Subgroups — #20283131

Order 6784.

564 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#11719912	C₂	1	1	yes	maximal, characteristic
2	C₂×M₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
3	#117×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#16810581	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810580	C₂	1	2	yes	maximal, complemented
6	#16810579	C₂	1	1	yes	maximal, characteristic, complemented
7	#16810578	C₂	1	1	yes	maximal, characteristic, complemented
8	#16810577	C₂	1	2	yes	maximal, complemented
9	#16810575	C₂	1	2	yes	maximal, complemented
10	C₄×#11719912	C₂	1	2	yes	maximal, complemented
11	C₅₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic
13	C₂×#11719917	C₂²	1	1	yes	characteristic
14	C₂×#11719912	C₄	1	8	yes	complemented
15	C₂×#11719912	C₂²	1	1	yes	characteristic
16	C₂×#11719912	C₂²	1	2	yes	—
17	C₂×#11719912	C₂²	1	1	yes	characteristic
18	C₂×#11719912	C₂²	1	1	yes	characteristic
19	C₂×#11719912	C₂²	1	1	yes	characteristic
20	M₁₆×D₅₃	C₄	1	8	yes	complemented
21	C₂²×#88322	C₂²	1	1	yes	characteristic
22	C₂×C₈×D₅₃	C₂²	1	2	yes	—
23	#13133176	C₂²	1	1	yes	characteristic
24	#13133175	C₄	1	2	yes	—
25	#13133174	C₂²	1	2	yes	—
26	#13133173	C₄	1	2	yes	—
27	C₄⋊C₄×D₅₃	C₄	1	2	yes	—
28	C₂²⋊C₄×D₅₃	C₄	1	2	yes	—
29	C₄²×D₅₃	C₂²	1	2	yes	—
30	#13133128	C₂²	1	2	yes	complemented
31	#13133127	C₂²	1	2	yes	complemented
32	#13133127	C₂²	1	2	yes	complemented
33	C₄×#88322	C₂²	1	2	yes	complemented
34	C₈×Dic₅₃	C₂²	1	2	yes	complemented
35	C₈×Dic₅₃	C₂²	1	2	yes	complemented
36	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
38	#117×C₅₃	C₂²	1	1	yes	characteristic
39	#97×C₅₃	C₂²	1	2	yes	complemented
40	C₄×C₈×C₅₃	C₂²	1	2	yes	complemented
41	C₂³×D₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
43	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₅₃	C₂×C₄	1	4	yes	—
45	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₅₃	C₂×C₄	1	4	yes	—
48	C₂×C₄×D₅₃	C₂×C₄	1	4	yes	—
49	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
50	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
51	M₁₆×C₅₃	C₂×C₄	1	4	yes	complemented
52	C₂×C₈×C₅₃	C₂×C₄	1	4	yes	complemented
53	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
54	C₂×C₈×C₅₃	C₂³	1	2	yes	—
55	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
56	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
57	C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
58	C₄²×C₅₃	C₂³	1	2	yes	—
59	C₂²⋊Dic₅₃	C₂×C₄	1	2	yes	—
60	#11719921	C₂×C₄	1	4	yes	—
61	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
62	#11719919	C₂×C₄	1	4	yes	—
63	C₄×Dic₅₃	C₂×C₄	1	4	yes	—
64	C₄×Dic₅₃	C₂³	1	2	yes	—
65	#11719917	C₂×C₄	1	4	yes	complemented
66	C₂×#88322	C₂×C₄	1	4	yes	complemented
67	C₂×#88322	C₂³	1	1	yes	characteristic
68	C₂×#88322	C₂³	1	2	yes	—
69	C₂×#88322	C₂³	1	1	yes	characteristic
70	#11719912	C₂×C₄	1	8	yes	—
71	#11719912	C₂×C₄	1	4	yes	—
72	#11719912	C₂×C₄	1	4	yes	—
73	#11719912	C₂×C₄	1	4	yes	complemented
74	#11719912	C₂×C₄	1	4	yes	complemented
75	C₈×D₅₃	C₂×C₄	1	8	yes	complemented
76	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×D₅₃	—	4	2	no	—
78	C₂²×D₅₃	C₄²	1	4	yes	—
79	C₂²×D₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₂²×D₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
82	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
86	C₂×Dic₅₃	C₄²	1	4	yes	—
87	C₂×Dic₅₃	C₂²×C₄	1	4	yes	—
88	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₄×D₅₃	—	4	4	no	—
91	C₄×D₅₃	C₄²	1	4	yes	—
92	C₄×D₅₃	C₄²	1	4	yes	—
93	C₄×D₅₃	C₂²×C₄	1	2	yes	—
94	C₄×D₅₃	C₂²×C₄	1	2	yes	—
95	C₄×D₅₃	C₂²×C₄	1	2	yes	—
96	C₄×D₅₃	C₂²×C₄	1	2	yes	—
97	C₈×C₅₃	C₂²×C₄	1	4	yes	—
98	C₈×C₅₃	C₂²×C₄	1	2	yes	—
99	C₈×C₅₃	C₂²×C₄	1	2	yes	—
100	#88322	C₂²×C₄	1	4	yes	—
101	#88322	C₂²×C₄	1	2	yes	—
102	#88322	C₂²×C₄	1	2	yes	—
103	C₂²×C₅₃	—	2	2	no	—
104	C₂²×C₅₃	C₂×C₄²	1	2	yes	—
105	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
106	C₂×D₅₃	—	2	2	no	—
107	C₂×D₅₃	—	2	2	no	—
108	C₂×D₅₃	—	4	4	no	—
109	C₂×D₅₃	C₂×C₄²	1	2	yes	—
110	C₂×D₅₃	C₂×C₄²	1	2	yes	—
111	C₄×C₅₃	—	2	4	no	—
112	C₄×C₅₃	C₂×C₄²	1	2	yes	—
113	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
114	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
115	Dic₅₃	—	2	4	no	—
116	Dic₅₃	C₂×C₄²	1	2	yes	—
117	Dic₅₃	C₂×C₄²	1	2	yes	—
118	#2873	—	53	1	no	maximal, Sylow
119	C₂×C₅₃	—	2	2	no	—
120	C₂×C₅₃	C₂²×C₄²	1	1	yes	characteristic
121	C₂×C₅₃	#568	1	2	yes	—
122	D₅₃	—	2	2	no	—
123	D₅₃	—	2	2	no	—
124	C₂²×M₁₆	—	53	1	no	—
125	C₂²×M₁₆	—	53	1	no	—
126	C₂×#117	—	53	1	no	—
127	#405	—	53	1	no	—
128	#405	—	53	1	no	—
129	#405	—	53	1	no	—
130	#405	—	53	2	no	—
131	#405	—	53	2	no	—
132	#405	D₅₃	1	1	yes	characteristic, complemented
133	C₄×M₁₆	—	53	2	no	—
134	C₄×M₁₆	—	53	2	no	—
135	C₅₃	#2873	1	1	yes	characteristic, complemented, Sylow
136	C₂³×C₄	—	53	1	no	—
137	C₂×M₁₆	—	53	1	no	—
138	C₂×M₁₆	—	53	8	no	—
139	C₂×M₁₆	—	53	8	no	—
140	C₂×M₁₆	—	53	1	no	—
141	C₂×M₁₆	—	53	2	no	—
142	C₂×M₁₆	—	53	1	no	—
143	C₂×M₁₆	—	53	1	no	—
144	C₂×M₁₆	—	53	1	no	—
145	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
146	C₂²×C₈	—	53	1	no	—
147	C₂²×C₈	—	53	2	no	—
148	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
149	#117	—	53	1	no	—
150	#117	—	53	2	no	—
151	#117	—	53	2	no	—
152	#117	—	53	2	no	—
153	#117	C₂×D₅₃	1	1	yes	characteristic
154	C₂×C₄⋊C₄	—	53	2	no	—
155	C₂×C₂²⋊C₄	—	53	2	no	—
156	C₂×C₄²	—	53	2	no	—
157	#97	—	53	2	no	—
158	#97	—	53	2	no	—
159	#97	—	53	2	no	—
160	#97	C₂×D₅₃	1	2	yes	complemented
161	C₄×C₈	—	53	2	no	—
162	C₄×C₈	—	53	2	no	—
163	C₄×C₈	—	53	2	no	—
164	C₄×C₈	C₂×D₅₃	1	2	yes	complemented
165	C₂⁴	—	53	1	no	—
166	C₂²×C₄	—	53	1	no	—
167	C₂²×C₄	—	53	1	no	—
168	C₂²×C₄	—	53	1	no	—
169	C₂²×C₄	—	53	4	no	—
170	C₂²×C₄	—	53	4	no	—
171	C₂²×C₄	—	53	1	no	—
172	C₂²×C₄	—	53	1	no	—
173	C₂²×C₄	—	53	4	no	—
174	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
175	M₁₆	—	53	4	no	—
176	M₁₆	—	53	8	no	—
177	M₁₆	—	53	4	no	—
178	M₁₆	—	53	4	no	—
179	M₁₆	—	53	4	no	—
180	M₁₆	—	53	4	no	—
181	M₁₆	C₄×D₅₃	1	4	yes	complemented
182	C₂×C₈	—	53	4	no	—
183	C₂×C₈	—	53	1	no	—
184	C₂×C₈	—	53	2	no	—
185	C₂×C₈	—	53	1	no	—
186	C₂×C₈	—	53	8	no	—
187	C₂×C₈	C₄×D₅₃	1	4	yes	complemented
188	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
189	C₂×C₈	C₂²×D₅₃	1	2	yes	—
190	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
191	C₄⋊C₄	—	53	4	no	—
192	C₄⋊C₄	—	53	2	no	—
193	C₄⋊C₄	C₄×D₅₃	1	2	yes	—
194	C₂²⋊C₄	—	53	2	no	—
195	C₂²⋊C₄	—	53	4	no	—
196	C₂²⋊C₄	C₄×D₅₃	1	2	yes	—
197	C₄²	—	53	4	no	—
198	C₄²	—	53	2	no	—
199	C₄²	C₂²×D₅₃	1	2	yes	—
200	C₂³	—	53	1	no	—
201	C₂³	—	53	1	no	—
202	C₂³	—	53	4	no	—
203	C₂³	—	212	2	no	—
204	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
205	C₂×C₄	—	53	4	no	—
206	C₂×C₄	—	53	1	no	—
207	C₂×C₄	—	53	1	no	—
208	C₂×C₄	—	53	4	no	—
209	C₂×C₄	—	53	4	no	—
210	C₂×C₄	—	53	2	no	—
211	C₂×C₄	—	53	2	no	—
212	C₂×C₄	—	53	2	no	—
213	C₂×C₄	—	53	2	no	—
214	C₂×C₄	—	53	4	no	—
215	C₂×C₄	—	212	4	no	—
216	C₂×C₄	C₂×C₄×D₅₃	1	4	yes	—
217	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
218	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
219	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
220	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
221	C₈	—	53	4	no	—
222	C₈	—	53	2	no	—
223	C₈	—	53	2	no	—
224	C₈	C₂×C₄×D₅₃	1	4	yes	—
225	C₈	C₂×C₄×D₅₃	1	2	yes	—
226	C₈	C₂×C₄×D₅₃	1	2	yes	—
227	C₂²	—	2	2	no	—
228	C₂²	—	53	2	no	—
229	C₂²	—	53	2	no	—
230	C₂²	—	106	2	no	—
231	C₂²	—	106	2	no	—
232	C₂²	—	212	4	no	—
233	C₂²	C₄²×D₅₃	1	2	yes	—
234	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
235	C₄	—	2	4	no	—
236	C₄	—	53	2	no	—
237	C₄	—	53	2	no	—
238	C₄	—	106	4	no	—
239	C₄	C₄²×D₅₃	1	2	yes	—
240	C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
241	C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
242	C₂	—	2	2	no	—
243	C₂	—	106	2	no	—
244	C₂	—	106	2	no	—
245	C₂	C₂×C₄²×D₅₃	1	1	yes	characteristic
246	C₂	#16811388	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator	C₂×C₅₃	C₂²×C₄²
Fitting	C₅₃×#405	C₂
socle	C₂²×C₅₃	C₂³×C₄
hypercenter	#405	D₅₃
hypocenter	C₅₃	#2873
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #405

Complements

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

N	H ≅ G/N	type
4. #16810581	C₂	semidirect
5. #16810580	C₂	semidirect
6. #16810579	C₂	semidirect
7. #16810578	C₂	semidirect
8. #16810577	C₂	semidirect
9. #16810575	C₂	semidirect
10. C₄×#11719912	C₂	semidirect
11. C₅₃×#405	C₂	semidirect
14. C₂×#11719912	C₄	semidirect
20. M₁₆×D₅₃	C₄	semidirect
30. #13133128	C₂²	semidirect
31. #13133127	C₂²	semidirect
32. #13133127	C₂²	semidirect
33. C₄×#88322	C₂²	semidirect
34. C₈×Dic₅₃	C₂²	semidirect
35. C₈×Dic₅₃	C₂²	semidirect
39. #97×C₅₃	C₂²	semidirect
40. C₄×C₈×C₅₃	C₂²	semidirect
51. M₁₆×C₅₃	C₂×C₄	semidirect
52. C₂×C₈×C₅₃	C₂×C₄	semidirect
65. #11719917	C₂×C₄	semidirect
66. C₂×#88322	C₂×C₄	semidirect
73. #11719912	C₂×C₄	semidirect
74. #11719912	C₂×C₄	semidirect
75. C₈×D₅₃	C₂×C₄	semidirect
132. #405	D₅₃	semidirect
135. C₅₃	#2873	semidirect
160. #97	C₂×D₅₃	semidirect
164. C₄×C₈	C₂×D₅₃	semidirect
181. M₁₆	C₄×D₅₃	semidirect
187. C₂×C₈	C₄×D₅₃	semidirect