Subgroups — #19144723

Order 5504.

380 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583453	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×D₄₃	C₂	1	1	yes	maximal, characteristic
3	#14583231	C₂	1	1	yes	maximal, characteristic
4	#14583228	C₂	1	1	yes	maximal, characteristic
5	#14583183	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583182	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583108	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583107	C₂	1	1	yes	maximal, characteristic, complemented
9	#14583056	C₂	1	1	yes	maximal, characteristic
10	#14583053	C₂	1	1	yes	maximal, characteristic
11	#14583007	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583004	C₂	1	1	yes	maximal, characteristic, complemented
13	#14583004	C₂	1	1	yes	maximal, characteristic, complemented
14	#14582954	C₂	1	1	yes	maximal, characteristic
15	C₄₃×#441	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₄₃	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₇₂	C₂²	1	1	yes	characteristic
18	C₂×#10621885	C₂²	1	1	yes	characteristic
19	C₂×#10621885	C₂²	1	1	yes	characteristic
20	#13063324	C₂²	1	1	yes	characteristic
21	#13063323	C₂²	1	1	yes	characteristic
22	#13063315	—	2	1	no	—
23	#13063305	C₂²	1	1	yes	characteristic
24	#13063303	—	2	1	no	—
25	#13063301	C₂²	1	1	yes	characteristic
26	Q₈×Dic₄₃	C₂²	1	1	yes	characteristic
27	C₄×Dic₈₆	—	2	1	no	—
28	C₄×Dic₈₆	—	2	1	no	—
29	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
30	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
31	#13063290	—	2	1	no	—
32	#13063290	C₂²	1	1	yes	characteristic
33	#13063282	—	2	1	no	—
34	C₄⋊C₄×D₄₃	—	2	1	no	—
35	C₄⋊C₄×D₄₃	C₂²	1	1	yes	characteristic
36	C₄²×D₄₃	—	2	1	no	—
37	C₄²×D₄₃	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	Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic
42	Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic
43	#13063250	C₂²	1	1	yes	characteristic
44	#13063250	C₂²	1	1	yes	characteristic
45	#13063249	C₂²	1	1	yes	characteristic
46	#13063249	C₂²	1	1	yes	characteristic
47	#13063248	C₂²	1	1	yes	characteristic
48	#13063248	C₂²	1	1	yes	characteristic
49	(C₂×D₄₃)⋊C₈	C₂²	1	1	yes	characteristic
50	(C₂×D₄₃)⋊C₈	C₂²	1	1	yes	characteristic
51	#13063236	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₄₃	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
54	C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
55	#106×C₄₃	C₂²	1	1	yes	characteristic
56	#103×C₄₃	C₂²	1	1	yes	characteristic
57	#103×C₄₃	C₂²	1	1	yes	characteristic
58	#97×C₄₃	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₄₃	C₂³	1	1	yes	characteristic
60	C₂×Q₈×C₄₃	C₂³	1	1	yes	characteristic
61	Q₈×D₄₃	—	2	2	no	—
62	Q₈×D₄₃	—	2	2	no	—
63	C₂×C₄×D₄₃	—	2	1	no	—
64	C₂×C₄×D₄₃	—	2	1	no	—
65	C₂×C₄×D₄₃	D₄	1	1	yes	characteristic
66	C₂×C₄×D₄₃	D₄	1	1	yes	characteristic
67	C₂×C₄×D₄₃	C₂³	1	1	yes	characteristic
68	C₂×Dic₈₆	—	2	1	no	—
69	C₂×Dic₈₆	—	2	1	no	—
70	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
71	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
72	Q₁₆×C₄₃	—	2	2	no	—
73	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
75	C₄⋊C₄×C₄₃	—	2	1	no	—
76	C₄⋊C₄×C₄₃	—	2	1	no	—
77	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
79	C₄²×C₄₃	—	2	1	no	—
80	C₄²×C₄₃	—	2	1	no	—
81	C₄²×C₄₃	C₂³	1	1	yes	characteristic
82	#10621885	—	2	2	no	—
83	#10621885	—	2	2	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	Dic₄₃⋊C₄	—	2	1	no	—
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	—	2	1	no	—
99	C₄×Dic₄₃	—	2	1	no	—
100	C₄×Dic₄₃	—	2	1	no	—
101	C₄×Dic₄₃	—	2	1	no	—
102	C₄×Dic₄₃	D₄	1	1	yes	characteristic
103	C₄×Dic₄₃	D₄	1	1	yes	characteristic
104	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
105	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
106	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
107	Dic₁₇₂	—	2	2	no	—
108	C₂²×D₄₃	C₂×D₄	1	1	yes	characteristic
109	Q₈×C₄₃	—	2	1	no	—
110	Q₈×C₄₃	—	2	1	no	—
111	Q₈×C₄₃	—	2	1	no	—
112	Q₈×C₄₃	#41	1	2	yes	—
113	C₂×C₄×C₄₃	—	2	1	no	—
114	C₂×C₄×C₄₃	—	2	1	no	—
115	C₂×C₄×C₄₃	—	2	1	no	—
116	C₂×C₄×C₄₃	—	2	1	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	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
127	C₄×D₄₃	—	2	4	no	—
128	C₄×D₄₃	—	2	2	no	—
129	C₄×D₄₃	—	2	1	no	—
130	C₄×D₄₃	—	2	1	no	—
131	C₄×D₄₃	—	2	2	no	—
132	C₄×D₄₃	—	4	2	no	—
133	Dic₈₆	—	2	4	no	—
134	Dic₈₆	—	2	1	no	—
135	Dic₈₆	—	2	1	no	—
136	Dic₈₆	—	2	1	no	—
137	Dic₈₆	—	4	2	no	—
138	Dic₈₆	#41	1	2	yes	—
139	C₈×C₄₃	—	2	1	no	—
140	C₈×C₄₃	—	2	1	no	—
141	C₄₃⋊C₈	—	2	1	no	—
142	C₄₃⋊C₈	—	2	1	no	—
143	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
144	C₂×D₄₃	—	2	1	no	—
145	C₂×D₄₃	—	2	1	no	—
146	C₂×D₄₃	#137	1	2	yes	—
147	C₄×C₄₃	—	2	2	no	—
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	—	4	1	no	—
153	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
154	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
155	Dic₄₃	—	2	1	no	—
156	Dic₄₃	—	2	2	no	—
157	Dic₄₃	—	2	1	no	—
158	Dic₄₃	—	4	1	no	—
159	Dic₄₃	—	4	1	no	—
160	Dic₄₃	—	4	1	no	—
161	Dic₄₃	#137	1	2	yes	—
162	#3113	—	43	1	no	maximal, Sylow
163	C₂×C₄₃	#525	1	1	yes	characteristic
164	C₂×C₄₃	C₂×#137	1	1	yes	characteristic
165	C₂×C₄₃	C₂×#137	1	1	yes	characteristic
166	D₄₃	—	2	2	no	—
167	#533	—	43	1	no	—
168	C₂×C₄×Q₈	—	43	1	no	—
169	#483	—	43	1	no	—
170	#483	—	43	1	no	—
171	#475	—	43	1	no	—
172	#475	—	43	1	no	—
173	#462	—	43	1	no	—
174	#462	—	43	1	no	—
175	#451	—	43	1	no	—
176	#451	—	43	1	no	—
177	#441	—	43	1	no	—
178	#441	—	43	1	no	—
179	#441	—	43	1	no	—
180	#441	D₄₃	1	1	yes	characteristic, complemented
181	#432	—	43	1	no	—
182	C₄₃	#3113	1	1	yes	characteristic, complemented, Sylow
183	C₂²×Q₈	—	43	1	no	—
184	C₂×Q₁₆	—	43	1	no	—
185	C₂×Q₁₆	—	43	1	no	—
186	C₂×Q₁₆	—	43	1	no	—
187	C₂×Q₁₆	C₂×D₄₃	1	1	yes	characteristic
188	C₄⋊Q₈	—	43	1	no	—
189	C₄⋊Q₈	—	43	1	no	—
190	#125	—	86	1	no	—
191	C₂²⋊Q₈	—	43	1	no	—
192	C₂²⋊Q₈	—	43	1	no	—
193	C₂²⋊Q₈	—	86	1	no	—
194	C₄×Q₈	—	43	1	no	—
195	C₄×Q₈	—	43	1	no	—
196	C₄×Q₈	—	43	1	no	—
197	C₄×Q₈	—	43	1	no	—
198	C₄×Q₈	—	86	1	no	—
199	C₄×Q₈	—	86	1	no	—
200	C₄×Q₈	—	86	1	no	—
201	C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
202	C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
203	#117	—	86	1	no	—
204	C₂×C₄⋊C₄	—	43	1	no	—
205	C₂×C₄⋊C₄	—	86	1	no	—
206	C₂×C₄²	—	43	1	no	—
207	C₂×C₄²	—	86	1	no	—
208	#106	—	43	1	no	—
209	#106	—	43	1	no	—
210	#106	—	43	1	no	—
211	#106	C₂×D₄₃	1	1	yes	characteristic
212	C₄⋊C₈	—	43	1	no	—
213	C₄⋊C₈	—	43	1	no	—
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	—	43	1	no	—
220	#103	C₂×D₄₃	1	1	yes	characteristic
221	#103	C₂×D₄₃	1	1	yes	characteristic
222	C₂²⋊C₈	—	43	1	no	—
223	C₂²⋊C₈	—	43	1	no	—
224	#97	—	43	1	no	—
225	#97	C₂×D₄₃	1	1	yes	characteristic
226	C₂×Q₈	—	43	1	no	—
227	C₂×Q₈	—	43	1	no	—
228	C₂×Q₈	—	86	1	no	—
229	C₂×Q₈	—	86	1	no	—
230	C₂×Q₈	—	86	2	no	—
231	C₂×Q₈	—	86	2	no	—
232	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
233	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
234	C₂²×C₄	—	43	1	no	—
235	C₂²×C₄	—	43	1	no	—
236	C₂²×C₄	—	43	1	no	—
237	C₂²×C₄	—	86	1	no	—
238	C₂²×C₄	—	86	1	no	—
239	Q₁₆	—	2	2	no	—
240	Q₁₆	—	86	2	no	—
241	Q₁₆	—	86	2	no	—
242	Q₁₆	—	86	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	—	2	1	no	—
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	43	1	no	—
250	C₄⋊C₄	—	43	1	no	—
251	C₄⋊C₄	—	86	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	C₂²⋊C₄	—	86	1	no	—
264	C₂²⋊C₄	—	86	1	no	—
265	C₄²	—	2	1	no	—
266	C₄²	—	2	1	no	—
267	C₄²	—	43	1	no	—
268	C₄²	—	43	1	no	—
269	C₄²	—	43	1	no	—
270	C₄²	—	86	1	no	—
271	C₄²	—	86	1	no	—
272	C₄²	—	86	1	no	—
273	C₄²	—	86	1	no	—
274	C₄²	C₂²×D₄₃	1	1	yes	characteristic
275	C₂³	—	43	1	no	—
276	Q₈	—	2	1	no	—
277	Q₈	—	2	1	no	—
278	Q₈	—	2	1	no	—
279	Q₈	—	43	2	no	—
280	Q₈	—	86	4	no	—
281	Q₈	—	86	1	no	—
282	Q₈	—	86	1	no	—
283	Q₈	—	86	1	no	—
284	Q₈	—	172	2	no	—
285	Q₈	#10621898	1	2	yes	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	43	1	no	—
291	C₂×C₄	—	43	1	no	—
292	C₂×C₄	—	43	1	no	—
293	C₂×C₄	—	86	2	no	—
294	C₂×C₄	—	86	2	no	—
295	C₂×C₄	—	86	4	no	—
296	C₂×C₄	—	86	1	no	—
297	C₂×C₄	—	86	1	no	—
298	C₂×C₄	—	86	1	no	—
299	C₂×C₄	—	86	1	no	—
300	C₂×C₄	—	86	1	no	—
301	C₂×C₄	—	86	1	no	—
302	C₂×C₄	—	172	2	no	—
303	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
304	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
305	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
306	C₈	—	2	1	no	—
307	C₈	—	2	1	no	—
308	C₈	—	86	1	no	—
309	C₈	—	86	1	no	—
310	C₂²	—	43	2	no	—
311	C₂²	—	86	1	no	—
312	C₂²	—	86	1	no	—
313	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
314	C₄	—	2	2	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	4	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	2	no	—
323	C₄	—	86	1	no	—
324	C₄	—	172	1	no	—
325	C₄	—	172	1	no	—
326	C₄	—	172	1	no	—
327	C₄	#41×D₄₃	1	1	yes	characteristic
328	C₄	C₂×#10621898	1	1	yes	characteristic
329	C₂	—	86	2	no	—
330	C₂	#14583408	1	1	yes	characteristic
331	C₂	#137×D₄₃	1	1	yes	characteristic
332	C₂	#137×D₄₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#441	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#441	D₄₃
hypocenter	C₄₃	#3113
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #441

Complements

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

N	H ≅ G/N	type
5. #14583183	C₂	semidirect
6. #14583182	C₂	semidirect
7. #14583108	C₂	semidirect
8. #14583107	C₂	semidirect
11. #14583007	C₂	semidirect
12. #14583004	C₂	semidirect
13. #14583004	C₂	semidirect
15. C₄₃×#441	C₂	semidirect
180. #441	D₄₃	semidirect
182. C₄₃	#3113	semidirect