Subgroups — C₅₃×#21734 #27229246

Order 13568.

678 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3160	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2960	C₂	1	2	yes	maximal
3	C₅₃×#1994	C₂	1	2	yes	maximal
4	C₅₃×#1946	C₂	1	4	yes	maximal
5	C₅₃×#1848	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#1773	C₂	1	4	yes	maximal
7	C₄×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
8	C₅₃×#533	C₂²	1	2	yes	—
9	C₅₃×#500	C₄	1	2	yes	complemented
10	C₅₃×#499	C₄	1	4	yes	complemented
11	C₅₃×#498	C₄	1	2	yes	complemented
12	C₅₃×#446	—	2	4	no	—
13	C₈×Q₈×C₅₃	—	2	4	no	—
14	C₂×#108×C₅₃	C₂²	1	4	yes	—
15	C₅₃×#427	C₂²	1	4	yes	—
16	C₅₃×#424	C₂²	1	4	yes	—
17	C₅₃×#423	C₂²	1	2	yes	—
18	C₅₃×#405	C₂²	1	4	yes	—
19	C₂×#97×C₅₃	C₂²	1	2	yes	—
20	C₂×#97×C₅₃	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₄³×C₅₃	C₂²	1	1	yes	characteristic
24	C₅₃×#339	C₂²	1	8	yes	—
25	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
26	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
28	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
29	C₄⋊Q₈×C₅₃	C₂×C₄	1	4	yes	—
30	#125×C₅₃	C₂×C₄	1	4	yes	—
31	C₂²⋊Q₈×C₅₃	—	2	4	no	—
32	C₄×Q₈×C₅₃	—	2	4	no	—
33	#117×C₅₃	C₂³	1	4	yes	—
34	C₂×C₄²×C₅₃	—	2	2	no	—
35	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
36	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
37	#108×C₅₃	—	2	8	no	—
38	#107×C₅₃	C₂×C₄	1	8	yes	—
39	#106×C₅₃	C₂×C₄	1	8	yes	—
40	C₄⋊C₈×C₅₃	—	2	8	no	—
41	C₄⋊C₈×C₅₃	—	2	4	no	—
42	#97×C₅₃	—	2	4	no	—
43	#97×C₅₃	—	2	4	no	—
44	#97×C₅₃	Q₈	1	8	yes	complemented
45	C₄×C₈×C₅₃	—	2	4	no	—
46	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₅₃	D₄	1	4	yes	—
50	C₂×Q₈×C₅₃	—	2	4	no	—
51	C₂²×C₄×C₅₃	—	2	2	no	—
52	C₂²×C₄×C₅₃	#41	1	2	yes	—
53	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
54	M₁₆×C₅₃	—	2	8	no	—
55	C₂×C₈×C₅₃	—	2	2	no	—
56	C₂×C₈×C₅₃	—	2	4	no	—
57	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
58	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
59	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
60	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
62	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
63	C₂×C₈×C₅₃	#41	1	4	yes	—
64	C₄⋊C₄×C₅₃	—	2	8	no	—
65	C₄⋊C₄×C₅₃	—	2	4	no	—
66	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
67	C₂²⋊C₄×C₅₃	—	2	4	no	—
68	C₄²×C₅₃	—	2	4	no	—
69	C₄²×C₅₃	—	2	2	no	—
70	C₄²×C₅₃	—	4	4	no	—
71	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
72	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₄²×C₅₃	C₂×D₄	1	2	yes	—
74	C₄²×C₅₃	C₂×Q₈	1	4	yes	—
75	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
76	Q₈×C₅₃	—	4	8	no	—
77	C₂×C₄×C₅₃	—	2	4	no	—
78	C₂×C₄×C₅₃	—	2	8	no	—
79	C₂×C₄×C₅₃	—	2	4	no	—
80	C₂×C₄×C₅₃	—	4	4	no	—
81	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
82	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
83	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
86	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
87	C₈×C₅₃	—	2	2	no	—
88	C₈×C₅₃	—	4	4	no	—
89	C₈×C₅₃	C₄×Q₈	1	4	yes	—
90	C₈×C₅₃	C₄×Q₈	1	4	yes	—
91	C₈×C₅₃	C₄⋊Q₈	1	4	yes	—
92	#21734	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂²×C₅₃	—	2	2	no	—
94	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
95	C₂²×C₅₃	#533	1	2	yes	—
96	C₄×C₅₃	—	2	2	no	—
97	C₄×C₅₃	—	4	4	no	—
98	C₄×C₅₃	—	4	8	no	—
99	C₄×C₅₃	#444	1	4	yes	—
100	C₄×C₅₃	C₂×C₄×Q₈	1	2	yes	—
101	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
102	C₄×C₅₃	#532	1	1	yes	characteristic
103	#3160	C₂×C₅₃	1	1	yes	characteristic
104	#2960	C₂×C₅₃	1	2	yes	—
105	#1994	C₂×C₅₃	1	2	yes	—
106	#1946	C₂×C₅₃	1	4	yes	—
107	#1848	C₂×C₅₃	1	1	yes	characteristic
108	#1773	C₂×C₅₃	1	4	yes	—
109	C₄×#97	C₂×C₅₃	1	1	yes	characteristic
110	C₂×C₅₃	—	2	2	no	—
111	C₂×C₅₃	C₄×C₄⋊Q₈	1	1	yes	characteristic
112	C₂×C₅₃	C₂×#444	1	2	yes	—
113	#533	C₂²×C₅₃	1	2	yes	—
114	#500	C₄×C₅₃	1	2	yes	complemented
115	#499	C₄×C₅₃	1	4	yes	complemented
116	#498	C₄×C₅₃	1	2	yes	complemented
117	#446	—	2	4	no	—
118	C₈×Q₈	—	2	4	no	—
119	C₂×#108	C₂²×C₅₃	1	4	yes	—
120	#427	C₂²×C₅₃	1	4	yes	—
121	#424	C₂²×C₅₃	1	4	yes	—
122	#423	C₂²×C₅₃	1	2	yes	—
123	#405	C₂²×C₅₃	1	4	yes	—
124	C₂×#97	C₂²×C₅₃	1	2	yes	—
125	C₂×#97	C₂²×C₅₃	1	2	yes	—
126	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
127	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
128	C₄³	C₂²×C₅₃	1	1	yes	characteristic
129	#339	C₂²×C₅₃	1	8	yes	—
130	C₅₃	#21734	1	1	yes	characteristic, direct product factor, Sylow
131	C₂×M₁₆	C₂³×C₅₃	1	4	yes	—
132	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
133	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
134	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
135	C₄⋊Q₈	C₂×C₄×C₅₃	1	4	yes	—
136	#125	C₂×C₄×C₅₃	1	4	yes	—
137	C₂²⋊Q₈	—	2	4	no	—
138	C₄×Q₈	—	2	4	no	—
139	#117	C₂³×C₅₃	1	4	yes	—
140	C₂×C₄²	—	2	2	no	—
141	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
142	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
143	#108	—	2	8	no	—
144	#107	C₂×C₄×C₅₃	1	8	yes	—
145	#106	C₂×C₄×C₅₃	1	8	yes	—
146	C₄⋊C₈	—	2	8	no	—
147	C₄⋊C₈	—	2	4	no	—
148	#97	—	2	4	no	—
149	#97	—	2	4	no	—
150	#97	Q₈×C₅₃	1	8	yes	complemented
151	C₄×C₈	—	2	4	no	—
152	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
153	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
154	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
155	C₄×C₈	D₄×C₅₃	1	4	yes	—
156	C₂×Q₈	—	2	4	no	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	#41×C₅₃	1	2	yes	—
159	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
160	M₁₆	—	2	8	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	2	4	no	—
163	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
164	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
165	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
166	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
167	C₂×C₈	C₂×Q₈×C₅₃	1	4	yes	—
168	C₂×C₈	C₂×Q₈×C₅₃	1	4	yes	—
169	C₂×C₈	#41×C₅₃	1	4	yes	—
170	C₄⋊C₄	—	2	8	no	—
171	C₄⋊C₄	—	2	4	no	—
172	C₄⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
173	C₂²⋊C₄	—	2	4	no	—
174	C₄²	—	2	4	no	—
175	C₄²	—	2	2	no	—
176	C₄²	—	4	4	no	—
177	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
178	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
179	C₄²	C₂×D₄×C₅₃	1	2	yes	—
180	C₄²	C₂×Q₈×C₅₃	1	4	yes	—
181	C₂³	C₂×#41×C₅₃	1	1	yes	characteristic
182	Q₈	—	4	8	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	8	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	4	4	no	—
187	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
188	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
189	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
190	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
191	C₂×C₄	C₂²×Q₈×C₅₃	1	2	yes	—
192	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
193	C₈	—	2	2	no	—
194	C₈	—	4	4	no	—
195	C₈	C₄×Q₈×C₅₃	1	4	yes	—
196	C₈	C₄×Q₈×C₅₃	1	4	yes	—
197	C₈	C₄⋊Q₈×C₅₃	1	4	yes	—
198	C₂²	—	2	2	no	—
199	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
200	C₂²	C₅₃×#533	1	2	yes	—
201	C₄	—	2	2	no	—
202	C₄	—	4	4	no	—
203	C₄	—	4	8	no	—
204	C₄	C₅₃×#444	1	4	yes	—
205	C₄	C₂×C₄×Q₈×C₅₃	1	2	yes	—
206	C₄	C₂×C₄⋊Q₈×C₅₃	1	1	yes	characteristic
207	C₄	C₅₃×#532	1	1	yes	characteristic
208	C₂	—	2	2	no	—
209	C₂	C₄×C₄⋊Q₈×C₅₃	1	1	yes	characteristic
210	C₂	C₂×C₅₃×#444	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	C₂³×C₅₃	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
9. C₅₃×#500	C₄	semidirect
10. C₅₃×#499	C₄	semidirect
11. C₅₃×#498	C₄	semidirect
44. #97×C₅₃	Q₈	semidirect
92. #21734	C₅₃	direct (N × H)
114. #500	C₄×C₅₃	semidirect
115. #499	C₄×C₅₃	semidirect
116. #498	C₄×C₅₃	semidirect
130. C₅₃	#21734	direct (N × H)
150. #97	Q₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3160	2
C₅₃×#2960	2
C₅₃×#1994	2
C₅₃×#1946	2
C₅₃×#1848	2
C₅₃×#1773	2
C₄×#97×C₅₃	2
#21734	53

Subgroups — C53×#21734 #27229246

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#21734 #27229246