Subgroups — C₅₃×#22282 #27229794

Order 13568.

810 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3147	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2932	C₂	1	2	yes	maximal, complemented
3	C₅₃×#1963	C₂	1	2	yes	maximal, complemented
4	C₅₃×#1876	C₂	1	8	yes	maximal, complemented
5	C₅₃×#1837	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1746	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#135×C₅₃	C₂²	1	4	yes	—
8	C₂×#131×C₅₃	C₂²	1	4	yes	—
9	C₅₃×#532	C₂²	1	2	yes	—
10	C₅₃×#495	C₄	1	4	yes	complemented
11	C₅₃×#494	C₄	1	1	yes	characteristic, complemented
12	C₅₃×#493	C₄	1	1	yes	characteristic, complemented
13	C₅₃×#492	C₄	1	2	yes	complemented
14	C₅₃×#435	—	2	8	no	—
15	C₅₃×#423	C₂²	1	2	yes	—
16	C₂×#104×C₅₃	C₂²	1	8	yes	—
17	C₅₃×#408	C₂²	1	4	yes	—
18	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₅₃×#376	C₂²	1	1	yes	characteristic
21	C₅₃×#341	C₂²	1	4	yes	—
22	C₅₃×#336	C₂²	1	4	yes	—
23	C₂×#41×C₅₃	C₂³	1	4	yes	—
24	#135×C₅₃	—	4	8	no	—
25	C₂×Q₁₆×C₅₃	C₂×C₄	1	4	yes	—
26	C₂×SD₁₆×C₅₃	C₂×C₄	1	8	yes	—
27	C₂×D₈×C₅₃	C₂×C₄	1	4	yes	—
28	#131×C₅₃	—	4	8	no	—
29	C₂×M₁₆×C₅₃	—	2	4	no	—
30	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
31	C₂²×C₈×C₅₃	—	2	4	no	—
32	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
33	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
34	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
35	#127×C₅₃	C₂×C₄	1	2	yes	—
36	#124×C₅₃	C₂×C₄	1	4	yes	—
37	#121×C₅₃	—	2	4	no	—
38	C₄×D₄×C₅₃	—	2	4	no	—
39	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
40	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₅₃	—	2	4	no	—
42	#104×C₅₃	—	4	8	no	—
43	C₂²⋊C₈×C₅₃	—	2	8	no	—
44	#97×C₅₃	D₄	1	4	yes	—
45	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
48	#95×C₅₃	—	2	2	no	—
49	#41×C₅₃	—	2	8	no	—
50	#41×C₅₃	—	2	8	no	—
51	C₂×Q₈×C₅₃	C₂²×C₄	1	4	yes	—
52	C₂×D₄×C₅₃	—	2	4	no	—
53	C₂×D₄×C₅₃	—	2	4	no	—
54	C₂×D₄×C₅₃	C₂²×C₄	1	4	yes	—
55	C₂²×C₄×C₅₃	—	2	4	no	—
56	C₂²×C₄×C₅₃	—	2	2	no	—
57	C₂²×C₄×C₅₃	#41	1	2	yes	—
58	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
59	Q₁₆×C₅₃	—	4	4	no	—
60	SD₁₆×C₅₃	—	4	8	no	—
61	D₈×C₅₃	—	4	4	no	—
62	M₁₆×C₅₃	—	2	8	no	—
63	M₁₆×C₅₃	—	4	8	no	—
64	C₂×C₈×C₅₃	—	2	4	no	—
65	C₂×C₈×C₅₃	—	2	4	no	—
66	C₂×C₈×C₅₃	—	2	4	no	—
67	C₂×C₈×C₅₃	—	4	8	no	—
68	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
69	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
71	C₄⋊C₄×C₅₃	—	2	4	no	—
72	C₂²⋊C₄×C₅₃	—	2	8	no	—
73	C₄²×C₅₃	—	2	4	no	—
74	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₅₃	C₂×D₄	1	2	yes	—
77	C₂³×C₅₃	—	2	4	no	—
78	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
79	Q₈×C₅₃	—	2	8	no	—
80	D₄×C₅₃	—	2	8	no	—
81	D₄×C₅₃	—	4	8	no	—
82	D₄×C₅₃	—	8	4	no	—
83	C₂×C₄×C₅₃	—	2	4	no	—
84	C₂×C₄×C₅₃	—	4	2	no	—
85	C₂×C₄×C₅₃	—	4	4	no	—
86	C₂×C₄×C₅₃	—	4	4	no	—
87	C₂×C₄×C₅₃	—	4	4	no	—
88	C₂×C₄×C₅₃	—	4	2	no	—
89	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
90	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
91	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
93	C₂×C₄×C₅₃	H(5,𝔽₂)	1	4	yes	—
94	C₈×C₅₃	—	2	2	no	—
95	C₈×C₅₃	—	2	2	no	—
96	C₈×C₅₃	—	2	4	no	—
97	C₈×C₅₃	—	4	4	no	—
98	#22282	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂²×C₅₃	—	2	2	no	—
100	C₂²×C₅₃	—	4	4	no	—
101	C₂²×C₅₃	—	4	8	no	—
102	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
103	C₂²×C₅₃	#540	1	2	yes	—
104	C₄×C₅₃	—	2	2	no	—
105	C₄×C₅₃	—	2	2	no	—
106	C₄×C₅₃	—	4	4	no	—
107	C₄×C₅₃	—	8	2	no	—
108	C₄×C₅₃	#518	1	2	yes	—
109	C₄×C₅₃	#535	1	2	yes	—
110	#3147	C₂×C₅₃	1	1	yes	characteristic
111	#2932	C₂×C₅₃	1	2	yes	complemented
112	#1963	C₂×C₅₃	1	2	yes	complemented
113	#1876	C₂×C₅₃	1	8	yes	complemented
114	#1837	C₂×C₅₃	1	1	yes	characteristic, complemented
115	#1746	C₂×C₅₃	1	1	yes	characteristic, complemented
116	C₂×C₅₃	—	2	2	no	—
117	C₂×C₅₃	—	8	4	no	—
118	C₂×C₅₃	#2328	1	1	yes	characteristic
119	C₂×C₅₃	#2957	1	2	yes	—
120	C₂×#135	C₂²×C₅₃	1	4	yes	—
121	C₂×#131	C₂²×C₅₃	1	4	yes	—
122	#532	C₂²×C₅₃	1	2	yes	—
123	#495	C₄×C₅₃	1	4	yes	complemented
124	#494	C₄×C₅₃	1	1	yes	characteristic, complemented
125	#493	C₄×C₅₃	1	1	yes	characteristic, complemented
126	#492	C₄×C₅₃	1	2	yes	complemented
127	#435	—	2	8	no	—
128	#423	C₂²×C₅₃	1	2	yes	—
129	C₂×#104	C₂²×C₅₃	1	8	yes	—
130	#408	C₂²×C₅₃	1	4	yes	—
131	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
132	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
133	#376	C₂²×C₅₃	1	1	yes	characteristic
134	#341	C₂²×C₅₃	1	4	yes	—
135	#336	C₂²×C₅₃	1	4	yes	—
136	C₅₃	#22282	1	1	yes	characteristic, direct product factor, Sylow
137	C₂×#41	C₂³×C₅₃	1	4	yes	—
138	#135	—	4	8	no	—
139	C₂×Q₁₆	C₂×C₄×C₅₃	1	4	yes	—
140	C₂×SD₁₆	C₂×C₄×C₅₃	1	8	yes	—
141	C₂×D₈	C₂×C₄×C₅₃	1	4	yes	—
142	#131	—	4	8	no	—
143	C₂×M₁₆	—	2	4	no	—
144	C₂×M₁₆	C₂³×C₅₃	1	4	yes	—
145	C₂²×C₈	—	2	4	no	—
146	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
147	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
148	C₄⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
149	#127	C₂×C₄×C₅₃	1	2	yes	—
150	#124	C₂×C₄×C₅₃	1	4	yes	—
151	#121	—	2	4	no	—
152	C₄×D₄	—	2	4	no	—
153	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
154	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
155	C₄⋊C₈	—	2	4	no	—
156	#104	—	4	8	no	—
157	C₂²⋊C₈	—	2	8	no	—
158	#97	D₄×C₅₃	1	4	yes	—
159	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
160	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
161	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
162	#95	—	2	2	no	—
163	#41	—	2	8	no	—
164	#41	—	2	8	no	—
165	C₂×Q₈	C₂²×C₄×C₅₃	1	4	yes	—
166	C₂×D₄	—	2	4	no	—
167	C₂×D₄	—	2	4	no	—
168	C₂×D₄	C₂²×C₄×C₅₃	1	4	yes	—
169	C₂²×C₄	—	2	4	no	—
170	C₂²×C₄	—	2	2	no	—
171	C₂²×C₄	#41×C₅₃	1	2	yes	—
172	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
173	Q₁₆	—	4	4	no	—
174	SD₁₆	—	4	8	no	—
175	D₈	—	4	4	no	—
176	M₁₆	—	2	8	no	—
177	M₁₆	—	4	8	no	—
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	2	4	no	—
180	C₂×C₈	—	2	4	no	—
181	C₂×C₈	—	4	8	no	—
182	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
183	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
184	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
185	C₄⋊C₄	—	2	4	no	—
186	C₂²⋊C₄	—	2	8	no	—
187	C₄²	—	2	4	no	—
188	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
189	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
190	C₄²	C₂×D₄×C₅₃	1	2	yes	—
191	C₂³	—	2	4	no	—
192	C₂³	C₂×#41×C₅₃	1	1	yes	characteristic
193	Q₈	—	2	8	no	—
194	D₄	—	2	8	no	—
195	D₄	—	4	8	no	—
196	D₄	—	8	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	—	4	4	no	—
201	C₂×C₄	—	4	4	no	—
202	C₂×C₄	—	4	2	no	—
203	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
204	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
205	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
206	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
207	C₂×C₄	H(5,𝔽₂)×C₅₃	1	4	yes	—
208	C₈	—	2	2	no	—
209	C₈	—	2	2	no	—
210	C₈	—	2	4	no	—
211	C₈	—	4	4	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	4	8	no	—
214	C₂²	—	4	4	no	—
215	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
216	C₂²	C₅₃×#540	1	2	yes	—
217	C₄	—	2	2	no	—
218	C₄	—	2	2	no	—
219	C₄	—	4	4	no	—
220	C₄	—	8	2	no	—
221	C₄	C₅₃×#518	1	2	yes	—
222	C₄	C₅₃×#535	1	2	yes	—
223	C₂	—	2	2	no	—
224	C₂	—	8	4	no	—
225	C₂	C₅₃×#2328	1	1	yes	characteristic
226	C₂	C₅₃×#2957	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₅₃×#3147	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×#97×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2932	C₂	semidirect
3. C₅₃×#1963	C₂	semidirect
4. C₅₃×#1876	C₂	semidirect
5. C₅₃×#1837	C₂	semidirect
6. C₅₃×#1746	C₂	semidirect
10. C₅₃×#495	C₄	semidirect
11. C₅₃×#494	C₄	semidirect
12. C₅₃×#493	C₄	semidirect
13. C₅₃×#492	C₄	semidirect
98. #22282	C₅₃	direct (N × H)
111. #2932	C₂×C₅₃	semidirect
112. #1963	C₂×C₅₃	semidirect
113. #1876	C₂×C₅₃	semidirect
114. #1837	C₂×C₅₃	semidirect
115. #1746	C₂×C₅₃	semidirect
123. #495	C₄×C₅₃	semidirect
124. #494	C₄×C₅₃	semidirect
125. #493	C₄×C₅₃	semidirect
126. #492	C₄×C₅₃	semidirect
136. C₅₃	#22282	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×#3147	2
C₅₃×#2932	2
C₅₃×#1963	2
C₅₃×#1876	2
C₅₃×#1837	2
C₅₃×#1746	2
#22282	53

Subgroups — C53×#22282 #27229794

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22282 #27229794