Subgroups — C₅₃×#7277 #27214789

Order 13568.

222 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1848	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1564	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₆×C₅₃	—	2	1	no	—
5	C₅₃×#431	—	2	1	no	—
6	C₂×#108×C₅₃	—	2	1	no	—
7	C₂×#108×C₅₃	C₄	1	2	yes	—
8	C₅₃×#423	—	2	1	no	—
9	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₅₃×C₄⋊C₁₆	—	2	1	no	—
11	C₅₃×C₄⋊C₁₆	C₄	1	2	yes	—
12	C₅₃×#335	C₄	1	2	yes	—
13	C₅₃×#334	—	2	1	no	—
14	C₂×M₁₆×C₅₃	—	2	2	no	—
15	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
17	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
18	C₂×C₁₆×C₅₃	—	2	2	no	—
19	C₂×C₁₆×C₅₃	—	4	1	no	—
20	C₂×C₁₆×C₅₃	—	4	1	no	—
21	#108×C₅₃	—	4	2	no	—
22	#108×C₅₃	—	4	1	no	—
23	#108×C₅₃	—	4	1	no	—
24	C₄⋊C₈×C₅₃	—	2	2	no	—
25	C₄⋊C₈×C₅₃	—	4	1	no	—
26	C₄⋊C₈×C₅₃	—	4	1	no	—
27	C₂²⋊C₈×C₅₃	—	4	1	no	—
28	C₄×C₈×C₅₃	—	4	1	no	—
29	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
32	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
36	M₁₆×C₅₃	—	4	2	no	—
37	M₁₆×C₅₃	—	4	2	no	—
38	C₂×C₈×C₅₃	—	2	1	no	—
39	C₂×C₈×C₅₃	—	2	1	no	—
40	C₂×C₈×C₅₃	—	4	2	no	—
41	C₂×C₈×C₅₃	—	4	2	no	—
42	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
43	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
47	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
48	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
49	C₂×C₈×C₅₃	Q₁₆	1	1	yes	characteristic
50	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
52	C₄²×C₅₃	D₈	1	1	yes	characteristic
53	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
54	C₁₆×C₅₃	—	4	2	no	—
55	C₂³×C₅₃	#103	1	1	yes	characteristic
56	C₂×C₄×C₅₃	—	2	2	no	—
57	C₂×C₄×C₅₃	—	4	1	no	—
58	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
59	C₂×C₄×C₅₃	#100	1	1	yes	characteristic
60	C₂×C₄×C₅₃	#101	1	1	yes	characteristic
61	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
62	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
63	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
64	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
65	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
66	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
67	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
68	C₈×C₅₃	—	2	1	no	—
69	C₈×C₅₃	—	2	1	no	—
70	C₈×C₅₃	—	8	2	no	—
71	C₈×C₅₃	—	8	2	no	—
72	C₈×C₅₃	#102	1	1	yes	characteristic
73	C₈×C₅₃	#103	1	1	yes	characteristic
74	C₈×C₅₃	#106	1	1	yes	characteristic
75	C₈×C₅₃	#107	1	1	yes	characteristic
76	#7277	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂²×C₅₃	—	2	1	no	—
78	C₂²×C₅₃	—	2	1	no	—
79	C₂²×C₅₃	#332	1	1	yes	characteristic
80	C₂²×C₅₃	#333	1	1	yes	characteristic
81	C₂²×C₅₃	#340	1	1	yes	characteristic
82	C₄×C₅₃	—	2	1	no	—
83	C₄×C₅₃	—	2	1	no	—
84	C₄×C₅₃	—	2	1	no	—
85	C₄×C₅₃	#331	1	1	yes	characteristic
86	C₄×C₅₃	#332	1	1	yes	characteristic
87	C₄×C₅₃	#340	1	1	yes	characteristic
88	C₄×C₅₃	#343	1	1	yes	characteristic
89	C₄×C₅₃	#359	1	2	yes	—
90	C₂×C₄⋊C₁₆	C₂×C₅₃	1	1	yes	characteristic
91	#1848	C₂×C₅₃	1	1	yes	characteristic
92	#1564	C₂×C₅₃	1	1	yes	characteristic
93	C₂×C₅₃	—	2	1	no	—
94	C₂×C₅₃	—	2	1	no	—
95	C₂×C₅₃	#1296	1	1	yes	characteristic
96	C₂×C₅₃	#1382	1	1	yes	characteristic
97	C₂×C₅₃	#1388	1	1	yes	characteristic
98	C₂²×C₁₆	—	2	1	no	—
99	#431	—	2	1	no	—
100	C₂×#108	—	2	1	no	—
101	C₂×#108	C₄×C₅₃	1	2	yes	—
102	#423	—	2	1	no	—
103	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
104	C₄⋊C₁₆	—	2	1	no	—
105	C₄⋊C₁₆	C₄×C₅₃	1	2	yes	—
106	#335	C₄×C₅₃	1	2	yes	—
107	#334	—	2	1	no	—
108	C₅₃	#7277	1	1	yes	characteristic, direct product factor, Sylow
109	C₂×M₁₆	—	2	2	no	—
110	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
111	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
112	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
113	C₂×C₁₆	—	2	2	no	—
114	C₂×C₁₆	—	4	1	no	—
115	C₂×C₁₆	—	4	1	no	—
116	#108	—	4	2	no	—
117	#108	—	4	1	no	—
118	#108	—	4	1	no	—
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	—	4	1	no	—
121	C₄⋊C₈	—	4	1	no	—
122	C₂²⋊C₈	—	4	1	no	—
123	C₄×C₈	—	4	1	no	—
124	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
125	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
126	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
127	C₄×C₈	Q₈×C₅₃	1	1	yes	characteristic
128	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
129	C₂²×C₄	SD₁₆×C₅₃	1	1	yes	characteristic
130	C₂²×C₄	Q₁₆×C₅₃	1	1	yes	characteristic
131	M₁₆	—	4	2	no	—
132	M₁₆	—	4	2	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	4	2	no	—
136	C₂×C₈	—	4	2	no	—
137	C₂×C₈	C₄²×C₅₃	1	1	yes	characteristic
138	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
139	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
140	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
141	C₂×C₈	D₈×C₅₃	1	1	yes	characteristic
142	C₂×C₈	SD₁₆×C₅₃	1	1	yes	characteristic
143	C₂×C₈	SD₁₆×C₅₃	1	1	yes	characteristic
144	C₂×C₈	Q₁₆×C₅₃	1	1	yes	characteristic
145	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
146	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
147	C₄²	D₈×C₅₃	1	1	yes	characteristic
148	C₄²	SD₁₆×C₅₃	1	1	yes	characteristic
149	C₁₆	—	4	2	no	—
150	C₂³	#103×C₅₃	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	#95×C₅₃	1	1	yes	characteristic
154	C₂×C₄	#100×C₅₃	1	1	yes	characteristic
155	C₂×C₄	#101×C₅₃	1	1	yes	characteristic
156	C₂×C₄	#102×C₅₃	1	1	yes	characteristic
157	C₂×C₄	#102×C₅₃	1	1	yes	characteristic
158	C₂×C₄	#102×C₅₃	1	1	yes	characteristic
159	C₂×C₄	#103×C₅₃	1	1	yes	characteristic
160	C₂×C₄	#103×C₅₃	1	1	yes	characteristic
161	C₂×C₄	#106×C₅₃	1	1	yes	characteristic
162	C₂×C₄	#107×C₅₃	1	1	yes	characteristic
163	C₈	—	2	1	no	—
164	C₈	—	2	1	no	—
165	C₈	—	8	2	no	—
166	C₈	—	8	2	no	—
167	C₈	#102×C₅₃	1	1	yes	characteristic
168	C₈	#103×C₅₃	1	1	yes	characteristic
169	C₈	#106×C₅₃	1	1	yes	characteristic
170	C₈	#107×C₅₃	1	1	yes	characteristic
171	C₂²	—	2	1	no	—
172	C₂²	—	2	1	no	—
173	C₂²	C₅₃×#332	1	1	yes	characteristic
174	C₂²	C₅₃×#333	1	1	yes	characteristic
175	C₂²	C₅₃×#340	1	1	yes	characteristic
176	C₄	—	2	1	no	—
177	C₄	—	2	1	no	—
178	C₄	—	2	1	no	—
179	C₄	C₅₃×#331	1	1	yes	characteristic
180	C₄	C₅₃×#332	1	1	yes	characteristic
181	C₄	C₅₃×#340	1	1	yes	characteristic
182	C₄	C₅₃×#343	1	1	yes	characteristic
183	C₄	C₅₃×#359	1	2	yes	—
184	C₂	—	2	1	no	—
185	C₂	—	2	1	no	—
186	C₂	C₅₃×#1296	1	1	yes	characteristic
187	C₂	C₅₃×#1382	1	1	yes	characteristic
188	C₂	C₅₃×#1388	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	#102
commutator	C₂×C₈	C₄²×C₅₃
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₅₃
socle	C₂²×C₅₃	#340
prime radical	C₂³×C₅₃	#103
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×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
76. #7277	C₅₃	direct (N × H)
108. C₅₃	#7277	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×C₄⋊C₁₆	2
C₅₃×#1848	2
C₅₃×#1564	2
#7277	53

Subgroups — C53×#7277 #27214789

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7277 #27214789