Subgroups — #19148490

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	#14583635	C₂	1	4	yes	maximal, complemented
2	#14583618	C₂	1	4	yes	maximal, complemented
3	#14583580	C₂	1	8	yes	maximal, complemented
4	#14583516	C₂	1	4	yes	maximal, complemented
5	#14583470	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583452	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583430	C₂	1	4	yes	maximal, complemented
8	C₂×#13063312	C₂	1	1	yes	maximal, characteristic, complemented
9	#14583376	C₂	1	1	yes	maximal, characteristic, complemented
10	#14583374	C₂	1	1	yes	maximal, characteristic, complemented
11	#41×Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄₃×#535	C₂	1	1	yes	maximal, characteristic
13	C₂²×Dic₈₆	C₂²	1	1	yes	characteristic
14	#13063324	C₂²	1	2	yes	complemented
15	#13063323	C₂²	1	2	yes	complemented
16	#13063318	C₂²	1	8	yes	complemented
17	#13063318	C₂²	1	8	yes	complemented
18	#13063316	C₂²	1	2	yes	complemented
19	#13063312	C₂²	1	4	yes	—
20	#13063312	C₂²	1	2	yes	—
21	#13063312	C₂²	1	2	yes	—
22	#13063312	C₂²	1	8	yes	complemented
23	#13063312	C₂²	1	2	yes	complemented
24	#13063306	C₂²	1	4	yes	—
25	#13063306	C₂²	1	8	yes	complemented
26	#13063306	C₂²	1	8	yes	complemented
27	#13063304	C₂²	1	4	yes	complemented
28	#13063302	C₂²	1	4	yes	—
29	#13063302	C₂²	1	8	yes	complemented
30	Q₈×Dic₄₃	C₂²	1	2	yes	complemented
31	C₄×Dic₈₆	C₂²	1	4	yes	complemented
32	C₄×Dic₈₆	C₂²	1	4	yes	complemented
33	#13063290	C₂²	1	4	yes	complemented
34	D₄×Dic₄₃	C₂²	1	4	yes	—
35	D₄×Dic₄₃	C₂²	1	4	yes	—
36	D₄×Dic₄₃	C₂²	1	4	yes	—
37	D₄×Dic₄₃	C₂²	1	8	yes	complemented
38	D₄×Dic₄₃	C₂²	1	2	yes	complemented
39	#13063284	C₂²	1	1	yes	characteristic
40	#13063284	C₂²	1	1	yes	characteristic
41	#13063284	C₂²	1	2	yes	complemented
42	#13063281	C₂²	1	4	yes	—
43	#13063281	C₂²	1	8	yes	complemented
44	C₂×C₂²⋊Dic₄₃	C₂²	1	4	yes	—
45	C₂×C₂²⋊Dic₄₃	C₂²	1	4	yes	—
46	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
47	C₂×C₄×Dic₄₃	C₂²	1	2	yes	complemented
48	C₂×#41×C₄₃	C₂²	1	1	yes	characteristic
49	C₂²×D₄×C₄₃	C₂²	1	1	yes	characteristic
50	#127×C₄₃	C₂²	1	2	yes	—
51	#124×C₄₃	C₂²	1	2	yes	—
52	#121×C₄₃	C₂²	1	4	yes	—
53	#120×C₄₃	C₂²	1	4	yes	—
54	#117×C₄₃	C₂²	1	1	yes	characteristic
55	C₂⁴×C₄₃	C₂³	1	2	yes	—
56	#41×C₄₃	—	2	4	no	—
57	C₂×Q₈×C₄₃	C₂³	1	1	yes	characteristic
58	C₂×D₄×C₄₃	—	2	4	no	—
59	C₂×D₄×C₄₃	C₂³	1	2	yes	—
60	C₂×D₄×C₄₃	C₂³	1	2	yes	—
61	C₂×D₄×C₄₃	C₂³	1	2	yes	—
62	C₂×D₄×C₄₃	C₂³	1	1	yes	characteristic
63	C₂×D₄×C₄₃	C₂³	1	8	yes	—
64	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
65	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
66	C₂²×Dic₄₃	C₂³	1	2	yes	—
67	C₂²×Dic₄₃	C₂³	1	2	yes	—
68	C₂²×Dic₄₃	C₂³	1	4	yes	—
69	C₂²×Dic₄₃	C₂³	1	8	yes	—
70	C₂×Dic₈₆	—	2	4	no	—
71	C₂×Dic₈₆	C₂³	1	2	yes	—
72	C₂×Dic₈₆	C₂³	1	2	yes	—
73	C₂×Dic₈₆	C₂³	1	4	yes	—
74	C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
75	C₂²⋊C₄×C₄₃	C₂³	1	2	yes	—
76	C₂²⋊C₄×C₄₃	C₂³	1	8	yes	—
77	C₄²×C₄₃	C₂³	1	2	yes	—
78	C₂²⋊Dic₄₃	C₂³	1	2	yes	—
79	C₂²⋊Dic₄₃	C₂³	1	2	yes	—
80	C₂²⋊Dic₄₃	C₂³	1	8	yes	—
81	C₂²⋊Dic₄₃	C₂³	1	8	yes	—
82	C₂²⋊Dic₄₃	C₂³	1	8	yes	—
83	C₂²⋊Dic₄₃	C₂³	1	4	yes	—
84	C₂²⋊Dic₄₃	C₂³	1	8	yes	—
85	C₄⋊Dic₄₃	C₂³	1	2	yes	—
86	C₄⋊Dic₄₃	C₂³	1	2	yes	—
87	C₄⋊Dic₄₃	C₂³	1	4	yes	—
88	C₄⋊Dic₄₃	C₂³	1	4	yes	—
89	Dic₄₃⋊C₄	C₂³	1	8	yes	—
90	Dic₄₃⋊C₄	C₂³	1	4	yes	—
91	Dic₄₃⋊C₄	C₂³	1	8	yes	—
92	Dic₄₃⋊C₄	C₂³	1	4	yes	—
93	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
94	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
95	C₄×Dic₄₃	C₂³	1	2	yes	—
96	C₄×Dic₄₃	C₂³	1	4	yes	—
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	2	yes	—
104	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
105	Q₈×C₄₃	—	2	2	no	—
106	D₄×C₄₃	—	2	4	no	—
107	D₄×C₄₃	—	2	4	no	—
108	D₄×C₄₃	—	2	4	no	—
109	D₄×C₄₃	—	2	2	no	—
110	D₄×C₄₃	—	4	8	no	—
111	C₂×C₄×C₄₃	—	2	4	no	—
112	C₂×C₄×C₄₃	#41	1	4	yes	—
113	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
114	C₂×C₄×C₄₃	C₂⁴	1	4	yes	—
115	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
116	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
117	C₂×Dic₄₃	—	2	4	no	—
118	C₂×Dic₄₃	—	4	2	no	—
119	C₂×Dic₄₃	—	4	8	no	—
120	C₂×Dic₄₃	—	4	4	no	—
121	C₂×Dic₄₃	C₂⁴	1	2	yes	—
122	C₂×Dic₄₃	C₂⁴	1	4	yes	—
123	C₂×Dic₄₃	C₂⁴	1	4	yes	—
124	C₂×Dic₄₃	C₂⁴	1	2	yes	—
125	C₂×Dic₄₃	C₂⁴	1	4	yes	—
126	Dic₈₆	—	2	4	no	—
127	Dic₈₆	—	2	4	no	—
128	Dic₈₆	—	4	4	no	—
129	C₂²×C₄₃	—	2	2	no	—
130	C₂²×C₄₃	—	2	4	no	—
131	C₂²×C₄₃	—	2	2	no	—
132	C₂²×C₄₃	—	2	8	no	—
133	C₂²×C₄₃	—	2	4	no	—
134	C₂²×C₄₃	—	8	4	no	—
135	C₂²×C₄₃	C₂×#41	1	2	yes	—
136	C₂²×C₄₃	C₂⁵	1	1	yes	characteristic
137	C₄×C₄₃	—	2	2	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	4	no	—
144	Dic₄₃	—	2	2	no	—
145	Dic₄₃	—	2	4	no	—
146	#3510	—	43	1	no	maximal, Sylow
147	C₂×C₄₃	—	2	2	no	—
148	C₂×C₄₃	—	4	2	no	—
149	C₂×C₄₃	—	4	4	no	—
150	C₂×C₄₃	C₂²×#41	1	1	yes	characteristic
151	C₂×C₄₃	#585	1	2	yes	—
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	D₄₃	1	1	yes	characteristic
158	#533	—	43	1	no	—
159	#529	—	43	4	no	—
160	C₂×#124	—	43	1	no	—
161	#519	—	43	1	no	—
162	#518	—	43	1	no	—
163	C₄×#41	—	43	1	no	—
164	C₄₃	#3510	1	1	yes	characteristic, complemented, Sylow
165	C₂×#41	C₂×D₄₃	1	1	yes	characteristic
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	C₂×D₄₃	1	2	yes	—
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	C₂×D₄₃	1	2	yes	—
180	#123	—	43	8	no	—
181	#123	—	43	4	no	—
182	#123	—	43	8	no	—
183	C₂²⋊Q₈	—	43	4	no	—
184	C₂²⋊Q₈	—	43	4	no	—
185	C₂²⋊Q₈	—	43	8	no	—
186	#121	C₂×D₄₃	1	4	yes	—
187	#120	C₂×D₄₃	1	4	yes	—
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	4	no	—
193	C₄×D₄	—	43	2	no	—
194	C₄×D₄	—	43	4	no	—
195	C₄×D₄	—	43	4	no	—
196	C₄×D₄	—	43	8	no	—
197	#117	—	43	1	no	—
198	#117	—	43	2	no	—
199	#117	—	43	1	no	—
200	#117	—	43	4	no	—
201	#117	—	43	8	no	—
202	#117	C₂×D₄₃	1	1	yes	characteristic
203	C₂×C₂²⋊C₄	—	43	4	no	—
204	C₂×C₂²⋊C₄	—	43	4	no	—
205	C₂×C₄²	—	43	1	no	—
206	C₂×C₄²	—	43	2	no	—
207	C₂⁴	C₂²×D₄₃	1	2	yes	—
208	#41	—	2	4	no	—
209	C₂×Q₈	—	43	2	no	—
210	C₂×Q₈	—	43	2	no	—
211	C₂×Q₈	—	43	4	no	—
212	C₂×Q₈	—	86	4	no	—
213	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
214	C₂×D₄	—	2	4	no	—
215	C₂×D₄	C₂²×D₄₃	1	2	yes	—
216	C₂×D₄	C₂²×D₄₃	1	2	yes	—
217	C₂×D₄	C₂²×D₄₃	1	2	yes	—
218	C₂×D₄	C₂²×D₄₃	1	1	yes	characteristic
219	C₂×D₄	C₂²×D₄₃	1	8	yes	—
220	C₂²×C₄	—	43	2	no	—
221	C₂²×C₄	—	43	2	no	—
222	C₂²×C₄	—	43	4	no	—
223	C₂²×C₄	—	43	8	no	—
224	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
225	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
226	C₄⋊C₄	—	43	4	no	—
227	C₄⋊C₄	—	43	2	no	—
228	C₄⋊C₄	—	43	2	no	—
229	C₄⋊C₄	—	43	4	no	—
230	C₄⋊C₄	—	43	8	no	—
231	C₄⋊C₄	—	43	8	no	—
232	C₄⋊C₄	—	43	4	no	—
233	C₄⋊C₄	—	43	4	no	—
234	C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
235	C₂²⋊C₄	—	43	2	no	—
236	C₂²⋊C₄	—	43	2	no	—
237	C₂²⋊C₄	—	43	8	no	—
238	C₂²⋊C₄	—	43	8	no	—
239	C₂²⋊C₄	—	43	8	no	—
240	C₂²⋊C₄	—	43	4	no	—
241	C₂²⋊C₄	—	43	8	no	—
242	C₂²⋊C₄	C₂²×D₄₃	1	2	yes	—
243	C₂²⋊C₄	C₂²×D₄₃	1	8	yes	—
244	C₄²	—	43	1	no	—
245	C₄²	—	43	2	no	—
246	C₄²	—	43	4	no	—
247	C₄²	—	43	1	no	—
248	C₄²	—	43	2	no	—
249	C₄²	—	43	8	no	—
250	C₄²	—	43	2	no	—
251	C₄²	C₂²×D₄₃	1	2	yes	—
252	C₂³	—	2	4	no	—
253	C₂³	—	4	4	no	—
254	C₂³	C₂³×D₄₃	1	4	yes	—
255	C₂³	C₂³×D₄₃	1	2	yes	—
256	C₂³	C₂³×D₄₃	1	1	yes	characteristic
257	Q₈	—	2	2	no	—
258	Q₈	—	86	4	no	—
259	Q₈	—	86	4	no	—
260	Q₈	—	172	4	no	—
261	D₄	—	2	4	no	—
262	D₄	—	2	4	no	—
263	D₄	—	2	4	no	—
264	D₄	—	2	2	no	—
265	D₄	—	4	8	no	—
266	C₂×C₄	—	2	4	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	4	no	—
272	C₂×C₄	—	86	4	no	—
273	C₂×C₄	—	172	4	no	—
274	C₂×C₄	—	172	8	no	—
275	C₂×C₄	—	172	2	no	—
276	C₂×C₄	#10621900	1	4	yes	—
277	C₂×C₄	C₂³×D₄₃	1	4	yes	—
278	C₂×C₄	C₂³×D₄₃	1	2	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	2	no	—
283	C₂²	—	2	4	no	—
284	C₂²	—	2	4	no	—
285	C₂²	—	2	8	no	—
286	C₂²	—	8	4	no	—
287	C₂²	C₂×#10621900	1	2	yes	—
288	C₂²	C₂⁴×D₄₃	1	1	yes	characteristic
289	C₄	—	2	2	no	—
290	C₄	—	2	4	no	—
291	C₄	—	86	4	no	—
292	C₄	—	86	2	no	—
293	C₄	—	86	4	no	—
294	C₄	—	86	4	no	—
295	C₄	—	86	2	no	—
296	C₄	C₂×#10621900	1	2	yes	—
297	C₄	C₂×#10621900	1	2	yes	—
298	C₂	—	2	2	no	—
299	C₂	—	4	4	no	—
300	C₂	—	4	2	no	—
301	C₂	C₂²×#10621900	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×#535	C₂
hypercenter	#535	D₄₃
hypocenter	C₄₃	#3510
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ #535

Complements

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

N	H ≅ G/N	type
1. #14583635	C₂	semidirect
2. #14583618	C₂	semidirect
3. #14583580	C₂	semidirect
4. #14583516	C₂	semidirect
5. #14583470	C₂	semidirect
6. #14583452	C₂	semidirect
7. #14583430	C₂	semidirect
8. C₂×#13063312	C₂	semidirect
9. #14583376	C₂	semidirect
10. #14583374	C₂	semidirect
11. #41×Dic₄₃	C₂	semidirect
14. #13063324	C₂²	semidirect
15. #13063323	C₂²	semidirect
16. #13063318	C₂²	semidirect
17. #13063318	C₂²	semidirect
18. #13063316	C₂²	semidirect
22. #13063312	C₂²	semidirect
23. #13063312	C₂²	semidirect
25. #13063306	C₂²	semidirect
26. #13063306	C₂²	semidirect
27. #13063304	C₂²	semidirect
29. #13063302	C₂²	semidirect
30. Q₈×Dic₄₃	C₂²	semidirect
31. C₄×Dic₈₆	C₂²	semidirect
32. C₄×Dic₈₆	C₂²	semidirect
33. #13063290	C₂²	semidirect
37. D₄×Dic₄₃	C₂²	semidirect
38. D₄×Dic₄₃	C₂²	semidirect
41. #13063284	C₂²	semidirect
43. #13063281	C₂²	semidirect
47. C₂×C₄×Dic₄₃	C₂²	semidirect
164. C₄₃	#3510	semidirect