Subgroups — #19145783

Order 5504.

368 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583446	C₂	1	1	yes	maximal, characteristic
2	#125×D₄₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#14583307	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583306	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583262	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583262	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583219	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583216	C₂	1	1	yes	maximal, characteristic, complemented
9	#14583214	C₂	1	1	yes	maximal, characteristic, complemented
10	#14583211	C₂	1	1	yes	maximal, characteristic, complemented
11	#14583200	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583199	C₂	1	1	yes	maximal, characteristic, complemented
13	#14583196	C₂	1	1	yes	maximal, characteristic, complemented
14	#14582955	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄₃×#478	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#10621898	—	2	1	no	—
17	#13063324	C₂²	1	1	yes	characteristic
18	#13063321	C₂²	1	1	yes	characteristic
19	#13063317	C₂²	1	1	yes	characteristic
20	#13063316	C₂²	1	1	yes	characteristic
21	#13063316	C₂²	1	1	yes	characteristic
22	#13063315	—	2	1	no	—
23	#13063315	—	2	1	no	—
24	#13063311	—	2	1	no	—
25	#13063300	C₂²	1	1	yes	characteristic
26	#13063298	—	2	1	no	—
27	#13063296	C₂²	1	1	yes	characteristic
28	C₄×D₁₇₂	C₂²	1	1	yes	characteristic
29	C₄×C₂²⋊D₄₃	—	2	1	no	—
30	C₄⋊C₄×D₄₃	—	2	1	no	—
31	C₄⋊C₄×D₄₃	—	2	1	no	—
32	C₄⋊C₄×D₄₃	C₂²	1	1	yes	characteristic
33	C₄⋊C₄×D₄₃	C₂²	1	1	yes	characteristic
34	C₄²×D₄₃	C₂²	1	1	yes	characteristic
35	#13063259	C₂²	1	1	yes	characteristic
36	#13063258	C₂²	1	1	yes	characteristic
37	#13063258	C₂²	1	1	yes	characteristic
38	#13063257	C₂²	1	1	yes	characteristic
39	#13063256	C₂²	1	1	yes	characteristic
40	#13063256	C₂²	1	1	yes	characteristic
41	C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
42	#13063247	C₂²	1	1	yes	characteristic
43	#13063247	C₂²	1	1	yes	characteristic
44	#13063246	C₂²	1	1	yes	characteristic
45	#13063246	C₂²	1	1	yes	characteristic
46	#13063245	C₂²	1	1	yes	characteristic
47	#13063245	C₂²	1	1	yes	characteristic
48	(C₂×D₄₃)⋊C₈	C₂²	1	1	yes	characteristic
49	(C₂×D₄₃)⋊C₈	C₂²	1	1	yes	characteristic
50	#13063235	C₂²	1	1	yes	characteristic
51	#13063235	C₂²	1	1	yes	characteristic
52	#125×C₄₃	C₂²	1	1	yes	characteristic
53	C₄×D₄×C₄₃	C₂²	1	1	yes	characteristic
54	#107×C₄₃	C₂²	1	1	yes	characteristic
55	#106×C₄₃	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
57	#102×C₄₃	C₂²	1	1	yes	characteristic
58	#102×C₄₃	C₂²	1	1	yes	characteristic
59	C₂×D₄×C₄₃	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₄₃	—	2	1	no	—
61	C₂×C₂²⋊D₄₃	—	2	1	no	—
62	C₂×C₂²⋊D₄₃	—	2	1	no	—
63	C₂×C₂²⋊D₄₃	—	2	1	no	—
64	#10621898	—	4	4	no	—
65	C₂×D₁₇₂	—	2	1	no	—
66	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
67	C₂×C₄×D₄₃	—	2	1	no	—
68	C₂×C₄×D₄₃	—	2	1	no	—
69	C₂×C₄×D₄₃	—	2	1	no	—
70	C₂×C₄×D₄₃	D₄	1	1	yes	characteristic
71	C₂×C₄×D₄₃	D₄	1	1	yes	characteristic
72	C₂×C₄×D₄₃	C₂³	1	1	yes	characteristic
73	C₂×Dic₈₆	—	2	1	no	—
74	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₄₃	—	2	1	no	—
77	C₄⋊C₄×C₄₃	—	2	1	no	—
78	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
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	C₂³	1	1	yes	characteristic
83	C₂²⋊Dic₄₃	—	2	1	no	—
84	(C₂×D₄₃)⋊C₄	—	2	1	no	—
85	(C₂×D₄₃)⋊C₄	—	2	1	no	—
86	C₄⋊Dic₄₃	—	2	1	no	—
87	C₄⋊Dic₄₃	—	2	1	no	—
88	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
89	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
90	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
91	Dic₄₃⋊C₄	—	2	1	no	—
92	Dic₄₃⋊C₄	—	2	1	no	—
93	Dic₄₃⋊C₄	—	2	1	no	—
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	C₄×Dic₄₃	D₄	1	1	yes	characteristic
99	C₄×Dic₄₃	D₄	1	1	yes	characteristic
100	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
101	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
102	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
103	C₂³×C₄₃	—	2	1	no	—
104	C₂²×D₄₃	—	2	1	no	—
105	C₂²×D₄₃	C₂×D₄	1	1	yes	characteristic
106	D₄×C₄₃	—	2	1	no	—
107	D₄×C₄₃	—	2	1	no	—
108	C₂×C₄×C₄₃	—	2	1	no	—
109	C₂×C₄×C₄₃	—	2	1	no	—
110	C₂×C₄×C₄₃	—	2	1	no	—
111	C₂×C₄×C₄₃	—	4	2	no	—
112	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
115	C₂²⋊D₄₃	—	4	2	no	—
116	C₂²⋊D₄₃	—	4	2	no	—
117	C₂²⋊D₄₃	—	4	2	no	—
118	C₂×Dic₄₃	—	2	1	no	—
119	C₂×Dic₄₃	—	2	1	no	—
120	C₂×Dic₄₃	—	2	1	no	—
121	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
124	D₁₇₂	—	2	1	no	—
125	D₁₇₂	—	2	1	no	—
126	D₁₇₂	—	4	2	no	—
127	C₄×D₄₃	—	2	2	no	—
128	C₄×D₄₃	—	2	1	no	—
129	C₄×D₄₃	—	2	1	no	—
130	C₄×D₄₃	—	4	1	no	—
131	C₄×D₄₃	—	4	2	no	—
132	C₄×D₄₃	—	4	2	no	—
133	C₄×D₄₃	—	4	2	no	—
134	Dic₈₆	—	4	2	no	—
135	C₈×C₄₃	—	2	1	no	—
136	C₈×C₄₃	—	2	1	no	—
137	C₄₃⋊C₈	—	2	1	no	—
138	C₄₃⋊C₈	—	2	1	no	—
139	C₂²×C₄₃	—	4	1	no	—
140	C₂²×C₄₃	—	4	1	no	—
141	C₂²×C₄₃	—	4	1	no	—
142	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
143	C₂×D₄₃	—	2	1	no	—
144	C₂×D₄₃	—	2	1	no	—
145	C₂×D₄₃	—	4	1	no	—
146	C₂×D₄₃	—	4	1	no	—
147	C₂×D₄₃	—	4	1	no	—
148	C₂×D₄₃	AGL(1,ℤ/8)	1	2	yes	—
149	C₄×C₄₃	—	2	1	no	—
150	C₄×C₄₃	—	2	1	no	—
151	C₄×C₄₃	—	4	1	no	—
152	C₄×C₄₃	—	4	1	no	—
153	C₄×C₄₃	—	4	1	no	—
154	C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
155	C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
156	Dic₄₃	—	2	1	no	—
157	Dic₄₃	—	2	1	no	—
158	Dic₄₃	—	2	1	no	—
159	Dic₄₃	—	4	1	no	—
160	Dic₄₃	—	4	1	no	—
161	Dic₄₃	—	4	1	no	—
162	#3234	—	43	1	no	maximal, Sylow
163	C₂×C₄₃	—	8	1	no	—
164	C₂×C₄₃	#536	1	1	yes	characteristic
165	C₂×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
166	C₂×C₄₃	#575	1	1	yes	characteristic
167	D₄₃	—	2	2	no	—
168	D₄₃	—	8	1	no	—
169	#532	—	43	1	no	—
170	C₂×#125	—	43	1	no	—
171	#501	—	43	1	no	—
172	#501	—	43	1	no	—
173	#490	—	43	1	no	—
174	#490	—	43	1	no	—
175	#481	—	43	1	no	—
176	#481	—	43	1	no	—
177	#480	—	43	1	no	—
178	#480	—	43	1	no	—
179	#478	—	43	1	no	—
180	#478	—	43	1	no	—
181	#478	—	43	1	no	—
182	#478	D₄₃	1	1	yes	characteristic, complemented
183	#432	—	43	1	no	—
184	C₄₃	#3234	1	1	yes	characteristic, complemented, Sylow
185	C₂×#41	—	86	1	no	—
186	C₄⋊Q₈	—	43	1	no	—
187	#127	—	43	1	no	—
188	#125	—	43	1	no	—
189	#125	—	43	1	no	—
190	#125	—	43	1	no	—
191	#125	—	86	1	no	—
192	#125	—	86	1	no	—
193	#125	C₂×D₄₃	1	1	yes	characteristic
194	#124	—	86	1	no	—
195	#121	—	43	1	no	—
196	#121	—	43	1	no	—
197	#121	—	86	1	no	—
198	C₄×D₄	—	43	1	no	—
199	C₄×D₄	—	86	1	no	—
200	C₄×D₄	C₂×D₄₃	1	1	yes	characteristic
201	C₂×C₄⋊C₄	—	43	1	no	—
202	C₂×C₄⋊C₄	—	43	1	no	—
203	C₂×C₄⋊C₄	—	86	1	no	—
204	C₂×C₄⋊C₄	—	86	1	no	—
205	C₂×C₄²	—	43	1	no	—
206	#107	—	43	1	no	—
207	#107	—	43	1	no	—
208	#107	—	43	1	no	—
209	#107	C₂×D₄₃	1	1	yes	characteristic
210	#106	—	43	1	no	—
211	#106	—	43	1	no	—
212	#106	—	43	1	no	—
213	#106	C₂×D₄₃	1	1	yes	characteristic
214	C₄⋊C₈	—	43	1	no	—
215	C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
216	#102	—	43	1	no	—
217	#102	—	43	1	no	—
218	#102	—	43	1	no	—
219	#102	—	43	1	no	—
220	#102	—	43	1	no	—
221	#102	—	43	1	no	—
222	#102	C₂×D₄₃	1	1	yes	characteristic
223	#102	C₂×D₄₃	1	1	yes	characteristic
224	C₂²⋊C₈	—	43	1	no	—
225	C₂²⋊C₈	—	43	1	no	—
226	#97	—	43	1	no	—
227	#97	—	43	1	no	—
228	#41	—	172	4	no	—
229	C₂×Q₈	—	86	1	no	—
230	C₂×D₄	—	43	1	no	—
231	C₂×D₄	—	86	1	no	—
232	C₂×D₄	—	86	1	no	—
233	C₂×D₄	—	86	1	no	—
234	C₂×D₄	—	86	1	no	—
235	C₂×D₄	C₂²×D₄₃	1	1	yes	characteristic
236	C₂²×C₄	—	2	1	no	—
237	C₂²×C₄	—	43	1	no	—
238	C₂²×C₄	—	43	1	no	—
239	C₂²×C₄	—	43	1	no	—
240	C₂²×C₄	—	86	1	no	—
241	C₂²×C₄	—	86	1	no	—
242	C₂²×C₄	—	86	1	no	—
243	C₂×C₈	—	43	1	no	—
244	C₂×C₈	—	43	1	no	—
245	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
246	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	43	1	no	—
250	C₄⋊C₄	—	43	1	no	—
251	C₄⋊C₄	—	43	1	no	—
252	C₄⋊C₄	—	86	1	no	—
253	C₄⋊C₄	—	86	1	no	—
254	C₄⋊C₄	—	86	1	no	—
255	C₄⋊C₄	—	86	1	no	—
256	C₄⋊C₄	—	86	1	no	—
257	C₄⋊C₄	—	86	1	no	—
258	C₄⋊C₄	—	86	1	no	—
259	C₄⋊C₄	—	86	1	no	—
260	C₄⋊C₄	—	86	1	no	—
261	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
262	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
263	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
264	C₂²⋊C₄	—	2	1	no	—
265	C₂²⋊C₄	—	86	1	no	—
266	C₂²⋊C₄	—	86	1	no	—
267	C₂²⋊C₄	—	86	1	no	—
268	C₄²	—	43	1	no	—
269	C₄²	—	43	1	no	—
270	C₄²	—	43	1	no	—
271	C₄²	C₂²×D₄₃	1	1	yes	characteristic
272	C₂³	—	2	1	no	—
273	C₂³	—	43	1	no	—
274	C₂³	—	86	1	no	—
275	Q₈	—	172	2	no	—
276	D₄	—	2	1	no	—
277	D₄	—	2	1	no	—
278	D₄	—	86	1	no	—
279	D₄	—	86	1	no	—
280	D₄	—	172	2	no	—
281	D₄	—	172	2	no	—
282	D₄	—	172	2	no	—
283	D₄	—	172	2	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	43	1	no	—
289	C₂×C₄	—	43	1	no	—
290	C₂×C₄	—	43	1	no	—
291	C₂×C₄	—	86	2	no	—
292	C₂×C₄	—	86	1	no	—
293	C₂×C₄	—	86	1	no	—
294	C₂×C₄	—	86	1	no	—
295	C₂×C₄	—	86	1	no	—
296	C₂×C₄	—	86	1	no	—
297	C₂×C₄	—	172	1	no	—
298	C₂×C₄	—	172	2	no	—
299	C₂×C₄	—	172	2	no	—
300	C₂×C₄	—	172	2	no	—
301	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
302	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
303	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
304	C₈	—	2	1	no	—
305	C₈	—	2	1	no	—
306	C₈	—	86	1	no	—
307	C₈	—	86	1	no	—
308	C₂²	—	4	1	no	—
309	C₂²	—	4	1	no	—
310	C₂²	—	4	1	no	—
311	C₂²	—	43	2	no	—
312	C₂²	—	86	1	no	—
313	C₂²	—	86	1	no	—
314	C₂²	—	172	1	no	—
315	C₂²	—	172	1	no	—
316	C₂²	—	172	1	no	—
317	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	4	1	no	—
323	C₄	—	86	1	no	—
324	C₄	—	86	1	no	—
325	C₄	—	86	1	no	—
326	C₄	—	172	1	no	—
327	C₄	—	172	1	no	—
328	C₄	—	172	1	no	—
329	C₄	#13063372	1	1	yes	characteristic
330	C₄	#13063373	1	1	yes	characteristic
331	C₂	—	8	1	no	—
332	C₂	—	86	2	no	—
333	C₂	—	344	1	no	—
334	C₂	#14583472	1	1	yes	characteristic
335	C₂	AGL(1,ℤ/8)×D₄₃	1	1	yes	characteristic
336	C₂	#14583696	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#478	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#478	D₄₃
hypocenter	C₄₃	#3234
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #478

Complements

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

N	H ≅ G/N	type
2. #125×D₄₃	C₂	semidirect
3. #14583307	C₂	semidirect
4. #14583306	C₂	semidirect
5. #14583262	C₂	semidirect
6. #14583262	C₂	semidirect
7. #14583219	C₂	semidirect
8. #14583216	C₂	semidirect
9. #14583214	C₂	semidirect
10. #14583211	C₂	semidirect
11. #14583200	C₂	semidirect
12. #14583199	C₂	semidirect
13. #14583196	C₂	semidirect
14. #14582955	C₂	semidirect
15. C₄₃×#478	C₂	semidirect
182. #478	D₄₃	semidirect
184. C₄₃	#3234	semidirect