Subgroups — #19143503

Order 5504.

458 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583696	C₂	1	1	yes	maximal, characteristic
2	#14583664	C₂	1	2	yes	maximal, complemented
3	#14583022	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583018	C₂	1	2	yes	maximal, complemented
5	#14583016	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583015	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583014	C₂	1	2	yes	maximal, complemented
8	#14582946	C₂	1	1	yes	maximal, characteristic, complemented
9	#104×D₄₃	C₂	1	2	yes	maximal, complemented
10	#14582851	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#444	C₂	1	1	yes	maximal, characteristic, complemented
12	#41×D₄₃	C₂²	1	2	yes	—
13	#13063357	C₄	1	2	yes	complemented
14	#13063354	C₄	1	2	yes	complemented
15	#13063352	C₄	1	2	yes	complemented
16	#13063349	C₄	1	2	yes	complemented
17	#13063347	C₂²	1	1	yes	characteristic
18	#13063345	C₂²	1	2	yes	—
19	#13063333	—	2	2	no	—
20	#13063333	C₂²	1	2	yes	complemented
21	#13063332	C₂²	1	2	yes	complemented
22	#13063331	—	2	2	no	—
23	#13063331	C₂²	1	2	yes	complemented
24	C₂×#10621872	—	2	2	no	—
25	C₂×#10621872	C₂²	1	1	yes	characteristic
26	C₂×#10621872	C₂²	1	1	yes	characteristic
27	M₁₆×D₄₃	—	2	2	no	—
28	M₁₆×D₄₃	C₂²	1	2	yes	—
29	C₄²×D₄₃	C₂²	1	1	yes	characteristic
30	#13063261	C₂²	1	1	yes	characteristic, complemented
31	#13063260	C₂²	1	2	yes	complemented
32	#13063253	C₂²	1	2	yes	complemented
33	#13063252	C₂²	1	2	yes	complemented
34	#13063251	C₂²	1	2	yes	complemented
35	#13063236	C₂²	1	1	yes	characteristic, complemented
36	C₈×Dic₄₃	C₂²	1	1	yes	characteristic, complemented
37	C₈×Dic₄₃	C₂²	1	1	yes	characteristic, complemented
38	#135×C₄₃	C₂²	1	1	yes	characteristic
39	#131×C₄₃	C₂²	1	2	yes	complemented
40	#108×C₄₃	C₂²	1	1	yes	characteristic, complemented
41	#104×C₄₃	C₂²	1	2	yes	complemented
42	#97×C₄₃	C₂²	1	1	yes	characteristic, complemented
43	#41×C₄₃	C₂³	1	2	yes	—
44	#10621902	C₂×C₄	1	2	yes	—
45	Q₈×D₄₃	C₂×C₄	1	2	yes	—
46	#10621900	—	2	2	no	—
47	#10621900	C₂×C₄	1	2	yes	—
48	D₄×D₄₃	—	2	2	no	—
49	D₄×D₄₃	C₂×C₄	1	2	yes	—
50	#10621898	—	2	2	no	—
51	#10621898	C₂³	1	2	yes	—
52	C₂×C₄×D₄₃	—	2	2	no	—
53	C₂×C₄×D₄₃	—	2	1	no	—
54	C₂×C₄×D₄₃	C₂³	1	1	yes	characteristic
55	Q₁₆×C₄₃	C₂×C₄	1	1	yes	characteristic, complemented
56	SD₁₆×C₄₃	C₂×C₄	1	2	yes	complemented
57	D₈×C₄₃	C₂×C₄	1	1	yes	characteristic, complemented
58	M₁₆×C₄₃	—	2	2	no	—
59	M₁₆×C₄₃	C₂³	1	2	yes	—
60	C₂×C₈×C₄₃	—	2	2	no	—
61	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
62	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
63	C₄²×C₄₃	C₂³	1	1	yes	characteristic
64	#10621885	C₂×C₄	1	2	yes	complemented
65	#10621884	C₂×C₄	1	2	yes	complemented
66	#10621883	C₂×C₄	1	2	yes	complemented
67	#10621882	C₂×C₄	1	2	yes	complemented
68	C₄×Dic₄₃	—	2	1	no	—
69	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
70	#10621877	—	2	2	no	—
71	#10621877	C₂³	1	2	yes	—
72	C₂×C₄₃⋊C₈	—	2	2	no	—
73	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
74	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
75	Dic₁₇₂	C₂×C₄	1	1	yes	characteristic, complemented
76	D₃₄₄	C₂×C₄	1	1	yes	characteristic, complemented
77	#10621873	C₂×C₄	1	2	yes	complemented
78	#10621872	—	2	2	no	—
79	#10621872	—	2	2	no	—
80	#10621872	—	2	2	no	—
81	#10621872	C₂×C₄	1	2	yes	—
82	#10621872	C₂×C₄	1	2	yes	—
83	#10621872	D₄	1	2	yes	—
84	#10621872	D₄	1	2	yes	—
85	C₈×D₄₃	—	2	2	no	—
86	C₈×D₄₃	—	2	2	no	—
87	C₈×D₄₃	—	2	2	no	—
88	C₂²×D₄₃	—	2	2	no	—
89	C₂²×D₄₃	#41	1	1	yes	characteristic
90	Q₈×C₄₃	C₂²×C₄	1	2	yes	—
91	D₄×C₄₃	—	2	2	no	—
92	D₄×C₄₃	C₂²×C₄	1	2	yes	—
93	C₂×C₄×C₄₃	—	2	2	no	—
94	C₂×C₄×C₄₃	—	2	1	no	—
95	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
96	C₂²⋊D₄₃	—	2	2	no	—
97	C₂²⋊D₄₃	—	2	2	no	—
98	C₂²⋊D₄₃	—	2	2	no	—
99	C₂×Dic₄₃	—	2	1	no	—
100	C₂×Dic₄₃	—	2	2	no	—
101	C₂×Dic₄₃	#41	1	1	yes	characteristic
102	D₁₇₂	—	2	2	no	—
103	D₁₇₂	C₂²×C₄	1	2	yes	—
104	C₄×D₄₃	—	2	2	no	—
105	C₄×D₄₃	—	2	2	no	—
106	C₄×D₄₃	—	2	2	no	—
107	C₄×D₄₃	—	4	2	no	—
108	C₄×D₄₃	C₂²×C₄	1	2	yes	—
109	C₄×D₄₃	C₂×D₄	1	2	yes	—
110	Dic₈₆	—	2	2	no	—
111	Dic₈₆	C₂²×C₄	1	2	yes	—
112	C₈×C₄₃	—	2	2	no	—
113	C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
114	C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
115	C₈×C₄₃	C₂×D₄	1	2	yes	—
116	C₄₃⋊C₈	—	2	2	no	—
117	C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
118	C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
119	C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄₃	—	2	2	no	—
121	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
122	C₂×D₄₃	—	2	2	no	—
123	C₂×D₄₃	—	4	1	no	—
124	C₂×D₄₃	—	8	2	no	—
125	C₂×D₄₃	C₄×D₄	1	2	yes	—
126	C₄×C₄₃	—	2	2	no	—
127	C₄×C₄₃	—	4	1	no	—
128	C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
129	C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
130	Dic₄₃	—	2	2	no	—
131	Dic₄₃	—	2	2	no	—
132	Dic₄₃	C₄×D₄	1	2	yes	—
133	#2956	—	43	1	no	maximal, Sylow
134	C₂×C₄₃	—	2	1	no	—
135	C₂×C₄₃	—	4	2	no	—
136	C₂×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
137	D₄₃	—	2	2	no	—
138	D₄₃	—	4	2	no	—
139	#575	—	43	1	no	—
140	#568	—	43	2	no	—
141	#444	—	43	2	no	—
142	#444	—	43	1	no	—
143	#444	D₄₃	1	1	yes	characteristic, complemented
144	#443	—	43	1	no	—
145	#443	—	43	2	no	—
146	#443	—	43	1	no	—
147	#430	—	43	1	no	—
148	C₂×#104	—	43	2	no	—
149	C₄×M₁₆	—	43	1	no	—
150	C₄₃	#2956	1	1	yes	characteristic, complemented, Sylow
151	C₂×#41	—	43	2	no	—
152	#137	—	43	2	no	—
153	#137	—	43	2	no	—
154	AGL(1,ℤ/8)	—	43	2	no	—
155	AGL(1,ℤ/8)	—	43	2	no	—
156	#135	—	43	2	no	—
157	#135	—	43	1	no	—
158	#135	C₂×D₄₃	1	1	yes	characteristic
159	#131	—	43	2	no	—
160	#131	—	43	2	no	—
161	#131	—	43	2	no	—
162	#131	—	86	2	no	—
163	#131	—	86	2	no	—
164	#131	C₂×D₄₃	1	2	yes	complemented
165	C₂×M₁₆	—	43	1	no	—
166	C₂×M₁₆	—	43	1	no	—
167	C₂×M₁₆	—	43	2	no	—
168	C₂×M₁₆	—	86	2	no	—
169	C₂×M₁₆	—	86	2	no	—
170	C₂×C₄²	—	43	1	no	—
171	#108	—	43	2	no	—
172	#108	—	43	1	no	—
173	#108	C₂×D₄₃	1	1	yes	characteristic, complemented
174	#104	—	43	2	no	—
175	#104	—	43	2	no	—
176	#104	—	43	2	no	—
177	#104	C₂×D₄₃	1	2	yes	complemented
178	#97	—	43	1	no	—
179	#97	C₂×D₄₃	1	1	yes	characteristic, complemented
180	C₄×C₈	—	43	1	no	—
181	C₄×C₈	—	43	1	no	—
182	#41	—	43	2	no	—
183	#41	—	43	2	no	—
184	#41	—	43	2	no	—
185	#41	—	86	2	no	—
186	#41	—	86	2	no	—
187	#41	C₂²×D₄₃	1	2	yes	—
188	C₂×Q₈	—	43	2	no	—
189	C₂×D₄	—	43	2	no	—
190	C₂×D₄	—	86	2	no	—
191	C₂²×C₄	—	43	1	no	—
192	C₂²×C₄	—	86	2	no	—
193	C₂²×C₄	—	86	1	no	—
194	Q₁₆	—	43	2	no	—
195	Q₁₆	—	43	1	no	—
196	Q₁₆	C₄×D₄₃	1	1	yes	characteristic, complemented
197	SD₁₆	—	43	2	no	—
198	SD₁₆	—	43	2	no	—
199	SD₁₆	—	43	2	no	—
200	SD₁₆	C₄×D₄₃	1	2	yes	complemented
201	D₈	—	43	2	no	—
202	D₈	—	43	1	no	—
203	D₈	C₄×D₄₃	1	1	yes	characteristic, complemented
204	M₁₆	—	2	2	no	—
205	M₁₆	—	43	2	no	—
206	M₁₆	—	43	2	no	—
207	M₁₆	—	43	2	no	—
208	M₁₆	—	43	2	no	—
209	M₁₆	—	43	2	no	—
210	M₁₆	—	86	2	no	—
211	M₁₆	—	86	2	no	—
212	M₁₆	—	86	2	no	—
213	M₁₆	—	86	2	no	—
214	M₁₆	C₂²×D₄₃	1	2	yes	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	43	1	no	—
217	C₂×C₈	—	43	1	no	—
218	C₂×C₈	—	86	2	no	—
219	C₂×C₈	—	86	2	no	—
220	C₂×C₈	—	86	2	no	—
221	C₂×C₈	—	86	2	no	—
222	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
223	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
224	C₄²	—	43	1	no	—
225	C₄²	—	86	1	no	—
226	C₄²	C₂²×D₄₃	1	1	yes	characteristic
227	C₂³	—	43	1	no	—
228	C₂³	—	86	2	no	—
229	Q₈	—	43	2	no	—
230	Q₈	—	86	2	no	—
231	Q₈	C₂×C₄×D₄₃	1	2	yes	—
232	D₄	—	2	2	no	—
233	D₄	—	43	2	no	—
234	D₄	—	86	2	no	—
235	D₄	—	86	2	no	—
236	D₄	—	86	2	no	—
237	D₄	—	86	2	no	—
238	D₄	C₂×C₄×D₄₃	1	2	yes	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	43	1	no	—
242	C₂×C₄	—	43	2	no	—
243	C₂×C₄	—	43	2	no	—
244	C₂×C₄	—	86	2	no	—
245	C₂×C₄	—	86	2	no	—
246	C₂×C₄	—	86	2	no	—
247	C₂×C₄	—	86	1	no	—
248	C₂×C₄	—	86	2	no	—
249	C₂×C₄	—	172	2	no	—
250	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	—	43	2	no	—
253	C₈	—	43	1	no	—
254	C₈	—	43	1	no	—
255	C₈	—	86	2	no	—
256	C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
257	C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
258	C₈	D₄×D₄₃	1	2	yes	—
259	C₂²	—	2	2	no	—
260	C₂²	—	43	2	no	—
261	C₂²	—	86	2	no	—
262	C₂²	—	344	2	no	—
263	C₂²	—	172	1	no	—
264	C₂²	#41×D₄₃	1	1	yes	characteristic
265	C₄	—	2	2	no	—
266	C₄	—	4	1	no	—
267	C₄	—	43	2	no	—
268	C₄	—	86	2	no	—
269	C₄	—	86	2	no	—
270	C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
271	C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
272	C₂	—	2	1	no	—
273	C₂	—	4	2	no	—
274	C₂	—	86	2	no	—
275	C₂	—	172	2	no	—
276	C₂	C₄×D₄×D₄₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₄₃
commutator	C₄×C₄₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#444	C₂
socle	C₂×C₄₃	C₂×C₄×D₄
hypercenter	#444	D₄₃
hypocenter	C₄₃	#2956
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #444

Complements

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

N	H ≅ G/N	type
2. #14583664	C₂	semidirect
3. #14583022	C₂	semidirect
4. #14583018	C₂	semidirect
5. #14583016	C₂	semidirect
6. #14583015	C₂	semidirect
7. #14583014	C₂	semidirect
8. #14582946	C₂	semidirect
9. #104×D₄₃	C₂	semidirect
10. #14582851	C₂	semidirect
11. C₄₃×#444	C₂	semidirect
13. #13063357	C₄	semidirect
14. #13063354	C₄	semidirect
15. #13063352	C₄	semidirect
16. #13063349	C₄	semidirect
20. #13063333	C₂²	semidirect
21. #13063332	C₂²	semidirect
23. #13063331	C₂²	semidirect
30. #13063261	C₂²	semidirect
31. #13063260	C₂²	semidirect
32. #13063253	C₂²	semidirect
33. #13063252	C₂²	semidirect
34. #13063251	C₂²	semidirect
35. #13063236	C₂²	semidirect
36. C₈×Dic₄₃	C₂²	semidirect
37. C₈×Dic₄₃	C₂²	semidirect
39. #131×C₄₃	C₂²	semidirect
40. #108×C₄₃	C₂²	semidirect
41. #104×C₄₃	C₂²	semidirect
42. #97×C₄₃	C₂²	semidirect
55. Q₁₆×C₄₃	C₂×C₄	semidirect
56. SD₁₆×C₄₃	C₂×C₄	semidirect
57. D₈×C₄₃	C₂×C₄	semidirect
64. #10621885	C₂×C₄	semidirect
65. #10621884	C₂×C₄	semidirect
66. #10621883	C₂×C₄	semidirect
67. #10621882	C₂×C₄	semidirect
75. Dic₁₇₂	C₂×C₄	semidirect
76. D₃₄₄	C₂×C₄	semidirect
77. #10621873	C₂×C₄	semidirect
143. #444	D₄₃	semidirect
150. C₄₃	#2956	semidirect
164. #131	C₂×D₄₃	semidirect
173. #108	C₂×D₄₃	semidirect
177. #104	C₂×D₄₃	semidirect
179. #97	C₂×D₄₃	semidirect
196. Q₁₆	C₄×D₄₃	semidirect
200. SD₁₆	C₄×D₄₃	semidirect
203. D₈	C₄×D₄₃	semidirect