Subgroups — #20283630

Order 6784.

452 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133237	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	C₂×#13133152	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810640	C₂	1	1	yes	maximal, characteristic, complemented
7	#16810639	C₂	1	1	yes	maximal, characteristic, complemented
8	#16810581	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	#13133237	C₄	1	2	yes	complemented
12	#13133237	C₄	1	2	yes	complemented
13	#13133237	C₂²	1	2	yes	—
14	#13133237	C₂²	1	2	yes	—
15	C₂×#11719925	C₄	1	1	yes	characteristic, complemented
16	C₂×#11719923	C₄	1	1	yes	characteristic, complemented
17	C₂×#11719924	C₄	1	1	yes	characteristic, complemented
18	C₂×#11719922	C₄	1	1	yes	characteristic, complemented
19	#13133225	—	2	2	no	—
20	#13133223	C₂²	1	4	yes	complemented
21	C₂×#11719917	C₂²	1	1	yes	characteristic
22	M₁₆×D₅₃	—	2	2	no	—
23	C₂²×#88322	C₂²	1	1	yes	characteristic
24	#13133176	C₂²	1	1	yes	characteristic
25	#13133152	C₂²	1	2	yes	complemented
26	#13133152	C₂²	1	2	yes	complemented
27	#13133144	C₂²	1	4	yes	complemented
28	#13133143	C₂²	1	4	yes	complemented
29	#13133127	C₂²	1	2	yes	complemented
30	C₈×Dic₅₃	C₂²	1	2	yes	complemented
31	C₂×#41×C₅₃	C₂²	1	1	yes	characteristic
32	#131×C₅₃	—	2	2	no	—
33	#131×C₅₃	C₂²	1	4	yes	complemented
34	C₂×M₁₆×C₅₃	—	2	2	no	—
35	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
36	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
37	#41×C₅₃	—	2	1	no	—
38	#41×C₅₃	—	2	1	no	—
39	#41×C₅₃	C₂×C₄	1	2	yes	—
40	#41×C₅₃	C₂³	1	2	yes	—
41	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂×D₄×C₅₃	—	2	1	no	—
43	C₂×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	—	2	1	no	—
45	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊D₅₃	—	2	1	no	—
47	#11719945	—	2	1	no	—
48	#11719945	—	2	1	no	—
49	#11719945	C₂×C₄	1	2	yes	—
50	#11719945	C₂³	1	2	yes	—
51	C₂×D₂₁₂	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₅₃	—	2	1	no	—
53	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
54	M₁₆×C₅₃	—	2	1	no	—
55	M₁₆×C₅₃	—	2	4	no	—
56	M₁₆×C₅₃	D₄	1	2	yes	—
57	M₁₆×C₅₃	D₄	1	2	yes	—
58	M₁₆×C₅₃	C₂³	1	2	yes	—
59	C₂×C₈×C₅₃	—	2	1	no	—
60	C₂×C₈×C₅₃	—	2	4	no	—
61	C₂×C₈×C₅₃	C₂³	1	2	yes	—
62	C₂²⋊Dic₅₃	—	2	1	no	—
63	#11719925	C₂×C₄	1	2	yes	—
64	#11719925	C₂×C₄	1	2	yes	—
65	#11719924	C₂×C₄	1	2	yes	—
66	#11719924	C₂×C₄	1	2	yes	—
67	#11719923	C₂×C₄	1	2	yes	—
68	#11719923	C₂×C₄	1	2	yes	—
69	#11719922	C₂×C₄	1	2	yes	—
70	#11719922	C₂×C₄	1	2	yes	—
71	C₄⋊Dic₅₃	—	2	1	no	—
72	C₄×Dic₅₃	C₂³	1	2	yes	—
73	#11719917	—	2	1	no	—
74	#11719917	C₂³	1	2	yes	—
75	C₂×#88322	—	2	1	no	—
76	C₂×#88322	C₂×C₄	1	1	yes	characteristic
77	C₂×#88322	C₂×C₄	1	1	yes	characteristic
78	C₂×#88322	C₂×C₄	1	1	yes	characteristic
79	C₂×#88322	C₂×C₄	1	1	yes	characteristic
80	C₂×#88322	C₂³	1	1	yes	characteristic
81	C₂×#88322	C₂³	1	1	yes	characteristic
82	#11719912	—	2	4	no	—
83	C₈×D₅₃	—	2	4	no	—
84	C₂³×C₅₃	—	2	1	no	—
85	C₂³×C₅₃	#41	1	1	yes	characteristic
86	C₂²×D₅₃	—	2	1	no	—
87	Q₈×C₅₃	—	2	1	no	—
88	Q₈×C₅₃	C₂²×C₄	1	2	yes	—
89	D₄×C₅₃	—	2	2	no	—
90	D₄×C₅₃	—	2	2	no	—
91	D₄×C₅₃	—	2	1	no	—
92	D₄×C₅₃	C₂²×C₄	1	2	yes	—
93	C₂×C₄×C₅₃	—	2	1	no	—
94	C₂×C₄×C₅₃	—	2	2	no	—
95	C₂×C₄×C₅₃	—	2	2	no	—
96	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
102	C₂²⋊D₅₃	—	2	2	no	—
103	C₂²⋊D₅₃	—	2	2	no	—
104	C₂×Dic₅₃	—	2	1	no	—
105	C₂×Dic₅₃	—	2	2	no	—
106	D₂₁₂	—	2	1	no	—
107	D₂₁₂	C₂²×C₄	1	2	yes	—
108	C₄×D₅₃	—	2	2	no	—
109	C₄×D₅₃	—	2	2	no	—
110	Dic₁₀₆	—	2	1	no	—
111	Dic₁₀₆	C₂²×C₄	1	2	yes	—
112	C₈×C₅₃	—	2	2	no	—
113	C₈×C₅₃	C₂×D₄	1	4	yes	—
114	#88322	—	2	2	no	—
115	#88322	C₂²×C₄	1	1	yes	characteristic
116	#88322	C₂²×C₄	1	1	yes	characteristic
117	#88322	C₂²×C₄	1	1	yes	characteristic
118	#88322	C₂²×C₄	1	1	yes	characteristic
119	C₂²×C₅₃	—	2	2	no	—
120	C₂²×C₅₃	—	4	1	no	—
121	C₂²×C₅₃	—	8	1	no	—
122	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
123	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
124	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
125	C₂×D₅₃	—	2	2	no	—
126	C₂×D₅₃	—	8	1	no	—
127	C₄×C₅₃	—	2	2	no	—
128	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
129	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
130	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
131	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
132	Dic₅₃	—	2	2	no	—
133	Dic₅₃	—	4	2	no	—
134	#2957	—	53	1	no	maximal, Sylow
135	C₂×C₅₃	—	2	1	no	—
136	C₂×C₅₃	—	2	1	no	—
137	C₂×C₅₃	—	2	1	no	—
138	C₂×C₅₃	—	4	2	no	—
139	C₂×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
140	D₅₃	—	4	2	no	—
141	C₂×#135	—	53	1	no	—
142	#568	—	53	1	no	—
143	#568	D₅₃	1	1	yes	characteristic, complemented
144	#444	—	53	4	no	—
145	#443	—	53	4	no	—
146	C₂×#108	—	53	1	no	—
147	#421	—	53	1	no	—
148	#421	—	53	1	no	—
149	#405	—	53	1	no	—
150	C₅₃	#2957	1	1	yes	characteristic, complemented, Sylow
151	C₂×#41	—	53	1	no	—
152	C₂×#41	C₂×D₅₃	1	1	yes	characteristic
153	#135	—	53	2	no	—
154	#135	—	53	2	no	—
155	#135	—	53	2	no	—
156	#135	—	53	2	no	—
157	C₂×Q₁₆	—	53	1	no	—
158	C₂×SD₁₆	—	53	1	no	—
159	C₂×SD₁₆	—	53	1	no	—
160	C₂×D₈	—	53	1	no	—
161	#131	—	2	2	no	—
162	#131	—	53	4	no	—
163	#131	—	106	2	no	—
164	#131	C₂×D₅₃	1	4	yes	complemented
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	—	53	1	no	—
167	C₂×M₁₆	—	106	2	no	—
168	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
169	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
170	C₂²×C₈	—	53	1	no	—
171	#117	—	53	1	no	—
172	#108	—	53	2	no	—
173	#108	—	53	2	no	—
174	#104	—	53	4	no	—
175	#104	—	53	4	no	—
176	#97	—	53	2	no	—
177	C₄×C₈	—	53	2	no	—
178	#41	—	2	1	no	—
179	#41	—	2	1	no	—
180	#41	—	53	2	no	—
181	#41	—	53	2	no	—
182	#41	—	106	1	no	—
183	#41	—	106	1	no	—
184	#41	C₄×D₅₃	1	2	yes	—
185	#41	C₂²×D₅₃	1	2	yes	—
186	C₂×Q₈	—	53	1	no	—
187	C₂×Q₈	C₄×D₅₃	1	1	yes	characteristic
188	C₂×D₄	—	2	1	no	—
189	C₂×D₄	—	53	1	no	—
190	C₂×D₄	—	106	1	no	—
191	C₂×D₄	C₄×D₅₃	1	1	yes	characteristic
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	106	1	no	—
194	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
195	Q₁₆	—	53	2	no	—
196	Q₁₆	—	53	2	no	—
197	SD₁₆	—	53	2	no	—
198	SD₁₆	—	53	2	no	—
199	SD₁₆	—	53	2	no	—
200	SD₁₆	—	53	2	no	—
201	D₈	—	53	2	no	—
202	D₈	—	53	2	no	—
203	M₁₆	—	2	1	no	—
204	M₁₆	—	2	4	no	—
205	M₁₆	—	53	2	no	—
206	M₁₆	—	106	1	no	—
207	M₁₆	—	106	4	no	—
208	M₁₆	C₂²⋊D₅₃	1	2	yes	—
209	M₁₆	C₂²⋊D₅₃	1	2	yes	—
210	M₁₆	C₂²×D₅₃	1	2	yes	—
211	C₂×C₈	—	2	1	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	53	1	no	—
214	C₂×C₈	—	53	1	no	—
215	C₂×C₈	—	53	1	no	—
216	C₂×C₈	—	53	1	no	—
217	C₂×C₈	—	53	1	no	—
218	C₂×C₈	—	53	1	no	—
219	C₂×C₈	—	106	1	no	—
220	C₂×C₈	—	106	4	no	—
221	C₂×C₈	C₂²×D₅₃	1	2	yes	—
222	C₄⋊C₄	—	106	1	no	—
223	C₂²⋊C₄	—	106	1	no	—
224	C₄²	—	53	2	no	—
225	C₂³	—	2	1	no	—
226	C₂³	—	106	1	no	—
227	C₂³	#11719945	1	1	yes	characteristic
228	Q₈	—	2	1	no	—
229	Q₈	—	53	2	no	—
230	Q₈	—	106	1	no	—
231	Q₈	C₂×C₄×D₅₃	1	2	yes	—
232	D₄	—	2	2	no	—
233	D₄	—	2	2	no	—
234	D₄	—	2	1	no	—
235	D₄	—	53	2	no	—
236	D₄	—	106	2	no	—
237	D₄	—	106	2	no	—
238	D₄	—	106	1	no	—
239	D₄	C₂×C₄×D₅₃	1	2	yes	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	106	1	no	—
244	C₂×C₄	—	106	2	no	—
245	C₂×C₄	—	106	2	no	—
246	C₂×C₄	—	106	2	no	—
247	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
248	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
249	C₂×C₄	#11719945	1	1	yes	characteristic
250	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
251	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
252	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
253	C₈	—	2	2	no	—
254	C₈	—	53	1	no	—
255	C₈	—	53	1	no	—
256	C₈	—	53	1	no	—
257	C₈	—	53	1	no	—
258	C₈	—	106	2	no	—
259	C₈	C₂×C₂²⋊D₅₃	1	4	yes	—
260	C₂²	—	2	2	no	—
261	C₂²	—	4	1	no	—
262	C₂²	—	8	1	no	—
263	C₂²	—	106	2	no	—
264	C₂²	—	424	1	no	—
265	C₂²	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
266	C₂²	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
267	C₂²	C₂×#11719945	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	106	2	no	—
270	C₄	—	212	2	no	—
271	C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
272	C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
273	C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
274	C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
275	C₂	—	2	1	no	—
276	C₂	—	2	1	no	—
277	C₂	—	2	1	no	—
278	C₂	—	4	2	no	—
279	C₂	—	212	2	no	—
280	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	#2957
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. C₂×#13133152	C₂	semidirect
6. #16810640	C₂	semidirect
7. #16810639	C₂	semidirect
8. #16810581	C₂	semidirect
9. C₅₃×#568	C₂	semidirect
11. #13133237	C₄	semidirect
12. #13133237	C₄	semidirect
15. C₂×#11719925	C₄	semidirect
16. C₂×#11719923	C₄	semidirect
17. C₂×#11719924	C₄	semidirect
18. C₂×#11719922	C₄	semidirect
20. #13133223	C₂²	semidirect
25. #13133152	C₂²	semidirect
26. #13133152	C₂²	semidirect
27. #13133144	C₂²	semidirect
28. #13133143	C₂²	semidirect
29. #13133127	C₂²	semidirect
30. C₈×Dic₅₃	C₂²	semidirect
33. #131×C₅₃	C₂²	semidirect
143. #568	D₅₃	semidirect
150. C₅₃	#2957	semidirect
164. #131	C₂×D₅₃	semidirect