	C₂×D₄	1	3	yes	—
139	Dic₈₆	C₂⁴	1	6	yes	—
140	C₈×C₄₃	C₂⁴	1	1	yes	characteristic
141	C₈×C₄₃	C₂⁴	1	3	yes	—
142	C₄₃⋊C₈	C₂⁴	1	1	yes	characteristic
143	C₄₃⋊C₈	C₂⁴	1	3	yes	—
144	C₂²×C₄₃	—	2	6	no	—
145	C₂²×C₄₃	—	8	6	no	—
146	C₂²×C₄₃	C₂²×D₄	1	3	yes	—
147	C₂×D₄₃	—	2	2	no	—
148	C₂×D₄₃	—	4	3	no	—
149	C₂×D₄₃	—	8	6	no	—
150	C₂×D₄₃	C₂²×D₄	1	3	yes	—
151	C₂×D₄₃	C₂²×D₄	1	1	yes	characteristic
152	C₄×C₄₃	—	2	2	no	—
153	C₄×C₄₃	C₂²×D₄	1	3	yes	—
154	C₄×C₄₃	C₂⁵	1	1	yes	characteristic
155	Dic₄₃	—	2	6	no	—
156	Dic₄₃	C₂²×D₄	1	3	yes	—
157	Dic₄₃	C₂²×D₄	1	1	yes	characteristic
158	#3584	—	43	1	no	maximal, Sylow
159	C₂×C₄₃	—	2	3	no	—
160	C₂×C₄₃	—	4	6	no	—
161	C₂×C₄₃	C₂³×D₄	1	1	yes	characteristic
162	D₄₃	—	2	1	no	—
163	D₄₃	—	2	3	no	—
164	D₄₃	—	4	2	no	—
165	#585	—	43	2	no	—
166	#578	—	43	3	no	—
167	#578	—	43	1	no	—
168	#577	—	43	2	no	—
169	#577	—	43	6	no	—
170	#576	—	43	3	no	—
171	#576	D₄₃	1	1	yes	characteristic, complemented
172	#575	—	43	6	no	—
173	C₂×#135	—	43	3	no	—
174	C₂×#135	—	43	3	no	—
175	C₂×#131	—	43	1	no	—
176	C₄₃	#3584	1	1	yes	characteristic, complemented, Sylow
177	2₋¹⁺⁴	—	43	6	no	—
178	2₋¹⁺⁴	—	43	2	no	—
179	2₋¹⁺⁴	—	86	2	no	—
180	H(5,𝔽₂)	—	43	6	no	—
181	H(5,𝔽₂)	—	86	6	no	—
182	H(5,𝔽₂)	C₂×D₄₃	1	2	yes	complemented
183	C₂×#41	—	43	1	no	—
184	C₂×#41	—	43	6	no	—
185	C₂×#41	—	43	6	no	—
186	C₂×#41	—	86	6	no	—
187	C₂×#41	—	86	2	no	—
188	#137	—	43	6	no	—
189	#137	—	43	12	no	—
190	#137	—	43	6	no	—
191	AGL(1,ℤ/8)	—	43	6	no	—
192	AGL(1,ℤ/8)	—	43	12	no	—
193	AGL(1,ℤ/8)	C₂×D₄₃	1	6	yes	complemented
194	#135	—	43	6	no	—
195	#135	—	43	1	no	—
196	#135	—	43	6	no	—
197	#135	—	43	3	no	—
198	#135	—	43	6	no	—
199	#135	—	43	6	no	—
200	#135	—	43	6	no	—
201	#135	—	43	3	no	—
202	#135	C₂×D₄₃	1	3	yes	complemented
203	C₂×Q₁₆	—	43	3	no	—
204	C₂×Q₁₆	—	43	3	no	—
205	C₂×SD₁₆	—	43	6	no	—
206	C₂×SD₁₆	—	43	6	no	—
207	C₂×D₈	—	43	3	no	—
208	C₂×D₈	C₂×D₄₃	1	3	yes	complemented
209	#131	—	43	1	no	—
210	#131	—	43	3	no	—
211	#131	—	43	3	no	—
212	#131	C₂×D₄₃	1	1	yes	characteristic, complemented
213	C₂×M₁₆	—	43	3	no	—
214	C₂²×C₈	—	43	3	no	—
215	#41	—	2	2	no	—
216	#41	—	43	1	no	—
217	#41	—	43	3	no	—
218	#41	—	43	3	no	—
219	#41	—	43	2	no	—
220	#41	—	43	6	no	—
221	#41	—	43	12	no	—
222	#41	—	43	6	no	—
223	#41	—	86	6	no	—
224	#41	—	86	6	no	—
225	#41	—	86	6	no	—
226	#41	—	86	6	no	—
227	#41	—	86	6	no	—
228	#41	—	86	12	no	—
229	#41	—	86	6	no	—
230	#41	—	86	6	no	—
231	#41	C₂²×D₄₃	1	6	yes	—
232	#41	C₂²×D₄₃	1	1	yes	characteristic
233	C₂×Q₈	—	43	1	no	—
234	C₂×Q₈	—	43	6	no	—
235	C₂×Q₈	—	43	6	no	—
236	C₂×Q₈	—	86	6	no	—
237	C₂×Q₈	—	86	2	no	—
238	C₂×D₄	—	2	6	no	—
239	C₂×D₄	—	43	3	no	—
240	C₂×D₄	—	43	6	no	—
241	C₂×D₄	—	86	6	no	—
242	C₂×D₄	—	86	6	no	—
243	C₂×D₄	—	86	12	no	—
244	C₂×D₄	—	86	6	no	—
245	C₂×D₄	C₂²×D₄₃	1	6	yes	—
246	C₂²×C₄	—	43	3	no	—
247	C₂²×C₄	—	86	6	no	—
248	C₂²×C₄	—	86	6	no	—
249	Q₁₆	—	43	6	no	—
250	Q₁₆	—	43	3	no	—
251	Q₁₆	—	43	6	no	—
252	Q₁₆	C₂²×D₄₃	1	1	yes	characteristic
253	SD₁₆	—	43	2	no	—
254	SD₁₆	—	43	6	no	—
255	SD₁₆	—	43	12	no	—
256	SD₁₆	—	43	6	no	—
257	SD₁₆	C₂²×D₄₃	1	6	yes	—
258	D₈	—	43	6	no	—
259	D₈	—	43	1	no	—
260	D₈	C₂²×D₄₃	1	3	yes	—
261	D₈	C₂²×D₄₃	1	6	yes	—
262	M₁₆	—	43	3	no	—
263	M₁₆	—	43	6	no	—
264	M₁₆	C₂²×D₄₃	1	3	yes	—
265	C₂×C₈	—	43	3	no	—
266	C₂×C₈	—	43	1	no	—
267	C₂×C₈	—	43	3	no	—
268	C₂×C₈	—	43	3	no	—
269	C₂×C₈	—	43	3	no	—
270	C₂×C₈	C₂²×D₄₃	1	3	yes	—
271	C₂³	—	2	6	no	—
272	C₂³	—	43	3	no	—
273	C₂³	—	86	6	no	—
274	Q₈	—	43	3	no	—
275	Q₈	—	43	6	no	—
276	Q₈	—	86	6	no	—
277	Q₈	—	86	6	no	—
278	Q₈	D₄×D₄₃	1	1	yes	characteristic
279	Q₈	C₂³×D₄₃	1	2	yes	—
280	D₄	—	2	6	no	—
281	D₄	—	2	6	no	—
282	D₄	—	43	6	no	—
283	D₄	—	43	3	no	—
284	D₄	—	43	2	no	—
285	D₄	—	86	2	no	—
286	D₄	—	86	6	no	—
287	D₄	—	86	6	no	—
288	D₄	—	86	12	no	—
289	D₄	—	86	6	no	—
290	D₄	D₄×D₄₃	1	3	yes	—
291	D₄	C₂³×D₄₃	1	6	yes	—
292	C₂×C₄	—	2	6	no	—
293	C₂×C₄	—	43	3	no	—
294	C₂×C₄	—	43	3	no	—
295	C₂×C₄	—	43	3	no	—
296	C₂×C₄	—	43	1	no	—
297	C₂×C₄	—	43	3	no	—
298	C₂×C₄	—	86	2	no	—
299	C₂×C₄	—	86	6	no	—
300	C₂×C₄	—	86	6	no	—
301	C₂×C₄	—	86	12	no	—
302	C₂×C₄	—	86	6	no	—
303	C₂×C₄	—	86	6	no	—
304	C₂×C₄	C₂³×D₄₃	1	3	yes	—
305	C₈	—	43	1	no	—
306	C₈	—	43	3	no	—
307	C₈	C₂³×D₄₃	1	1	yes	characteristic
308	C₈	C₂³×D₄₃	1	3	yes	—
309	C₂²	—	2	6	no	—
310	C₂²	—	8	6	no	—
311	C₂²	—	43	1	no	—
312	C₂²	—	43	3	no	—
313	C₂²	—	86	2	no	—
314	C₂²	—	344	6	no	—
315	C₂²	—	172	3	no	—
316	C₂²	C₂×D₄×D₄₃	1	3	yes	—
317	C₄	—	2	2	no	—
318	C₄	—	43	1	no	—
319	C₄	—	43	3	no	—
320	C₄	—	86	6	no	—
321	C₄	C₂×D₄×D₄₃	1	3	yes	—
322	C₄	C₂⁴×D₄₃	1	1	yes	characteristic
323	C₂	—	2	3	no	—
324	C₂	—	4	6	no	—
325	C₂	—	86	3	no	—
326	C₂	—	86	1	no	—
327	C₂	—	172	2	no	—
328	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	#3584
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. #14583734	C₂	semidirect
2. #14583716	C₂	semidirect
3. #14583713	C₂	semidirect
4. #14583710	C₂	semidirect
5. #14583709	C₂	semidirect
6. #14583700	C₂	semidirect
7. #14583695	C₂	semidirect
8. C₂×#13063344	C₂	semidirect
9. #135×D₄₃	C₂	semidirect
10. #131×D₄₃	C₂	semidirect
11. C₄₃×#576	C₂	semidirect
13. #13063372	C₂²	semidirect
14. #13063372	C₂²	semidirect
16. #13063370	C₂²	semidirect
18. C₂×#10621900	C₂²	semidirect
21. #41×D₄₃	C₂²	semidirect
22. #13063358	C₂²	semidirect
23. #13063355	C₂²	semidirect
24. #13063354	C₂²	semidirect
25. #13063351	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. #13063344	C₂²	semidirect
33. #13063344	C₂²	semidirect
34. #13063344	C₂²	semidirect
35. C₂×Dic₁₇₂	C₂²	semidirect
36. Q₁₆×D₄₃	C₂²	semidirect
37. C₂×#10621883	C₂²	semidirect
38. SD₁₆×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
171. #576	D₄₃	semidirect
176. C₄₃	#3584	semidirect
182. H(5,𝔽₂)	C₂×D₄₃	semidirect
193. AGL(1,ℤ/8)	C₂×D₄₃	semidirect
202. #135	C₂×D₄₃	semidirect
208. C₂×D₈	C₂×D₄₃	semidirect
212. #131	C₂×D₄₃	semidirect