Subgroups — C₅₃×#2570 #20271507

Order 6784.

380 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#400	C₂	1	2	yes	maximal
2	C₅₃×#398	C₂	1	2	yes	maximal
3	C₅₃×#389	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#389	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#387	C₂	1	2	yes	maximal
6	C₅₃×#387	C₂	1	2	yes	maximal
7	C₅₃×#384	C₂	1	1	yes	maximal, characteristic
8	C₅₃×#382	C₂	1	1	yes	maximal, characteristic
9	C₅₃×#382	C₂	1	1	yes	maximal, characteristic
10	C₄×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
11	C₄×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
22	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
25	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
26	#95×C₅₃	C₂²	1	1	yes	characteristic
27	#95×C₅₃	C₂²	1	1	yes	characteristic
28	#95×C₅₃	C₂²	1	2	yes	—
29	#95×C₅₃	C₂²	1	2	yes	—
30	#95×C₅₃	C₂²	1	2	yes	—
31	#95×C₅₃	C₂²	1	2	yes	—
32	#95×C₅₃	C₂²	1	2	yes	—
33	#95×C₅₃	C₂²	1	2	yes	—
34	#95×C₅₃	C₂²	1	2	yes	—
35	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
37	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
39	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
40	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
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×C₄×C₅₃	C₂³	1	1	yes	characteristic
46	C₄⋊C₄×C₅₃	—	2	2	no	—
47	C₄⋊C₄×C₅₃	—	2	2	no	—
48	C₄⋊C₄×C₅₃	—	2	4	no	—
49	C₄⋊C₄×C₅₃	—	2	4	no	—
50	C₄⋊C₄×C₅₃	—	2	2	no	—
51	C₄⋊C₄×C₅₃	—	2	2	no	—
52	C₄⋊C₄×C₅₃	—	2	2	no	—
53	C₄⋊C₄×C₅₃	—	2	2	no	—
54	C₄⋊C₄×C₅₃	—	4	2	no	—
55	C₄⋊C₄×C₅₃	—	4	2	no	—
56	C₄²×C₅₃	—	2	2	no	—
57	C₄²×C₅₃	—	2	2	no	—
58	C₄²×C₅₃	—	4	2	no	—
59	C₄²×C₅₃	Q₈	1	4	yes	—
60	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
61	C₂×C₄×C₅₃	—	2	1	no	—
62	C₂×C₄×C₅₃	—	2	1	no	—
63	C₂×C₄×C₅₃	—	2	1	no	—
64	C₂×C₄×C₅₃	—	2	1	no	—
65	C₂×C₄×C₅₃	—	2	1	no	—
66	C₂×C₄×C₅₃	—	2	1	no	—
67	C₂×C₄×C₅₃	—	2	1	no	—
68	C₂×C₄×C₅₃	—	2	1	no	—
69	C₂×C₄×C₅₃	—	2	1	no	—
70	C₂×C₄×C₅₃	—	2	1	no	—
71	C₂×C₄×C₅₃	—	2	1	no	—
72	C₂×C₄×C₅₃	—	2	2	no	—
73	C₂×C₄×C₅₃	—	2	1	no	—
74	C₂×C₄×C₅₃	—	2	2	no	—
75	C₂×C₄×C₅₃	—	2	2	no	—
76	C₂×C₄×C₅₃	—	2	2	no	—
77	C₂×C₄×C₅₃	—	2	2	no	—
78	C₂×C₄×C₅₃	—	2	2	no	—
79	C₂×C₄×C₅₃	—	2	2	no	—
80	C₂×C₄×C₅₃	—	2	2	no	—
81	C₂×C₄×C₅₃	—	2	2	no	—
82	C₂×C₄×C₅₃	—	2	2	no	—
83	C₂×C₄×C₅₃	—	2	1	no	—
84	C₂×C₄×C₅₃	—	2	2	no	—
85	C₂×C₄×C₅₃	—	2	1	no	—
86	C₂×C₄×C₅₃	—	2	2	no	—
87	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
88	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
89	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
90	C₂×C₄×C₅₃	#41	1	2	yes	—
91	C₂×C₄×C₅₃	#41	1	2	yes	—
92	C₂×C₄×C₅₃	#41	1	2	yes	—
93	C₂×C₄×C₅₃	#41	1	2	yes	—
94	C₂²×C₅₃	C₂²×Q₈	1	1	yes	characteristic
95	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
96	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
97	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
98	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
99	C₂²×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
100	C₂²×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
101	C₄×C₅₃	—	2	2	no	—
102	C₄×C₅₃	—	2	2	no	—
103	C₄×C₅₃	—	2	2	no	—
104	C₄×C₅₃	—	2	2	no	—
105	C₄×C₅₃	—	4	1	no	—
106	C₄×C₅₃	—	4	1	no	—
107	C₄×C₅₃	—	4	2	no	—
108	C₄×C₅₃	—	4	2	no	—
109	C₄×C₅₃	—	4	2	no	—
110	C₄×C₅₃	—	4	2	no	—
111	C₄×C₅₃	#125	1	4	yes	—
112	#2570	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂×C₅₃	C₂×#125	1	1	yes	characteristic
114	C₂×C₅₃	#533	1	1	yes	characteristic
115	C₂×C₅₃	#544	1	1	yes	characteristic
116	C₂×C₅₃	#552	1	1	yes	characteristic
117	C₂×C₅₃	#553	1	1	yes	characteristic
118	C₂×C₅₃	#556	1	1	yes	characteristic
119	C₂×C₅₃	#559	1	1	yes	characteristic
120	#400	C₂×C₅₃	1	2	yes	—
121	#398	C₂×C₅₃	1	2	yes	—
122	#389	C₂×C₅₃	1	1	yes	characteristic
123	#389	C₂×C₅₃	1	1	yes	characteristic
124	#387	C₂×C₅₃	1	2	yes	—
125	#387	C₂×C₅₃	1	2	yes	—
126	#384	C₂×C₅₃	1	1	yes	characteristic
127	#382	C₂×C₅₃	1	1	yes	characteristic
128	#382	C₂×C₅₃	1	1	yes	characteristic
129	C₄×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic
130	C₄×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic
131	C₅₃	#2570	1	1	yes	characteristic, direct product factor, Sylow
132	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
133	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
134	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
135	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
136	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
137	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
138	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
139	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
140	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
141	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
142	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
143	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
144	C₂×C₄²	C₂²×C₅₃	1	2	yes	—
145	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
146	#95	C₂²×C₅₃	1	1	yes	characteristic
147	#95	C₂²×C₅₃	1	1	yes	characteristic
148	#95	C₂²×C₅₃	1	2	yes	—
149	#95	C₂²×C₅₃	1	2	yes	—
150	#95	C₂²×C₅₃	1	2	yes	—
151	#95	C₂²×C₅₃	1	2	yes	—
152	#95	C₂²×C₅₃	1	2	yes	—
153	#95	C₂²×C₅₃	1	2	yes	—
154	#95	C₂²×C₅₃	1	2	yes	—
155	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
156	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
157	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
158	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
159	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
160	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
161	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
162	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
163	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
164	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
165	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	2	4	no	—
169	C₄⋊C₄	—	2	4	no	—
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	4	2	no	—
175	C₄⋊C₄	—	4	2	no	—
176	C₄²	—	2	2	no	—
177	C₄²	—	2	2	no	—
178	C₄²	—	4	2	no	—
179	C₄²	Q₈×C₅₃	1	4	yes	—
180	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	C₂×Q₈×C₅₃	1	2	yes	—
208	C₂×C₄	C₂×Q₈×C₅₃	1	2	yes	—
209	C₂×C₄	C₂×Q₈×C₅₃	1	2	yes	—
210	C₂×C₄	#41×C₅₃	1	2	yes	—
211	C₂×C₄	#41×C₅₃	1	2	yes	—
212	C₂×C₄	#41×C₅₃	1	2	yes	—
213	C₂×C₄	#41×C₅₃	1	2	yes	—
214	C₂²	C₂²×Q₈×C₅₃	1	1	yes	characteristic
215	C₂²	C₂×#41×C₅₃	1	1	yes	characteristic
216	C₂²	C₂×#41×C₅₃	1	1	yes	characteristic
217	C₂²	C₂×#41×C₅₃	1	1	yes	characteristic
218	C₂²	C₂×#41×C₅₃	1	1	yes	characteristic
219	C₂²	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
220	C₂²	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
221	C₄	—	2	2	no	—
222	C₄	—	2	2	no	—
223	C₄	—	2	2	no	—
224	C₄	—	2	2	no	—
225	C₄	—	4	1	no	—
226	C₄	—	4	1	no	—
227	C₄	—	4	2	no	—
228	C₄	—	4	2	no	—
229	C₄	—	4	2	no	—
230	C₄	—	4	2	no	—
231	C₄	#125×C₅₃	1	4	yes	—
232	C₂	C₂×#125×C₅₃	1	1	yes	characteristic
233	C₂	C₅₃×#533	1	1	yes	characteristic
234	C₂	C₅₃×#544	1	1	yes	characteristic
235	C₂	C₅₃×#552	1	1	yes	characteristic
236	C₂	C₅₃×#553	1	1	yes	characteristic
237	C₂	C₅₃×#556	1	1	yes	characteristic
238	C₂	C₅₃×#559	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₅₃
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
112. #2570	C₅₃	direct (N × H)
131. C₅₃	#2570	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₅₃×#400	2
C₅₃×#398	2
C₅₃×#389	2
C₅₃×#387	2
C₅₃×#384	2
C₅₃×#382	2
C₄×C₄⋊C₄×C₅₃	2
#2570	53

Subgroups — C53×#2570 #20271507

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2570 #20271507