Subgroups — #20279115

Order 6784.

412 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃⋊#401	C₂	1	2	yes	maximal, complemented
2	#16810551	C₂	1	2	yes	maximal
3	#16810521	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810521	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810517	C₂	1	2	yes	maximal
6	#16810505	C₂	1	2	yes	maximal, complemented
7	#16810489	C₂	1	1	yes	maximal, characteristic, complemented
8	#16810483	C₂	1	1	yes	maximal, characteristic
9	#16810483	C₂	1	1	yes	maximal, characteristic
10	C₄×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₂²⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
16	C₂×#11719919	C₂²	1	2	yes	—
17	C₂×#11719919	C₂²	1	2	yes	—
18	C₂×C₂²⋊Dic₅₃	C₂²	1	2	yes	—
19	C₂×C₂²⋊Dic₅₃	C₂²	1	2	yes	—
20	C₂×C₄×Dic₅₃	C₂²	1	2	yes	—
21	#13133124	C₂²	1	2	yes	—
22	#13133124	C₂²	1	2	yes	—
23	#13133124	C₂²	1	2	yes	—
24	#13133124	C₂²	1	2	yes	—
25	#13133124	C₂²	1	2	yes	—
26	#13133124	C₂²	1	2	yes	—
27	#13133124	C₂²	1	2	yes	—
28	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
29	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
31	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
33	#95×C₅₃	C₂²	1	1	yes	characteristic
34	#95×C₅₃	C₂²	1	1	yes	characteristic
35	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	—	2	2	no	—
37	C₂²×C₄×C₅₃	—	2	2	no	—
38	C₂²×C₄×C₅₃	—	2	2	no	—
39	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
45	C₂²×Dic₅₃	C₂³	1	2	yes	—
46	C₂²×Dic₅₃	C₂³	1	2	yes	—
47	C₂²×Dic₅₃	C₂³	1	2	yes	—
48	C₂²×Dic₅₃	C₂³	1	2	yes	—
49	C₂²⋊C₄×C₅₃	—	2	2	no	—
50	C₂²⋊C₄×C₅₃	—	2	2	no	—
51	C₄²×C₅₃	—	2	2	no	—
52	C₄²×C₅₃	—	2	2	no	—
53	C₂²⋊Dic₅₃	—	2	4	no	—
54	C₂²⋊Dic₅₃	—	4	2	no	—
55	C₄⋊Dic₅₃	—	2	2	no	—
56	C₄⋊Dic₅₃	—	2	2	no	—
57	C₄⋊Dic₅₃	—	2	2	no	—
58	C₄⋊Dic₅₃	—	2	2	no	—
59	#11719919	—	2	4	no	—
60	#11719919	—	4	2	no	—
61	C₄×Dic₅₃	—	4	2	no	—
62	C₂³×C₅₃	—	2	1	no	—
63	C₂³×C₅₃	—	2	1	no	—
64	C₂³×C₅₃	—	2	1	no	—
65	C₂³×C₅₃	—	2	1	no	—
66	C₂³×C₅₃	—	2	1	no	—
67	C₂³×C₅₃	—	2	1	no	—
68	C₂³×C₅₃	#41	1	2	yes	—
69	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
70	C₂×C₄×C₅₃	—	2	1	no	—
71	C₂×C₄×C₅₃	—	2	1	no	—
72	C₂×C₄×C₅₃	—	2	1	no	—
73	C₂×C₄×C₅₃	—	2	1	no	—
74	C₂×C₄×C₅₃	—	2	1	no	—
75	C₂×C₄×C₅₃	—	2	1	no	—
76	C₂×C₄×C₅₃	—	2	1	no	—
77	C₂×C₄×C₅₃	—	2	1	no	—
78	C₂×C₄×C₅₃	—	2	1	no	—
79	C₂×C₄×C₅₃	—	2	1	no	—
80	C₂×C₄×C₅₃	—	2	1	no	—
81	C₂×C₄×C₅₃	—	2	1	no	—
82	C₂×C₄×C₅₃	—	2	1	no	—
83	C₂×C₄×C₅₃	—	2	1	no	—
84	C₂×C₄×C₅₃	—	4	4	no	—
85	C₂×C₄×C₅₃	#41	1	2	yes	—
86	C₂×C₄×C₅₃	#41	1	2	yes	—
87	C₂×C₄×C₅₃	#41	1	2	yes	—
88	C₂×C₄×C₅₃	#41	1	2	yes	—
89	C₂×Dic₅₃	—	2	2	no	—
90	C₂×Dic₅₃	—	2	2	no	—
91	C₂×Dic₅₃	—	2	2	no	—
92	C₂×Dic₅₃	—	2	2	no	—
93	C₂×Dic₅₃	—	2	2	no	—
94	C₂×Dic₅₃	—	2	2	no	—
95	C₂×Dic₅₃	—	2	2	no	—
96	C₂×Dic₅₃	—	2	2	no	—
97	C₂×Dic₅₃	—	2	2	no	—
98	C₂×Dic₅₃	—	2	2	no	—
99	C₂×Dic₅₃	—	2	2	no	—
100	C₂×Dic₅₃	—	2	2	no	—
101	C₂²×C₅₃	—	2	2	no	—
102	C₂²×C₅₃	—	2	2	no	—
103	C₂²×C₅₃	—	2	2	no	—
104	C₂²×C₅₃	—	4	1	no	—
105	C₂²×C₅₃	—	4	1	no	—
106	C₂²×C₅₃	—	4	1	no	—
107	C₂²×C₅₃	—	4	1	no	—
108	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
109	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
110	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
111	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
112	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
113	C₂²×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
114	C₂²×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
115	C₄×C₅₃	—	2	2	no	—
116	C₄×C₅₃	—	4	1	no	—
117	C₄×C₅₃	—	4	1	no	—
118	C₄×C₅₃	—	4	1	no	—
119	C₄×C₅₃	—	4	1	no	—
120	C₄×C₅₃	#126	1	4	yes	—
121	Dic₅₃	—	4	2	no	—
122	Dic₅₃	—	4	2	no	—
123	Dic₅₃	—	4	2	no	—
124	Dic₅₃	—	4	2	no	—
125	#2494	—	53	1	no	maximal, Sylow
126	C₂×C₅₃	—	4	2	no	—
127	C₂×C₅₃	C₂×#126	1	1	yes	characteristic
128	C₂×C₅₃	#529	1	1	yes	characteristic
129	C₂×C₅₃	#542	1	1	yes	characteristic
130	C₂×C₅₃	#552	1	1	yes	characteristic
131	C₂×C₅₃	#553	1	1	yes	characteristic
132	C₂×C₅₃	#553	1	1	yes	characteristic
133	C₂×C₅₃	#556	1	1	yes	characteristic
134	#401	—	53	2	no	—
135	#397	—	53	2	no	—
136	#389	—	53	1	no	—
137	#389	—	53	1	no	—
138	#388	—	53	2	no	—
139	#387	—	53	2	no	—
140	#382	—	53	1	no	—
141	#380	—	53	1	no	—
142	#380	—	53	1	no	—
143	C₄×C₄⋊C₄	—	53	1	no	—
144	C₄×C₂²⋊C₄	D₅₃	1	1	yes	characteristic
145	C₅₃	#2494	1	1	yes	characteristic, complemented, Sylow
146	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
147	C₂×C₄⋊C₄	—	53	2	no	—
148	C₂×C₄⋊C₄	—	53	1	no	—
149	C₂×C₄⋊C₄	—	53	1	no	—
150	C₂×C₄⋊C₄	—	53	1	no	—
151	C₂×C₄⋊C₄	—	53	1	no	—
152	C₂×C₄⋊C₄	—	53	2	no	—
153	C₂×C₂²⋊C₄	—	53	2	no	—
154	C₂×C₂²⋊C₄	—	53	2	no	—
155	C₂×C₂²⋊C₄	C₂×D₅₃	1	1	yes	characteristic
156	C₂×C₂²⋊C₄	C₂×D₅₃	1	1	yes	characteristic
157	C₂×C₄²	—	53	2	no	—
158	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
159	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
160	#95	—	53	2	no	—
161	#95	—	53	2	no	—
162	#95	—	53	2	no	—
163	#95	—	53	2	no	—
164	#95	—	53	2	no	—
165	#95	—	53	2	no	—
166	#95	—	53	2	no	—
167	#95	C₂×D₅₃	1	1	yes	characteristic
168	#95	C₂×D₅₃	1	1	yes	characteristic
169	C₂⁴	C₂²×D₅₃	1	1	yes	characteristic
170	C₂²×C₄	—	2	2	no	—
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	53	2	no	—
174	C₂²×C₄	—	53	2	no	—
175	C₂²×C₄	—	53	2	no	—
176	C₂²×C₄	—	53	2	no	—
177	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
178	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
179	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
180	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
181	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
182	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
183	C₄⋊C₄	—	106	4	no	—
184	C₄⋊C₄	—	106	2	no	—
185	C₄⋊C₄	—	106	2	no	—
186	C₄⋊C₄	—	106	2	no	—
187	C₄⋊C₄	—	106	2	no	—
188	C₄⋊C₄	—	212	2	no	—
189	C₂²⋊C₄	—	2	2	no	—
190	C₂²⋊C₄	—	2	2	no	—
191	C₂²⋊C₄	—	106	4	no	—
192	C₂²⋊C₄	—	212	2	no	—
193	C₄²	—	2	2	no	—
194	C₄²	—	2	2	no	—
195	C₄²	—	212	2	no	—
196	C₂³	—	2	1	no	—
197	C₂³	—	2	1	no	—
198	C₂³	—	2	1	no	—
199	C₂³	—	2	1	no	—
200	C₂³	—	2	1	no	—
201	C₂³	—	2	1	no	—
202	C₂³	#11719945	1	2	yes	—
203	C₂³	C₂³×D₅₃	1	1	yes	characteristic
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	4	4	no	—
219	C₂×C₄	—	106	2	no	—
220	C₂×C₄	—	106	2	no	—
221	C₂×C₄	—	106	2	no	—
222	C₂×C₄	—	106	2	no	—
223	C₂×C₄	—	106	2	no	—
224	C₂×C₄	—	106	2	no	—
225	C₂×C₄	—	106	2	no	—
226	C₂×C₄	—	106	2	no	—
227	C₂×C₄	—	106	2	no	—
228	C₂×C₄	—	106	2	no	—
229	C₂×C₄	—	106	2	no	—
230	C₂×C₄	—	106	2	no	—
231	C₂×C₄	#11719945	1	2	yes	—
232	C₂×C₄	#11719945	1	2	yes	—
233	C₂×C₄	#11719947	1	2	yes	—
234	C₂×C₄	#11719947	1	2	yes	—
235	C₂²	—	2	2	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	2	2	no	—
238	C₂²	—	4	1	no	—
239	C₂²	—	4	1	no	—
240	C₂²	—	4	1	no	—
241	C₂²	—	4	1	no	—
242	C₂²	C₂×#11719947	1	1	yes	characteristic
243	C₂²	C₂×#11719947	1	1	yes	characteristic
244	C₂²	C₂×#11719945	1	1	yes	characteristic
245	C₂²	C₂×#11719945	1	1	yes	characteristic
246	C₂²	C₂×#11719945	1	1	yes	characteristic
247	C₂²	#13133263	1	1	yes	characteristic
248	C₂²	#13133264	1	1	yes	characteristic
249	C₄	—	2	2	no	—
250	C₄	—	4	1	no	—
251	C₄	—	4	1	no	—
252	C₄	—	4	1	no	—
253	C₄	—	4	1	no	—
254	C₄	—	212	2	no	—
255	C₄	—	212	2	no	—
256	C₄	—	212	2	no	—
257	C₄	—	212	2	no	—
258	C₄	#13133210	1	4	yes	—
259	C₂	—	4	2	no	—
260	C₂	C₂×#13133210	1	1	yes	characteristic
261	C₂	#16811157	1	1	yes	characteristic
262	C₂	#16811244	1	1	yes	characteristic
263	C₂	#16811313	1	1	yes	characteristic
264	C₂	#16811319	1	1	yes	characteristic
265	C₂	#16811319	1	1	yes	characteristic
266	C₂	#16811343	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂⁴
Fitting	C₄×C₂²⋊C₄×C₅₃	C₂
hypercenter	C₄×C₂²⋊C₄	D₅₃
hypocenter	C₅₃	#2494
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₅₃⋊#401	C₂	semidirect
3. #16810521	C₂	semidirect
4. #16810521	C₂	semidirect
6. #16810505	C₂	semidirect
7. #16810489	C₂	semidirect
10. C₄×C₄⋊Dic₅₃	C₂	semidirect
145. C₅₃	#2494	semidirect