Subgroups — #20283622

Order 6784.

458 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811421	C₂	1	1	yes	maximal, characteristic
2	#16811389	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810743	C₂	1	4	yes	maximal, complemented
4	#16810738	C₂	1	4	yes	maximal, complemented
5	#16810671	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13133144	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#13133143	C₂	1	1	yes	maximal, characteristic, complemented
8	M₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#568	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#11719945	C₂²	1	1	yes	characteristic
11	#13133248	C₄	1	2	yes	complemented
12	#13133247	C₄	1	2	yes	complemented
13	#13133243	C₄	1	2	yes	complemented
14	#13133242	C₄	1	2	yes	complemented
15	#13133237	C₂²	1	4	yes	—
16	#13133225	—	2	2	no	—
17	#13133223	C₂²	1	4	yes	complemented
18	C₂×#11719917	C₂²	1	1	yes	characteristic
19	C₂×#11719917	C₂²	1	1	yes	characteristic
20	M₁₆×D₅₃	—	2	2	no	—
21	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
22	#13133152	C₂²	1	4	yes	complemented
23	#13133144	C₂²	1	2	yes	complemented
24	#13133144	C₂²	1	2	yes	complemented
25	#13133143	C₂²	1	2	yes	complemented
26	#13133143	C₂²	1	2	yes	complemented
27	#13133127	C₂²	1	2	yes	complemented
28	C₈×Dic₅₃	C₂²	1	2	yes	complemented
29	C₂×#41×C₅₃	C₂²	1	1	yes	characteristic
30	#131×C₅₃	—	2	2	no	—
31	#131×C₅₃	C₂²	1	4	yes	complemented
32	C₂×M₁₆×C₅₃	—	2	2	no	—
33	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
34	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
35	#41×C₅₃	—	2	1	no	—
36	#41×C₅₃	—	2	1	no	—
37	#41×C₅₃	C₂×C₄	1	2	yes	—
38	#41×C₅₃	C₂³	1	2	yes	—
39	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×D₄×C₅₃	—	2	1	no	—
41	C₂×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	—	2	1	no	—
43	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
44	C₂×C₂²⋊D₅₃	—	2	1	no	—
45	C₂²×Dic₅₃	—	2	1	no	—
46	#11719945	—	2	1	no	—
47	#11719945	—	2	1	no	—
48	#11719945	C₂×C₄	1	2	yes	—
49	#11719945	C₂³	1	2	yes	—
50	C₂×D₂₁₂	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₅₃	—	2	1	no	—
52	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
53	M₁₆×C₅₃	—	2	1	no	—
54	M₁₆×C₅₃	—	2	4	no	—
55	M₁₆×C₅₃	D₄	1	2	yes	—
56	M₁₆×C₅₃	D₄	1	2	yes	—
57	M₁₆×C₅₃	C₂³	1	2	yes	—
58	C₂×C₈×C₅₃	—	2	1	no	—
59	C₂×C₈×C₅₃	—	2	4	no	—
60	C₂×C₈×C₅₃	C₂³	1	2	yes	—
61	#11719925	C₂×C₄	1	4	yes	complemented
62	#11719924	C₂×C₄	1	4	yes	complemented
63	#11719923	C₂×C₄	1	4	yes	complemented
64	#11719922	C₂×C₄	1	4	yes	complemented
65	C₄×Dic₅₃	—	2	1	no	—
66	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
67	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
68	#11719917	—	2	1	no	—
69	#11719917	C₂×C₄	1	2	yes	—
70	#11719917	C₂×C₄	1	2	yes	—
71	#11719917	C₂³	1	2	yes	—
72	C₂×#88322	—	2	1	no	—
73	C₂×#88322	C₂³	1	2	yes	—
74	#11719912	—	2	4	no	—
75	C₈×D₅₃	—	2	4	no	—
76	C₂³×C₅₃	—	2	1	no	—
77	C₂³×C₅₃	#41	1	1	yes	characteristic
78	C₂²×D₅₃	—	2	1	no	—
79	Q₈×C₅₃	—	2	1	no	—
80	Q₈×C₅₃	C₂²×C₄	1	2	yes	—
81	D₄×C₅₃	—	2	2	no	—
82	D₄×C₅₃	—	2	2	no	—
83	D₄×C₅₃	—	2	1	no	—
84	D₄×C₅₃	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₅₃	—	2	1	no	—
86	C₂×C₄×C₅₃	—	2	2	no	—
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
94	C₂²⋊D₅₃	—	2	2	no	—
95	C₂²⋊D₅₃	—	2	2	no	—
96	C₂×Dic₅₃	—	2	1	no	—
97	C₂×Dic₅₃	—	2	1	no	—
98	C₂×Dic₅₃	—	2	1	no	—
99	C₂×Dic₅₃	—	4	2	no	—
100	D₂₁₂	—	2	1	no	—
101	D₂₁₂	C₂²×C₄	1	2	yes	—
102	C₄×D₅₃	—	2	2	no	—
103	C₄×D₅₃	—	2	2	no	—
104	Dic₁₀₆	—	2	1	no	—
105	Dic₁₀₆	C₂²×C₄	1	2	yes	—
106	C₈×C₅₃	—	2	2	no	—
107	C₈×C₅₃	C₂×D₄	1	4	yes	—
108	#88322	—	2	2	no	—
109	#88322	C₂²×C₄	1	4	yes	—
110	C₂²×C₅₃	—	2	2	no	—
111	C₂²×C₅₃	—	4	1	no	—
112	C₂²×C₅₃	—	8	1	no	—
113	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
114	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
115	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
116	C₂×D₅₃	—	2	2	no	—
117	C₂×D₅₃	—	8	1	no	—
118	C₄×C₅₃	—	2	2	no	—
119	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
120	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
121	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
122	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
123	Dic₅₃	—	2	2	no	—
124	Dic₅₃	—	2	2	no	—
125	Dic₅₃	—	4	1	no	—
126	#2956	—	53	1	no	maximal, Sylow
127	C₂×C₅₃	—	2	1	no	—
128	C₂×C₅₃	—	2	1	no	—
129	C₂×C₅₃	—	2	1	no	—
130	C₂×C₅₃	—	4	2	no	—
131	C₂×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
132	D₅₃	—	4	2	no	—
133	#575	—	53	1	no	—
134	#568	—	53	1	no	—
135	#568	D₅₃	1	1	yes	characteristic, complemented
136	#444	—	53	4	no	—
137	#443	—	53	4	no	—
138	#430	—	53	1	no	—
139	C₂×#104	—	53	1	no	—
140	C₂×#104	—	53	1	no	—
141	C₄×M₁₆	—	53	1	no	—
142	C₅₃	#2956	1	1	yes	characteristic, complemented, Sylow
143	C₂×#41	—	53	1	no	—
144	C₂×#41	C₂×D₅₃	1	1	yes	characteristic
145	#137	—	53	2	no	—
146	#137	—	53	2	no	—
147	AGL(1,ℤ/8)	—	53	2	no	—
148	AGL(1,ℤ/8)	—	53	2	no	—
149	#135	—	53	4	no	—
150	#131	—	2	2	no	—
151	#131	—	53	4	no	—
152	#131	—	106	2	no	—
153	#131	C₂×D₅₃	1	4	yes	complemented
154	C₂×M₁₆	—	2	2	no	—
155	C₂×M₁₆	—	53	1	no	—
156	C₂×M₁₆	—	53	1	no	—
157	C₂×M₁₆	—	106	2	no	—
158	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
159	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
160	C₂×C₄²	—	53	1	no	—
161	#108	—	53	4	no	—
162	#104	—	53	2	no	—
163	#104	—	53	2	no	—
164	#104	—	53	2	no	—
165	#104	—	53	2	no	—
166	#97	—	53	2	no	—
167	C₄×C₈	—	53	2	no	—
168	#41	—	2	1	no	—
169	#41	—	2	1	no	—
170	#41	—	53	2	no	—
171	#41	—	53	2	no	—
172	#41	—	106	1	no	—
173	#41	—	106	1	no	—
174	#41	C₄×D₅₃	1	2	yes	—
175	#41	C₂²×D₅₃	1	2	yes	—
176	C₂×Q₈	—	53	1	no	—
177	C₂×Q₈	C₄×D₅₃	1	1	yes	characteristic
178	C₂×D₄	—	2	1	no	—
179	C₂×D₄	—	53	1	no	—
180	C₂×D₄	—	106	1	no	—
181	C₂×D₄	C₄×D₅₃	1	1	yes	characteristic
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	106	1	no	—
184	C₂²×C₄	—	106	1	no	—
185	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
186	Q₁₆	—	53	4	no	—
187	SD₁₆	—	53	4	no	—
188	SD₁₆	—	53	4	no	—
189	D₈	—	53	4	no	—
190	M₁₆	—	2	1	no	—
191	M₁₆	—	2	4	no	—
192	M₁₆	—	53	2	no	—
193	M₁₆	—	53	2	no	—
194	M₁₆	—	53	2	no	—
195	M₁₆	—	106	1	no	—
196	M₁₆	—	106	4	no	—
197	M₁₆	C₂²⋊D₅₃	1	2	yes	—
198	M₁₆	C₂²⋊D₅₃	1	2	yes	—
199	M₁₆	C₂²×D₅₃	1	2	yes	—
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	—	53	2	no	—
203	C₂×C₈	—	106	1	no	—
204	C₂×C₈	—	106	4	no	—
205	C₂×C₈	C₂²×D₅₃	1	2	yes	—
206	C₄²	—	53	1	no	—
207	C₄²	—	53	1	no	—
208	C₄²	—	106	1	no	—
209	C₂³	—	2	1	no	—
210	C₂³	—	106	1	no	—
211	C₂³	#11719945	1	1	yes	characteristic
212	Q₈	—	2	1	no	—
213	Q₈	—	53	2	no	—
214	Q₈	—	106	1	no	—
215	Q₈	C₂×C₄×D₅₃	1	2	yes	—
216	D₄	—	2	2	no	—
217	D₄	—	2	2	no	—
218	D₄	—	2	1	no	—
219	D₄	—	53	2	no	—
220	D₄	—	106	2	no	—
221	D₄	—	106	2	no	—
222	D₄	—	106	1	no	—
223	D₄	C₂×C₄×D₅₃	1	2	yes	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	106	1	no	—
228	C₂×C₄	—	106	1	no	—
229	C₂×C₄	—	106	2	no	—
230	C₂×C₄	—	106	2	no	—
231	C₂×C₄	—	106	1	no	—
232	C₂×C₄	—	212	2	no	—
233	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
234	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
235	C₂×C₄	#11719945	1	1	yes	characteristic
236	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
237	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
238	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
239	C₈	—	2	2	no	—
240	C₈	—	53	4	no	—
241	C₈	—	106	2	no	—
242	C₈	C₂×C₂²⋊D₅₃	1	4	yes	—
243	C₂²	—	2	2	no	—
244	C₂²	—	4	1	no	—
245	C₂²	—	8	1	no	—
246	C₂²	—	106	2	no	—
247	C₂²	—	424	1	no	—
248	C₂²	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
249	C₂²	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
250	C₂²	C₂×#11719945	1	1	yes	characteristic
251	C₄	—	2	2	no	—
252	C₄	—	106	2	no	—
253	C₄	—	106	2	no	—
254	C₄	—	212	1	no	—
255	C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
256	C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
257	C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
258	C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
259	C₂	—	2	1	no	—
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	—	4	2	no	—
263	C₂	—	212	2	no	—
264	C₂	C₂×C₄×C₂²⋊D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#568	C₂
socle	C₂×C₅₃	C₂×C₄×D₄
hypercenter	#568	D₅₃
hypocenter	C₅₃	#2956
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #568

Complements

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

N	H ≅ G/N	type
2. #16811389	C₂	semidirect
3. #16810743	C₂	semidirect
4. #16810738	C₂	semidirect
5. #16810671	C₂	semidirect
6. C₂×#13133144	C₂	semidirect
7. C₂×#13133143	C₂	semidirect
8. M₁₆×Dic₅₃	C₂	semidirect
9. C₅₃×#568	C₂	semidirect
11. #13133248	C₄	semidirect
12. #13133247	C₄	semidirect
13. #13133243	C₄	semidirect
14. #13133242	C₄	semidirect
17. #13133223	C₂²	semidirect
22. #13133152	C₂²	semidirect
23. #13133144	C₂²	semidirect
24. #13133144	C₂²	semidirect
25. #13133143	C₂²	semidirect
26. #13133143	C₂²	semidirect
27. #13133127	C₂²	semidirect
28. C₈×Dic₅₃	C₂²	semidirect
31. #131×C₅₃	C₂²	semidirect
61. #11719925	C₂×C₄	semidirect
62. #11719924	C₂×C₄	semidirect
63. #11719923	C₂×C₄	semidirect
64. #11719922	C₂×C₄	semidirect
135. #568	D₅₃	semidirect
142. C₅₃	#2956	semidirect
153. #131	C₂×D₅₃	semidirect