Subgroups — #19147308

Order 5504.

372 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	#14583618	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×Dic₈₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#14583258	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583257	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583248	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583195	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583192	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583177	C₂	1	1	yes	maximal, characteristic, complemented
9	#14583112	C₂	1	1	yes	maximal, characteristic
10	#14583102	C₂	1	1	yes	maximal, characteristic, complemented
11	#14583028	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583000	C₂	1	1	yes	maximal, characteristic, complemented
13	#14582999	C₂	1	1	yes	maximal, characteristic, complemented
14	D₈×Dic₄₃	C₂	1	1	yes	maximal, characteristic
15	C₄₃×#463	C₂	1	1	yes	maximal, characteristic
16	C₂²×Dic₈₆	—	2	1	no	—
17	C₂×#10621883	C₂²	1	1	yes	characteristic
18	#13063325	C₂²	1	1	yes	characteristic
19	#13063324	C₂²	1	1	yes	characteristic
20	#13063324	C₂²	1	1	yes	characteristic
21	#13063323	C₂²	1	1	yes	characteristic
22	#13063318	—	2	1	no	—
23	#13063318	—	2	1	no	—
24	#13063312	C₂²	1	1	yes	characteristic
25	#13063304	—	2	1	no	—
26	#13063302	—	2	1	no	—
27	#13063302	—	2	1	no	—
28	#13063302	—	2	1	no	—
29	Q₈×Dic₄₃	C₂²	1	1	yes	characteristic
30	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
31	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
32	D₄×Dic₄₃	C₂²	1	1	yes	characteristic
33	D₄×Dic₄₃	C₂²	1	1	yes	characteristic
34	D₄×Dic₄₃	C₂²	1	1	yes	characteristic
35	#13063281	—	2	1	no	—
36	#13063259	C₂²	1	1	yes	characteristic
37	#13063259	C₂²	1	1	yes	characteristic
38	#13063258	C₂²	1	1	yes	characteristic
39	Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic
40	Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic
41	#13063250	C₂²	1	1	yes	characteristic
42	#13063250	C₂²	1	1	yes	characteristic
43	#13063249	C₂²	1	1	yes	characteristic
44	#13063249	C₂²	1	1	yes	characteristic
45	#13063248	C₂²	1	1	yes	characteristic
46	#13063246	C₂²	1	1	yes	characteristic
47	#13063246	C₂²	1	1	yes	characteristic
48	#13063246	C₂²	1	1	yes	characteristic
49	#13063236	C₂²	1	1	yes	characteristic
50	#13063235	C₂²	1	1	yes	characteristic
51	C₈×Dic₄₃	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₄₃	C₂²	1	1	yes	characteristic
53	C₂×D₈×C₄₃	C₂²	1	1	yes	characteristic
54	#124×C₄₃	C₂²	1	1	yes	characteristic
55	C₄×D₄×C₄₃	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
57	#103×C₄₃	C₂²	1	1	yes	characteristic
58	#102×C₄₃	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₄₃	C₂³	1	1	yes	characteristic
60	C₂×D₄×C₄₃	C₂³	1	1	yes	characteristic
61	C₂×D₄×C₄₃	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₄₃	—	2	1	no	—
63	C₂²×Dic₄₃	—	2	1	no	—
64	C₂²×Dic₄₃	—	2	1	no	—
65	C₂²×Dic₄₃	—	2	1	no	—
66	C₂×Dic₈₆	—	2	1	no	—
67	C₂×Dic₈₆	—	2	1	no	—
68	C₂×Dic₈₆	—	2	1	no	—
69	C₂×Dic₈₆	—	4	4	no	—
70	C₂×Dic₈₆	D₄	1	1	yes	characteristic
71	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
72	SD₁₆×C₄₃	—	2	2	no	—
73	D₈×C₄₃	—	2	2	no	—
74	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
77	C₂²⋊C₄×C₄₃	—	2	1	no	—
78	C₂²⋊C₄×C₄₃	—	2	1	no	—
79	C₂²⋊C₄×C₄₃	—	2	1	no	—
80	C₄²×C₄₃	C₂³	1	1	yes	characteristic
81	C₂²⋊Dic₄₃	—	2	1	no	—
82	C₂²⋊Dic₄₃	—	2	1	no	—
83	C₂²⋊Dic₄₃	—	2	1	no	—
84	C₂²⋊Dic₄₃	—	2	1	no	—
85	C₂²⋊Dic₄₃	—	2	1	no	—
86	#10621883	—	2	2	no	—
87	C₄⋊Dic₄₃	—	2	1	no	—
88	C₄⋊Dic₄₃	—	2	1	no	—
89	C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
90	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
91	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
92	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
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	Dic₄₃⋊C₄	D₄	1	1	yes	characteristic
99	Dic₄₃⋊C₄	D₄	1	1	yes	characteristic
100	C₄×Dic₄₃	—	2	1	no	—
101	C₄×Dic₄₃	—	2	1	no	—
102	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
103	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
104	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
105	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
106	C₂³×C₄₃	—	2	1	no	—
107	C₂³×C₄₃	—	2	1	no	—
108	Q₈×C₄₃	—	2	1	no	—
109	Q₈×C₄₃	—	2	1	no	—
110	D₄×C₄₃	—	2	1	no	—
111	D₄×C₄₃	—	2	1	no	—
112	D₄×C₄₃	—	2	1	no	—
113	D₄×C₄₃	#41	1	2	yes	—
114	C₂×C₄×C₄₃	—	2	1	no	—
115	C₂×C₄×C₄₃	—	2	1	no	—
116	C₂×C₄×C₄₃	—	4	2	no	—
117	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
120	C₂×Dic₄₃	—	2	1	no	—
121	C₂×Dic₄₃	—	2	1	no	—
122	C₂×Dic₄₃	—	2	1	no	—
123	C₂×Dic₄₃	—	2	1	no	—
124	C₂×Dic₄₃	—	4	2	no	—
125	C₂×Dic₄₃	—	4	2	no	—
126	C₂×Dic₄₃	—	4	2	no	—
127	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
128	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
131	Dic₈₆	—	2	2	no	—
132	Dic₈₆	—	2	1	no	—
133	Dic₈₆	—	2	1	no	—
134	Dic₈₆	—	4	2	no	—
135	Dic₈₆	—	4	2	no	—
136	Dic₈₆	—	4	2	no	—
137	C₈×C₄₃	—	2	1	no	—
138	C₈×C₄₃	—	2	1	no	—
139	C₄₃⋊C₈	—	2	1	no	—
140	C₄₃⋊C₈	—	2	1	no	—
141	C₂²×C₄₃	—	2	2	no	—
142	C₂²×C₄₃	—	4	1	no	—
143	C₂²×C₄₃	—	4	1	no	—
144	C₂²×C₄₃	—	4	1	no	—
145	C₂²×C₄₃	—	4	1	no	—
146	C₂²×C₄₃	—	4	1	no	—
147	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
148	C₄×C₄₃	—	2	1	no	—
149	C₄×C₄₃	—	2	1	no	—
150	C₄×C₄₃	—	4	1	no	—
151	C₄×C₄₃	—	4	1	no	—
152	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
153	C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
154	Dic₄₃	—	2	1	no	—
155	Dic₄₃	—	2	1	no	—
156	Dic₄₃	—	2	1	no	—
157	Dic₄₃	—	4	1	no	—
158	Dic₄₃	—	4	1	no	—
159	Dic₄₃	—	4	1	no	—
160	Dic₄₃	—	4	1	no	—
161	Dic₄₃	AGL(1,ℤ/8)	1	2	yes	—
162	#3367	—	43	1	no	maximal, Sylow
163	C₂×C₄₃	—	4	2	no	—
164	C₂×C₄₃	—	8	1	no	—
165	C₂×C₄₃	#548	1	1	yes	characteristic
166	C₂×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
167	C₂×C₄₃	#578	1	1	yes	characteristic
168	#556	—	43	1	no	—
169	D₄×Q₈	—	43	1	no	—
170	#489	—	43	1	no	—
171	#489	—	43	1	no	—
172	#487	—	43	1	no	—
173	#477	—	43	1	no	—
174	#477	—	43	1	no	—
175	#474	—	43	1	no	—
176	#463	—	43	1	no	—
177	#463	D₄₃	1	1	yes	characteristic
178	#461	—	43	1	no	—
179	#446	—	43	1	no	—
180	#440	—	43	1	no	—
181	#440	—	43	1	no	—
182	C₄×D₈	—	43	1	no	—
183	C₄₃	#3367	1	1	yes	characteristic, complemented, Sylow
184	C₂²×Q₈	—	86	1	no	—
185	C₂×SD₁₆	—	43	1	no	—
186	C₂×SD₁₆	C₂×D₄₃	1	1	yes	characteristic
187	C₂×D₈	C₂×D₄₃	1	1	yes	characteristic
188	C₄⋊Q₈	—	43	1	no	—
189	C₄⋊Q₈	—	43	1	no	—
190	C₄⋊Q₈	—	43	1	no	—
191	C₄⋊Q₈	—	43	1	no	—
192	#126	—	86	1	no	—
193	#126	—	86	1	no	—
194	#124	—	43	1	no	—
195	#124	C₂×D₄₃	1	1	yes	characteristic
196	C₂²⋊Q₈	—	86	1	no	—
197	C₂²⋊Q₈	—	86	1	no	—
198	C₂²⋊Q₈	—	86	1	no	—
199	C₂²⋊Q₈	—	86	1	no	—
200	C₄×Q₈	—	43	1	no	—
201	C₄×Q₈	—	43	1	no	—
202	C₄×Q₈	—	43	1	no	—
203	C₄×D₄	—	43	1	no	—
204	C₄×D₄	—	43	1	no	—
205	C₄×D₄	—	43	1	no	—
206	C₄×D₄	C₂×D₄₃	1	1	yes	characteristic
207	#117	—	86	1	no	—
208	#107	—	43	1	no	—
209	#107	—	43	1	no	—
210	#107	—	43	1	no	—
211	C₄⋊C₈	—	43	1	no	—
212	C₄⋊C₈	—	43	1	no	—
213	C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
214	#103	—	43	1	no	—
215	#103	—	43	1	no	—
216	#103	—	43	1	no	—
217	#103	—	43	1	no	—
218	#103	—	43	1	no	—
219	#103	C₂×D₄₃	1	1	yes	characteristic
220	#102	—	43	1	no	—
221	#102	—	43	1	no	—
222	#102	—	43	1	no	—
223	#102	C₂×D₄₃	1	1	yes	characteristic
224	#97	—	43	1	no	—
225	#97	—	43	1	no	—
226	C₄×C₈	—	43	1	no	—
227	C₂×Q₈	—	43	1	no	—
228	C₂×Q₈	—	43	1	no	—
229	C₂×Q₈	—	86	1	no	—
230	C₂×Q₈	—	86	1	no	—
231	C₂×Q₈	—	86	1	no	—
232	C₂×Q₈	—	172	4	no	—
233	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
234	C₂×D₄	C₂²×D₄₃	1	1	yes	characteristic
235	C₂×D₄	C₂²×D₄₃	1	1	yes	characteristic
236	C₂²×C₄	—	2	1	no	—
237	C₂²×C₄	—	86	1	no	—
238	C₂²×C₄	—	86	1	no	—
239	C₂²×C₄	—	86	1	no	—
240	SD₁₆	—	2	2	no	—
241	SD₁₆	—	86	2	no	—
242	D₈	—	2	2	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	—	43	1	no	—
248	C₄⋊C₄	—	43	1	no	—
249	C₄⋊C₄	—	43	1	no	—
250	C₄⋊C₄	—	43	1	no	—
251	C₄⋊C₄	—	43	1	no	—
252	C₄⋊C₄	—	43	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	C₂²×D₄₃	1	1	yes	characteristic
261	C₂²⋊C₄	—	2	1	no	—
262	C₂²⋊C₄	—	2	1	no	—
263	C₂²⋊C₄	—	2	1	no	—
264	C₂²⋊C₄	—	86	1	no	—
265	C₂²⋊C₄	—	86	1	no	—
266	C₂²⋊C₄	—	86	1	no	—
267	C₂²⋊C₄	—	86	1	no	—
268	C₂²⋊C₄	—	86	1	no	—
269	C₄²	—	43	1	no	—
270	C₄²	—	43	1	no	—
271	C₄²	—	86	1	no	—
272	C₄²	—	86	1	no	—
273	C₄²	C₂²×D₄₃	1	1	yes	characteristic
274	C₂³	—	2	1	no	—
275	C₂³	—	2	1	no	—
276	Q₈	—	2	1	no	—
277	Q₈	—	2	1	no	—
278	Q₈	—	86	2	no	—
279	Q₈	—	86	1	no	—
280	Q₈	—	86	1	no	—
281	Q₈	—	172	2	no	—
282	Q₈	—	172	2	no	—
283	Q₈	—	172	2	no	—
284	D₄	—	2	1	no	—
285	D₄	—	2	1	no	—
286	D₄	—	2	1	no	—
287	D₄	#10621900	1	2	yes	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	43	1	no	—
292	C₂×C₄	—	43	1	no	—
293	C₂×C₄	—	43	1	no	—
294	C₂×C₄	—	43	1	no	—
295	C₂×C₄	—	86	1	no	—
296	C₂×C₄	—	86	1	no	—
297	C₂×C₄	—	86	1	no	—
298	C₂×C₄	—	86	1	no	—
299	C₂×C₄	—	172	2	no	—
300	C₂×C₄	—	172	2	no	—
301	C₂×C₄	—	172	2	no	—
302	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
303	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
304	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	—	86	1	no	—
308	C₈	—	86	1	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	4	1	no	—
311	C₂²	—	4	1	no	—
312	C₂²	—	4	1	no	—
313	C₂²	—	4	1	no	—
314	C₂²	—	4	1	no	—
315	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	43	2	no	—
321	C₄	—	86	1	no	—
322	C₄	—	86	1	no	—
323	C₄	—	86	1	no	—
324	C₄	—	172	1	no	—
325	C₄	—	172	1	no	—
326	C₄	—	172	1	no	—
327	C₄	—	172	1	no	—
328	C₄	C₂×#10621900	1	1	yes	characteristic
329	C₄	#13063372	1	1	yes	characteristic
330	C₂	—	4	2	no	—
331	C₂	—	8	1	no	—
332	C₂	#14583555	1	1	yes	characteristic
333	C₂	AGL(1,ℤ/8)×D₄₃	1	1	yes	characteristic
334	C₂	#14583713	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#463	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#463	D₄₃
hypocenter	C₄₃	#3367
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #463

Complements

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

N	H ≅ G/N	type
1. #14583618	C₂	semidirect
2. D₄×Dic₈₆	C₂	semidirect
3. #14583258	C₂	semidirect
4. #14583257	C₂	semidirect
5. #14583248	C₂	semidirect
6. #14583195	C₂	semidirect
7. #14583192	C₂	semidirect
8. #14583177	C₂	semidirect
10. #14583102	C₂	semidirect
11. #14583028	C₂	semidirect
12. #14583000	C₂	semidirect
13. #14582999	C₂	semidirect
183. C₄₃	#3367	semidirect