Subgroups — C₅₃×#20451 #27227963

Order 13568.

1142 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	C₂×C₅₃×#427	C₂	1	2	yes	maximal
2	C₂²×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₄×#117×C₅₃	C₂	1	2	yes	maximal
4	C₅₃×#1776	C₂	1	8	yes	maximal, complemented
5	C₅₃×#1735	C₂	1	2	yes	maximal
6	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×#117×C₅₃	C₂²	1	4	yes	—
9	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
10	C₅₃×#427	C₄	1	16	yes	complemented
11	C₂×#107×C₅₃	C₂²	1	4	yes	—
12	C₂×#106×C₅₃	C₂²	1	4	yes	—
13	C₂×#97×C₅₃	C₄	1	8	yes	complemented
14	C₂×#97×C₅₃	C₂²	1	4	yes	—
15	C₄×C₄⋊C₄×C₅₃	C₂²	1	8	yes	—
16	C₄×C₂²⋊C₄×C₅₃	—	2	4	no	—
17	C₅₃×#376	—	2	2	no	—
18	C₄³×C₅₃	—	2	2	no	—
19	C₅₃×#340	C₂²	1	8	yes	—
20	C₂³×C₄×C₅₃	D₄	1	2	yes	—
21	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
22	C₂²×C₈×C₅₃	—	2	4	no	—
23	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
24	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₅₃	C₂×C₄	1	8	yes	—
26	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
27	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
28	#117×C₅₃	—	2	8	no	—
29	#117×C₅₃	C₂×C₄	1	16	yes	—
30	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
31	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
32	C₂×C₄²×C₅₃	—	2	8	no	—
33	C₂×C₄²×C₅₃	—	2	4	no	—
34	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
35	C₂×C₄²×C₅₃	D₄	1	2	yes	—
36	C₂×C₄²×C₅₃	Q₈	1	4	yes	—
37	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
38	#107×C₅₃	C₂×C₄	1	16	yes	complemented
39	#106×C₅₃	C₂×C₄	1	16	yes	complemented
40	#97×C₅₃	C₂×C₄	1	16	yes	—
41	#95×C₅₃	—	2	8	no	—
42	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	—	2	8	no	—
44	C₂²×C₄×C₅₃	—	4	4	no	—
45	C₂²×C₄×C₅₃	C₄⋊C₄	1	8	yes	—
46	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
47	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
48	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
49	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
51	C₂²×C₄×C₅₃	#41	1	4	yes	—
52	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
53	C₂×C₈×C₅₃	—	2	8	no	—
54	C₂×C₈×C₅₃	—	2	4	no	—
55	C₂×C₈×C₅₃	C₄²	1	16	yes	complemented
56	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
57	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
58	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
59	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
60	C₄⋊C₄×C₅₃	—	2	8	no	—
61	C₄⋊C₄×C₅₃	C₂²×C₄	1	16	yes	—
62	C₂²⋊C₄×C₅₃	—	2	16	no	—
63	C₄²×C₅₃	—	2	16	no	—
64	C₄²×C₅₃	—	4	16	no	—
65	C₄²×C₅₃	C₄⋊C₄	1	8	yes	—
66	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
67	C₄²×C₅₃	C₂×Q₈	1	4	yes	—
68	C₂³×C₅₃	—	2	4	no	—
69	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
70	C₂³×C₅₃	C₄×D₄	1	4	yes	—
71	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
72	C₂×C₄×C₅₃	—	2	8	no	—
73	C₂×C₄×C₅₃	—	2	16	no	—
74	C₂×C₄×C₅₃	—	4	8	no	—
75	C₂×C₄×C₅₃	—	4	16	no	—
76	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
77	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	8	yes	—
78	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
79	C₂×C₄×C₅₃	#117	1	4	yes	—
80	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
81	C₂×C₄×C₅₃	C₄×Q₈	1	8	yes	—
82	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
84	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
86	C₈×C₅₃	—	2	4	no	—
87	C₈×C₅₃	C₂×C₄²	1	8	yes	—
88	#20451	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂²×C₅₃	—	2	4	no	—
90	C₂²×C₅₃	—	2	8	no	—
91	C₂²×C₅₃	C₄×C₄⋊C₄	1	4	yes	—
92	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
93	C₂²×C₅₃	C₂×C₄×D₄	1	2	yes	—
94	C₂²×C₅₃	#575	1	4	yes	—
95	C₄×C₅₃	—	2	8	no	—
96	C₄×C₅₃	—	4	16	no	—
97	C₄×C₅₃	C₄×C₄⋊C₄	1	4	yes	—
98	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
99	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
100	C₄×C₅₃	C₂×C₄×Q₈	1	2	yes	—
101	C₂×#427	C₂×C₅₃	1	2	yes	—
102	C₂²×#97	C₂×C₅₃	1	1	yes	characteristic
103	C₄×#117	C₂×C₅₃	1	2	yes	—
104	#1776	C₂×C₅₃	1	8	yes	complemented
105	#1735	C₂×C₅₃	1	2	yes	—
106	C₂×C₅₃	—	2	4	no	—
107	C₂×C₅₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₅₃	#2911	1	2	yes	—
109	C₂×C₅₃	#2943	1	4	yes	—
110	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
111	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
112	C₂×#117	C₂²×C₅₃	1	4	yes	—
113	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
114	#427	C₄×C₅₃	1	16	yes	complemented
115	C₂×#107	C₂²×C₅₃	1	4	yes	—
116	C₂×#106	C₂²×C₅₃	1	4	yes	—
117	C₂×#97	C₄×C₅₃	1	8	yes	complemented
118	C₂×#97	C₂²×C₅₃	1	4	yes	—
119	C₄×C₄⋊C₄	C₂²×C₅₃	1	8	yes	—
120	C₄×C₂²⋊C₄	—	2	4	no	—
121	#376	—	2	2	no	—
122	C₄³	—	2	2	no	—
123	#340	C₂²×C₅₃	1	8	yes	—
124	C₅₃	#20451	1	1	yes	characteristic, direct product factor, Sylow
125	C₂³×C₄	D₄×C₅₃	1	2	yes	—
126	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
127	C₂²×C₈	—	2	4	no	—
128	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
129	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
130	C₂²×C₈	C₂×C₄×C₅₃	1	8	yes	—
131	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
132	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
133	#117	—	2	8	no	—
134	#117	C₂×C₄×C₅₃	1	16	yes	—
135	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
136	C₂×C₂²⋊C₄	—	2	4	no	—
137	C₂×C₄²	—	2	4	no	—
138	C₂×C₄²	—	2	8	no	—
139	C₂×C₄²	C₂×C₄×C₅₃	1	4	yes	—
140	C₂×C₄²	D₄×C₅₃	1	2	yes	—
141	C₂×C₄²	Q₈×C₅₃	1	4	yes	—
142	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
143	#107	C₂×C₄×C₅₃	1	16	yes	complemented
144	#106	C₂×C₄×C₅₃	1	16	yes	complemented
145	#97	C₂×C₄×C₅₃	1	16	yes	—
146	#95	—	2	8	no	—
147	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
148	C₂²×C₄	—	2	8	no	—
149	C₂²×C₄	—	4	4	no	—
150	C₂²×C₄	C₄⋊C₄×C₅₃	1	8	yes	—
151	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
152	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
153	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
154	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
155	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
156	C₂²×C₄	#41×C₅₃	1	4	yes	—
157	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
158	C₂×C₈	—	2	8	no	—
159	C₂×C₈	—	2	4	no	—
160	C₂×C₈	C₄²×C₅₃	1	16	yes	complemented
161	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
162	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
163	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
164	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
165	C₄⋊C₄	—	2	8	no	—
166	C₄⋊C₄	C₂²×C₄×C₅₃	1	16	yes	—
167	C₂²⋊C₄	—	2	16	no	—
168	C₄²	—	2	16	no	—
169	C₄²	—	4	16	no	—
170	C₄²	C₄⋊C₄×C₅₃	1	8	yes	—
171	C₄²	C₂²×C₄×C₅₃	1	4	yes	—
172	C₄²	C₂×Q₈×C₅₃	1	4	yes	—
173	C₂³	—	2	4	no	—
174	C₂³	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
175	C₂³	C₄×D₄×C₅₃	1	4	yes	—
176	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
177	C₂×C₄	—	2	8	no	—
178	C₂×C₄	—	2	16	no	—
179	C₂×C₄	—	4	8	no	—
180	C₂×C₄	—	4	16	no	—
181	C₂×C₄	C₂×C₄²×C₅₃	1	4	yes	—
182	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	8	yes	—
183	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
184	C₂×C₄	#117×C₅₃	1	4	yes	—
185	C₂×C₄	C₄×D₄×C₅₃	1	4	yes	—
186	C₂×C₄	C₄×Q₈×C₅₃	1	8	yes	—
187	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
188	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
189	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
190	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
191	C₈	—	2	4	no	—
192	C₈	C₂×C₄²×C₅₃	1	8	yes	—
193	C₂²	—	2	8	no	—
194	C₂²	—	2	4	no	—
195	C₂²	C₄×C₄⋊C₄×C₅₃	1	4	yes	—
196	C₂²	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
197	C₂²	C₂×C₄×D₄×C₅₃	1	2	yes	—
198	C₂²	C₅₃×#575	1	4	yes	—
199	C₄	—	2	8	no	—
200	C₄	—	4	16	no	—
201	C₄	C₄×C₄⋊C₄×C₅₃	1	4	yes	—
202	C₄	C₂²×C₄²×C₅₃	1	1	yes	characteristic
203	C₄	C₂×#117×C₅₃	1	1	yes	characteristic
204	C₄	C₂×C₄×Q₈×C₅₃	1	2	yes	—
205	C₂	—	2	4	no	—
206	C₂	C₂×C₄×C₄⋊C₄×C₅₃	1	1	yes	characteristic
207	C₂	C₅₃×#2911	1	2	yes	—
208	C₂	C₅₃×#2943	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₅₃×#1776	C₂	semidirect
10. C₅₃×#427	C₄	semidirect
13. C₂×#97×C₅₃	C₄	semidirect
38. #107×C₅₃	C₂×C₄	semidirect
39. #106×C₅₃	C₂×C₄	semidirect
55. C₂×C₈×C₅₃	C₄²	semidirect
88. #20451	C₅₃	direct (N × H)
104. #1776	C₂×C₅₃	semidirect
114. #427	C₄×C₅₃	semidirect
117. C₂×#97	C₄×C₅₃	semidirect
124. C₅₃	#20451	direct (N × H)
143. #107	C₂×C₄×C₅₃	semidirect
144. #106	C₂×C₄×C₅₃	semidirect
160. C₂×C₈	C₄²×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#427	2
C₂²×#97×C₅₃	2
C₄×#117×C₅₃	2
C₅₃×#1776	2
C₅₃×#1735	2
#20451	53

Subgroups — C53×#20451 #27227963

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#20451 #27227963