Subgroups — C₅₃×#13170 #27220682

Order 13568.

278 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#368	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2158	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#346	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1945	C₂	1	2	yes	maximal
5	C₅₃×#1382	C₂	1	2	yes	maximal
6	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
7	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×#108×C₅₃	C₂²	1	2	yes	—
9	C₅₃×#427	C₂²	1	2	yes	—
10	C₅₃×#424	—	2	2	no	—
11	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#368	—	2	2	no	—
13	C₅₃×#367	C₄	1	2	yes	complemented
14	C₅₃×#366	C₄	1	2	yes	complemented
15	C₅₃×#346	—	2	2	no	—
16	C₅₃×#339	—	2	2	no	—
17	C₂×M₁₆×C₅₃	—	2	2	no	—
18	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
20	#117×C₅₃	—	2	2	no	—
21	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
22	C₂×C₁₆×C₅₃	—	2	1	no	—
23	C₂×C₁₆×C₅₃	—	2	1	no	—
24	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
27	C₂×C₁₆×C₅₃	Q₈	1	2	yes	—
28	C₂×C₁₆×C₅₃	Q₈	1	2	yes	—
29	#108×C₅₃	—	2	2	no	—
30	#108×C₅₃	—	2	2	no	—
31	#107×C₅₃	C₂×C₄	1	4	yes	—
32	#106×C₅₃	—	2	2	no	—
33	C₄⋊C₈×C₅₃	—	4	4	no	—
34	C₄×C₈×C₅₃	—	2	1	no	—
35	C₄×C₈×C₅₃	—	2	1	no	—
36	C₂²×C₄×C₅₃	—	2	1	no	—
37	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
38	M₁₆×C₅₃	—	4	2	no	—
39	M₁₆×C₅₃	—	4	2	no	—
40	C₂×C₈×C₅₃	—	2	1	no	—
41	C₂×C₈×C₅₃	—	2	1	no	—
42	C₂×C₈×C₅₃	—	4	2	no	—
43	C₂×C₈×C₅₃	D₈	1	2	yes	—
44	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
49	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
50	C₄⋊C₄×C₅₃	—	2	4	no	—
51	C₂²⋊C₄×C₅₃	—	4	2	no	—
52	C₄²×C₅₃	—	2	1	no	—
53	C₄²×C₅₃	—	4	2	no	—
54	C₄²×C₅₃	Q₁₆	1	2	yes	—
55	C₁₆×C₅₃	—	2	2	no	—
56	C₁₆×C₅₃	C₄⋊C₄	1	2	yes	—
57	C₁₆×C₅₃	C₄⋊C₄	1	2	yes	—
58	C₂³×C₅₃	#135	1	1	yes	characteristic
59	C₂×C₄×C₅₃	—	2	2	no	—
60	C₂×C₄×C₅₃	—	4	2	no	—
61	C₂×C₄×C₅₃	—	4	4	no	—
62	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
63	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
67	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
68	C₈×C₅₃	—	2	2	no	—
69	C₈×C₅₃	—	8	2	no	—
70	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₈×C₅₃	C₄×Q₈	1	1	yes	characteristic
72	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
73	C₈×C₅₃	#125	1	1	yes	characteristic
74	#13170	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂²×C₅₃	—	2	2	no	—
76	C₂²×C₅₃	C₄×D₈	1	1	yes	characteristic
77	C₂²×C₅₃	#467	1	1	yes	characteristic
78	C₂²×C₅₃	#499	1	1	yes	characteristic
79	C₄×C₅₃	—	4	2	no	—
80	C₄×C₅₃	—	8	4	no	—
81	C₄×C₅₃	#389	1	1	yes	characteristic
82	C₄×C₅₃	C₄×Q₁₆	1	1	yes	characteristic
83	C₄×C₅₃	#466	1	1	yes	characteristic
84	C₄×C₅₃	#500	1	1	yes	characteristic
85	C₂×#368	C₂×C₅₃	1	1	yes	characteristic
86	#2158	C₂×C₅₃	1	1	yes	characteristic
87	C₂×#346	C₂×C₅₃	1	1	yes	characteristic
88	#1945	C₂×C₅₃	1	2	yes	—
89	#1382	C₂×C₅₃	1	2	yes	—
90	C₂×C₅₃	—	2	1	no	—
91	C₂×C₅₃	—	2	1	no	—
92	C₂×C₅₃	#1942	1	1	yes	characteristic
93	C₂×C₅₃	#2179	1	2	yes	—
94	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
95	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
96	C₂×#108	C₂²×C₅₃	1	2	yes	—
97	#427	C₂²×C₅₃	1	2	yes	—
98	#424	—	2	2	no	—
99	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
100	#368	—	2	2	no	—
101	#367	C₄×C₅₃	1	2	yes	complemented
102	#366	C₄×C₅₃	1	2	yes	complemented
103	#346	—	2	2	no	—
104	#339	—	2	2	no	—
105	C₅₃	#13170	1	1	yes	characteristic, direct product factor, Sylow
106	C₂×M₁₆	—	2	2	no	—
107	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
108	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
109	#117	—	2	2	no	—
110	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
111	C₂×C₁₆	—	2	1	no	—
112	C₂×C₁₆	—	2	1	no	—
113	C₂×C₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
114	C₂×C₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
115	C₂×C₁₆	D₄×C₅₃	1	2	yes	—
116	C₂×C₁₆	Q₈×C₅₃	1	2	yes	—
117	C₂×C₁₆	Q₈×C₅₃	1	2	yes	—
118	#108	—	2	2	no	—
119	#108	—	2	2	no	—
120	#107	C₂×C₄×C₅₃	1	4	yes	—
121	#106	—	2	2	no	—
122	C₄⋊C₈	—	4	4	no	—
123	C₄×C₈	—	2	1	no	—
124	C₄×C₈	—	2	1	no	—
125	C₂²×C₄	—	2	1	no	—
126	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
127	M₁₆	—	4	2	no	—
128	M₁₆	—	4	2	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	4	2	no	—
132	C₂×C₈	D₈×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	1	yes	characteristic
136	C₂×C₈	C₂×Q₈×C₅₃	1	1	yes	characteristic
137	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
138	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
139	C₄⋊C₄	—	2	4	no	—
140	C₂²⋊C₄	—	4	2	no	—
141	C₄²	—	2	1	no	—
142	C₄²	—	4	2	no	—
143	C₄²	Q₁₆×C₅₃	1	2	yes	—
144	C₁₆	—	2	2	no	—
145	C₁₆	C₄⋊C₄×C₅₃	1	2	yes	—
146	C₁₆	C₄⋊C₄×C₅₃	1	2	yes	—
147	C₂³	#135×C₅₃	1	1	yes	characteristic
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	—	4	4	no	—
151	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
152	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
153	C₂×C₄	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
154	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
155	C₂×C₄	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
156	C₂×C₄	#135×C₅₃	1	1	yes	characteristic
157	C₈	—	2	2	no	—
158	C₈	—	8	2	no	—
159	C₈	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
160	C₈	C₄×Q₈×C₅₃	1	1	yes	characteristic
161	C₈	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
162	C₈	#125×C₅₃	1	1	yes	characteristic
163	C₂²	—	2	2	no	—
164	C₂²	C₄×D₈×C₅₃	1	1	yes	characteristic
165	C₂²	C₅₃×#467	1	1	yes	characteristic
166	C₂²	C₅₃×#499	1	1	yes	characteristic
167	C₄	—	4	2	no	—
168	C₄	—	8	4	no	—
169	C₄	C₅₃×#389	1	1	yes	characteristic
170	C₄	C₄×Q₁₆×C₅₃	1	1	yes	characteristic
171	C₄	C₅₃×#466	1	1	yes	characteristic
172	C₄	C₅₃×#500	1	1	yes	characteristic
173	C₂	—	2	1	no	—
174	C₂	—	2	1	no	—
175	C₂	C₅₃×#1942	1	1	yes	characteristic
176	C₂	C₅₃×#2179	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
socle	C₂²×C₅₃	C₄×D₈
prime radical	C₂³×C₅₃	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂²×C₈×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₅₃×#367	C₄	semidirect
14. C₅₃×#366	C₄	semidirect
74. #13170	C₅₃	direct (N × H)
101. #367	C₄×C₅₃	semidirect
102. #366	C₄×C₅₃	semidirect
105. C₅₃	#13170	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×#368	2
C₅₃×#2158	2
C₂×C₅₃×#346	2
C₅₃×#1945	2
C₅₃×#1382	2
#13170	53

Subgroups — C53×#13170 #27220682

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#13170 #27220682