Subgroups — C₅₃×#7470 #27214982

Order 13568.

178 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×M₆₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2108	C₂	1	1	yes	maximal, characteristic
4	C₂×M₃₂×C₅₃	C₄	1	1	yes	characteristic
5	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
6	C₅₃×#405	C₄	1	1	yes	characteristic
7	C₅₃×M₆₄	—	2	1	no	—
8	C₅₃×M₆₄	C₄	1	2	yes	complemented
9	C₂×C₃₂×C₅₃	—	2	1	no	—
10	C₂×C₃₂×C₅₃	—	2	1	no	—
11	C₂×C₃₂×C₅₃	—	2	1	no	—
12	C₂×C₃₂×C₅₃	C₄	1	2	yes	complemented
13	C₅₃×#346	—	2	1	no	—
14	C₄×C₁₆×C₅₃	—	2	1	no	—
15	C₂×M₁₆×C₅₃	C₈	1	1	yes	characteristic
16	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
17	#117×C₅₃	C₈	1	1	yes	characteristic
18	M₃₂×C₅₃	—	2	1	no	—
19	M₃₂×C₅₃	C₈	1	2	yes	—
20	C₂×C₁₆×C₅₃	—	2	1	no	—
21	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
24	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
25	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
26	C₂×C₁₆×C₅₃	Q₈	1	1	yes	characteristic
27	#97×C₅₃	—	2	1	no	—
28	C₄×C₈×C₅₃	—	2	1	no	—
29	C₃₂×C₅₃	—	2	2	no	—
30	C₃₂×C₅₃	—	2	2	no	—
31	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
32	M₁₆×C₅₃	—	2	1	no	—
33	M₁₆×C₅₃	C₁₆	1	2	yes	—
34	C₂×C₈×C₅₃	—	2	1	no	—
35	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
36	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
39	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
40	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
41	C₄⋊C₄×C₅₃	C₁₆	1	2	yes	—
42	C₂²⋊C₄×C₅₃	—	2	1	no	—
43	C₄²×C₅₃	—	2	1	no	—
44	C₁₆×C₅₃	—	2	2	no	—
45	C₁₆×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₁₆×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₁₆×C₅₃	C₄⋊C₄	1	1	yes	characteristic
48	C₁₆×C₅₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂³×C₅₃	M₃₂	1	1	yes	characteristic
50	C₂×C₄×C₅₃	—	2	2	no	—
51	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
56	C₂×C₄×C₅₃	M₃₂	1	1	yes	characteristic
57	C₈×C₅₃	—	2	2	no	—
58	C₈×C₅₃	#95	1	1	yes	characteristic
59	C₈×C₅₃	#97	1	1	yes	characteristic
60	C₈×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
61	C₈×C₅₃	C₄⋊C₈	1	1	yes	characteristic
62	#7470	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂²×C₅₃	—	2	2	no	—
64	C₂²×C₅₃	#346	1	1	yes	characteristic
65	C₂²×C₅₃	C₂²⋊C₁₆	1	1	yes	characteristic
66	C₂²×C₅₃	C₄⋊C₁₆	1	1	yes	characteristic
67	C₄×C₅₃	—	4	2	no	—
68	C₄×C₅₃	#336	1	1	yes	characteristic
69	C₄×C₅₃	C₄×C₁₆	1	1	yes	characteristic
70	C₄×C₅₃	C₂²⋊C₁₆	1	1	yes	characteristic
71	C₄×C₅₃	C₄⋊C₁₆	1	1	yes	characteristic
72	C₂×M₆₄	C₂×C₅₃	1	1	yes	characteristic
73	C₂²×C₃₂	C₂×C₅₃	1	1	yes	characteristic
74	#2108	C₂×C₅₃	1	1	yes	characteristic
75	C₂×C₅₃	—	2	1	no	—
76	C₂×C₅₃	—	2	1	no	—
77	C₂×C₅₃	#1374	1	1	yes	characteristic
78	C₂×C₅₃	#1401	1	2	yes	—
79	C₂×M₃₂	C₄×C₅₃	1	1	yes	characteristic
80	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
81	#405	C₄×C₅₃	1	1	yes	characteristic
82	M₆₄	—	2	1	no	—
83	M₆₄	C₄×C₅₃	1	2	yes	complemented
84	C₂×C₃₂	—	2	1	no	—
85	C₂×C₃₂	—	2	1	no	—
86	C₂×C₃₂	—	2	1	no	—
87	C₂×C₃₂	C₄×C₅₃	1	2	yes	complemented
88	#346	—	2	1	no	—
89	C₄×C₁₆	—	2	1	no	—
90	C₅₃	#7470	1	1	yes	characteristic, direct product factor, Sylow
91	C₂×M₁₆	C₈×C₅₃	1	1	yes	characteristic
92	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
93	#117	C₈×C₅₃	1	1	yes	characteristic
94	M₃₂	—	2	1	no	—
95	M₃₂	C₈×C₅₃	1	2	yes	—
96	C₂×C₁₆	—	2	1	no	—
97	C₂×C₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
98	C₂×C₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
99	C₂×C₁₆	D₄×C₅₃	1	1	yes	characteristic
100	C₂×C₁₆	D₄×C₅₃	1	1	yes	characteristic
101	C₂×C₁₆	D₄×C₅₃	1	1	yes	characteristic
102	C₂×C₁₆	Q₈×C₅₃	1	1	yes	characteristic
103	#97	—	2	1	no	—
104	C₄×C₈	—	2	1	no	—
105	C₃₂	—	2	2	no	—
106	C₃₂	—	2	2	no	—
107	C₂²×C₄	C₂×C₈×C₅₃	1	1	yes	characteristic
108	M₁₆	—	2	1	no	—
109	M₁₆	C₁₆×C₅₃	1	2	yes	—
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	C₄²×C₅₃	1	1	yes	characteristic
112	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
113	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
114	C₂×C₈	C₂×C₈×C₅₃	1	1	yes	characteristic
115	C₂×C₈	M₁₆×C₅₃	1	1	yes	characteristic
116	C₂×C₈	M₁₆×C₅₃	1	1	yes	characteristic
117	C₄⋊C₄	C₁₆×C₅₃	1	2	yes	—
118	C₂²⋊C₄	—	2	1	no	—
119	C₄²	—	2	1	no	—
120	C₁₆	—	2	2	no	—
121	C₁₆	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
122	C₁₆	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
123	C₁₆	C₄⋊C₄×C₅₃	1	1	yes	characteristic
124	C₁₆	C₄⋊C₄×C₅₃	1	1	yes	characteristic
125	C₂³	M₃₂×C₅₃	1	1	yes	characteristic
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	C₄×C₈×C₅₃	1	1	yes	characteristic
128	C₂×C₄	C₂²⋊C₈×C₅₃	1	1	yes	characteristic
129	C₂×C₄	C₄⋊C₈×C₅₃	1	1	yes	characteristic
130	C₂×C₄	C₂×C₁₆×C₅₃	1	1	yes	characteristic
131	C₂×C₄	C₂×C₁₆×C₅₃	1	1	yes	characteristic
132	C₂×C₄	M₃₂×C₅₃	1	1	yes	characteristic
133	C₈	—	2	2	no	—
134	C₈	#95×C₅₃	1	1	yes	characteristic
135	C₈	#97×C₅₃	1	1	yes	characteristic
136	C₈	C₂²⋊C₈×C₅₃	1	1	yes	characteristic
137	C₈	C₄⋊C₈×C₅₃	1	1	yes	characteristic
138	C₂²	—	2	2	no	—
139	C₂²	C₅₃×#346	1	1	yes	characteristic
140	C₂²	C₅₃×C₂²⋊C₁₆	1	1	yes	characteristic
141	C₂²	C₅₃×C₄⋊C₁₆	1	1	yes	characteristic
142	C₄	—	4	2	no	—
143	C₄	C₅₃×#336	1	1	yes	characteristic
144	C₄	C₄×C₁₆×C₅₃	1	1	yes	characteristic
145	C₄	C₅₃×C₂²⋊C₁₆	1	1	yes	characteristic
146	C₄	C₅₃×C₄⋊C₁₆	1	1	yes	characteristic
147	C₂	—	2	1	no	—
148	C₂	—	2	1	no	—
149	C₂	C₅₃×#1374	1	1	yes	characteristic
150	C₂	C₅₃×#1401	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₆×C₅₃	D₄
commutator	C₄	C₄×C₁₆×C₅₃
Frattini, cosocle	C₂²×C₁₆	C₂²×C₅₃
socle	C₂²×C₅₃	C₂²⋊C₁₆
prime radical	C₂³×C₅₃	M₃₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆×C₅₃ ◁ C₂²×C₁₆×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₅₃×M₆₄	C₄	semidirect
12. C₂×C₃₂×C₅₃	C₄	semidirect
62. #7470	C₅₃	direct (N × H)
83. M₆₄	C₄×C₅₃	semidirect
87. C₂×C₃₂	C₄×C₅₃	semidirect
90. C₅₃	#7470	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×M₆₄	2
C₂²×C₃₂×C₅₃	2
C₅₃×#2108	2
#7470	53

Subgroups — C53×#7470 #27214982

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7470 #27214982