Subgroups — #20283460

Order 6784.

492 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133225	C₂	1	2	yes	maximal, complemented
2	#16811170	C₂	1	1	yes	maximal, characteristic
3	#16810700	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810692	C₂	1	2	yes	maximal, complemented
5	#16810691	C₂	1	2	yes	maximal, complemented
6	C₈×D₂₁₂	C₂	1	1	yes	maximal, characteristic, complemented
7	D₄×#88322	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄⋊C₈×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#16810588	C₂	1	2	yes	maximal, complemented
10	C₄×#11719912	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11719945	C₂²	1	2	yes	—
13	#13133225	—	2	8	no	—
14	C₂×#11719912	C₂²	1	1	yes	characteristic
15	C₂×#11719912	C₂²	1	2	yes	—
16	C₂×#11719912	C₂²	1	1	yes	characteristic
17	C₂²×#88322	C₂²	1	2	yes	—
18	C₂×C₈×D₅₃	C₂²	1	2	yes	—
19	C₂×C₈×D₅₃	C₂²	1	2	yes	—
20	#13133216	C₄	1	1	yes	characteristic
21	#13133213	C₄	1	1	yes	characteristic
22	#13133203	C₄	1	2	yes	—
23	#13133192	C₄	1	1	yes	characteristic
24	#13133190	C₄	1	2	yes	—
25	#13133188	C₄	1	1	yes	characteristic
26	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
27	C₄×C₂²⋊D₅₃	C₂²	1	2	yes	—
28	C₄²×D₅₃	C₂²	1	1	yes	characteristic
29	C₄⋊#88322	C₂²	1	1	yes	characteristic
30	#13133146	C₂²	1	2	yes	—
31	C₂²⋊#88322	C₂²	1	2	yes	—
32	#13133129	C₂²	1	2	yes	—
33	#13133129	C₂²	1	2	yes	—
34	#13133127	C₂²	1	1	yes	characteristic
35	#13133127	C₂²	1	1	yes	characteristic
36	C₄×#88322	C₂²	1	1	yes	characteristic
37	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
38	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
39	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
40	C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
41	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
42	C₂×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
44	C₂×C₂²⋊D₅₃	C₂×C₄	1	2	yes	—
45	C₂×C₂²⋊D₅₃	C₂×C₄	1	2	yes	—
46	C₂×C₂²⋊D₅₃	C₂×C₄	1	2	yes	—
47	#11719945	—	2	4	no	—
48	#11719945	—	2	4	no	—
49	C₂×D₂₁₂	C₂×C₄	1	1	yes	characteristic
50	C₂×D₂₁₂	C₂×C₄	1	2	yes	—
51	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×D₅₃	C₂³	1	2	yes	—
54	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
55	C₂×Dic₁₀₆	C₂×C₄	1	2	yes	—
56	M₁₆×C₅₃	—	2	4	no	—
57	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
58	C₂×C₈×C₅₃	C₂³	1	2	yes	—
59	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
60	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
61	C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
62	C₄²×C₅₃	C₂³	1	1	yes	characteristic
63	C₂²⋊Dic₅₃	C₂×C₄	1	2	yes	—
64	#11719921	C₂×C₄	1	2	yes	—
65	#11719921	C₂×C₄	1	2	yes	—
66	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
67	#11719919	C₂×C₄	1	2	yes	—
68	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
69	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
70	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
71	C₂×#88322	—	2	4	no	—
72	C₂×#88322	C₂³	1	1	yes	characteristic
73	C₂×#88322	C₂³	1	2	yes	—
74	C₂×#88322	C₂³	1	1	yes	characteristic
75	#11719912	—	2	2	no	—
76	#11719912	—	2	4	no	—
77	#11719912	D₄	1	4	yes	complemented
78	C₈×D₅₃	—	2	4	no	—
79	C₈×D₅₃	—	2	4	no	—
80	C₂³×C₅₃	C₂²×C₄	1	2	yes	—
81	C₂²×D₅₃	C₂²×C₄	1	2	yes	—
82	C₂²×D₅₃	C₂²×C₄	1	1	yes	characteristic
83	D₄×C₅₃	—	2	2	no	—
84	C₂×C₄×C₅₃	—	2	2	no	—
85	C₂×C₄×C₅₃	—	2	2	no	—
86	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
87	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
90	C₂²⋊D₅₃	—	2	4	no	—
91	C₂²⋊D₅₃	—	2	4	no	—
92	C₂²⋊D₅₃	—	4	2	no	—
93	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
94	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
97	D₂₁₂	—	2	4	no	—
98	D₂₁₂	—	2	2	no	—
99	C₄×D₅₃	—	2	2	no	—
100	C₄×D₅₃	—	2	2	no	—
101	C₄×D₅₃	—	2	2	no	—
102	C₄×D₅₃	—	4	1	no	—
103	C₄×D₅₃	C₂×D₄	1	2	yes	—
104	C₄×D₅₃	#41	1	2	yes	—
105	Dic₁₀₆	—	2	4	no	—
106	C₈×C₅₃	—	2	2	no	—
107	C₈×C₅₃	—	2	1	no	—
108	C₈×C₅₃	C₂×D₄	1	2	yes	—
109	#88322	—	2	2	no	—
110	#88322	—	2	1	no	—
111	#88322	C₂×D₄	1	2	yes	—
112	C₂²×C₅₃	—	2	2	no	—
113	C₂²×C₅₃	—	2	2	no	—
114	C₂²×C₅₃	—	2	2	no	—
115	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
116	C₂×D₅₃	—	2	1	no	—
117	C₂×D₅₃	—	2	1	no	—
118	C₂×D₅₃	—	2	2	no	—
119	C₂×D₅₃	—	2	2	no	—
120	C₂×D₅₃	—	2	2	no	—
121	C₂×D₅₃	C₄×D₄	1	2	yes	—
122	C₄×C₅₃	—	2	2	no	—
123	C₄×C₅₃	—	2	1	no	—
124	C₄×C₅₃	#131	1	2	yes	complemented
125	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
126	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
127	Dic₅₃	—	2	2	no	—
128	Dic₅₃	—	2	1	no	—
129	Dic₅₃	—	2	1	no	—
130	Dic₅₃	C₄×D₄	1	2	yes	—
131	#2935	—	53	1	no	maximal, Sylow
132	C₂×C₅₃	—	4	2	no	—
133	C₂×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
134	C₂×C₅₃	C₂×#131	1	1	yes	characteristic
135	C₂×C₅₃	#568	1	1	yes	characteristic
136	D₅₃	—	2	2	no	—
137	D₅₃	—	4	2	no	—
138	C₂×#131	—	53	2	no	—
139	#532	—	53	1	no	—
140	#436	—	53	1	no	—
141	#436	D₅₃	1	1	yes	characteristic, complemented
142	#435	—	53	2	no	—
143	#435	—	53	2	no	—
144	C₈×D₄	—	53	1	no	—
145	C₈×D₄	—	53	1	no	—
146	C₂×C₄⋊C₈	—	53	1	no	—
147	#408	—	53	2	no	—
148	C₄×M₁₆	—	53	1	no	—
149	C₅₃	#2935	1	1	yes	characteristic, complemented, Sylow
150	C₂×#41	—	53	2	no	—
151	#131	—	106	8	no	—
152	C₂×M₁₆	—	53	1	no	—
153	C₂×M₁₆	—	53	2	no	—
154	C₂×M₁₆	—	53	1	no	—
155	C₂×M₁₆	C₂×D₅₃	1	2	yes	—
156	C₂²×C₈	—	53	2	no	—
157	C₂²×C₈	—	53	2	no	—
158	C₂²×C₈	—	53	2	no	—
159	C₄⋊Q₈	—	53	1	no	—
160	#127	—	53	1	no	—
161	#124	—	53	2	no	—
162	#121	—	53	1	no	—
163	#121	—	53	1	no	—
164	#121	—	53	2	no	—
165	C₄×D₄	—	53	2	no	—
166	C₄×D₄	—	53	1	no	—
167	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
168	C₂×C₄²	—	53	1	no	—
169	C₄⋊C₈	—	53	2	no	—
170	C₄⋊C₈	—	53	1	no	—
171	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
172	C₂²⋊C₈	—	53	2	no	—
173	C₂²⋊C₈	—	53	2	no	—
174	C₂²⋊C₈	—	53	2	no	—
175	C₂²⋊C₈	C₂×D₅₃	1	2	yes	—
176	#97	—	53	1	no	—
177	#97	—	53	1	no	—
178	C₄×C₈	—	53	1	no	—
179	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
180	#41	—	106	4	no	—
181	#41	—	106	4	no	—
182	C₂×Q₈	—	53	2	no	—
183	C₂×D₄	—	53	2	no	—
184	C₂×D₄	—	53	2	no	—
185	C₂×D₄	—	53	2	no	—
186	C₂×D₄	—	53	1	no	—
187	C₂×D₄	—	53	2	no	—
188	C₂×D₄	C₄×D₅₃	1	1	yes	characteristic
189	C₂²×C₄	—	53	1	no	—
190	C₂²×C₄	—	53	1	no	—
191	C₂²×C₄	—	53	2	no	—
192	C₂²×C₄	—	53	1	no	—
193	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
194	M₁₆	—	2	4	no	—
195	M₁₆	—	53	4	no	—
196	M₁₆	—	106	4	no	—
197	M₁₆	—	106	2	no	—
198	C₂×C₈	—	53	1	no	—
199	C₂×C₈	—	53	1	no	—
200	C₂×C₈	—	53	2	no	—
201	C₂×C₈	—	106	4	no	—
202	C₂×C₈	—	106	4	no	—
203	C₂×C₈	—	106	4	no	—
204	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
205	C₂×C₈	C₂²×D₅₃	1	2	yes	—
206	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
207	C₄⋊C₄	—	53	2	no	—
208	C₄⋊C₄	—	53	1	no	—
209	C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
210	C₂²⋊C₄	—	53	2	no	—
211	C₂²⋊C₄	—	53	2	no	—
212	C₂²⋊C₄	—	53	2	no	—
213	C₂²⋊C₄	C₄×D₅₃	1	2	yes	—
214	C₄²	—	53	1	no	—
215	C₄²	—	53	1	no	—
216	C₄²	—	53	1	no	—
217	C₄²	C₂²×D₅₃	1	1	yes	characteristic
218	C₂³	—	53	1	no	—
219	C₂³	—	53	2	no	—
220	C₂³	C₂×C₄×D₅₃	1	2	yes	—
221	Q₈	—	106	4	no	—
222	D₄	—	2	2	no	—
223	D₄	—	106	4	no	—
224	D₄	—	106	4	no	—
225	D₄	—	106	4	no	—
226	D₄	—	106	2	no	—
227	D₄	—	212	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	53	1	no	—
231	C₂×C₄	—	53	2	no	—
232	C₂×C₄	—	53	2	no	—
233	C₂×C₄	—	53	1	no	—
234	C₂×C₄	—	53	1	no	—
235	C₂×C₄	—	53	2	no	—
236	C₂×C₄	—	106	2	no	—
237	C₂×C₄	—	106	2	no	—
238	C₂×C₄	—	106	2	no	—
239	C₂×C₄	—	212	1	no	—
240	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
241	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
242	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
243	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
244	C₈	—	2	2	no	—
245	C₈	—	2	1	no	—
246	C₈	—	53	2	no	—
247	C₈	—	106	2	no	—
248	C₈	—	106	1	no	—
249	C₈	D₄×D₅₃	1	2	yes	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	53	2	no	—
254	C₂²	—	106	1	no	—
255	C₂²	—	106	1	no	—
256	C₂²	—	106	2	no	—
257	C₂²	—	106	2	no	—
258	C₂²	—	106	2	no	—
259	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	2	1	no	—
262	C₄	—	53	2	no	—
263	C₄	—	106	2	no	—
264	C₄	—	106	1	no	—
265	C₄	—	106	1	no	—
266	C₄	#13133225	1	2	yes	complemented
267	C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
268	C₄	#41×D₅₃	1	1	yes	characteristic
269	C₂	—	4	2	no	—
270	C₂	—	106	2	no	—
271	C₂	—	212	2	no	—
272	C₂	C₄×D₄×D₅₃	1	1	yes	characteristic
273	C₂	C₂×#13133225	1	1	yes	characteristic
274	C₂	#16811388	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₅₃×#436	C₂
hypercenter	#436	D₅₃
hypocenter	C₅₃	#2935
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #436

Complements

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

N	H ≅ G/N	type
1. C₂×#13133225	C₂	semidirect
3. #16810700	C₂	semidirect
4. #16810692	C₂	semidirect
5. #16810691	C₂	semidirect
6. C₈×D₂₁₂	C₂	semidirect
7. D₄×#88322	C₂	semidirect
8. C₄⋊C₈×D₅₃	C₂	semidirect
9. #16810588	C₂	semidirect
10. C₄×#11719912	C₂	semidirect
11. C₅₃×#436	C₂	semidirect
77. #11719912	D₄	semidirect
124. C₄×C₅₃	#131	semidirect
141. #436	D₅₃	semidirect
149. C₅₃	#2935	semidirect
266. C₄	#13133225	semidirect