Subgroups — C₅₃×#17788 #27225300

Order 13568.

734 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2988	C₂	1	2	yes	maximal
2	C₄×#131×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1565	C₂	1	6	yes	maximal, complemented
4	C₅₃×#1564	C₂	1	3	yes	maximal, complemented
5	C₂×C₅₃×#334	C₂	1	3	yes	maximal, complemented
6	C₂×#131×C₅₃	C₄	1	1	yes	characteristic
7	C₂×#131×C₅₃	C₄	1	1	yes	characteristic
8	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
9	C₈×Q₈×C₅₃	—	2	2	no	—
10	C₈×D₄×C₅₃	—	2	6	no	—
11	C₅₃×#431	C₂²	1	6	yes	—
12	C₂×C₄⋊C₈×C₅₃	C₂²	1	6	yes	—
13	C₅₃×#405	C₄	1	3	yes	—
14	C₅₃×#405	C₄	1	3	yes	—
15	C₄×M₁₆×C₅₃	C₂²	1	3	yes	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	3	yes	—
17	C₅₃×#335	C₂²	1	6	yes	—
18	C₅₃×#334	C₂²	1	3	yes	—
19	C₅₃×#334	C₂²	1	6	yes	—
20	C₅₃×#334	C₂²	1	1	yes	characteristic
21	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
22	#131×C₅₃	—	2	4	no	—
23	#131×C₅₃	C₈	1	8	yes	complemented
24	C₂×M₁₆×C₅₃	C₂×C₄	1	3	yes	—
25	C₂×M₁₆×C₅₃	C₂×C₄	1	3	yes	—
26	C₂²×C₈×C₅₃	—	2	6	no	—
27	C₂²×C₈×C₅₃	C₂×C₄	1	3	yes	—
28	C₂²×C₈×C₅₃	C₂×C₄	1	3	yes	—
29	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
30	C₄×Q₈×C₅₃	Q₈	1	1	yes	characteristic
31	C₄×D₄×C₅₃	D₄	1	3	yes	—
32	C₄×D₄×C₅₃	Q₈	1	3	yes	—
33	#117×C₅₃	C₂×C₄	1	3	yes	—
34	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
35	C₄⋊C₈×C₅₃	—	2	6	no	—
36	C₄⋊C₈×C₅₃	C₂³	1	6	yes	—
37	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
38	C₂²⋊C₈×C₅₃	—	2	6	no	—
39	#97×C₅₃	C₂×C₄	1	6	yes	—
40	C₄×C₈×C₅₃	—	2	6	no	—
41	C₄×C₈×C₅₃	C₂×C₄	1	3	yes	—
42	C₄×C₈×C₅₃	C₂×C₄	1	3	yes	—
43	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
44	C₄×C₈×C₅₃	C₂³	1	3	yes	—
45	#41×C₅₃	C₂×C₈	1	4	yes	—
46	#41×C₅₃	M₁₆	1	4	yes	—
47	C₂×Q₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂×D₄×C₅₃	C₄⋊C₄	1	3	yes	—
49	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
50	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
51	C₂²×C₄×C₅₃	C₂×Q₈	1	3	yes	—
52	M₁₆×C₅₃	—	2	6	no	—
53	M₁₆×C₅₃	C₂×C₈	1	12	yes	complemented
54	C₂×C₈×C₅₃	—	2	6	no	—
55	C₂×C₈×C₅₃	—	2	2	no	—
56	C₂×C₈×C₅₃	—	2	6	no	—
57	C₂×C₈×C₅₃	—	8	6	no	—
58	C₂×C₈×C₅₃	C₂×C₈	1	12	yes	complemented
59	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
61	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
63	C₄⋊C₄×C₅₃	C₄⋊C₄	1	3	yes	—
64	C₄⋊C₄×C₅₃	C₄⋊C₄	1	3	yes	—
65	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	6	yes	—
66	C₄²×C₅₃	C₂²×C₄	1	3	yes	—
67	C₄²×C₅₃	C₂×D₄	1	3	yes	—
68	C₄²×C₅₃	C₂×Q₈	1	3	yes	—
69	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
70	C₂³×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
71	Q₈×C₅₃	C₄⋊C₈	1	4	yes	—
72	D₄×C₅₃	C₄⋊C₈	1	12	yes	—
73	C₂×C₄×C₅₃	—	2	6	no	—
74	C₂×C₄×C₅₃	—	4	3	no	—
75	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
76	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
77	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
78	C₂×C₄×C₅₃	C₂²×C₈	1	6	yes	—
79	C₂×C₄×C₅₃	C₂×M₁₆	1	6	yes	—
80	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
83	C₈×C₅₃	—	2	1	no	—
84	C₈×C₅₃	—	2	3	no	—
85	C₈×C₅₃	—	4	2	no	—
86	C₈×C₅₃	—	4	6	no	—
87	C₈×C₅₃	C₂²×C₈	1	2	yes	—
88	C₈×C₅₃	C₂²×C₈	1	6	yes	—
89	#17788	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂²×C₅₃	—	2	3	no	—
91	C₂²×C₅₃	—	2	3	no	—
92	C₂²×C₅₃	C₂×C₄⋊C₈	1	6	yes	—
93	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
94	C₄×C₅₃	—	2	3	no	—
95	C₄×C₅₃	—	2	1	no	—
96	C₄×C₅₃	—	2	3	no	—
97	C₄×C₅₃	C₂×C₄⋊C₈	1	6	yes	—
98	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
99	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
100	C₄×C₅₃	#577	1	2	yes	—
101	#2988	C₂×C₅₃	1	2	yes	—
102	C₄×#131	C₂×C₅₃	1	1	yes	characteristic
103	#1565	C₂×C₅₃	1	6	yes	complemented
104	#1564	C₂×C₅₃	1	3	yes	complemented
105	C₂×#334	C₂×C₅₃	1	3	yes	complemented
106	C₂×C₅₃	—	2	6	no	—
107	C₂×C₅₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
108	C₂×C₅₃	#2912	1	1	yes	characteristic
109	C₂×C₅₃	#2916	1	1	yes	characteristic
110	C₂×#131	C₄×C₅₃	1	1	yes	characteristic
111	C₂×#131	C₄×C₅₃	1	1	yes	characteristic
112	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
113	C₈×Q₈	—	2	2	no	—
114	C₈×D₄	—	2	6	no	—
115	#431	C₂²×C₅₃	1	6	yes	—
116	C₂×C₄⋊C₈	C₂²×C₅₃	1	6	yes	—
117	#405	C₄×C₅₃	1	3	yes	—
118	#405	C₄×C₅₃	1	3	yes	—
119	C₄×M₁₆	C₂²×C₅₃	1	3	yes	—
120	C₂×C₄×C₈	C₂²×C₅₃	1	3	yes	—
121	#335	C₂²×C₅₃	1	6	yes	—
122	#334	C₂²×C₅₃	1	3	yes	—
123	#334	C₂²×C₅₃	1	6	yes	—
124	#334	C₂²×C₅₃	1	1	yes	characteristic
125	C₅₃	#17788	1	1	yes	characteristic, direct product factor, Sylow
126	C₂×#41	C₂×C₄×C₅₃	1	1	yes	characteristic
127	#131	—	2	4	no	—
128	#131	C₈×C₅₃	1	8	yes	complemented
129	C₂×M₁₆	C₂×C₄×C₅₃	1	3	yes	—
130	C₂×M₁₆	C₂×C₄×C₅₃	1	3	yes	—
131	C₂²×C₈	—	2	6	no	—
132	C₂²×C₈	C₂×C₄×C₅₃	1	3	yes	—
133	C₂²×C₈	C₂×C₄×C₅₃	1	3	yes	—
134	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
135	C₄×Q₈	Q₈×C₅₃	1	1	yes	characteristic
136	C₄×D₄	D₄×C₅₃	1	3	yes	—
137	C₄×D₄	Q₈×C₅₃	1	3	yes	—
138	#117	C₂×C₄×C₅₃	1	3	yes	—
139	C₂×C₄²	C₂³×C₅₃	1	3	yes	—
140	C₄⋊C₈	—	2	6	no	—
141	C₄⋊C₈	C₂³×C₅₃	1	6	yes	—
142	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
143	C₂²⋊C₈	—	2	6	no	—
144	#97	C₂×C₄×C₅₃	1	6	yes	—
145	C₄×C₈	—	2	6	no	—
146	C₄×C₈	C₂×C₄×C₅₃	1	3	yes	—
147	C₄×C₈	C₂×C₄×C₅₃	1	3	yes	—
148	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
149	C₄×C₈	C₂³×C₅₃	1	3	yes	—
150	#41	C₂×C₈×C₅₃	1	4	yes	—
151	#41	M₁₆×C₅₃	1	4	yes	—
152	C₂×Q₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
153	C₂×D₄	C₄⋊C₄×C₅₃	1	3	yes	—
154	C₂²×C₄	C₂²×C₄×C₅₃	1	3	yes	—
155	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
156	C₂²×C₄	C₂×Q₈×C₅₃	1	3	yes	—
157	M₁₆	—	2	6	no	—
158	M₁₆	C₂×C₈×C₅₃	1	12	yes	complemented
159	C₂×C₈	—	2	6	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	6	no	—
162	C₂×C₈	—	8	6	no	—
163	C₂×C₈	C₂×C₈×C₅₃	1	12	yes	complemented
164	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
165	C₂×C₈	C₂²×C₄×C₅₃	1	3	yes	—
166	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
167	C₂×C₈	C₂²×C₄×C₅₃	1	3	yes	—
168	C₄⋊C₄	C₄⋊C₄×C₅₃	1	3	yes	—
169	C₄⋊C₄	C₄⋊C₄×C₅₃	1	3	yes	—
170	C₂²⋊C₄	C₄⋊C₄×C₅₃	1	6	yes	—
171	C₄²	C₂²×C₄×C₅₃	1	3	yes	—
172	C₄²	C₂×D₄×C₅₃	1	3	yes	—
173	C₄²	C₂×Q₈×C₅₃	1	3	yes	—
174	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
175	C₂³	C₂×C₄⋊C₄×C₅₃	1	3	yes	—
176	Q₈	C₄⋊C₈×C₅₃	1	4	yes	—
177	D₄	C₄⋊C₈×C₅₃	1	12	yes	—
178	C₂×C₄	—	2	6	no	—
179	C₂×C₄	—	4	3	no	—
180	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	3	yes	—
181	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	3	yes	—
182	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	3	yes	—
183	C₂×C₄	C₂²×C₈×C₅₃	1	6	yes	—
184	C₂×C₄	C₂×M₁₆×C₅₃	1	6	yes	—
185	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
186	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
187	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
188	C₈	—	2	1	no	—
189	C₈	—	2	3	no	—
190	C₈	—	4	2	no	—
191	C₈	—	4	6	no	—
192	C₈	C₂²×C₈×C₅₃	1	2	yes	—
193	C₈	C₂²×C₈×C₅₃	1	6	yes	—
194	C₂²	—	2	3	no	—
195	C₂²	—	2	3	no	—
196	C₂²	C₂×C₄⋊C₈×C₅₃	1	6	yes	—
197	C₂²	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
198	C₄	—	2	3	no	—
199	C₄	—	2	1	no	—
200	C₄	—	2	3	no	—
201	C₄	C₂×C₄⋊C₈×C₅₃	1	6	yes	—
202	C₄	C₂³×C₈×C₅₃	1	1	yes	characteristic
203	C₄	C₂²×M₁₆×C₅₃	1	1	yes	characteristic
204	C₄	C₅₃×#577	1	2	yes	—
205	C₂	—	2	6	no	—
206	C₂	C₂²×C₄⋊C₈×C₅₃	1	1	yes	characteristic
207	C₂	C₅₃×#2912	1	1	yes	characteristic
208	C₂	C₅₃×#2916	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂²×D₄
commutator	C₄	C₂³×C₈×C₅₃
Frattini, cosocle	C₄²	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₅₃	C₂×C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₄×#41×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1565	C₂	semidirect
4. C₅₃×#1564	C₂	semidirect
5. C₂×C₅₃×#334	C₂	semidirect
23. #131×C₅₃	C₈	semidirect
53. M₁₆×C₅₃	C₂×C₈	semidirect
58. C₂×C₈×C₅₃	C₂×C₈	semidirect
89. #17788	C₅₃	direct (N × H)
103. #1565	C₂×C₅₃	semidirect
104. #1564	C₂×C₅₃	semidirect
105. C₂×#334	C₂×C₅₃	semidirect
125. C₅₃	#17788	direct (N × H)
128. #131	C₈×C₅₃	semidirect
158. M₁₆	C₂×C₈×C₅₃	semidirect
163. C₂×C₈	C₂×C₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#2988	2
C₄×#131×C₅₃	2
C₅₃×#1565	2
C₅₃×#1564	2
C₂×C₅₃×#334	2
#17788	53

Subgroups — C53×#17788 #27225300

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#17788 #27225300