Subgroups — #568×D₅₃ #20289422

Order 6784.

1460 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×D₅₃	C₂	1	1	yes	maximal, characteristic
2	#16811390	C₂	1	1	yes	maximal, characteristic, complemented
3	#16811389	C₂	1	6	yes	maximal, complemented
4	#16811388	C₂	1	8	yes	maximal, complemented
5	#131×D₅₃	C₂	1	8	yes	maximal, complemented
6	C₂×M₁₆×D₅₃	C₂	1	6	yes	maximal, complemented
7	C₅₃×#568	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11719949	C₄	1	1	yes	characteristic
9	C₂×#11719945	C₂²	1	3	yes	—
10	C₂×#11719947	C₄	1	3	yes	—
11	#41×D₅₃	C₄	1	4	yes	—
12	#41×D₅₃	C₄	1	4	yes	—
13	#41×D₅₃	C₂²	1	4	yes	—
14	#41×D₅₃	C₂²	1	4	yes	—
15	C₂×Q₈×D₅₃	C₄	1	1	yes	characteristic
16	C₂×D₄×D₅₃	C₄	1	3	yes	—
17	C₂²×C₄×D₅₃	C₂²	1	3	yes	—
18	#13133225	C₂²	1	24	yes	complemented
19	#13133224	C₂²	1	8	yes	complemented
20	#13133223	C₂²	1	24	yes	complemented
21	C₂×#11719917	C₂²	1	6	yes	complemented
22	C₂×#11719912	C₂²	1	12	yes	complemented
23	M₁₆×D₅₃	C₂²	1	4	yes	—
24	M₁₆×D₅₃	C₂²	1	12	yes	complemented
25	M₁₆×D₅₃	C₂²	1	12	yes	complemented
26	M₁₆×D₅₃	C₂²	1	12	yes	complemented
27	C₂×C₈×D₅₃	C₂²	1	12	yes	complemented
28	C₂×#41×C₅₃	C₂²	1	1	yes	characteristic
29	#131×C₅₃	C₂²	1	8	yes	complemented
30	C₂×M₁₆×C₅₃	C₂²	1	6	yes	complemented
31	C₂³×D₅₃	C₂×C₄	1	3	yes	—
32	#41×C₅₃	C₂×C₄	1	4	yes	—
33	#41×C₅₃	C₂³	1	4	yes	—
34	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×D₄×C₅₃	C₂×C₄	1	3	yes	—
36	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
37	C₂×C₂²⋊D₅₃	C₂×C₄	1	6	yes	—
38	C₂²×Dic₅₃	C₂×C₄	1	3	yes	—
39	#11719949	C₂×C₄	1	4	yes	—
40	#11719949	C₂×C₄	1	4	yes	—
41	Q₈×D₅₃	C₂×C₄	1	4	yes	—
42	Q₈×D₅₃	C₂×C₄	1	4	yes	—
43	#11719947	C₂×C₄	1	12	yes	—
44	#11719947	C₂×C₄	1	12	yes	—
45	D₄×D₅₃	C₂×C₄	1	12	yes	—
46	D₄×D₅₃	C₂×C₄	1	12	yes	—
47	#11719945	C₂×C₄	1	12	yes	—
48	#11719945	C₂³	1	12	yes	—
49	C₂×D₂₁₂	C₂×C₄	1	3	yes	—
50	C₂×C₄×D₅₃	C₂×C₄	1	3	yes	—
51	C₂×C₄×D₅₃	C₂×C₄	1	6	yes	—
52	C₂×C₄×D₅₃	C₂×C₄	1	6	yes	—
53	C₂×C₄×D₅₃	C₂×C₄	1	3	yes	—
54	C₂×C₄×D₅₃	C₂³	1	6	yes	—
55	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
56	C₂×C₄×D₅₃	C₂³	1	3	yes	—
57	C₂×C₄×D₅₃	C₂³	1	6	yes	—
58	C₂×Dic₁₀₆	C₂×C₄	1	3	yes	—
59	M₁₆×C₅₃	C₂³	1	4	yes	—
60	M₁₆×C₅₃	C₂³	1	12	yes	complemented
61	C₂×C₈×C₅₃	C₂³	1	12	yes	complemented
62	#11719917	C₂³	1	4	yes	—
63	#11719917	C₂³	1	12	yes	complemented
64	C₂×#88322	C₂³	1	12	yes	complemented
65	#11719912	C₂³	1	8	yes	—
66	#11719912	C₂³	1	24	yes	complemented
67	C₈×D₅₃	C₂³	1	8	yes	—
68	C₈×D₅₃	C₂³	1	24	yes	complemented
69	C₂³×C₅₃	C₂²×C₄	1	3	yes	—
70	C₂²×D₅₃	—	4	6	no	—
71	C₂²×D₅₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×D₅₃	C₂²×C₄	1	6	yes	—
73	C₂²×D₅₃	C₂²×C₄	1	3	yes	—
74	C₂²×D₅₃	C₂²×C₄	1	6	yes	—
75	Q₈×C₅₃	C₂²×C₄	1	4	yes	—
76	D₄×C₅₃	C₂²×C₄	1	12	yes	—
77	C₂×C₄×C₅₃	C₂²×C₄	1	3	yes	—
78	C₂×C₄×C₅₃	C₂²×C₄	1	6	yes	—
79	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂⁴	1	6	yes	—
81	C₂²⋊D₅₃	C₂²×C₄	1	24	yes	—
82	C₂×Dic₅₃	C₂²×C₄	1	3	yes	—
83	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×Dic₅₃	C₂²×C₄	1	6	yes	—
85	C₂×Dic₅₃	C₂²×C₄	1	6	yes	—
86	D₂₁₂	C₂²×C₄	1	12	yes	—
87	C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
89	C₄×D₅₃	C₂²×C₄	1	6	yes	—
90	C₄×D₅₃	C₂²×C₄	1	6	yes	—
91	C₄×D₅₃	C₂²×C₄	1	6	yes	—
92	C₄×D₅₃	C₂²×C₄	1	6	yes	—
93	C₄×D₅₃	C₂²×C₄	1	6	yes	—
94	C₄×D₅₃	C₂⁴	1	1	yes	characteristic
95	C₄×D₅₃	C₂⁴	1	1	yes	characteristic
96	C₄×D₅₃	C₂⁴	1	6	yes	—
97	Dic₁₀₆	C₂²×C₄	1	12	yes	—
98	C₈×C₅₃	C₂⁴	1	8	yes	—
99	#88322	C₂⁴	1	8	yes	—
100	C₂²×C₅₃	—	4	3	no	—
101	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
102	C₂²×C₅₃	C₂³×C₄	1	6	yes	—
103	C₂×D₅₃	—	2	2	no	—
104	C₂×D₅₃	—	2	12	no	—
105	C₂×D₅₃	—	4	6	no	—
106	C₂×D₅₃	—	4	3	no	—
107	C₂×D₅₃	C₂³×C₄	1	1	yes	characteristic
108	C₂×D₅₃	C₂³×C₄	1	6	yes	—
109	C₂×D₅₃	C₂³×C₄	1	1	yes	characteristic
110	C₄×C₅₃	C₂³×C₄	1	6	yes	—
111	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
112	C₄×C₅₃	C₂⁵	1	1	yes	characteristic
113	Dic₅₃	C₂³×C₄	1	6	yes	—
114	Dic₅₃	C₂³×C₄	1	1	yes	characteristic
115	Dic₅₃	C₂³×C₄	1	1	yes	characteristic
116	C₂×#568	—	53	1	no	maximal, Sylow
117	C₂×C₅₃	—	2	1	no	—
118	C₂×C₅₃	—	2	6	no	—
119	C₂×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
120	D₅₃	—	2	1	no	—
121	D₅₃	—	2	6	no	—
122	D₅₃	#568	1	2	yes	direct product factor
123	C₂²×#41	—	53	1	no	—
124	#568	—	53	1	no	—
125	#568	—	53	8	no	—
126	#568	—	53	6	no	—
127	#568	D₅₃	1	1	yes	characteristic, direct product factor
128	C₂×#131	—	53	8	no	—
129	C₂²×M₁₆	—	53	6	no	—
130	C₅₃	C₂×#568	1	1	yes	characteristic, complemented, Sylow
131	C₂×#41	—	53	4	no	—
132	C₂×#41	—	53	4	no	—
133	C₂×#41	—	53	4	no	—
134	C₂×#41	—	53	4	no	—
135	C₂×#41	—	53	1	no	—
136	C₂×#41	—	53	3	no	—
137	C₂×#41	—	53	3	no	—
138	C₂×#41	C₂×D₅₃	1	1	yes	characteristic
139	C₂²×Q₈	—	53	1	no	—
140	C₂²×D₄	—	53	3	no	—
141	C₂³×C₄	—	53	3	no	—
142	#131	—	53	8	no	—
143	#131	—	53	24	no	—
144	#131	—	53	24	no	—
145	#131	C₂×D₅₃	1	8	yes	complemented
146	C₂×M₁₆	—	53	6	no	—
147	C₂×M₁₆	—	53	4	no	—
148	C₂×M₁₆	—	53	12	no	—
149	C₂×M₁₆	—	53	12	no	—
150	C₂×M₁₆	—	53	12	no	—
151	C₂×M₁₆	—	53	12	no	—
152	C₂×M₁₆	C₂×D₅₃	1	6	yes	complemented
153	C₂²×C₈	—	53	12	no	—
154	C₂⁴	—	53	3	no	—
155	#41	—	53	4	no	—
156	#41	—	53	4	no	—
157	#41	—	53	12	no	—
158	#41	—	53	12	no	—
159	#41	—	53	12	no	—
160	#41	—	53	12	no	—
161	#41	C₄×D₅₃	1	4	yes	—
162	#41	C₂²×D₅₃	1	4	yes	—
163	C₂×Q₈	—	53	4	no	—
164	C₂×Q₈	—	53	4	no	—
165	C₂×Q₈	—	53	3	no	—
166	C₂×Q₈	C₄×D₅₃	1	1	yes	characteristic
167	C₂×D₄	—	53	6	no	—
168	C₂×D₄	—	53	12	no	—
169	C₂×D₄	—	53	12	no	—
170	C₂×D₄	—	53	3	no	—
171	C₂×D₄	C₄×D₅₃	1	3	yes	—
172	C₂²×C₄	—	53	3	no	—
173	C₂²×C₄	—	53	3	no	—
174	C₂²×C₄	—	53	3	no	—
175	C₂²×C₄	—	53	6	no	—
176	C₂²×C₄	—	53	6	no	—
177	C₂²×C₄	—	53	6	no	—
178	C₂²×C₄	—	53	1	no	—
179	C₂²×C₄	—	53	3	no	—
180	C₂²×C₄	—	53	6	no	—
181	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
182	M₁₆	—	53	4	no	—
183	M₁₆	—	53	12	no	—
184	M₁₆	—	53	8	no	—
185	M₁₆	—	53	24	no	—
186	M₁₆	C₂²×D₅₃	1	4	yes	—
187	M₁₆	C₂²×D₅₃	1	12	yes	complemented
188	C₂×C₈	—	53	12	no	—
189	C₂×C₈	—	53	8	no	—
190	C₂×C₈	—	53	24	no	—
191	C₂×C₈	C₂²×D₅₃	1	12	yes	complemented
192	C₂³	—	53	1	no	—
193	C₂³	—	53	6	no	—
194	C₂³	—	53	6	no	—
195	C₂³	—	53	3	no	—
196	C₂³	—	212	6	no	—
197	C₂³	C₂×C₄×D₅₃	1	3	yes	—
198	Q₈	—	53	12	no	—
199	Q₈	C₂×C₄×D₅₃	1	4	yes	—
200	D₄	—	53	24	no	—
201	D₄	—	53	12	no	—
202	D₄	C₂×C₄×D₅₃	1	12	yes	—
203	C₂×C₄	—	53	1	no	—
204	C₂×C₄	—	53	3	no	—
205	C₂×C₄	—	53	6	no	—
206	C₂×C₄	—	53	6	no	—
207	C₂×C₄	—	53	1	no	—
208	C₂×C₄	—	53	1	no	—
209	C₂×C₄	—	53	6	no	—
210	C₂×C₄	—	53	6	no	—
211	C₂×C₄	—	53	6	no	—
212	C₂×C₄	—	53	6	no	—
213	C₂×C₄	—	53	6	no	—
214	C₂×C₄	—	53	1	no	—
215	C₂×C₄	—	53	6	no	—
216	C₂×C₄	—	53	1	no	—
217	C₂×C₄	C₂×C₄×D₅₃	1	3	yes	—
218	C₂×C₄	C₂×C₄×D₅₃	1	6	yes	—
219	C₂×C₄	C₂³×D₅₃	1	6	yes	—
220	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
221	C₈	—	53	8	no	—
222	C₈	C₂³×D₅₃	1	8	yes	—
223	C₂²	—	4	3	no	—
224	C₂²	—	53	1	no	—
225	C₂²	—	53	1	no	—
226	C₂²	—	53	6	no	—
227	C₂²	—	106	2	no	—
228	C₂²	—	106	12	no	—
229	C₂²	—	212	3	no	—
230	C₂²	—	212	6	no	—
231	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
232	C₂²	C₂²×C₄×D₅₃	1	6	yes	—
233	C₄	—	53	1	no	—
234	C₄	—	53	1	no	—
235	C₄	—	53	6	no	—
236	C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
237	C₄	C₂²×C₄×D₅₃	1	6	yes	—
238	C₄	C₂⁴×D₅₃	1	1	yes	characteristic
239	C₂	—	2	1	no	—
240	C₂	—	2	6	no	—
241	C₂	—	53	2	no	—
242	C₂	—	106	1	no	—
243	C₂	—	106	6	no	—
244	C₂	C₂³×C₄×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂⁴×D₅₃
commutator, socle	C₂×C₅₃	C₂⁴×C₄
Fitting	C₅₃×#568	C₂
hypercenter	#568	D₅₃
hypocenter	C₅₃	C₂×#568
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ #568

Complements

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

N	H ≅ G/N	type
2. #16811390	C₂	semidirect
3. #16811389	C₂	semidirect
4. #16811388	C₂	semidirect
5. #131×D₅₃	C₂	semidirect
6. C₂×M₁₆×D₅₃	C₂	semidirect
7. C₅₃×#568	C₂	semidirect
18. #13133225	C₂²	semidirect
19. #13133224	C₂²	semidirect
20. #13133223	C₂²	semidirect
21. C₂×#11719917	C₂²	semidirect
22. C₂×#11719912	C₂²	semidirect
24. M₁₆×D₅₃	C₂²	semidirect
25. M₁₆×D₅₃	C₂²	semidirect
26. M₁₆×D₅₃	C₂²	semidirect
27. C₂×C₈×D₅₃	C₂²	semidirect
29. #131×C₅₃	C₂²	semidirect
30. C₂×M₁₆×C₅₃	C₂²	semidirect
60. M₁₆×C₅₃	C₂³	semidirect
61. C₂×C₈×C₅₃	C₂³	semidirect
63. #11719917	C₂³	semidirect
64. C₂×#88322	C₂³	semidirect
66. #11719912	C₂³	semidirect
68. C₈×D₅₃	C₂³	semidirect
122. D₅₃	#568	direct (N × H)
127. #568	D₅₃	direct (N × H)
130. C₅₃	C₂×#568	semidirect
145. #131	C₂×D₅₃	semidirect
152. C₂×M₁₆	C₂×D₅₃	semidirect
187. M₁₆	C₂²×D₅₃	semidirect
191. C₂×C₈	C₂²×D₅₃	semidirect

Subgroups — #568×D53 #20289422

Conjugacy classes of subgroups

Complements

Subgroups — #568×D₅₃ #20289422