Subgroups — C₅₃×#7296 #27214808

Order 13568.

242 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2162	C₂	1	1	yes	maximal, characteristic
2	C₄×#107×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1567	C₂	1	1	yes	maximal, characteristic
4	C₂×#107×C₅₃	—	2	1	no	—
5	C₅₃×#423	—	2	1	no	—
6	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
7	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
8	C₅₃×C₄⋊C₁₆	C₄	1	2	yes	complemented
9	C₅₃×C₂²⋊C₁₆	—	2	1	no	—
10	C₄×C₁₆×C₅₃	—	2	1	no	—
11	C₅₃×#340	C₄	1	2	yes	—
12	C₅₃×#335	—	2	1	no	—
13	C₅₃×#335	C₄	1	2	yes	complemented
14	C₂×M₁₆×C₅₃	—	4	1	no	—
15	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
18	C₂×C₄²×C₅₃	—	4	1	no	—
19	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
20	C₂×C₁₆×C₅₃	—	2	2	no	—
21	#107×C₅₃	—	2	4	no	—
22	C₄⋊C₈×C₅₃	—	2	2	no	—
23	C₄⋊C₈×C₅₃	—	4	1	no	—
24	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
27	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
28	#95×C₅₃	—	4	1	no	—
29	C₂²×C₄×C₅₃	—	4	2	no	—
30	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
33	M₁₆×C₅₃	—	8	1	no	—
34	M₁₆×C₅₃	—	8	1	no	—
35	C₂×C₈×C₅₃	—	2	2	no	—
36	C₂×C₈×C₅₃	—	4	1	no	—
37	C₂×C₈×C₅₃	—	4	2	no	—
38	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
39	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
42	C₄⋊C₄×C₅₃	—	2	4	no	—
43	C₄⋊C₄×C₅₃	—	4	2	no	—
44	C₄²×C₅₃	—	8	1	no	—
45	C₄²×C₅₃	—	8	1	no	—
46	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
48	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
49	C₄²×C₅₃	Q₁₆	1	1	yes	characteristic
50	C₁₆×C₅₃	—	4	2	no	—
51	C₂³×C₅₃	#102	1	1	yes	characteristic
52	C₂×C₄×C₅₃	—	2	1	no	—
53	C₂×C₄×C₅₃	—	2	1	no	—
54	C₂×C₄×C₅₃	—	4	1	no	—
55	C₂×C₄×C₅₃	—	4	4	no	—
56	C₂×C₄×C₅₃	—	8	2	no	—
57	C₂×C₄×C₅₃	—	8	2	no	—
58	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
59	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
60	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
61	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
62	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
63	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
64	C₂×C₄×C₅₃	D₁₆	1	1	yes	characteristic
65	C₂×C₄×C₅₃	SD₃₂	1	1	yes	characteristic
66	C₂×C₄×C₅₃	SD₃₂	1	1	yes	characteristic
67	C₂×C₄×C₅₃	Q₃₂	1	1	yes	characteristic
68	C₈×C₅₃	—	2	1	no	—
69	C₈×C₅₃	—	2	1	no	—
70	C₈×C₅₃	—	2	1	no	—
71	C₈×C₅₃	—	8	2	no	—
72	C₈×C₅₃	#104	1	2	yes	—
73	#7296	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂²×C₅₃	—	2	1	no	—
75	C₂²×C₅₃	—	2	1	no	—
76	C₂²×C₅₃	#340	1	1	yes	characteristic
77	C₂²×C₅₃	#357	1	1	yes	characteristic
78	C₂²×C₅₃	#358	1	1	yes	characteristic
79	C₄×C₅₃	—	2	1	no	—
80	C₄×C₅₃	—	2	1	no	—
81	C₄×C₅₃	—	2	1	no	—
82	C₄×C₅₃	—	8	4	no	—
83	C₄×C₅₃	#339	1	1	yes	characteristic
84	C₄×C₅₃	#344	1	1	yes	characteristic
85	C₄×C₅₃	#357	1	1	yes	characteristic
86	C₄×C₅₃	#358	1	1	yes	characteristic
87	C₄×C₅₃	#366	1	1	yes	characteristic
88	C₄×C₅₃	#367	1	1	yes	characteristic
89	#2162	C₂×C₅₃	1	1	yes	characteristic
90	C₄×#107	C₂×C₅₃	1	1	yes	characteristic
91	#1567	C₂×C₅₃	1	1	yes	characteristic
92	C₂×C₅₃	—	2	1	no	—
93	C₂×C₅₃	—	2	1	no	—
94	C₂×C₅₃	#1303	1	1	yes	characteristic
95	C₂×C₅₃	#1380	1	1	yes	characteristic
96	C₂×C₅₃	#1389	1	1	yes	characteristic
97	C₂×#107	—	2	1	no	—
98	#423	—	2	1	no	—
99	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
100	C₄×C₄⋊C₄	—	2	1	no	—
101	C₄⋊C₁₆	C₄×C₅₃	1	2	yes	complemented
102	C₂²⋊C₁₆	—	2	1	no	—
103	C₄×C₁₆	—	2	1	no	—
104	#340	C₄×C₅₃	1	2	yes	—
105	#335	—	2	1	no	—
106	#335	C₄×C₅₃	1	2	yes	complemented
107	C₅₃	#7296	1	1	yes	characteristic, direct product factor, Sylow
108	C₂×M₁₆	—	4	1	no	—
109	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
110	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
111	C₂×C₄⋊C₄	—	2	2	no	—
112	C₂×C₄²	—	4	1	no	—
113	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
114	C₂×C₁₆	—	2	2	no	—
115	#107	—	2	4	no	—
116	C₄⋊C₈	—	2	2	no	—
117	C₄⋊C₈	—	4	1	no	—
118	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
119	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
120	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
121	C₄×C₈	Q₈×C₅₃	1	1	yes	characteristic
122	#95	—	4	1	no	—
123	C₂²×C₄	—	4	2	no	—
124	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
125	C₂²×C₄	D₈×C₅₃	1	1	yes	characteristic
126	C₂²×C₄	SD₁₆×C₅₃	1	1	yes	characteristic
127	M₁₆	—	8	1	no	—
128	M₁₆	—	8	1	no	—
129	C₂×C₈	—	2	2	no	—
130	C₂×C₈	—	4	1	no	—
131	C₂×C₈	—	4	2	no	—
132	C₂×C₈	C₄²×C₅₃	1	1	yes	characteristic
133	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
134	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
135	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
136	C₄⋊C₄	—	2	4	no	—
137	C₄⋊C₄	—	4	2	no	—
138	C₄²	—	8	1	no	—
139	C₄²	—	8	1	no	—
140	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
141	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
142	C₄²	SD₁₆×C₅₃	1	1	yes	characteristic
143	C₄²	Q₁₆×C₅₃	1	1	yes	characteristic
144	C₁₆	—	4	2	no	—
145	C₂³	#102×C₅₃	1	1	yes	characteristic
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	4	4	no	—
150	C₂×C₄	—	8	2	no	—
151	C₂×C₄	—	8	2	no	—
152	C₂×C₄	#95×C₅₃	1	1	yes	characteristic
153	C₂×C₄	#102×C₅₃	1	1	yes	characteristic
154	C₂×C₄	#103×C₅₃	1	1	yes	characteristic
155	C₂×C₄	#103×C₅₃	1	1	yes	characteristic
156	C₂×C₄	#106×C₅₃	1	1	yes	characteristic
157	C₂×C₄	#107×C₅₃	1	1	yes	characteristic
158	C₂×C₄	D₁₆×C₅₃	1	1	yes	characteristic
159	C₂×C₄	SD₃₂×C₅₃	1	1	yes	characteristic
160	C₂×C₄	SD₃₂×C₅₃	1	1	yes	characteristic
161	C₂×C₄	Q₃₂×C₅₃	1	1	yes	characteristic
162	C₈	—	2	1	no	—
163	C₈	—	2	1	no	—
164	C₈	—	2	1	no	—
165	C₈	—	8	2	no	—
166	C₈	#104×C₅₃	1	2	yes	—
167	C₂²	—	2	1	no	—
168	C₂²	—	2	1	no	—
169	C₂²	C₅₃×#340	1	1	yes	characteristic
170	C₂²	C₅₃×#357	1	1	yes	characteristic
171	C₂²	C₅₃×#358	1	1	yes	characteristic
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	—	8	4	no	—
176	C₄	C₅₃×#339	1	1	yes	characteristic
177	C₄	C₅₃×#344	1	1	yes	characteristic
178	C₄	C₅₃×#357	1	1	yes	characteristic
179	C₄	C₅₃×#358	1	1	yes	characteristic
180	C₄	C₅₃×#366	1	1	yes	characteristic
181	C₄	C₅₃×#367	1	1	yes	characteristic
182	C₂	—	2	1	no	—
183	C₂	—	2	1	no	—
184	C₂	C₅₃×#1303	1	1	yes	characteristic
185	C₂	C₅₃×#1380	1	1	yes	characteristic
186	C₂	C₅₃×#1389	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	#102
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
8. C₅₃×C₄⋊C₁₆	C₄	semidirect
13. C₅₃×#335	C₄	semidirect
73. #7296	C₅₃	direct (N × H)
101. C₄⋊C₁₆	C₄×C₅₃	semidirect
106. #335	C₄×C₅₃	semidirect
107. C₅₃	#7296	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×#2162	2
C₄×#107×C₅₃	2
C₅₃×#1567	2
#7296	53

Subgroups — C53×#7296 #27214808

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7296 #27214808