Subgroups — #19145158

Order 5504.

380 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583435	C₂	1	1	yes	maximal, characteristic, complemented
2	#14583430	C₂	1	1	yes	maximal, characteristic, complemented
3	#14583299	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583280	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583255	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583252	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583138	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583133	C₂	1	1	yes	maximal, characteristic
9	#14583128	C₂	1	1	yes	maximal, characteristic, complemented
10	#14582991	C₂	1	1	yes	maximal, characteristic, complemented
11	SD₁₆×Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#14582986	C₂	1	1	yes	maximal, characteristic, complemented
13	#14582979	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×C₈×Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄₃×#465	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#10621885	C₂²	1	1	yes	characteristic
17	C₂×#10621873	C₂²	1	1	yes	characteristic
18	C₂×#10621882	C₂²	1	1	yes	characteristic
19	C₂²×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
20	#13063320	—	2	1	no	—
21	#13063318	—	2	1	no	—
22	#13063316	C₂²	1	1	yes	characteristic
23	#13063316	C₂²	1	1	yes	characteristic
24	#13063314	C₂²	1	1	yes	characteristic
25	#13063312	C₂²	1	1	yes	characteristic
26	#13063309	—	2	1	no	—
27	#13063306	—	2	1	no	—
28	#13063304	C₂²	1	1	yes	characteristic
29	#13063299	C₂²	1	1	yes	characteristic
30	Q₈×Dic₄₃	C₂²	1	1	yes	characteristic
31	#13063290	C₂²	1	1	yes	characteristic
32	D₄×Dic₄₃	—	2	1	no	—
33	D₄×Dic₄₃	C₂²	1	1	yes	characteristic
34	#13063287	C₂²	1	1	yes	characteristic
35	(C₂²⋊D₄₃)⋊C₄	—	2	1	no	—
36	#13063283	—	2	1	no	—
37	#13063281	—	2	1	no	—
38	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
39	#13063258	C₂²	1	1	yes	characteristic
40	#13063258	C₂²	1	1	yes	characteristic
41	#13063257	C₂²	1	1	yes	characteristic
42	#13063250	C₂²	1	1	yes	characteristic
43	#13063249	C₂²	1	1	yes	characteristic
44	#13063248	C₂²	1	1	yes	characteristic
45	#13063247	C₂²	1	1	yes	characteristic
46	#13063246	C₂²	1	1	yes	characteristic
47	#13063245	C₂²	1	1	yes	characteristic
48	C₈×Dic₄₃	C₂²	1	1	yes	characteristic
49	C₈×Dic₄₃	C₂²	1	1	yes	characteristic
50	C₈×Dic₄₃	C₂²	1	1	yes	characteristic
51	C₈×Dic₄₃	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₄₃	C₂²	1	1	yes	characteristic
53	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic
54	C₂²⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
55	#121×C₄₃	C₂²	1	1	yes	characteristic
56	#106×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	—	2	1	no	—
61	C₂×D₄×C₄₃	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₄₃	—	2	1	no	—
64	C₂²×Dic₄₃	—	2	1	no	—
65	C₂²×Dic₄₃	D₄	1	1	yes	characteristic
66	C₂²×Dic₄₃	D₄	1	1	yes	characteristic
67	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
68	C₂×C₄×D₄₃	—	2	1	no	—
69	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
70	SD₁₆×C₄₃	—	2	2	no	—
71	C₂×C₈×C₄₃	—	2	2	no	—
72	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
73	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₄₃	—	2	1	no	—
75	C₄⋊C₄×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⋊Dic₄₃	—	2	1	no	—
80	C₂²⋊Dic₄₃	—	2	1	no	—
81	C₂²⋊Dic₄₃	—	2	1	no	—
82	C₂²⋊Dic₄₃	—	2	1	no	—
83	#10621885	—	2	2	no	—
84	#10621882	—	2	2	no	—
85	(C₂×D₄₃)⋊C₄	—	2	1	no	—
86	(C₂×D₄₃)⋊C₄	—	2	1	no	—
87	(C₂×D₄₃)⋊C₄	—	2	1	no	—
88	(C₂×D₄₃)⋊C₄	—	2	1	no	—
89	C₄⋊Dic₄₃	—	2	1	no	—
90	C₄⋊Dic₄₃	—	2	1	no	—
91	C₄⋊Dic₄₃	—	2	1	no	—
92	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
93	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
99	C₄×Dic₄₃	—	2	1	no	—
100	C₄×Dic₄₃	D₄	1	1	yes	characteristic
101	C₄×Dic₄₃	D₄	1	1	yes	characteristic
102	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
103	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
104	C₂×C₄₃⋊C₈	—	2	2	no	—
105	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
106	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
107	#10621873	—	2	2	no	—
108	C₂³×C₄₃	—	2	1	no	—
109	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
110	C₂²×D₄₃	—	2	1	no	—
111	Q₈×C₄₃	—	2	1	no	—
112	Q₈×C₄₃	—	2	1	no	—
113	D₄×C₄₃	—	2	1	no	—
114	D₄×C₄₃	—	2	1	no	—
115	D₄×C₄₃	—	4	2	no	—
116	C₂×C₄×C₄₃	—	2	1	no	—
117	C₂×C₄×C₄₃	—	2	1	no	—
118	C₂×C₄×C₄₃	—	2	1	no	—
119	C₂×C₄×C₄₃	—	2	1	no	—
120	C₂×C₄×C₄₃	—	2	1	no	—
121	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
123	C₂²⋊D₄₃	—	4	2	no	—
124	C₂×Dic₄₃	—	2	2	no	—
125	C₂×Dic₄₃	—	2	2	no	—
126	C₂×Dic₄₃	—	2	1	no	—
127	C₂×Dic₄₃	—	2	1	no	—
128	C₂×Dic₄₃	—	2	1	no	—
129	C₂×Dic₄₃	—	4	2	no	—
130	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
131	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
132	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
133	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
134	D₁₇₂	—	2	1	no	—
135	D₁₇₂	—	2	1	no	—
136	C₄×D₄₃	—	4	2	no	—
137	Dic₈₆	—	2	1	no	—
138	Dic₈₆	—	2	1	no	—
139	C₈×C₄₃	—	2	1	no	—
140	C₈×C₄₃	#41	1	2	yes	—
141	C₄₃⋊C₈	—	2	1	no	—
142	C₄₃⋊C₈	#41	1	2	yes	—
143	C₂²×C₄₃	—	2	1	no	—
144	C₂²×C₄₃	—	2	1	no	—
145	C₂²×C₄₃	—	4	1	no	—
146	C₂²×C₄₃	—	4	1	no	—
147	C₂²×C₄₃	—	4	1	no	—
148	C₂²×C₄₃	#135	1	2	yes	—
149	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
150	C₂×D₄₃	—	4	1	no	—
151	C₂×D₄₃	—	4	1	no	—
152	C₂×D₄₃	—	4	1	no	—
153	C₄×C₄₃	—	2	1	no	—
154	C₄×C₄₃	—	4	1	no	—
155	C₄×C₄₃	—	4	1	no	—
156	C₄×C₄₃	—	4	1	no	—
157	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
158	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
159	Dic₄₃	—	2	1	no	—
160	Dic₄₃	—	2	1	no	—
161	Dic₄₃	—	2	1	no	—
162	Dic₄₃	—	4	1	no	—
163	Dic₄₃	—	4	1	no	—
164	Dic₄₃	—	4	1	no	—
165	Dic₄₃	#135	1	2	yes	—
166	#3168	—	43	1	no	maximal, Sylow
167	C₂×C₄₃	—	2	2	no	—
168	C₂×C₄₃	—	8	1	no	—
169	C₂×C₄₃	#532	1	1	yes	characteristic
170	C₂×C₄₃	C₂×#135	1	1	yes	characteristic
171	C₂×C₄₃	C₂×#135	1	1	yes	characteristic
172	D₄₃	—	8	1	no	—
173	#529	—	43	1	no	—
174	#529	—	43	1	no	—
175	#499	—	43	1	no	—
176	#495	—	43	1	no	—
177	#488	—	43	1	no	—
178	#488	—	43	1	no	—
179	#467	—	43	1	no	—
180	#466	—	43	1	no	—
181	#465	—	43	1	no	—
182	#465	D₄₃	1	1	yes	characteristic, complemented
183	C₄×Q₁₆	—	43	1	no	—
184	C₄×SD₁₆	—	43	1	no	—
185	C₄×SD₁₆	—	43	1	no	—
186	C₄×D₈	—	43	1	no	—
187	C₂×C₄×C₈	—	43	1	no	—
188	C₄₃	#3168	1	1	yes	characteristic, complemented, Sylow
189	C₂×Q₁₆	—	43	1	no	—
190	C₂×SD₁₆	—	43	1	no	—
191	C₂×SD₁₆	C₂×D₄₃	1	1	yes	characteristic
192	C₂×D₈	—	43	1	no	—
193	C₂²×C₈	—	43	1	no	—
194	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic
195	#126	—	86	1	no	—
196	#126	—	86	1	no	—
197	#125	—	43	1	no	—
198	#125	—	43	1	no	—
199	#124	—	43	1	no	—
200	#124	—	43	1	no	—
201	#123	—	86	1	no	—
202	#123	—	86	1	no	—
203	C₂²⋊Q₈	—	43	1	no	—
204	C₂²⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
205	#121	—	43	1	no	—
206	#121	C₂×D₄₃	1	1	yes	characteristic
207	C₄×Q₈	—	43	1	no	—
208	C₄×Q₈	—	43	1	no	—
209	C₄×D₄	—	43	1	no	—
210	C₄×D₄	—	43	1	no	—
211	C₄×D₄	—	86	1	no	—
212	C₄×D₄	—	86	1	no	—
213	#117	—	86	1	no	—
214	#117	—	86	1	no	—
215	C₂×C₄²	—	43	1	no	—
216	#107	—	43	1	no	—
217	#107	—	43	1	no	—
218	#106	—	43	1	no	—
219	#106	C₂×D₄₃	1	1	yes	characteristic
220	#103	—	43	1	no	—
221	#103	—	43	1	no	—
222	#103	—	43	1	no	—
223	#103	C₂×D₄₃	1	1	yes	characteristic
224	#102	—	43	1	no	—
225	#102	—	43	1	no	—
226	#102	—	43	1	no	—
227	#102	C₂×D₄₃	1	1	yes	characteristic
228	C₄×C₈	—	43	1	no	—
229	C₄×C₈	—	43	1	no	—
230	C₄×C₈	—	43	1	no	—
231	C₄×C₈	—	43	1	no	—
232	C₂×Q₈	—	43	1	no	—
233	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
234	C₂×D₄	—	2	1	no	—
235	C₂×D₄	—	43	1	no	—
236	C₂×D₄	—	86	1	no	—
237	C₂×D₄	C₂²×D₄₃	1	1	yes	characteristic
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	C₂²×D₄₃	1	1	yes	characteristic
243	Q₁₆	—	86	2	no	—
244	SD₁₆	—	2	2	no	—
245	SD₁₆	—	86	2	no	—
246	D₈	—	86	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	43	1	no	—
249	C₂×C₈	—	43	1	no	—
250	C₂×C₈	—	86	2	no	—
251	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
252	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	43	1	no	—
255	C₄⋊C₄	—	43	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	—	86	1	no	—
262	C₄⋊C₄	—	86	1	no	—
263	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
264	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	2	1	no	—
267	C₂²⋊C₄	—	86	1	no	—
268	C₂²⋊C₄	—	86	1	no	—
269	C₂²⋊C₄	—	86	1	no	—
270	C₂²⋊C₄	—	86	1	no	—
271	C₂²⋊C₄	—	86	1	no	—
272	C₂²⋊C₄	—	86	1	no	—
273	C₂²⋊C₄	—	86	1	no	—
274	C₂²⋊C₄	—	86	1	no	—
275	C₄²	—	43	1	no	—
276	C₄²	—	43	1	no	—
277	C₄²	—	43	1	no	—
278	C₄²	—	43	1	no	—
279	C₄²	—	86	1	no	—
280	C₄²	—	86	1	no	—
281	C₂³	—	2	1	no	—
282	C₂³	—	86	1	no	—
283	C₂³	D₄×D₄₃	1	1	yes	characteristic
284	Q₈	—	2	1	no	—
285	Q₈	—	2	1	no	—
286	Q₈	—	86	1	no	—
287	Q₈	—	86	1	no	—
288	D₄	—	2	1	no	—
289	D₄	—	2	1	no	—
290	D₄	—	4	2	no	—
291	D₄	—	86	1	no	—
292	D₄	—	86	1	no	—
293	D₄	—	172	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	43	1	no	—
300	C₂×C₄	—	43	1	no	—
301	C₂×C₄	—	43	1	no	—
302	C₂×C₄	—	43	1	no	—
303	C₂×C₄	—	86	2	no	—
304	C₂×C₄	—	86	2	no	—
305	C₂×C₄	—	86	1	no	—
306	C₂×C₄	—	86	1	no	—
307	C₂×C₄	—	86	1	no	—
308	C₂×C₄	—	172	2	no	—
309	C₂×C₄	—	172	2	no	—
310	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
311	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
312	C₈	—	2	1	no	—
313	C₈	—	43	2	no	—
314	C₈	—	86	1	no	—
315	C₈	#10621900	1	2	yes	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	4	1	no	—
321	C₂²	—	172	1	no	—
322	C₂²	—	172	1	no	—
323	C₂²	—	172	1	no	—
324	C₂²	#13063346	1	2	yes	—
325	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	4	1	no	—
328	C₄	—	4	1	no	—
329	C₄	—	4	1	no	—
330	C₄	—	43	2	no	—
331	C₄	—	86	1	no	—
332	C₄	—	86	1	no	—
333	C₄	—	86	1	no	—
334	C₄	—	172	1	no	—
335	C₄	—	172	1	no	—
336	C₄	—	172	1	no	—
337	C₄	#41×D₄₃	1	1	yes	characteristic
338	C₄	C₂×#10621900	1	1	yes	characteristic
339	C₂	—	2	2	no	—
340	C₂	—	8	1	no	—
341	C₂	—	344	1	no	—
342	C₂	#14583444	1	1	yes	characteristic
343	C₂	#135×D₄₃	1	1	yes	characteristic
344	C₂	C₂×#13063346	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#465	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#465	D₄₃
hypocenter	C₄₃	#3168
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #465

Complements

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

N	H ≅ G/N	type
1. #14583435	C₂	semidirect
2. #14583430	C₂	semidirect
3. #14583299	C₂	semidirect
4. #14583280	C₂	semidirect
5. #14583255	C₂	semidirect
6. #14583252	C₂	semidirect
7. #14583138	C₂	semidirect
9. #14583128	C₂	semidirect
10. #14582991	C₂	semidirect
11. SD₁₆×Dic₄₃	C₂	semidirect
12. #14582986	C₂	semidirect
13. #14582979	C₂	semidirect
14. C₂×C₈×Dic₄₃	C₂	semidirect
15. C₄₃×#465	C₂	semidirect
182. #465	D₄₃	semidirect
188. C₄₃	#3168	semidirect