Subgroups — C₅₃×#20827 #27228339

Order 13568.

758 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2920	C₂	1	2	yes	maximal
2	C₂×C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1849	C₂	1	2	yes	maximal, complemented
4	C₅₃×#1838	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1811	C₂	1	4	yes	maximal
6	C₄×#108×C₅₃	C₂	1	4	yes	maximal, complemented
7	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
8	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
9	C₅₃×#432	—	2	4	no	—
10	C₅₃×#431	—	2	4	no	—
11	C₂×#108×C₅₃	C₂²	1	8	yes	—
12	C₅₃×#423	C₂²	1	4	yes	—
13	C₅₃×#407	C₂²	1	4	yes	—
14	C₄×M₁₆×C₅₃	C₄	1	8	yes	—
15	C₄×M₁₆×C₅₃	C₂²	1	4	yes	—
16	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
19	C₅₃×#341	C₂²	1	8	yes	—
20	C₂³×C₄×C₅₃	Q₈	1	2	yes	—
21	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
22	C₂×M₁₆×C₅₃	—	2	8	no	—
23	C₂×M₁₆×C₅₃	C₂×C₄	1	8	yes	—
24	C₂×M₁₆×C₅₃	C₂³	1	8	yes	—
25	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
26	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
27	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₅₃	D₄	1	2	yes	—
29	C₂×C₄²×C₅₃	D₄	1	2	yes	—
30	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
31	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
33	#108×C₅₃	—	2	16	no	—
34	C₄⋊C₈×C₅₃	—	2	8	no	—
35	C₂²⋊C₈×C₅₃	—	2	8	no	—
36	#97×C₅₃	—	2	4	no	—
37	#97×C₅₃	C₂×C₄	1	4	yes	—
38	#97×C₅₃	C₂×C₄	1	4	yes	—
39	C₄×C₈×C₅₃	—	2	4	no	—
40	C₄×C₈×C₅₃	C₂×C₄	1	8	yes	—
41	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	—	2	4	no	—
43	C₂²×C₄×C₅₃	—	4	4	no	—
44	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
45	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
46	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
47	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
50	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
51	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
52	M₁₆×C₅₃	—	2	16	no	—
53	M₁₆×C₅₃	—	2	16	no	—
54	C₂×C₈×C₅₃	—	2	4	no	—
55	C₂×C₈×C₅₃	—	2	8	no	—
56	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
57	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
58	C₂×C₈×C₅₃	#41	1	8	yes	—
59	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
60	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
61	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
62	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
63	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₅₃	C₂×D₄	1	2	yes	—
65	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₂³×C₅₃	—	2	2	no	—
67	C₂³×C₅₃	—	4	2	no	—
68	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
69	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
70	C₂×C₄×C₅₃	—	2	2	no	—
71	C₂×C₄×C₅₃	—	2	4	no	—
72	C₂×C₄×C₅₃	—	2	4	no	—
73	C₂×C₄×C₅₃	—	2	8	no	—
74	C₂×C₄×C₅₃	—	2	4	no	—
75	C₂×C₄×C₅₃	—	4	4	no	—
76	C₂×C₄×C₅₃	—	4	2	no	—
77	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
78	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
79	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
80	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
81	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
84	C₈×C₅₃	—	2	4	no	—
85	C₈×C₅₃	—	4	8	no	—
86	C₈×C₅₃	#117	1	8	yes	—
87	#20827	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂²×C₅₃	—	2	2	no	—
89	C₂²×C₅₃	—	2	4	no	—
90	C₂²×C₅₃	—	2	2	no	—
91	C₂²×C₅₃	—	4	4	no	—
92	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
93	C₂²×C₅₃	#533	1	2	yes	—
94	C₄×C₅₃	—	2	4	no	—
95	C₄×C₅₃	—	2	2	no	—
96	C₄×C₅₃	—	2	2	no	—
97	C₄×C₅₃	#430	1	4	yes	—
98	C₄×C₅₃	C₂×#117	1	2	yes	—
99	C₄×C₅₃	#532	1	2	yes	—
100	#2920	C₂×C₅₃	1	2	yes	—
101	C₂×C₄×M₁₆	C₂×C₅₃	1	1	yes	characteristic
102	#1849	C₂×C₅₃	1	2	yes	complemented
103	#1838	C₂×C₅₃	1	2	yes	complemented
104	#1811	C₂×C₅₃	1	4	yes	—
105	C₄×#108	C₂×C₅₃	1	4	yes	complemented
106	C₂×C₅₃	—	2	2	no	—
107	C₂×C₅₃	—	4	2	no	—
108	C₂×C₅₃	#2285	1	1	yes	characteristic
109	C₂×C₅₃	C₂×#430	1	2	yes	—
110	C₂²×M₁₆	C₂²×C₅₃	1	2	yes	—
111	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
112	#432	—	2	4	no	—
113	#431	—	2	4	no	—
114	C₂×#108	C₂²×C₅₃	1	8	yes	—
115	#423	C₂²×C₅₃	1	4	yes	—
116	#407	C₂²×C₅₃	1	4	yes	—
117	C₄×M₁₆	C₄×C₅₃	1	8	yes	—
118	C₄×M₁₆	C₂²×C₅₃	1	4	yes	—
119	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
120	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
121	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
122	#341	C₂²×C₅₃	1	8	yes	—
123	C₅₃	#20827	1	1	yes	characteristic, direct product factor, Sylow
124	C₂³×C₄	Q₈×C₅₃	1	2	yes	—
125	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
126	C₂×M₁₆	—	2	8	no	—
127	C₂×M₁₆	C₂×C₄×C₅₃	1	8	yes	—
128	C₂×M₁₆	C₂³×C₅₃	1	8	yes	—
129	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
130	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
131	C₂×C₄²	C₂×C₄×C₅₃	1	4	yes	—
132	C₂×C₄²	D₄×C₅₃	1	2	yes	—
133	C₂×C₄²	D₄×C₅₃	1	2	yes	—
134	C₂×C₄²	Q₈×C₅₃	1	2	yes	—
135	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
136	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
137	#108	—	2	16	no	—
138	C₄⋊C₈	—	2	8	no	—
139	C₂²⋊C₈	—	2	8	no	—
140	#97	—	2	4	no	—
141	#97	C₂×C₄×C₅₃	1	4	yes	—
142	#97	C₂×C₄×C₅₃	1	4	yes	—
143	C₄×C₈	—	2	4	no	—
144	C₄×C₈	C₂×C₄×C₅₃	1	8	yes	—
145	C₂⁴	C₂×Q₈×C₅₃	1	1	yes	characteristic
146	C₂²×C₄	—	2	4	no	—
147	C₂²×C₄	—	4	4	no	—
148	C₂²×C₄	C₄⋊C₄×C₅₃	1	4	yes	—
149	C₂²×C₄	C₄⋊C₄×C₅₃	1	4	yes	—
150	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
151	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
152	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
153	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
154	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
155	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
156	M₁₆	—	2	16	no	—
157	M₁₆	—	2	16	no	—
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	—	2	8	no	—
160	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
161	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
162	C₂×C₈	#41×C₅₃	1	8	yes	—
163	C₄²	C₄⋊C₄×C₅₃	1	4	yes	—
164	C₄²	C₄⋊C₄×C₅₃	1	4	yes	—
165	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
166	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
167	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
168	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
169	C₄²	C₂×D₄×C₅₃	1	2	yes	—
170	C₂³	—	2	2	no	—
171	C₂³	—	4	2	no	—
172	C₂³	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
173	C₂³	C₂²×Q₈×C₅₃	1	1	yes	characteristic
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	8	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	—	4	4	no	—
181	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
182	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
183	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
184	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
185	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
186	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
187	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
188	C₈	—	2	4	no	—
189	C₈	—	4	8	no	—
190	C₈	#117×C₅₃	1	8	yes	—
191	C₂²	—	2	2	no	—
192	C₂²	—	2	2	no	—
193	C₂²	—	2	4	no	—
194	C₂²	—	4	4	no	—
195	C₂²	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
196	C₂²	C₅₃×#533	1	2	yes	—
197	C₄	—	2	2	no	—
198	C₄	—	2	2	no	—
199	C₄	—	2	4	no	—
200	C₄	C₅₃×#430	1	4	yes	—
201	C₄	C₂×#117×C₅₃	1	2	yes	—
202	C₄	C₅₃×#532	1	2	yes	—
203	C₂	—	2	2	no	—
204	C₂	—	4	2	no	—
205	C₂	C₅₃×#2285	1	1	yes	characteristic
206	C₂	C₂×C₅₃×#430	1	2	yes	—

	subgroup	G/H
center	C₄²×C₅₃	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1849	C₂	semidirect
4. C₅₃×#1838	C₂	semidirect
6. C₄×#108×C₅₃	C₂	semidirect
87. #20827	C₅₃	direct (N × H)
102. #1849	C₂×C₅₃	semidirect
103. #1838	C₂×C₅₃	semidirect
105. C₄×#108	C₂×C₅₃	semidirect
123. C₅₃	#20827	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×#2920	2
C₂×C₄×M₁₆×C₅₃	2
C₅₃×#1849	2
C₅₃×#1838	2
C₅₃×#1811	2
C₄×#108×C₅₃	2
#20827	53

Subgroups — C53×#20827 #27228339

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#20827 #27228339