Subgroups — C₅₃×#13628 #27221140

Order 13568.

206 conjugacy classes of subgroups.

The subgroup lattice has 124 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	C₅₃×C₄⋊C₃₂	C₂	1	4	yes	maximal, complemented
5	C₂×M₃₂×C₅₃	C₄	1	1	yes	characteristic
6	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
7	C₅₃×#405	C₄	1	1	yes	characteristic
8	C₅₃×M₆₄	—	2	2	no	—
9	C₂×C₃₂×C₅₃	—	2	2	no	—
10	C₂×C₃₂×C₅₃	C₂²	1	2	yes	—
11	C₂×C₃₂×C₅₃	C₂²	1	2	yes	—
12	C₅₃×#346	C₄	1	2	yes	—
13	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
14	C₂×M₁₆×C₅₃	C₈	1	1	yes	characteristic
15	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
16	#117×C₅₃	C₈	1	1	yes	characteristic
17	M₃₂×C₅₃	—	2	2	no	—
18	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
19	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
21	C₂×C₁₆×C₅₃	Q₈	1	2	yes	—
22	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
23	#97×C₅₃	C₈	1	2	yes	—
24	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
25	C₃₂×C₅₃	—	2	2	no	—
26	C₃₂×C₅₃	—	2	2	no	—
27	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
28	M₁₆×C₅₃	—	2	2	no	—
29	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
30	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
31	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
33	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
34	C₄⋊C₄×C₅₃	C₁₆	1	2	yes	—
35	C₂²⋊C₄×C₅₃	C₁₆	1	2	yes	—
36	C₄²×C₅₃	C₂×C₈	1	2	yes	—
37	C₁₆×C₅₃	—	2	2	no	—
38	C₁₆×C₅₃	C₄⋊C₄	1	2	yes	—
39	C₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₁₆×C₅₃	C₂×Q₈	1	1	yes	characteristic
41	C₂³×C₅₃	C₂×C₁₆	1	1	yes	characteristic
42	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
43	C₂×C₄×C₅₃	C₂×C₁₆	1	4	yes	—
44	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
45	C₂×C₄×C₅₃	M₃₂	1	2	yes	—
46	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
47	C₈×C₅₃	—	2	2	no	—
48	C₈×C₅₃	C₄⋊C₈	1	2	yes	—
49	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
51	#13628	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂²×C₅₃	—	2	2	no	—
53	C₂²×C₅₃	C₄⋊C₁₆	1	2	yes	—
54	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
55	C₄×C₅₃	—	2	4	no	—
56	C₄×C₅₃	C₄⋊C₁₆	1	2	yes	—
57	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
58	C₄×C₅₃	C₂×M₃₂	1	1	yes	characteristic
59	C₂×M₆₄	C₂×C₅₃	1	1	yes	characteristic
60	C₂²×C₃₂	C₂×C₅₃	1	1	yes	characteristic
61	#2108	C₂×C₅₃	1	1	yes	characteristic
62	C₄⋊C₃₂	C₂×C₅₃	1	4	yes	complemented
63	C₂×C₅₃	—	2	2	no	—
64	C₂×C₅₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
65	C₂×C₅₃	#2258	1	2	yes	—
66	C₂×M₃₂	C₄×C₅₃	1	1	yes	characteristic
67	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
68	#405	C₄×C₅₃	1	1	yes	characteristic
69	M₆₄	—	2	2	no	—
70	C₂×C₃₂	—	2	2	no	—
71	C₂×C₃₂	C₂²×C₅₃	1	2	yes	—
72	C₂×C₃₂	C₂²×C₅₃	1	2	yes	—
73	#346	C₄×C₅₃	1	2	yes	—
74	C₄×C₁₆	C₂²×C₅₃	1	2	yes	—
75	C₅₃	#13628	1	1	yes	characteristic, direct product factor, Sylow
76	C₂×M₁₆	C₈×C₅₃	1	1	yes	characteristic
77	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
78	#117	C₈×C₅₃	1	1	yes	characteristic
79	M₃₂	—	2	2	no	—
80	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
81	C₂×C₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
82	C₂×C₁₆	D₄×C₅₃	1	2	yes	—
83	C₂×C₁₆	Q₈×C₅₃	1	2	yes	—
84	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
85	#97	C₈×C₅₃	1	2	yes	—
86	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
87	C₃₂	—	2	2	no	—
88	C₃₂	—	2	2	no	—
89	C₂²×C₄	C₂×C₈×C₅₃	1	1	yes	characteristic
90	M₁₆	—	2	2	no	—
91	C₂×C₈	C₄⋊C₄×C₅₃	1	2	yes	—
92	C₂×C₈	C₂×C₈×C₅₃	1	2	yes	—
93	C₂×C₈	C₂×C₈×C₅₃	1	1	yes	characteristic
94	C₂×C₈	M₁₆×C₅₃	1	2	yes	—
95	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
96	C₄⋊C₄	C₁₆×C₅₃	1	2	yes	—
97	C₂²⋊C₄	C₁₆×C₅₃	1	2	yes	—
98	C₄²	C₂×C₈×C₅₃	1	2	yes	—
99	C₁₆	—	2	2	no	—
100	C₁₆	C₄⋊C₄×C₅₃	1	2	yes	—
101	C₁₆	C₂×D₄×C₅₃	1	1	yes	characteristic
102	C₁₆	C₂×Q₈×C₅₃	1	1	yes	characteristic
103	C₂³	C₂×C₁₆×C₅₃	1	1	yes	characteristic
104	C₂×C₄	C₄⋊C₈×C₅₃	1	2	yes	—
105	C₂×C₄	C₂×C₁₆×C₅₃	1	4	yes	—
106	C₂×C₄	C₂×C₁₆×C₅₃	1	1	yes	characteristic
107	C₂×C₄	M₃₂×C₅₃	1	2	yes	—
108	C₂×C₄	C₂²×C₈×C₅₃	1	1	yes	characteristic
109	C₈	—	2	2	no	—
110	C₈	C₄⋊C₈×C₅₃	1	2	yes	—
111	C₈	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
112	C₈	C₂×M₁₆×C₅₃	1	1	yes	characteristic
113	C₂²	—	2	2	no	—
114	C₂²	C₅₃×C₄⋊C₁₆	1	2	yes	—
115	C₂²	C₂²×C₁₆×C₅₃	1	1	yes	characteristic
116	C₄	—	2	4	no	—
117	C₄	C₅₃×C₄⋊C₁₆	1	2	yes	—
118	C₄	C₂×C₄⋊C₈×C₅₃	1	1	yes	characteristic
119	C₄	C₂×M₃₂×C₅₃	1	1	yes	characteristic
120	C₂	—	2	2	no	—
121	C₂	C₂×C₅₃×C₄⋊C₁₆	1	1	yes	characteristic
122	C₂	C₅₃×#2258	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₅₃×C₄⋊C₃₂	C₂	semidirect
51. #13628	C₅₃	direct (N × H)
62. C₄⋊C₃₂	C₂×C₅₃	semidirect
75. C₅₃	#13628	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
C₅₃×C₄⋊C₃₂	2
#13628	53

Subgroups — C53×#13628 #27221140

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#13628 #27221140