Subgroups — C₅₃×#13284 #27220796

Order 13568.

374 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	C₂×C₅₃×#499	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#346	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1845	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1380	C₂	1	4	yes	maximal
5	C₂×#125×C₅₃	—	2	1	no	—
6	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
7	C₅₃×#499	—	2	2	no	—
8	C₅₃×#499	C₄	1	4	yes	complemented
9	C₂×#107×C₅₃	—	2	1	no	—
10	C₂×#107×C₅₃	C₂²	1	2	yes	—
11	C₂×#107×C₅₃	C₂²	1	2	yes	—
12	C₂×#106×C₅₃	—	2	1	no	—
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#384	—	2	1	no	—
15	C₅₃×#346	—	2	2	no	—
16	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
18	#125×C₅₃	—	2	4	no	—
19	#125×C₅₃	—	4	2	no	—
20	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
21	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
22	C₂×C₄⋊C₄×C₅₃	—	4	2	no	—
23	C₂×C₄⋊C₄×C₅₃	—	4	2	no	—
24	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
25	C₂×C₁₆×C₅₃	—	2	2	no	—
26	C₂×C₁₆×C₅₃	—	2	2	no	—
27	C₂×C₁₆×C₅₃	Q₈	1	4	yes	—
28	#107×C₅₃	—	2	2	no	—
29	#107×C₅₃	—	2	4	no	—
30	#107×C₅₃	—	4	2	no	—
31	#107×C₅₃	C₂×C₄	1	4	yes	—
32	#106×C₅₃	—	2	4	no	—
33	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₅₃	D₄	1	2	yes	—
35	C₂²×C₄×C₅₃	—	4	2	no	—
36	C₂²×C₄×C₅₃	—	4	2	no	—
37	C₂²×C₄×C₅₃	D₈	1	2	yes	—
38	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
40	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
41	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
42	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
45	C₂×C₈×C₅₃	#41	1	2	yes	—
46	C₄⋊C₄×C₅₃	—	2	4	no	—
47	C₄⋊C₄×C₅₃	—	4	8	no	—
48	C₄⋊C₄×C₅₃	—	4	2	no	—
49	C₄⋊C₄×C₅₃	—	4	2	no	—
50	C₄⋊C₄×C₅₃	—	4	2	no	—
51	C₄⋊C₄×C₅₃	—	8	4	no	—
52	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
53	C₄²×C₅₃	SD₁₆	1	2	yes	—
54	C₁₆×C₅₃	—	2	4	no	—
55	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
56	C₂×C₄×C₅₃	—	2	2	no	—
57	C₂×C₄×C₅₃	—	2	2	no	—
58	C₂×C₄×C₅₃	—	4	4	no	—
59	C₂×C₄×C₅₃	—	8	2	no	—
60	C₂×C₄×C₅₃	—	8	2	no	—
61	C₂×C₄×C₅₃	—	8	2	no	—
62	C₂×C₄×C₅₃	—	8	4	no	—
63	C₂×C₄×C₅₃	#102	1	4	yes	—
64	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
70	C₈×C₅₃	#103	1	4	yes	—
71	C₈×C₅₃	C₄×Q₈	1	2	yes	—
72	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
73	#13284	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
75	C₂²×C₅₃	#486	1	1	yes	characteristic
76	C₂²×C₅₃	#500	1	1	yes	characteristic
77	C₂²×C₅₃	#509	1	2	yes	—
78	C₂²×C₅₃	#510	1	2	yes	—
79	C₄×C₅₃	—	2	4	no	—
80	C₄×C₅₃	—	8	4	no	—
81	C₄×C₅₃	—	16	2	no	—
82	C₄×C₅₃	#391	1	1	yes	characteristic
83	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
84	C₄×C₅₃	#487	1	1	yes	characteristic
85	C₄×C₅₃	#498	1	1	yes	characteristic
86	C₂×#499	C₂×C₅₃	1	1	yes	characteristic
87	C₂×#346	C₂×C₅₃	1	1	yes	characteristic
88	#1845	C₂×C₅₃	1	1	yes	characteristic
89	#1380	C₂×C₅₃	1	4	yes	—
90	C₂×C₅₃	#1990	1	1	yes	characteristic
91	C₂×C₅₃	#2139	1	2	yes	—
92	C₂×C₅₃	#2244	1	1	yes	characteristic
93	C₂×C₅₃	#2245	1	1	yes	characteristic
94	C₂×C₅₃	#2255	1	2	yes	—
95	C₂×#125	—	2	1	no	—
96	C₂²×C₁₆	C₂²×C₅₃	1	2	yes	—
97	#499	—	2	2	no	—
98	#499	C₄×C₅₃	1	4	yes	complemented
99	C₂×#107	—	2	1	no	—
100	C₂×#107	C₂²×C₅₃	1	2	yes	—
101	C₂×#107	C₂²×C₅₃	1	2	yes	—
102	C₂×#106	—	2	1	no	—
103	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
104	#384	—	2	1	no	—
105	#346	—	2	2	no	—
106	C₅₃	#13284	1	1	yes	characteristic, direct product factor, Sylow
107	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
108	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
109	#125	—	2	4	no	—
110	#125	—	4	2	no	—
111	C₂×C₄⋊C₄	—	2	2	no	—
112	C₂×C₄⋊C₄	—	2	2	no	—
113	C₂×C₄⋊C₄	—	4	2	no	—
114	C₂×C₄⋊C₄	—	4	2	no	—
115	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
116	C₂×C₁₆	—	2	2	no	—
117	C₂×C₁₆	—	2	2	no	—
118	C₂×C₁₆	Q₈×C₅₃	1	4	yes	—
119	#107	—	2	2	no	—
120	#107	—	2	4	no	—
121	#107	—	4	2	no	—
122	#107	C₂×C₄×C₅₃	1	4	yes	—
123	#106	—	2	4	no	—
124	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
125	C₄×C₈	D₄×C₅₃	1	2	yes	—
126	C₂²×C₄	—	4	2	no	—
127	C₂²×C₄	—	4	2	no	—
128	C₂²×C₄	D₈×C₅₃	1	2	yes	—
129	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
130	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
131	C₂×C₈	SD₁₆×C₅₃	1	2	yes	—
132	C₂×C₈	Q₁₆×C₅₃	1	2	yes	—
133	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
134	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
135	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
136	C₂×C₈	#41×C₅₃	1	2	yes	—
137	C₄⋊C₄	—	2	4	no	—
138	C₄⋊C₄	—	4	8	no	—
139	C₄⋊C₄	—	4	2	no	—
140	C₄⋊C₄	—	4	2	no	—
141	C₄⋊C₄	—	4	2	no	—
142	C₄⋊C₄	—	8	4	no	—
143	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
144	C₄²	SD₁₆×C₅₃	1	2	yes	—
145	C₁₆	—	2	4	no	—
146	C₂³	C₂×D₈×C₅₃	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	4	4	no	—
150	C₂×C₄	—	8	2	no	—
151	C₂×C₄	—	8	2	no	—
152	C₂×C₄	—	8	2	no	—
153	C₂×C₄	—	8	4	no	—
154	C₂×C₄	#102×C₅₃	1	4	yes	—
155	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
156	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
157	C₂×C₄	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
158	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
159	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
160	C₂×C₄	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
161	C₈	#103×C₅₃	1	4	yes	—
162	C₈	C₄×Q₈×C₅₃	1	2	yes	—
163	C₈	C₂²⋊Q₈×C₅₃	1	2	yes	—
164	C₂²	C₂×#102×C₅₃	1	1	yes	characteristic
165	C₂²	C₅₃×#486	1	1	yes	characteristic
166	C₂²	C₅₃×#500	1	1	yes	characteristic
167	C₂²	C₅₃×#509	1	2	yes	—
168	C₂²	C₅₃×#510	1	2	yes	—
169	C₄	—	2	4	no	—
170	C₄	—	8	4	no	—
171	C₄	—	16	2	no	—
172	C₄	C₅₃×#391	1	1	yes	characteristic
173	C₄	C₂×#103×C₅₃	1	1	yes	characteristic
174	C₄	C₅₃×#487	1	1	yes	characteristic
175	C₄	C₅₃×#498	1	1	yes	characteristic
176	C₂	C₅₃×#1990	1	1	yes	characteristic
177	C₂	C₅₃×#2139	1	2	yes	—
178	C₂	C₅₃×#2244	1	1	yes	characteristic
179	C₂	C₅₃×#2245	1	1	yes	characteristic
180	C₂	C₅₃×#2255	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₅₃×#499	C₄	semidirect
73. #13284	C₅₃	direct (N × H)
98. #499	C₄×C₅₃	semidirect
106. C₅₃	#13284	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#499	2
C₂×C₅₃×#346	2
C₅₃×#1845	2
C₅₃×#1380	2
#13284	53

Subgroups — C53×#13284 #27220796

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#13284 #27220796