Subgroups — #19148372

Order 5504.

1748 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13063371	C₂	1	2	yes	maximal, complemented
2	#14583571	C₂	1	1	yes	maximal, characteristic, complemented
3	#14583568	C₂	1	2	yes	maximal, complemented
4	D₄×D₁₇₂	C₂	1	3	yes	maximal, complemented
5	#14583534	C₂	1	3	yes	maximal, complemented
6	#14583533	C₂	1	6	yes	maximal, complemented
7	#14583465	C₂	1	6	yes	maximal, complemented
8	#14583446	C₂	1	3	yes	maximal, complemented
9	#127×D₄₃	C₂	1	3	yes	maximal, complemented
10	C₄×#10621902	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#550	C₂	1	1	yes	maximal, characteristic, complemented
12	#13063371	—	2	16	no	—
13	C₂×#10621902	C₂²	1	1	yes	characteristic
14	C₂×#10621902	C₂²	1	2	yes	complemented
15	C₂×#10621898	C₂²	1	6	yes	complemented
16	C₂²×D₁₇₂	C₂²	1	6	yes	complemented
17	C₂×D₄×D₄₃	C₂²	1	6	yes	complemented
18	C₂×D₄×D₄₃	C₂²	1	6	yes	complemented
19	C₂×D₄×D₄₃	C₂²	1	6	yes	complemented
20	C₂×D₄×D₄₃	C₂²	1	6	yes	complemented
21	#13063324	C₂²	1	1	yes	characteristic, complemented
22	#13063322	C₂²	1	3	yes	complemented
23	#13063321	C₂²	1	6	yes	complemented
24	#13063321	C₂²	1	3	yes	complemented
25	#13063321	C₂²	1	6	yes	complemented
26	#13063311	C₂²	1	6	yes	complemented
27	#13063300	C₂²	1	6	yes	complemented
28	#13063300	C₂²	1	3	yes	complemented
29	#13063299	C₂²	1	6	yes	complemented
30	#13063298	C₂²	1	6	yes	complemented
31	#13063296	C₂²	1	6	yes	complemented
32	#13063296	C₂²	1	3	yes	complemented
33	#13063295	C₂²	1	12	yes	complemented
34	#13063293	C₂²	1	12	yes	complemented
35	Q₈×Dic₄₃	C₂²	1	1	yes	characteristic, complemented
36	C₄×D₁₇₂	C₂²	1	3	yes	complemented
37	C₄×D₁₇₂	C₂²	1	3	yes	complemented
38	#13063287	C₂²	1	3	yes	complemented
39	C₄×C₂²⋊D₄₃	C₂²	1	6	yes	complemented
40	#13063283	C₂²	1	3	yes	complemented
41	C₄²×D₄₃	C₂²	1	3	yes	complemented
42	C₂×#41×C₄₃	C₂²	1	2	yes	complemented
43	#127×C₄₃	C₂²	1	3	yes	complemented
44	#121×C₄₃	C₂²	1	6	yes	complemented
45	C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic, complemented
46	C₄×D₄×C₄₃	C₂²	1	3	yes	complemented
47	C₂³×D₄₃	C₂³	1	12	yes	—
48	#41×C₄₃	—	2	8	no	—
49	C₂×Q₈×C₄₃	C₂³	1	1	yes	characteristic
50	C₂×D₄×C₄₃	C₂³	1	3	yes	—
51	C₂×D₄×C₄₃	C₂³	1	6	yes	—
52	C₂×D₄×C₄₃	C₂³	1	6	yes	—
53	C₂²×C₄×C₄₃	C₂³	1	6	yes	—
54	C₂×C₂²⋊D₄₃	C₂³	1	6	yes	—
55	C₂×C₂²⋊D₄₃	C₂³	1	6	yes	—
56	C₂×C₂²⋊D₄₃	C₂³	1	6	yes	—
57	C₂×C₂²⋊D₄₃	C₂³	1	12	yes	—
58	#10621902	—	2	8	no	—
59	#10621902	D₄	1	8	yes	complemented
60	D₄×D₄₃	—	2	24	no	—
61	D₄×D₄₃	—	2	24	no	—
62	D₄×D₄₃	—	4	12	no	—
63	D₄×D₄₃	—	4	12	no	—
64	#10621898	—	2	24	no	—
65	C₂×D₁₇₂	—	2	24	no	—
66	C₂×D₁₇₂	C₂³	1	3	yes	—
67	C₂×D₁₇₂	C₂³	1	6	yes	—
68	C₂×D₁₇₂	C₂³	1	6	yes	—
69	C₂×D₁₇₂	C₂³	1	6	yes	—
70	C₂×D₁₇₂	C₂³	1	3	yes	—
71	C₂×C₄×D₄₃	C₂³	1	3	yes	—
72	C₂×C₄×D₄₃	C₂³	1	3	yes	—
73	C₂×C₄×D₄₃	C₂³	1	6	yes	—
74	C₂×C₄×D₄₃	C₂³	1	3	yes	—
75	C₂×Dic₈₆	C₂³	1	2	yes	—
76	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₄₃	C₂³	1	3	yes	—
78	C₂²⋊C₄×C₄₃	C₂³	1	6	yes	—
79	C₄²×C₄₃	C₂³	1	3	yes	—
80	C₂²⋊Dic₄₃	C₂³	1	6	yes	—
81	(C₂×D₄₃)⋊C₄	C₂³	1	6	yes	—
82	(C₂×D₄₃)⋊C₄	C₂³	1	6	yes	—
83	(C₂×D₄₃)⋊C₄	C₂³	1	6	yes	—
84	C₄⋊Dic₄₃	C₂³	1	3	yes	—
85	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
86	Dic₄₃⋊C₄	C₂³	1	2	yes	—
87	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
88	C₄×Dic₄₃	C₂³	1	3	yes	—
89	C₄×Dic₄₃	C₂³	1	3	yes	—
90	C₂³×C₄₃	C₂⁴	1	6	yes	—
91	C₂²×D₄₃	—	2	24	no	—
92	C₂²×D₄₃	—	4	12	no	—
93	C₂²×D₄₃	—	4	12	no	—
94	C₂²×D₄₃	C₂⁴	1	6	yes	—
95	C₂²×D₄₃	C₂⁴	1	3	yes	—
96	Q₈×C₄₃	C₂×D₄	1	4	yes	complemented
97	D₄×C₄₃	—	2	12	no	—
98	D₄×C₄₃	—	2	6	no	—
99	D₄×C₄₃	—	4	6	no	—
100	C₂×C₄×C₄₃	—	2	12	no	—
101	C₂×C₄×C₄₃	C₂⁴	1	3	yes	—
102	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
103	C₂×C₄×C₄₃	C₂⁴	1	3	yes	—
104	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
105	C₂²⋊D₄₃	—	2	12	no	—
106	C₂²⋊D₄₃	—	2	12	no	—
107	C₂²⋊D₄₃	—	2	12	no	—
108	C₂²⋊D₄₃	—	4	12	no	—
109	C₂×Dic₄₃	C₂⁴	1	3	yes	—
110	C₂×Dic₄₃	C₂⁴	1	2	yes	—
111	C₂×Dic₄₃	C₂⁴	1	1	yes	characteristic
112	C₂×Dic₄₃	C₂⁴	1	1	yes	characteristic
113	D₁₇₂	—	2	12	no	—
114	D₁₇₂	—	2	12	no	—
115	D₁₇₂	—	2	6	no	—
116	D₁₇₂	—	4	6	no	—
117	D₁₇₂	C₂×D₄	1	12	yes	complemented
118	C₄×D₄₃	—	2	12	no	—
119	C₄×D₄₃	—	2	6	no	—
120	C₄×D₄₃	—	2	6	no	—
121	C₄×D₄₃	C₂×D₄	1	12	yes	complemented
122	Dic₈₆	—	2	4	no	—
123	C₂²×C₄₃	—	2	6	no	—
124	C₂²×C₄₃	—	2	6	no	—
125	C₂²×C₄₃	—	2	6	no	—
126	C₂²×C₄₃	C₂⁵	1	1	yes	characteristic
127	C₂×D₄₃	—	2	3	no	—
128	C₂×D₄₃	—	2	3	no	—
129	C₂×D₄₃	—	2	6	no	—
130	C₂×D₄₃	—	2	6	no	—
131	C₂×D₄₃	—	2	6	no	—
132	C₂×D₄₃	—	8	24	no	—
133	C₂×D₄₃	C₂²×D₄	1	6	yes	—
134	C₄×C₄₃	—	2	2	no	—
135	C₄×C₄₃	—	2	3	no	—
136	C₄×C₄₃	C₂²×D₄	1	6	yes	—
137	C₄×C₄₃	H(5,𝔽₂)	1	2	yes	complemented
138	Dic₄₃	—	2	2	no	—
139	Dic₄₃	—	2	3	no	—
140	Dic₄₃	C₂²×D₄	1	2	yes	—
141	Dic₄₃	H(5,𝔽₂)	1	2	yes	complemented
142	#3498	—	43	1	no	maximal, Sylow
143	C₂×C₄₃	—	4	6	no	—
144	C₂×C₄₃	C₂³×D₄	1	1	yes	characteristic
145	C₂×C₄₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
146	C₂×C₄₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
147	D₄₃	—	2	6	no	—
148	D₄₃	—	4	6	no	—
149	C₂×H(5,𝔽₂)	—	43	2	no	—
150	#550	—	43	2	no	—
151	#550	—	43	1	no	—
152	#550	D₄₃	1	1	yes	characteristic, complemented
153	D₄²	—	43	3	no	—
154	D₄²	—	43	6	no	—
155	D₄²	—	43	3	no	—
156	#535	—	43	6	no	—
157	#532	—	43	3	no	—
158	C₂×#127	—	43	3	no	—
159	C₄×#41	—	43	1	no	—
160	C₄₃	#3498	1	1	yes	characteristic, complemented, Sylow
161	H(5,𝔽₂)	—	86	16	no	—
162	C₂×#41	—	43	1	no	—
163	C₂×#41	—	43	2	no	—
164	C₂×#41	—	43	6	no	—
165	C₂×#41	C₂×D₄₃	1	2	yes	complemented
166	C₂²×D₄	—	43	6	no	—
167	C₂²×D₄	—	43	6	no	—
168	C₂²×D₄	—	43	6	no	—
169	C₂²×D₄	—	43	6	no	—
170	C₂²×D₄	—	43	6	no	—
171	C₄⋊Q₈	—	43	1	no	—
172	#127	—	43	3	no	—
173	#127	—	43	6	no	—
174	#127	—	43	6	no	—
175	#127	—	43	3	no	—
176	#127	C₂×D₄₃	1	3	yes	complemented
177	#124	—	43	6	no	—
178	#121	—	43	3	no	—
179	#121	—	43	6	no	—
180	#121	—	43	6	no	—
181	#121	—	43	3	no	—
182	#121	—	43	6	no	—
183	#121	—	43	6	no	—
184	#121	C₂×D₄₃	1	6	yes	complemented
185	#120	—	43	12	no	—
186	#120	—	43	12	no	—
187	C₄×Q₈	—	43	1	no	—
188	C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic, complemented
189	C₄×D₄	—	43	6	no	—
190	C₄×D₄	—	43	3	no	—
191	C₄×D₄	—	43	3	no	—
192	C₄×D₄	—	43	3	no	—
193	C₄×D₄	C₂×D₄₃	1	3	yes	complemented
194	#117	—	43	3	no	—
195	C₂×C₄²	—	43	3	no	—
196	C₂⁴	—	43	12	no	—
197	#41	—	2	8	no	—
198	#41	—	43	8	no	—
199	#41	—	86	24	no	—
200	#41	—	86	8	no	—
201	C₂×Q₈	—	43	2	no	—
202	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
203	C₂×D₄	—	43	6	no	—
204	C₂×D₄	—	43	6	no	—
205	C₂×D₄	—	43	6	no	—
206	C₂×D₄	—	43	12	no	—
207	C₂×D₄	—	43	3	no	—
208	C₂×D₄	—	43	3	no	—
209	C₂×D₄	—	43	6	no	—
210	C₂×D₄	—	43	6	no	—
211	C₂×D₄	—	43	6	no	—
212	C₂×D₄	—	86	24	no	—
213	C₂×D₄	—	86	24	no	—
214	C₂×D₄	—	86	24	no	—
215	C₂×D₄	—	172	12	no	—
216	C₂×D₄	—	172	12	no	—
217	C₂×D₄	C₂²×D₄₃	1	3	yes	—
218	C₂×D₄	C₂²×D₄₃	1	6	yes	—
219	C₂×D₄	C₂²×D₄₃	1	6	yes	—
220	C₂²×C₄	—	43	3	no	—
221	C₂²×C₄	—	43	3	no	—
222	C₂²×C₄	—	43	3	no	—
223	C₂²×C₄	—	43	6	no	—
224	C₂²×C₄	C₂²×D₄₃	1	6	yes	—
225	C₄⋊C₄	—	43	3	no	—
226	C₄⋊C₄	—	43	2	no	—
227	C₄⋊C₄	—	43	1	no	—
228	C₄⋊C₄	C₂²×D₄₃	1	3	yes	—
229	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
230	C₂²⋊C₄	—	43	6	no	—
231	C₂²⋊C₄	—	43	6	no	—
232	C₂²⋊C₄	—	43	6	no	—
233	C₂²⋊C₄	—	43	6	no	—
234	C₂²⋊C₄	C₂²×D₄₃	1	6	yes	—
235	C₄²	—	43	1	no	—
236	C₄²	—	43	3	no	—
237	C₄²	—	43	3	no	—
238	C₄²	C₂²×D₄₃	1	3	yes	—
239	C₂³	—	43	3	no	—
240	C₂³	—	43	6	no	—
241	C₂³	—	86	24	no	—
242	C₂³	—	172	12	no	—
243	C₂³	—	172	12	no	—
244	C₂³	C₂³×D₄₃	1	6	yes	—
245	Q₈	—	86	4	no	—
246	Q₈	D₄×D₄₃	1	4	yes	complemented
247	D₄	—	2	12	no	—
248	D₄	—	2	6	no	—
249	D₄	—	4	6	no	—
250	D₄	—	43	12	no	—
251	D₄	—	86	12	no	—
252	D₄	—	86	12	no	—
253	D₄	—	86	12	no	—
254	D₄	—	86	12	no	—
255	D₄	—	86	12	no	—
256	D₄	—	86	6	no	—
257	D₄	—	172	6	no	—
258	D₄	—	172	12	no	—
259	C₂×C₄	—	2	12	no	—
260	C₂×C₄	—	43	1	no	—
261	C₂×C₄	—	43	1	no	—
262	C₂×C₄	—	43	12	no	—
263	C₂×C₄	—	43	3	no	—
264	C₂×C₄	—	43	2	no	—
265	C₂×C₄	—	86	6	no	—
266	C₂×C₄	—	86	12	no	—
267	C₂×C₄	—	86	6	no	—
268	C₂×C₄	C₂³×D₄₃	1	3	yes	—
269	C₂×C₄	C₂³×D₄₃	1	2	yes	—
270	C₂×C₄	C₂³×D₄₃	1	3	yes	—
271	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
272	C₂²	—	2	6	no	—
273	C₂²	—	2	6	no	—
274	C₂²	—	2	6	no	—
275	C₂²	—	43	6	no	—
276	C₂²	—	86	3	no	—
277	C₂²	—	86	3	no	—
278	C₂²	—	86	6	no	—
279	C₂²	—	86	6	no	—
280	C₂²	—	86	6	no	—
281	C₂²	—	344	24	no	—
282	C₂²	C₂⁴×D₄₃	1	1	yes	characteristic
283	C₄	—	2	2	no	—
284	C₄	—	2	3	no	—
285	C₄	—	43	2	no	—
286	C₄	—	43	2	no	—
287	C₄	—	86	2	no	—
288	C₄	—	86	3	no	—
289	C₄	C₂×D₄×D₄₃	1	6	yes	—
290	C₄	#13063371	1	2	yes	complemented
291	C₂	—	4	6	no	—
292	C₂	—	86	6	no	—
293	C₂	—	172	6	no	—
294	C₂	C₂²×D₄×D₄₃	1	1	yes	characteristic
295	C₂	H(5,𝔽₂)×D₄₃	1	1	yes	characteristic
296	C₂	C₂×#13063371	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₄₃
commutator, socle	C₂²×C₄₃	C₂⁵
Fitting	C₄₃×#550	C₂
hypercenter	#550	D₄₃
hypocenter	C₄₃	#3498
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ #550

Complements

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

N	H ≅ G/N	type
1. C₂×#13063371	C₂	semidirect
2. #14583571	C₂	semidirect
3. #14583568	C₂	semidirect
4. D₄×D₁₇₂	C₂	semidirect
5. #14583534	C₂	semidirect
6. #14583533	C₂	semidirect
7. #14583465	C₂	semidirect
8. #14583446	C₂	semidirect
9. #127×D₄₃	C₂	semidirect
10. C₄×#10621902	C₂	semidirect
11. C₄₃×#550	C₂	semidirect
14. C₂×#10621902	C₂²	semidirect
15. C₂×#10621898	C₂²	semidirect
16. C₂²×D₁₇₂	C₂²	semidirect
17. C₂×D₄×D₄₃	C₂²	semidirect
18. C₂×D₄×D₄₃	C₂²	semidirect
19. C₂×D₄×D₄₃	C₂²	semidirect
20. C₂×D₄×D₄₃	C₂²	semidirect
21. #13063324	C₂²	semidirect
22. #13063322	C₂²	semidirect
23. #13063321	C₂²	semidirect
24. #13063321	C₂²	semidirect
25. #13063321	C₂²	semidirect
26. #13063311	C₂²	semidirect
27. #13063300	C₂²	semidirect
28. #13063300	C₂²	semidirect
29. #13063299	C₂²	semidirect
30. #13063298	C₂²	semidirect
31. #13063296	C₂²	semidirect
32. #13063296	C₂²	semidirect
33. #13063295	C₂²	semidirect
34. #13063293	C₂²	semidirect
35. Q₈×Dic₄₃	C₂²	semidirect
36. C₄×D₁₇₂	C₂²	semidirect
37. C₄×D₁₇₂	C₂²	semidirect
38. #13063287	C₂²	semidirect
39. C₄×C₂²⋊D₄₃	C₂²	semidirect
40. #13063283	C₂²	semidirect
41. C₄²×D₄₃	C₂²	semidirect
42. C₂×#41×C₄₃	C₂²	semidirect
43. #127×C₄₃	C₂²	semidirect
44. #121×C₄₃	C₂²	semidirect
45. C₄×Q₈×C₄₃	C₂²	semidirect
46. C₄×D₄×C₄₃	C₂²	semidirect
59. #10621902	D₄	semidirect
96. Q₈×C₄₃	C₂×D₄	semidirect
117. D₁₇₂	C₂×D₄	semidirect
121. C₄×D₄₃	C₂×D₄	semidirect
137. C₄×C₄₃	H(5,𝔽₂)	semidirect
141. Dic₄₃	H(5,𝔽₂)	semidirect
152. #550	D₄₃	semidirect
160. C₄₃	#3498	semidirect
165. C₂×#41	C₂×D₄₃	semidirect
176. #127	C₂×D₄₃	semidirect
184. #121	C₂×D₄₃	semidirect
188. C₄×Q₈	C₂×D₄₃	semidirect
193. C₄×D₄	C₂×D₄₃	semidirect
246. Q₈	D₄×D₄₃	semidirect
290. C₄	#13063371	semidirect