Subgroups — C₅₃×#22258 #27229770

Order 13568.

578 conjugacy classes of subgroups.

The subgroup lattice has 182 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×#2961	C₂	1	2	yes	maximal
3	C₅₃×#1995	C₂	1	2	yes	maximal
4	C₅₃×#1922	C₂	1	8	yes	maximal
5	C₅₃×#1837	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#1746	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#533	C₂²	1	2	yes	—
8	C₅₃×#500	C₄	1	2	yes	complemented
9	C₅₃×#499	C₄	1	4	yes	complemented
10	C₅₃×#498	C₄	1	2	yes	complemented
11	C₅₃×#446	—	2	8	no	—
12	C₅₃×#427	C₂²	1	4	yes	—
13	C₅₃×#424	C₂²	1	4	yes	—
14	C₅₃×#423	C₂²	1	2	yes	—
15	C₅₃×#405	C₂²	1	4	yes	—
16	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#376	C₂²	1	1	yes	characteristic
19	C₅₃×#341	C₂²	1	4	yes	—
20	C₅₃×#339	C₂²	1	8	yes	—
21	C₅₃×#336	C₂²	1	4	yes	—
22	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
23	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
24	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
25	C₄⋊Q₈×C₅₃	C₂×C₄	1	4	yes	—
26	#125×C₅₃	C₂×C₄	1	4	yes	—
27	C₂²⋊Q₈×C₅₃	—	2	4	no	—
28	C₄×Q₈×C₅₃	—	2	4	no	—
29	#117×C₅₃	C₂³	1	4	yes	—
30	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
31	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
32	#107×C₅₃	C₂×C₄	1	8	yes	—
33	#106×C₅₃	C₂×C₄	1	8	yes	—
34	C₄⋊C₈×C₅₃	—	2	8	no	—
35	C₄⋊C₈×C₅₃	—	2	4	no	—
36	#97×C₅₃	—	2	4	no	—
37	#97×C₅₃	D₄	1	4	yes	—
38	C₄×C₈×C₅₃	—	2	4	no	—
39	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	#95×C₅₃	—	2	2	no	—
43	C₂×Q₈×C₅₃	—	2	4	no	—
44	C₂²×C₄×C₅₃	—	2	2	no	—
45	C₂²×C₄×C₅₃	#41	1	2	yes	—
46	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
47	M₁₆×C₅₃	—	2	8	no	—
48	C₂×C₈×C₅₃	—	2	4	no	—
49	C₂×C₈×C₅₃	—	2	4	no	—
50	C₂×C₈×C₅₃	—	2	4	no	—
51	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
52	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
53	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
54	C₄⋊C₄×C₅₃	—	2	8	no	—
55	C₄⋊C₄×C₅₃	—	2	4	no	—
56	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
57	C₂²⋊C₄×C₅₃	—	2	4	no	—
58	C₄²×C₅₃	—	2	4	no	—
59	C₄²×C₅₃	—	2	4	no	—
60	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₄²×C₅₃	C₂×D₄	1	2	yes	—
63	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
64	Q₈×C₅₃	—	8	4	no	—
65	C₂×C₄×C₅₃	—	2	8	no	—
66	C₂×C₄×C₅₃	—	4	2	no	—
67	C₂×C₄×C₅₃	—	4	4	no	—
68	C₂×C₄×C₅₃	—	4	2	no	—
69	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
70	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
71	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
73	C₂×C₄×C₅₃	2₋¹⁺⁴	1	4	yes	—
74	C₈×C₅₃	—	2	2	no	—
75	C₈×C₅₃	—	2	2	no	—
76	C₈×C₅₃	—	2	4	no	—
77	C₈×C₅₃	—	4	4	no	—
78	#22258	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₂²×C₅₃	—	2	2	no	—
80	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
81	C₂²×C₅₃	#541	1	2	yes	—
82	C₄×C₅₃	—	2	2	no	—
83	C₄×C₅₃	—	2	2	no	—
84	C₄×C₅₃	—	4	8	no	—
85	C₄×C₅₃	—	8	2	no	—
86	C₄×C₅₃	#519	1	2	yes	—
87	C₄×C₅₃	#536	1	2	yes	—
88	#3160	C₂×C₅₃	1	1	yes	characteristic
89	#2961	C₂×C₅₃	1	2	yes	—
90	#1995	C₂×C₅₃	1	2	yes	—
91	#1922	C₂×C₅₃	1	8	yes	—
92	#1837	C₂×C₅₃	1	1	yes	characteristic
93	#1746	C₂×C₅₃	1	1	yes	characteristic
94	C₂×C₅₃	—	2	2	no	—
95	C₂×C₅₃	#2327	1	1	yes	characteristic
96	C₂×C₅₃	#2957	1	2	yes	—
97	#533	C₂²×C₅₃	1	2	yes	—
98	#500	C₄×C₅₃	1	2	yes	complemented
99	#499	C₄×C₅₃	1	4	yes	complemented
100	#498	C₄×C₅₃	1	2	yes	complemented
101	#446	—	2	8	no	—
102	#427	C₂²×C₅₃	1	4	yes	—
103	#424	C₂²×C₅₃	1	4	yes	—
104	#423	C₂²×C₅₃	1	2	yes	—
105	#405	C₂²×C₅₃	1	4	yes	—
106	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
107	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
108	#376	C₂²×C₅₃	1	1	yes	characteristic
109	#341	C₂²×C₅₃	1	4	yes	—
110	#339	C₂²×C₅₃	1	8	yes	—
111	#336	C₂²×C₅₃	1	4	yes	—
112	C₅₃	#22258	1	1	yes	characteristic, direct product factor, Sylow
113	C₂×M₁₆	C₂³×C₅₃	1	4	yes	—
114	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
115	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
116	C₄⋊Q₈	C₂×C₄×C₅₃	1	4	yes	—
117	#125	C₂×C₄×C₅₃	1	4	yes	—
118	C₂²⋊Q₈	—	2	4	no	—
119	C₄×Q₈	—	2	4	no	—
120	#117	C₂³×C₅₃	1	4	yes	—
121	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
122	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
123	#107	C₂×C₄×C₅₃	1	8	yes	—
124	#106	C₂×C₄×C₅₃	1	8	yes	—
125	C₄⋊C₈	—	2	8	no	—
126	C₄⋊C₈	—	2	4	no	—
127	#97	—	2	4	no	—
128	#97	D₄×C₅₃	1	4	yes	—
129	C₄×C₈	—	2	4	no	—
130	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
131	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
132	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
133	#95	—	2	2	no	—
134	C₂×Q₈	—	2	4	no	—
135	C₂²×C₄	—	2	2	no	—
136	C₂²×C₄	#41×C₅₃	1	2	yes	—
137	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
138	M₁₆	—	2	8	no	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	—	2	4	no	—
141	C₂×C₈	—	2	4	no	—
142	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
143	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
144	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
145	C₄⋊C₄	—	2	8	no	—
146	C₄⋊C₄	—	2	4	no	—
147	C₄⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
148	C₂²⋊C₄	—	2	4	no	—
149	C₄²	—	2	4	no	—
150	C₄²	—	2	4	no	—
151	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
152	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
153	C₄²	C₂×D₄×C₅₃	1	2	yes	—
154	C₂³	C₂×#41×C₅₃	1	1	yes	characteristic
155	Q₈	—	8	4	no	—
156	C₂×C₄	—	2	8	no	—
157	C₂×C₄	—	4	2	no	—
158	C₂×C₄	—	4	4	no	—
159	C₂×C₄	—	4	2	no	—
160	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
161	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
162	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
163	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
164	C₂×C₄	2₋¹⁺⁴×C₅₃	1	4	yes	—
165	C₈	—	2	2	no	—
166	C₈	—	2	2	no	—
167	C₈	—	2	4	no	—
168	C₈	—	4	4	no	—
169	C₂²	—	2	2	no	—
170	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
171	C₂²	C₅₃×#541	1	2	yes	—
172	C₄	—	2	2	no	—
173	C₄	—	2	2	no	—
174	C₄	—	4	8	no	—
175	C₄	—	8	2	no	—
176	C₄	C₅₃×#519	1	2	yes	—
177	C₄	C₅₃×#536	1	2	yes	—
178	C₂	—	2	2	no	—
179	C₂	C₅₃×#2327	1	1	yes	characteristic
180	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	C₂³×C₅₃	C₂×#41
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
8. C₅₃×#500	C₄	semidirect
9. C₅₃×#499	C₄	semidirect
10. C₅₃×#498	C₄	semidirect
78. #22258	C₅₃	direct (N × H)
98. #500	C₄×C₅₃	semidirect
99. #499	C₄×C₅₃	semidirect
100. #498	C₄×C₅₃	semidirect
112. C₅₃	#22258	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×#3160	2
C₅₃×#2961	2
C₅₃×#1995	2
C₅₃×#1922	2
C₅₃×#1837	2
C₅₃×#1746	2
#22258	53

Subgroups — C53×#22258 #27229770

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22258 #27229770