Subgroups — #19148486

Order 5504.

1040 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583632	C₂	1	4	yes	maximal, complemented
2	#14583619	C₂	1	4	yes	maximal, complemented
3	#14583575	C₂	1	8	yes	maximal, complemented
4	#14583514	C₂	1	4	yes	maximal, complemented
5	#14583475	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583466	C₂	1	1	yes	maximal, characteristic
7	#14583452	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583431	C₂	1	4	yes	maximal, complemented
9	C₂×#13063312	C₂	1	1	yes	maximal, characteristic, complemented
10	#14583376	C₂	1	1	yes	maximal, characteristic, complemented
11	#14583367	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄₃×#518	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#10621898	C₂²	1	1	yes	characteristic
14	C₂²×Dic₈₆	C₂²	1	1	yes	characteristic
15	#13063325	C₂²	1	2	yes	complemented
16	#13063324	C₂²	1	2	yes	complemented
17	#13063321	C₂²	1	2	yes	—
18	#13063318	C₂²	1	8	yes	complemented
19	#13063318	C₂²	1	8	yes	complemented
20	#13063316	C₂²	1	2	yes	complemented
21	#13063313	C₂²	1	2	yes	complemented
22	#13063312	C₂²	1	2	yes	—
23	#13063312	C₂²	1	4	yes	—
24	#13063312	C₂²	1	2	yes	—
25	#13063312	C₂²	1	2	yes	—
26	#13063311	C₂²	1	8	yes	complemented
27	#13063309	C₂²	1	4	yes	—
28	#13063308	C₂²	1	8	yes	complemented
29	#13063307	C₂²	1	8	yes	complemented
30	#13063305	C₂²	1	4	yes	complemented
31	#13063304	C₂²	1	8	yes	complemented
32	#13063302	C₂²	1	4	yes	—
33	#13063296	C₂²	1	4	yes	—
34	#13063294	C₂²	1	4	yes	—
35	C₄×Dic₈₆	C₂²	1	4	yes	complemented
36	C₄×Dic₈₆	C₂²	1	4	yes	complemented
37	#13063290	C₂²	1	2	yes	complemented
38	#13063290	C₂²	1	4	yes	complemented
39	#13063287	C₂²	1	2	yes	complemented
40	C₄×C₂²⋊D₄₃	C₂²	1	8	yes	complemented
41	(C₂²⋊D₄₃)⋊C₄	C₂²	1	4	yes	—
42	#13063284	C₂²	1	1	yes	characteristic
43	#13063284	C₂²	1	8	yes	complemented
44	#13063282	C₂²	1	2	yes	complemented
45	#13063281	C₂²	1	4	yes	—
46	C₂×C₂²⋊Dic₄₃	C₂²	1	4	yes	—
47	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
48	C₄²×D₄₃	C₂²	1	2	yes	complemented
49	C₂²×D₄×C₄₃	C₂²	1	1	yes	characteristic
50	C₄×D₄×C₄₃	C₂²	1	4	yes	—
51	C₄×D₄×C₄₃	C₂²	1	4	yes	—
52	#117×C₄₃	C₂²	1	1	yes	characteristic
53	#117×C₄₃	C₂²	1	1	yes	characteristic
54	C₂×C₂²⋊C₄×C₄₃	C₂²	1	4	yes	—
55	C₂⁴×C₄₃	C₂³	1	2	yes	—
56	C₂×D₄×C₄₃	—	2	4	no	—
57	C₂×D₄×C₄₃	C₂³	1	2	yes	—
58	C₂×D₄×C₄₃	C₂³	1	2	yes	—
59	C₂²×C₄×C₄₃	C₂³	1	8	yes	—
60	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
61	C₂×C₂²⋊D₄₃	C₂³	1	2	yes	—
62	C₂×C₂²⋊D₄₃	C₂³	1	8	yes	—
63	C₂²×Dic₄₃	C₂³	1	2	yes	—
64	C₂²×Dic₄₃	C₂³	1	2	yes	—
65	#10621898	—	2	4	no	—
66	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
67	C₂×C₄×D₄₃	C₂³	1	2	yes	—
68	C₂×C₄×D₄₃	C₂³	1	4	yes	—
69	C₂×Dic₈₆	—	2	4	no	—
70	C₂×Dic₈₆	C₂³	1	2	yes	—
71	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
72	C₂×Dic₈₆	C₂³	1	2	yes	—
73	C₂×Dic₈₆	C₂³	1	4	yes	—
74	C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
75	C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
76	C₂²⋊C₄×C₄₃	C₂³	1	2	yes	—
77	C₂²⋊C₄×C₄₃	C₂³	1	2	yes	—
78	C₂²⋊C₄×C₄₃	C₂³	1	8	yes	—
79	C₄²×C₄₃	C₂³	1	2	yes	—
80	C₄²×C₄₃	C₂³	1	2	yes	—
81	C₂²⋊Dic₄₃	C₂³	1	8	yes	—
82	C₂²⋊Dic₄₃	C₂³	1	8	yes	—
83	C₂²⋊Dic₄₃	C₂³	1	2	yes	—
84	C₂²⋊Dic₄₃	C₂³	1	8	yes	—
85	(C₂×D₄₃)⋊C₄	C₂³	1	4	yes	—
86	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	—
87	C₄⋊Dic₄₃	C₂³	1	2	yes	—
88	C₄⋊Dic₄₃	C₂³	1	4	yes	—
89	C₄⋊Dic₄₃	C₂³	1	8	yes	—
90	Dic₄₃⋊C₄	C₂³	1	8	yes	—
91	Dic₄₃⋊C₄	C₂³	1	4	yes	—
92	Dic₄₃⋊C₄	C₂³	1	4	yes	—
93	Dic₄₃⋊C₄	C₂³	1	4	yes	—
94	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
95	C₄×Dic₄₃	C₂³	1	4	yes	—
96	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
97	C₄×Dic₄₃	C₂³	1	2	yes	—
98	C₄×Dic₄₃	C₂³	1	8	yes	—
99	C₄×Dic₄₃	C₂³	1	2	yes	—
100	C₂³×C₄₃	—	2	4	no	—
101	C₂³×C₄₃	—	4	4	no	—
102	C₂³×C₄₃	C₂⁴	1	4	yes	—
103	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
104	C₂²×D₄₃	C₂⁴	1	2	yes	—
105	D₄×C₄₃	—	2	4	no	—
106	D₄×C₄₃	—	2	4	no	—
107	C₂×C₄×C₄₃	—	4	8	no	—
108	C₂×C₄×C₄₃	#41	1	4	yes	—
109	C₂×C₄×C₄₃	C₂⁴	1	4	yes	—
110	C₂×C₄×C₄₃	C₂⁴	1	4	yes	—
111	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
112	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
113	C₂²⋊D₄₃	—	2	4	no	—
114	C₂²⋊D₄₃	—	4	8	no	—
115	C₂×Dic₄₃	—	2	4	no	—
116	C₂×Dic₄₃	—	4	2	no	—
117	C₂×Dic₄₃	C₂⁴	1	2	yes	—
118	C₂×Dic₄₃	C₂⁴	1	4	yes	—
119	C₂×Dic₄₃	C₂⁴	1	2	yes	—
120	C₂×Dic₄₃	C₂⁴	1	2	yes	—
121	C₂×Dic₄₃	C₂⁴	1	4	yes	—
122	D₁₇₂	—	2	2	no	—
123	C₄×D₄₃	—	2	4	no	—
124	C₄×D₄₃	—	4	4	no	—
125	Dic₈₆	—	2	4	no	—
126	Dic₈₆	—	2	4	no	—
127	Dic₈₆	—	2	2	no	—
128	Dic₈₆	—	4	4	no	—
129	C₂²×C₄₃	—	2	2	no	—
130	C₂²×C₄₃	—	2	4	no	—
131	C₂²×C₄₃	—	2	8	no	—
132	C₂²×C₄₃	—	8	4	no	—
133	C₂²×C₄₃	C₂×#41	1	2	yes	—
134	C₂²×C₄₃	C₂⁵	1	1	yes	characteristic
135	C₂×D₄₃	—	2	2	no	—
136	C₂×D₄₃	—	2	4	no	—
137	C₄×C₄₃	—	2	4	no	—
138	C₄×C₄₃	—	2	4	no	—
139	C₄×C₄₃	C₂×#41	1	2	yes	—
140	C₄×C₄₃	C₂×#41	1	2	yes	—
141	Dic₄₃	—	2	4	no	—
142	Dic₄₃	—	2	2	no	—
143	Dic₄₃	—	2	2	no	—
144	Dic₄₃	—	2	4	no	—
145	Dic₄₃	—	2	2	no	—
146	#3510	—	43	1	no	maximal, Sylow
147	C₂×C₄₃	—	2	2	no	—
148	C₂×C₄₃	—	4	4	no	—
149	C₂×C₄₃	C₂²×#41	1	1	yes	characteristic
150	C₂×C₄₃	#585	1	2	yes	—
151	D₄₃	—	4	2	no	—
152	#559	—	43	4	no	—
153	#556	—	43	4	no	—
154	#551	—	43	8	no	—
155	#542	—	43	4	no	—
156	#536	—	43	1	no	—
157	#535	—	43	1	no	—
158	#533	—	43	1	no	—
159	#529	—	43	4	no	—
160	C₂×#124	—	43	1	no	—
161	#519	—	43	1	no	—
162	#518	D₄₃	1	1	yes	characteristic, complemented
163	C₄×#41	—	43	1	no	—
164	C₄₃	#3510	1	1	yes	characteristic, complemented, Sylow
165	C₂×#41	—	43	1	no	—
166	C₂²×Q₈	—	43	1	no	—
167	C₂²×D₄	C₂×D₄₃	1	1	yes	characteristic
168	C₄⋊Q₈	—	43	2	no	—
169	C₄⋊Q₈	—	43	2	no	—
170	#127	—	43	2	no	—
171	#126	—	43	8	no	—
172	#126	—	43	8	no	—
173	#125	—	43	2	no	—
174	#124	—	43	2	no	—
175	#124	—	43	2	no	—
176	#124	—	43	4	no	—
177	#124	—	43	2	no	—
178	#124	—	43	8	no	—
179	#124	—	43	2	no	—
180	#123	—	43	8	no	—
181	#123	—	43	8	no	—
182	#123	—	43	4	no	—
183	C₂²⋊Q₈	—	43	8	no	—
184	C₂²⋊Q₈	—	43	4	no	—
185	C₂²⋊Q₈	—	43	4	no	—
186	#121	—	43	4	no	—
187	#120	—	43	4	no	—
188	C₄×Q₈	—	43	2	no	—
189	C₄×Q₈	—	43	4	no	—
190	C₄×Q₈	—	43	4	no	—
191	C₄×Q₈	—	43	4	no	—
192	C₄×D₄	—	43	8	no	—
193	C₄×D₄	—	43	2	no	—
194	C₄×D₄	—	43	4	no	—
195	C₄×D₄	C₂×D₄₃	1	4	yes	—
196	C₄×D₄	C₂×D₄₃	1	4	yes	—
197	#117	—	43	1	no	—
198	#117	—	43	8	no	—
199	#117	—	43	4	no	—
200	#117	—	43	2	no	—
201	#117	C₂×D₄₃	1	1	yes	characteristic
202	#117	C₂×D₄₃	1	1	yes	characteristic
203	C₂×C₂²⋊C₄	—	43	4	no	—
204	C₂×C₂²⋊C₄	C₂×D₄₃	1	4	yes	—
205	C₂×C₄²	—	43	1	no	—
206	C₂×C₄²	—	43	2	no	—
207	C₂⁴	C₂²×D₄₃	1	2	yes	—
208	#41	—	86	4	no	—
209	C₂×Q₈	—	43	2	no	—
210	C₂×Q₈	—	43	2	no	—
211	C₂×Q₈	—	43	1	no	—
212	C₂×Q₈	—	43	4	no	—
213	C₂×Q₈	—	86	4	no	—
214	C₂×D₄	—	2	4	no	—
215	C₂×D₄	—	43	2	no	—
216	C₂×D₄	—	43	8	no	—
217	C₂×D₄	—	43	1	no	—
218	C₂×D₄	C₂²×D₄₃	1	2	yes	—
219	C₂×D₄	C₂²×D₄₃	1	2	yes	—
220	C₂²×C₄	—	43	2	no	—
221	C₂²×C₄	—	43	2	no	—
222	C₂²×C₄	—	43	2	no	—
223	C₂²×C₄	—	43	4	no	—
224	C₂²×C₄	C₂²×D₄₃	1	8	yes	—
225	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
226	C₄⋊C₄	—	43	4	no	—
227	C₄⋊C₄	—	43	4	no	—
228	C₄⋊C₄	—	43	2	no	—
229	C₄⋊C₄	—	43	8	no	—
230	C₄⋊C₄	—	43	8	no	—
231	C₄⋊C₄	—	43	4	no	—
232	C₄⋊C₄	—	43	4	no	—
233	C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
234	C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
235	C₂²⋊C₄	—	43	8	no	—
236	C₂²⋊C₄	—	43	8	no	—
237	C₂²⋊C₄	—	43	2	no	—
238	C₂²⋊C₄	—	43	8	no	—
239	C₂²⋊C₄	—	43	4	no	—
240	C₂²⋊C₄	—	43	8	no	—
241	C₂²⋊C₄	C₂²×D₄₃	1	2	yes	—
242	C₂²⋊C₄	C₂²×D₄₃	1	2	yes	—
243	C₂²⋊C₄	C₂²×D₄₃	1	8	yes	—
244	C₄²	—	43	1	no	—
245	C₄²	—	43	4	no	—
246	C₄²	—	43	1	no	—
247	C₄²	—	43	2	no	—
248	C₄²	—	43	8	no	—
249	C₄²	—	43	2	no	—
250	C₄²	C₂²×D₄₃	1	2	yes	—
251	C₄²	C₂²×D₄₃	1	2	yes	—
252	C₂³	—	2	4	no	—
253	C₂³	—	4	4	no	—
254	C₂³	—	43	2	no	—
255	C₂³	C₂³×D₄₃	1	4	yes	—
256	C₂³	C₂³×D₄₃	1	1	yes	characteristic
257	Q₈	—	86	4	no	—
258	Q₈	—	86	4	no	—
259	Q₈	—	86	2	no	—
260	Q₈	—	172	4	no	—
261	D₄	—	2	4	no	—
262	D₄	—	2	4	no	—
263	D₄	—	86	4	no	—
264	D₄	—	86	2	no	—
265	D₄	—	172	8	no	—
266	C₂×C₄	—	4	8	no	—
267	C₂×C₄	—	43	2	no	—
268	C₂×C₄	—	43	4	no	—
269	C₂×C₄	—	43	4	no	—
270	C₂×C₄	—	43	2	no	—
271	C₂×C₄	—	43	2	no	—
272	C₂×C₄	—	86	4	no	—
273	C₂×C₄	—	86	4	no	—
274	C₂×C₄	—	172	4	no	—
275	C₂×C₄	—	172	2	no	—
276	C₂×C₄	#10621898	1	4	yes	—
277	C₂×C₄	C₂³×D₄₃	1	4	yes	—
278	C₂×C₄	C₂³×D₄₃	1	4	yes	—
279	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
280	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
281	C₂²	—	2	2	no	—
282	C₂²	—	2	4	no	—
283	C₂²	—	2	8	no	—
284	C₂²	—	8	4	no	—
285	C₂²	—	86	2	no	—
286	C₂²	—	86	4	no	—
287	C₂²	C₂×#10621898	1	2	yes	—
288	C₂²	C₂⁴×D₄₃	1	1	yes	characteristic
289	C₄	—	2	4	no	—
290	C₄	—	2	4	no	—
291	C₄	—	86	2	no	—
292	C₄	—	86	4	no	—
293	C₄	—	86	2	no	—
294	C₄	—	86	2	no	—
295	C₄	—	86	4	no	—
296	C₄	C₂×#10621898	1	2	yes	—
297	C₄	C₂×#10621898	1	2	yes	—
298	C₂	—	2	2	no	—
299	C₂	—	4	4	no	—
300	C₂	—	172	2	no	—
301	C₂	C₂²×#10621898	1	1	yes	characteristic
302	C₂	#14583734	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₄₃
commutator, socle	C₂²×C₄₃	C₂⁵
Fitting	C₄₃×#518	C₂
hypercenter	#518	D₄₃
hypocenter	C₄₃	#3510
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ #518

Complements

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

N	H ≅ G/N	type
1. #14583632	C₂	semidirect
2. #14583619	C₂	semidirect
3. #14583575	C₂	semidirect
4. #14583514	C₂	semidirect
5. #14583475	C₂	semidirect
7. #14583452	C₂	semidirect
8. #14583431	C₂	semidirect
9. C₂×#13063312	C₂	semidirect
10. #14583376	C₂	semidirect
11. #14583367	C₂	semidirect
12. C₄₃×#518	C₂	semidirect
15. #13063325	C₂²	semidirect
16. #13063324	C₂²	semidirect
18. #13063318	C₂²	semidirect
19. #13063318	C₂²	semidirect
20. #13063316	C₂²	semidirect
21. #13063313	C₂²	semidirect
26. #13063311	C₂²	semidirect
28. #13063308	C₂²	semidirect
29. #13063307	C₂²	semidirect
30. #13063305	C₂²	semidirect
31. #13063304	C₂²	semidirect
35. C₄×Dic₈₆	C₂²	semidirect
36. C₄×Dic₈₆	C₂²	semidirect
37. #13063290	C₂²	semidirect
38. #13063290	C₂²	semidirect
39. #13063287	C₂²	semidirect
40. C₄×C₂²⋊D₄₃	C₂²	semidirect
43. #13063284	C₂²	semidirect
44. #13063282	C₂²	semidirect
48. C₄²×D₄₃	C₂²	semidirect
162. #518	D₄₃	semidirect
164. C₄₃	#3510	semidirect