Subgroups — C₅₃×#8814 #27216326

Order 13568.

454 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#1845	C₂	1	2	yes	maximal
2	C₅₃×#1844	C₂	1	2	yes	maximal
3	C₅₃×#1834	C₂	1	2	yes	maximal
4	C₂×C₈²×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#107×C₅₃	—	2	4	no	—
6	C₂×#106×C₅₃	—	2	4	no	—
7	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₅₃×#384	C₂²	1	4	yes	—
11	C₅₃×#340	—	2	4	no	—
12	C₈²×C₅₃	C₄	1	4	yes	complemented
13	C₂²×C₈×C₅₃	D₄	1	2	yes	—
14	C₂²×C₈×C₅₃	D₄	1	2	yes	—
15	C₂²×C₈×C₅₃	D₄	1	2	yes	—
16	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
17	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
18	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
19	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
20	#107×C₅₃	—	4	8	no	—
21	#106×C₅₃	—	4	8	no	—
22	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
24	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₅₃	Q₈	1	2	yes	—
26	C₄×C₈×C₅₃	Q₈	1	2	yes	—
27	C₄×C₈×C₅₃	Q₈	1	2	yes	—
28	C₂²×C₄×C₅₃	—	4	4	no	—
29	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
32	C₂×C₈×C₅₃	—	2	4	no	—
33	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
34	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
35	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
36	C₂×C₈×C₅₃	D₈	1	4	yes	—
37	C₂×C₈×C₅₃	SD₁₆	1	8	yes	—
38	C₂×C₈×C₅₃	Q₁₆	1	4	yes	—
39	C₄⋊C₄×C₅₃	—	4	8	no	—
40	C₄⋊C₄×C₅₃	—	4	8	no	—
41	C₄⋊C₄×C₅₃	—	4	8	no	—
42	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
44	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
45	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
46	C₂³×C₅₃	#127	1	1	yes	characteristic
47	C₂×C₄×C₅₃	—	8	4	no	—
48	C₂×C₄×C₅₃	—	8	4	no	—
49	C₂×C₄×C₅₃	—	8	4	no	—
50	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
56	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
57	C₂×C₄×C₅₃	C₂×SD₁₆	1	4	yes	—
58	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
59	C₂×C₄×C₅₃	#135	1	4	yes	—
60	C₈×C₅₃	—	2	4	no	—
61	C₈×C₅₃	#106	1	8	yes	complemented
62	C₈×C₅₃	#107	1	8	yes	complemented
63	#8814	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₂²×C₅₃	#384	1	1	yes	characteristic
65	C₂²×C₅₃	#492	1	2	yes	—
66	C₂²×C₅₃	#493	1	1	yes	characteristic
67	C₂²×C₅₃	#494	1	1	yes	characteristic
68	C₂²×C₅₃	#495	1	2	yes	—
69	C₄×C₅₃	—	16	4	no	—
70	C₄×C₅₃	C₂×#106	1	2	yes	—
71	C₄×C₅₃	C₂×#107	1	2	yes	—
72	C₄×C₅₃	#427	1	2	yes	—
73	C₄×C₅₃	#498	1	2	yes	—
74	C₄×C₅₃	#499	1	2	yes	—
75	C₄×C₅₃	#500	1	2	yes	—
76	#1845	C₂×C₅₃	1	2	yes	—
77	#1844	C₂×C₅₃	1	2	yes	—
78	#1834	C₂×C₅₃	1	2	yes	—
79	C₂×C₈²	C₂×C₅₃	1	1	yes	characteristic
80	C₂×C₅₃	#1706	1	2	yes	—
81	C₂×C₅₃	#1707	1	2	yes	—
82	C₂×C₅₃	#1844	1	1	yes	characteristic
83	C₂×C₅₃	#1845	1	1	yes	characteristic
84	C₂×C₅₃	#1846	1	1	yes	characteristic
85	C₂×#107	—	2	4	no	—
86	C₂×#106	—	2	4	no	—
87	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
88	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
89	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
90	#384	C₂²×C₅₃	1	4	yes	—
91	#340	—	2	4	no	—
92	C₈²	C₄×C₅₃	1	4	yes	complemented
93	C₅₃	#8814	1	1	yes	characteristic, direct product factor, Sylow
94	C₂²×C₈	D₄×C₅₃	1	2	yes	—
95	C₂²×C₈	D₄×C₅₃	1	2	yes	—
96	C₂²×C₈	D₄×C₅₃	1	2	yes	—
97	C₂×C₄⋊C₄	—	2	4	no	—
98	C₂×C₄⋊C₄	—	2	4	no	—
99	C₂×C₄⋊C₄	—	2	4	no	—
100	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
101	#107	—	4	8	no	—
102	#106	—	4	8	no	—
103	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
104	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
105	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
106	C₄×C₈	Q₈×C₅₃	1	2	yes	—
107	C₄×C₈	Q₈×C₅₃	1	2	yes	—
108	C₄×C₈	Q₈×C₅₃	1	2	yes	—
109	C₂²×C₄	—	4	4	no	—
110	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
111	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
112	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
113	C₂×C₈	—	2	4	no	—
114	C₂×C₈	C₄⋊C₄×C₅₃	1	4	yes	—
115	C₂×C₈	C₄⋊C₄×C₅₃	1	4	yes	—
116	C₂×C₈	C₄⋊C₄×C₅₃	1	4	yes	—
117	C₂×C₈	D₈×C₅₃	1	4	yes	—
118	C₂×C₈	SD₁₆×C₅₃	1	8	yes	—
119	C₂×C₈	Q₁₆×C₅₃	1	4	yes	—
120	C₄⋊C₄	—	4	8	no	—
121	C₄⋊C₄	—	4	8	no	—
122	C₄⋊C₄	—	4	8	no	—
123	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
124	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
125	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
126	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
127	C₂³	#127×C₅₃	1	1	yes	characteristic
128	C₂×C₄	—	8	4	no	—
129	C₂×C₄	—	8	4	no	—
130	C₂×C₄	—	8	4	no	—
131	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
132	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
133	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
134	C₂×C₄	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
135	C₂×C₄	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
136	C₂×C₄	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
137	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
138	C₂×C₄	C₂×SD₁₆×C₅₃	1	4	yes	—
139	C₂×C₄	C₂×Q₁₆×C₅₃	1	2	yes	—
140	C₂×C₄	#135×C₅₃	1	4	yes	—
141	C₈	—	2	4	no	—
142	C₈	#106×C₅₃	1	8	yes	complemented
143	C₈	#107×C₅₃	1	8	yes	complemented
144	C₂²	C₅₃×#384	1	1	yes	characteristic
145	C₂²	C₅₃×#492	1	2	yes	—
146	C₂²	C₅₃×#493	1	1	yes	characteristic
147	C₂²	C₅₃×#494	1	1	yes	characteristic
148	C₂²	C₅₃×#495	1	2	yes	—
149	C₄	—	16	4	no	—
150	C₄	C₂×#106×C₅₃	1	2	yes	—
151	C₄	C₂×#107×C₅₃	1	2	yes	—
152	C₄	C₅₃×#427	1	2	yes	—
153	C₄	C₅₃×#498	1	2	yes	—
154	C₄	C₅₃×#499	1	2	yes	—
155	C₄	C₅₃×#500	1	2	yes	—
156	C₂	C₅₃×#1706	1	2	yes	—
157	C₂	C₅₃×#1707	1	2	yes	—
158	C₂	C₅₃×#1844	1	1	yes	characteristic
159	C₂	C₅₃×#1845	1	1	yes	characteristic
160	C₂	C₅₃×#1846	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	#127
commutator	C₄²	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₈²×C₅₃	C₄	semidirect
61. C₈×C₅₃	#106	semidirect
62. C₈×C₅₃	#107	semidirect
63. #8814	C₅₃	direct (N × H)
92. C₈²	C₄×C₅₃	semidirect
93. C₅₃	#8814	direct (N × H)
142. C₈	#106×C₅₃	semidirect
143. C₈	#107×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#1845	2
C₅₃×#1844	2
C₅₃×#1834	2
C₂×C₈²×C₅₃	2
#8814	53

Subgroups — C53×#8814 #27216326

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8814 #27216326