Subgroups — C₅₃×#2495 #20271432

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×#400	C₂	1	2	yes	maximal, complemented
2	C₅₃×#397	C₂	1	2	yes	maximal
3	C₅₃×#389	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#389	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#388	C₂	1	2	yes	maximal
6	C₅₃×#383	C₂	1	2	yes	maximal, complemented
7	C₅₃×#382	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#380	C₂	1	1	yes	maximal, characteristic
9	C₅₃×#380	C₂	1	1	yes	maximal, characteristic
10	C₄×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₂²⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
12	C₂³×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	2	yes	—
18	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
23	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
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	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	2	yes	—
42	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
43	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
46	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
47	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
49	C₄⋊C₄×C₅₃	—	2	2	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	—	4	2	no	—
54	C₄⋊C₄×C₅₃	—	4	2	no	—
55	C₂²⋊C₄×C₅₃	—	2	2	no	—
56	C₂²⋊C₄×C₅₃	—	2	4	no	—
57	C₂²⋊C₄×C₅₃	—	2	2	no	—
58	C₂²⋊C₄×C₅₃	—	4	2	no	—
59	C₄²×C₅₃	—	2	2	no	—
60	C₄²×C₅₃	—	2	4	no	—
61	C₄²×C₅₃	—	2	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	2	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	2	no	—
80	C₂×C₄×C₅₃	—	2	2	no	—
81	C₂×C₄×C₅₃	—	2	1	no	—
82	C₂×C₄×C₅₃	—	2	1	no	—
83	C₂×C₄×C₅₃	—	2	2	no	—
84	C₂×C₄×C₅₃	—	2	1	no	—
85	C₂×C₄×C₅₃	—	2	2	no	—
86	C₂×C₄×C₅₃	—	2	2	no	—
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	—	2	2	no	—
89	C₂×C₄×C₅₃	—	2	2	no	—
90	C₂×C₄×C₅₃	—	2	2	no	—
91	C₂×C₄×C₅₃	—	2	1	no	—
92	C₂×C₄×C₅₃	—	2	2	no	—
93	C₂×C₄×C₅₃	—	2	1	no	—
94	C₂×C₄×C₅₃	—	2	2	no	—
95	C₂×C₄×C₅₃	—	2	1	no	—
96	C₂×C₄×C₅₃	—	4	4	no	—
97	C₂×C₄×C₅₃	#41	1	2	yes	—
98	C₂×C₄×C₅₃	#41	1	2	yes	—
99	C₂×C₄×C₅₃	#41	1	2	yes	—
100	C₂×C₄×C₅₃	#41	1	2	yes	—
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	2	no	—
118	C₄×C₅₃	—	4	2	no	—
119	C₄×C₅₃	—	4	1	no	—
120	C₄×C₅₃	—	4	1	no	—
121	C₄×C₅₃	—	4	2	no	—
122	C₄×C₅₃	—	4	1	no	—
123	C₄×C₅₃	—	4	2	no	—
124	C₄×C₅₃	#126	1	4	yes	—
125	#2495	C₅₃	1	1	yes	maximal, characteristic, direct product factor, 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	#552	1	1	yes	characteristic
132	C₂×C₅₃	#553	1	1	yes	characteristic
133	C₂×C₅₃	#555	1	1	yes	characteristic
134	#400	C₂×C₅₃	1	2	yes	complemented
135	#397	C₂×C₅₃	1	2	yes	—
136	#389	C₂×C₅₃	1	1	yes	characteristic, complemented
137	#389	C₂×C₅₃	1	1	yes	characteristic, complemented
138	#388	C₂×C₅₃	1	2	yes	—
139	#383	C₂×C₅₃	1	2	yes	complemented
140	#382	C₂×C₅₃	1	1	yes	characteristic, complemented
141	#380	C₂×C₅₃	1	1	yes	characteristic
142	#380	C₂×C₅₃	1	1	yes	characteristic
143	C₄×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic, complemented
144	C₄×C₂²⋊C₄	C₂×C₅₃	1	1	yes	characteristic
145	C₅₃	#2495	1	1	yes	characteristic, direct product factor, Sylow
146	C₂³×C₄	C₂²×C₅₃	1	1	yes	characteristic
147	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
148	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
149	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
150	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
151	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
152	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
153	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
154	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
155	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
156	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
157	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
158	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
159	C₂×C₄²	C₂²×C₅₃	1	2	yes	—
160	#95	C₂²×C₅₃	1	1	yes	characteristic
161	#95	C₂²×C₅₃	1	1	yes	characteristic
162	#95	C₂²×C₅₃	1	2	yes	—
163	#95	C₂²×C₅₃	1	2	yes	—
164	#95	C₂²×C₅₃	1	2	yes	—
165	#95	C₂²×C₅₃	1	2	yes	—
166	#95	C₂²×C₅₃	1	2	yes	—
167	#95	C₂²×C₅₃	1	2	yes	—
168	#95	C₂²×C₅₃	1	2	yes	—
169	C₂⁴	C₂³×C₅₃	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	C₂³×C₅₃	1	1	yes	characteristic
174	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
175	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
176	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
177	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
178	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
179	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
180	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
181	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
182	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	2	2	no	—
186	C₄⋊C₄	—	2	2	no	—
187	C₄⋊C₄	—	4	2	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₂²⋊C₄	—	2	2	no	—
190	C₂²⋊C₄	—	2	4	no	—
191	C₂²⋊C₄	—	2	2	no	—
192	C₂²⋊C₄	—	4	2	no	—
193	C₄²	—	2	2	no	—
194	C₄²	—	2	4	no	—
195	C₄²	—	2	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	#41×C₅₃	1	2	yes	—
203	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	#41×C₅₃	1	2	yes	—
232	C₂×C₄	#41×C₅₃	1	2	yes	—
233	C₂×C₄	#41×C₅₃	1	2	yes	—
234	C₂×C₄	#41×C₅₃	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×#41×C₅₃	1	1	yes	characteristic
243	C₂²	C₂×#41×C₅₃	1	1	yes	characteristic
244	C₂²	C₂×#41×C₅₃	1	1	yes	characteristic
245	C₂²	C₂×#41×C₅₃	1	1	yes	characteristic
246	C₂²	C₂×#41×C₅₃	1	1	yes	characteristic
247	C₂²	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
248	C₂²	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
249	C₄	—	2	2	no	—
250	C₄	—	4	1	no	—
251	C₄	—	4	2	no	—
252	C₄	—	4	2	no	—
253	C₄	—	4	1	no	—
254	C₄	—	4	1	no	—
255	C₄	—	4	2	no	—
256	C₄	—	4	1	no	—
257	C₄	—	4	2	no	—
258	C₄	#126×C₅₃	1	4	yes	—
259	C₂	—	4	2	no	—
260	C₂	C₂×#126×C₅₃	1	1	yes	characteristic
261	C₂	C₅₃×#529	1	1	yes	characteristic
262	C₂	C₅₃×#542	1	1	yes	characteristic
263	C₂	C₅₃×#552	1	1	yes	characteristic
264	C₂	C₅₃×#552	1	1	yes	characteristic
265	C₂	C₅₃×#553	1	1	yes	characteristic
266	C₂	C₅₃×#555	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₅₃
prime radical	C₂⁴×C₅₃	C₂³
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
1. C₅₃×#400	C₂	semidirect
3. C₅₃×#389	C₂	semidirect
4. C₅₃×#389	C₂	semidirect
6. C₅₃×#383	C₂	semidirect
7. C₅₃×#382	C₂	semidirect
10. C₄×C₄⋊C₄×C₅₃	C₂	semidirect
125. #2495	C₅₃	direct (N × H)
134. #400	C₂×C₅₃	semidirect
136. #389	C₂×C₅₃	semidirect
137. #389	C₂×C₅₃	semidirect
139. #383	C₂×C₅₃	semidirect
140. #382	C₂×C₅₃	semidirect
143. C₄×C₄⋊C₄	C₂×C₅₃	semidirect
145. C₅₃	#2495	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×#397	2
C₅₃×#389	2
C₅₃×#388	2
C₅₃×#383	2
C₅₃×#382	2
C₅₃×#380	2
C₄×C₄⋊C₄×C₅₃	2
C₄×C₂²⋊C₄×C₅₃	2
#2495	53

Subgroups — C53×#2495 #20271432

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2495 #20271432