Subgroups — #19143507

Order 5504.

452 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#135×D₄₃	C₂	1	1	yes	maximal, characteristic
2	#14583664	C₂	1	2	yes	maximal, complemented
3	#14583019	C₂	1	2	yes	maximal, complemented
4	#14583018	C₂	1	2	yes	maximal, complemented
5	#14583017	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583016	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583015	C₂	1	1	yes	maximal, characteristic, complemented
8	#108×D₄₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#14582914	C₂	1	2	yes	maximal, complemented
10	#14582853	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#443	C₂	1	1	yes	maximal, characteristic, complemented
12	#41×D₄₃	C₂²	1	2	yes	—
13	#13063348	C₄	1	1	yes	characteristic, complemented
14	#13063347	C₂²	1	1	yes	characteristic
15	#13063346	C₄	1	2	yes	complemented
16	#13063345	C₂²	1	2	yes	—
17	#13063344	C₄	1	1	yes	characteristic, complemented
18	Q₁₆×D₄₃	C₄	1	1	yes	characteristic, complemented
19	SD₁₆×D₄₃	C₄	1	2	yes	complemented
20	D₈×D₄₃	C₄	1	1	yes	characteristic, complemented
21	#13063333	—	2	2	no	—
22	#13063333	C₂²	1	2	yes	complemented
23	#13063332	C₂²	1	2	yes	complemented
24	#13063331	—	2	2	no	—
25	#13063331	C₂²	1	2	yes	complemented
26	C₂×#10621872	—	2	2	no	—
27	C₂×#10621872	C₂²	1	1	yes	characteristic
28	M₁₆×D₄₃	—	2	2	no	—
29	M₁₆×D₄₃	C₂²	1	2	yes	—
30	C₂×C₈×D₄₃	C₂²	1	1	yes	characteristic
31	#13063282	C₂²	1	1	yes	characteristic
32	#13063261	C₂²	1	1	yes	characteristic, complemented
33	#13063260	C₂²	1	2	yes	complemented
34	#13063253	C₂²	1	2	yes	complemented
35	#13063252	C₂²	1	2	yes	complemented
36	#13063251	C₂²	1	2	yes	complemented
37	#13063236	C₂²	1	1	yes	characteristic, complemented
38	#13063235	C₂²	1	1	yes	characteristic, complemented
39	C₈×Dic₄₃	C₂²	1	1	yes	characteristic, complemented
40	#135×C₄₃	C₂²	1	1	yes	characteristic
41	#131×C₄₃	C₂²	1	2	yes	complemented
42	#108×C₄₃	C₂²	1	1	yes	characteristic, complemented
43	#104×C₄₃	C₂²	1	2	yes	complemented
44	C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic, complemented
45	#41×C₄₃	C₂³	1	2	yes	—
46	#10621902	C₂×C₄	1	2	yes	—
47	Q₈×D₄₃	C₂×C₄	1	2	yes	—
48	#10621900	—	2	2	no	—
49	#10621900	C₂×C₄	1	2	yes	—
50	D₄×D₄₃	—	2	2	no	—
51	D₄×D₄₃	C₂×C₄	1	2	yes	—
52	#10621898	—	2	2	no	—
53	#10621898	C₂³	1	2	yes	—
54	C₂×C₄×D₄₃	—	2	2	no	—
55	C₂×C₄×D₄₃	C₂³	1	1	yes	characteristic
56	Q₁₆×C₄₃	C₂×C₄	1	1	yes	characteristic
57	SD₁₆×C₄₃	C₂×C₄	1	2	yes	—
58	D₈×C₄₃	C₂×C₄	1	1	yes	characteristic
59	M₁₆×C₄₃	—	2	2	no	—
60	M₁₆×C₄₃	C₂³	1	2	yes	—
61	C₂×C₈×C₄₃	—	2	2	no	—
62	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
63	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
64	C₄²×C₄₃	C₂³	1	1	yes	characteristic
65	#10621885	C₂×C₄	1	2	yes	—
66	#10621884	C₂×C₄	1	2	yes	—
67	#10621883	C₂×C₄	1	2	yes	—
68	#10621882	C₂×C₄	1	2	yes	—
69	(C₂×D₄₃)⋊C₄	—	2	1	no	—
70	Dic₄₃⋊C₄	—	2	1	no	—
71	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
72	#10621877	—	2	2	no	—
73	#10621877	C₂³	1	2	yes	—
74	C₂×C₄₃⋊C₈	—	2	2	no	—
75	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
76	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
77	Dic₁₇₂	C₂×C₄	1	1	yes	characteristic
78	D₃₄₄	C₂×C₄	1	1	yes	characteristic
79	#10621873	C₂×C₄	1	2	yes	—
80	#10621872	—	2	2	no	—
81	#10621872	—	2	2	no	—
82	#10621872	—	2	2	no	—
83	#10621872	D₄	1	1	yes	characteristic
84	#10621872	D₄	1	1	yes	characteristic
85	#10621872	D₄	1	1	yes	characteristic
86	#10621872	D₄	1	1	yes	characteristic
87	C₈×D₄₃	—	2	2	no	—
88	C₈×D₄₃	—	2	2	no	—
89	C₈×D₄₃	—	2	2	no	—
90	C₈×D₄₃	C₂×C₄	1	1	yes	characteristic
91	C₈×D₄₃	C₂×C₄	1	1	yes	characteristic
92	C₈×D₄₃	C₂×C₄	1	1	yes	characteristic
93	C₈×D₄₃	C₂×C₄	1	1	yes	characteristic
94	C₂²×D₄₃	—	2	2	no	—
95	C₂²×D₄₃	#41	1	1	yes	characteristic
96	Q₈×C₄₃	C₂²×C₄	1	2	yes	—
97	D₄×C₄₃	—	2	2	no	—
98	D₄×C₄₃	C₂²×C₄	1	2	yes	—
99	C₂×C₄×C₄₃	—	2	2	no	—
100	C₂×C₄×C₄₃	—	2	1	no	—
101	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
102	C₂²⋊D₄₃	—	2	2	no	—
103	C₂²⋊D₄₃	—	2	2	no	—
104	C₂²⋊D₄₃	—	2	2	no	—
105	C₂×Dic₄₃	—	2	2	no	—
106	C₂×Dic₄₃	—	2	1	no	—
107	C₂×Dic₄₃	#41	1	1	yes	characteristic
108	D₁₇₂	—	2	2	no	—
109	D₁₇₂	C₂²×C₄	1	2	yes	—
110	C₄×D₄₃	—	2	2	no	—
111	C₄×D₄₃	—	2	2	no	—
112	C₄×D₄₃	—	2	2	no	—
113	C₄×D₄₃	C₂²×C₄	1	1	yes	characteristic
114	C₄×D₄₃	C₂²×C₄	1	1	yes	characteristic
115	C₄×D₄₃	C₂×D₄	1	1	yes	characteristic
116	C₄×D₄₃	C₂×D₄	1	1	yes	characteristic
117	Dic₈₆	—	2	2	no	—
118	Dic₈₆	C₂²×C₄	1	2	yes	—
119	C₈×C₄₃	—	2	2	no	—
120	C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
121	C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
122	C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
123	C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
124	C₄₃⋊C₈	—	2	2	no	—
125	C₄₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
126	C₄₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
127	C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
128	C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₄₃	—	2	2	no	—
130	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
131	C₂×D₄₃	—	2	2	no	—
132	C₂×D₄₃	—	4	1	no	—
133	C₂×D₄₃	—	8	2	no	—
134	C₂×D₄₃	C₄×D₄	1	1	yes	characteristic
135	C₂×D₄₃	C₄×D₄	1	1	yes	characteristic
136	C₄×C₄₃	—	2	2	no	—
137	C₄×C₄₃	—	4	1	no	—
138	C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
139	C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
140	Dic₄₃	—	2	2	no	—
141	Dic₄₃	—	4	1	no	—
142	Dic₄₃	C₄×D₄	1	1	yes	characteristic
143	Dic₄₃	C₄×D₄	1	1	yes	characteristic
144	#2957	—	43	1	no	maximal, Sylow
145	C₂×C₄₃	—	2	1	no	—
146	C₂×C₄₃	—	4	2	no	—
147	C₂×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
148	D₄₃	—	2	1	no	—
149	D₄₃	—	2	1	no	—
150	D₄₃	—	4	2	no	—
151	C₂×#135	—	43	1	no	—
152	#568	—	43	2	no	—
153	#444	—	43	2	no	—
154	#444	—	43	2	no	—
155	#443	—	43	1	no	—
156	#443	—	43	1	no	—
157	#443	—	43	1	no	—
158	#443	D₄₃	1	1	yes	characteristic, complemented
159	C₂×#108	—	43	1	no	—
160	#421	—	43	2	no	—
161	#405	—	43	1	no	—
162	C₄₃	#2957	1	1	yes	characteristic, complemented, Sylow
163	C₂×#41	—	43	2	no	—
164	#135	—	43	2	no	—
165	#135	—	43	1	no	—
166	#135	—	43	2	no	—
167	#135	—	43	1	no	—
168	#135	—	43	1	no	—
169	#135	C₂×D₄₃	1	1	yes	characteristic
170	C₂×Q₁₆	—	43	1	no	—
171	C₂×SD₁₆	—	43	2	no	—
172	C₂×D₈	—	43	1	no	—
173	#131	—	43	2	no	—
174	#131	—	43	2	no	—
175	#131	—	43	2	no	—
176	#131	—	86	2	no	—
177	#131	—	86	2	no	—
178	#131	C₂×D₄₃	1	2	yes	complemented
179	C₂×M₁₆	—	43	1	no	—
180	C₂×M₁₆	—	43	2	no	—
181	C₂×M₁₆	—	86	2	no	—
182	C₂×M₁₆	—	86	2	no	—
183	C₂²×C₈	—	43	1	no	—
184	#117	—	43	1	no	—
185	#108	—	43	2	no	—
186	#108	—	43	1	no	—
187	#108	C₂×D₄₃	1	1	yes	characteristic, complemented
188	#104	—	43	2	no	—
189	#104	—	43	2	no	—
190	#104	—	43	2	no	—
191	#104	C₂×D₄₃	1	2	yes	complemented
192	#97	—	43	1	no	—
193	#97	—	43	1	no	—
194	C₄×C₈	—	43	1	no	—
195	C₄×C₈	C₂×D₄₃	1	1	yes	characteristic, complemented
196	#41	—	43	2	no	—
197	#41	—	43	2	no	—
198	#41	—	43	2	no	—
199	#41	—	86	2	no	—
200	#41	—	86	2	no	—
201	#41	C₂²×D₄₃	1	2	yes	—
202	C₂×Q₈	—	43	2	no	—
203	C₂×D₄	—	43	2	no	—
204	C₂×D₄	—	86	2	no	—
205	C₂²×C₄	—	43	1	no	—
206	C₂²×C₄	—	86	2	no	—
207	Q₁₆	—	43	2	no	—
208	Q₁₆	—	43	1	no	—
209	Q₁₆	C₄×D₄₃	1	1	yes	characteristic
210	SD₁₆	—	43	2	no	—
211	SD₁₆	—	43	2	no	—
212	SD₁₆	—	43	2	no	—
213	SD₁₆	C₄×D₄₃	1	2	yes	—
214	D₈	—	43	2	no	—
215	D₈	—	43	1	no	—
216	D₈	C₄×D₄₃	1	1	yes	characteristic
217	M₁₆	—	2	2	no	—
218	M₁₆	—	43	1	no	—
219	M₁₆	—	43	1	no	—
220	M₁₆	—	43	1	no	—
221	M₁₆	—	43	1	no	—
222	M₁₆	—	43	2	no	—
223	M₁₆	—	86	2	no	—
224	M₁₆	—	86	2	no	—
225	M₁₆	—	86	2	no	—
226	M₁₆	—	86	2	no	—
227	M₁₆	C₂²×D₄₃	1	2	yes	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	43	1	no	—
230	C₂×C₈	—	43	1	no	—
231	C₂×C₈	—	43	1	no	—
232	C₂×C₈	—	43	1	no	—
233	C₂×C₈	—	43	1	no	—
234	C₂×C₈	—	43	1	no	—
235	C₂×C₈	—	86	2	no	—
236	C₂×C₈	—	86	2	no	—
237	C₂×C₈	—	86	2	no	—
238	C₂×C₈	—	86	2	no	—
239	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
240	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
241	C₄⋊C₄	—	86	1	no	—
242	C₂²⋊C₄	—	86	1	no	—
243	C₄²	—	43	1	no	—
244	C₄²	C₂²×D₄₃	1	1	yes	characteristic
245	C₂³	—	43	1	no	—
246	C₂³	—	86	2	no	—
247	Q₈	—	43	2	no	—
248	Q₈	—	86	2	no	—
249	Q₈	C₂×C₄×D₄₃	1	2	yes	—
250	D₄	—	2	2	no	—
251	D₄	—	43	2	no	—
252	D₄	—	86	2	no	—
253	D₄	—	86	2	no	—
254	D₄	—	86	2	no	—
255	D₄	—	86	2	no	—
256	D₄	C₂×C₄×D₄₃	1	2	yes	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	43	1	no	—
260	C₂×C₄	—	43	1	no	—
261	C₂×C₄	—	43	1	no	—
262	C₂×C₄	—	43	1	no	—
263	C₂×C₄	—	43	1	no	—
264	C₂×C₄	—	86	2	no	—
265	C₂×C₄	—	86	2	no	—
266	C₂×C₄	—	86	2	no	—
267	C₂×C₄	—	86	2	no	—
268	C₂×C₄	—	86	1	no	—
269	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
270	C₈	—	2	2	no	—
271	C₈	—	43	1	no	—
272	C₈	—	43	1	no	—
273	C₈	—	43	1	no	—
274	C₈	—	43	1	no	—
275	C₈	—	86	2	no	—
276	C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
277	C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
278	C₈	D₄×D₄₃	1	1	yes	characteristic
279	C₈	D₄×D₄₃	1	1	yes	characteristic
280	C₂²	—	2	2	no	—
281	C₂²	—	43	1	no	—
282	C₂²	—	43	1	no	—
283	C₂²	—	86	2	no	—
284	C₂²	—	344	2	no	—
285	C₂²	—	172	1	no	—
286	C₂²	#41×D₄₃	1	1	yes	characteristic
287	C₄	—	2	2	no	—
288	C₄	—	4	1	no	—
289	C₄	—	43	1	no	—
290	C₄	—	43	1	no	—
291	C₄	—	86	2	no	—
292	C₄	—	172	1	no	—
293	C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
294	C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
295	C₂	—	2	1	no	—
296	C₂	—	4	2	no	—
297	C₂	—	86	1	no	—
298	C₂	—	86	1	no	—
299	C₂	—	172	2	no	—
300	C₂	C₄×D₄×D₄₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₄₃
commutator	C₄×C₄₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#443	C₂
socle	C₂×C₄₃	C₂×C₄×D₄
hypercenter	#443	D₄₃
hypocenter	C₄₃	#2957
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #443

Complements

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

N	H ≅ G/N	type
2. #14583664	C₂	semidirect
3. #14583019	C₂	semidirect
4. #14583018	C₂	semidirect
5. #14583017	C₂	semidirect
6. #14583016	C₂	semidirect
7. #14583015	C₂	semidirect
8. #108×D₄₃	C₂	semidirect
9. #14582914	C₂	semidirect
10. #14582853	C₂	semidirect
11. C₄₃×#443	C₂	semidirect
13. #13063348	C₄	semidirect
15. #13063346	C₄	semidirect
17. #13063344	C₄	semidirect
18. Q₁₆×D₄₃	C₄	semidirect
19. SD₁₆×D₄₃	C₄	semidirect
20. D₈×D₄₃	C₄	semidirect
22. #13063333	C₂²	semidirect
23. #13063332	C₂²	semidirect
25. #13063331	C₂²	semidirect
32. #13063261	C₂²	semidirect
33. #13063260	C₂²	semidirect
34. #13063253	C₂²	semidirect
35. #13063252	C₂²	semidirect
36. #13063251	C₂²	semidirect
37. #13063236	C₂²	semidirect
38. #13063235	C₂²	semidirect
39. C₈×Dic₄₃	C₂²	semidirect
41. #131×C₄₃	C₂²	semidirect
42. #108×C₄₃	C₂²	semidirect
43. #104×C₄₃	C₂²	semidirect
44. C₄×C₈×C₄₃	C₂²	semidirect
158. #443	D₄₃	semidirect
162. C₄₃	#2957	semidirect
178. #131	C₂×D₄₃	semidirect
187. #108	C₂×D₄₃	semidirect
191. #104	C₂×D₄₃	semidirect
195. C₄×C₈	C₂×D₄₃	semidirect