Subgroups — C₈×M₃₂×C₅₃ #27218910

Order 13568.

354 conjugacy classes of subgroups.

The subgroup lattice has 116 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×M₃₂×C₅₃	C₂	1	2	yes	maximal
2	C₂×C₈²×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1311	C₂	1	2	yes	maximal, complemented
4	C₈×C₁₆×C₅₃	C₂	1	2	yes	maximal, complemented
5	C₂×M₃₂×C₅₃	C₄	1	4	yes	—
6	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₅₃×#346	C₄	1	4	yes	—
9	C₄×C₁₆×C₅₃	C₂²	1	4	yes	—
10	C₈²×C₅₃	C₄	1	2	yes	—
11	C₈²×C₅₃	C₂²	1	1	yes	characteristic
12	C₈²×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₅₃	C₈	1	4	yes	—
14	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
15	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
16	M₃₂×C₅₃	C₈	1	16	yes	direct product factor
17	C₂×C₁₆×C₅₃	C₂×C₄	1	8	yes	—
18	C₄×C₈×C₅₃	C₈	1	4	yes	—
19	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
20	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
24	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
26	C₂×C₈×C₅₃	—	2	8	no	—
27	C₂×C₈×C₅₃	C₄²	1	2	yes	—
28	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
29	C₂×C₈×C₅₃	C₂×C₈	1	8	yes	—
30	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
31	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
32	C₄²×C₅₃	C₄²	1	1	yes	characteristic
33	C₄²×C₅₃	C₂×C₈	1	2	yes	—
34	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
35	C₁₆×C₅₃	C₂×C₈	1	16	yes	complemented
36	C₂³×C₅₃	C₄×C₈	1	1	yes	characteristic
37	C₂×C₄×C₅₃	—	2	4	no	—
38	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
39	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
40	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
41	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
42	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
43	C₈×C₅₃	—	2	4	no	—
44	C₈×C₅₃	C₄×C₈	1	4	yes	—
45	C₈×C₅₃	M₃₂	1	8	yes	direct product factor
46	C₈×C₅₃	C₂²×C₈	1	4	yes	—
47	C₈×M₃₂	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
48	C₂²×C₅₃	—	2	2	no	—
49	C₂²×C₅₃	C₈²	1	2	yes	—
50	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
51	C₄×C₅₃	—	2	2	no	—
52	C₄×C₅₃	C₈²	1	2	yes	—
53	C₄×C₅₃	C₂×C₄×C₈	1	2	yes	—
54	C₄×C₅₃	C₂×M₃₂	1	4	yes	—
55	C₄×M₃₂	C₂×C₅₃	1	2	yes	—
56	C₂×C₈²	C₂×C₅₃	1	1	yes	characteristic
57	#1311	C₂×C₅₃	1	2	yes	complemented
58	C₈×C₁₆	C₂×C₅₃	1	2	yes	complemented
59	C₂×C₅₃	—	2	2	no	—
60	C₂×C₅₃	C₂×C₈²	1	1	yes	characteristic
61	C₂×C₅₃	C₄×M₃₂	1	2	yes	—
62	C₂×M₃₂	C₄×C₅₃	1	4	yes	—
63	C₂×C₄×C₈	C₄×C₅₃	1	2	yes	—
64	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
65	#346	C₄×C₅₃	1	4	yes	—
66	C₄×C₁₆	C₂²×C₅₃	1	4	yes	—
67	C₈²	C₄×C₅₃	1	2	yes	—
68	C₈²	C₂²×C₅₃	1	1	yes	characteristic
69	C₈²	C₂²×C₅₃	1	1	yes	characteristic
70	C₅₃	C₈×M₃₂	1	1	yes	characteristic, direct product factor, Sylow
71	C₂²×C₈	C₈×C₅₃	1	4	yes	—
72	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
73	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
74	M₃₂	C₈×C₅₃	1	16	yes	direct product factor
75	C₂×C₁₆	C₂×C₄×C₅₃	1	8	yes	—
76	C₄×C₈	C₈×C₅₃	1	4	yes	—
77	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
78	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
79	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
80	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
81	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
82	C₂²×C₄	C₄²×C₅₃	1	1	yes	characteristic
83	C₂²×C₄	C₂×C₈×C₅₃	1	2	yes	—
84	C₂×C₈	—	2	8	no	—
85	C₂×C₈	C₄²×C₅₃	1	2	yes	—
86	C₂×C₈	C₂×C₈×C₅₃	1	4	yes	—
87	C₂×C₈	C₂×C₈×C₅₃	1	8	yes	—
88	C₂×C₈	C₂×C₈×C₅₃	1	4	yes	—
89	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
90	C₄²	C₄²×C₅₃	1	1	yes	characteristic
91	C₄²	C₂×C₈×C₅₃	1	2	yes	—
92	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
93	C₁₆	C₂×C₈×C₅₃	1	16	yes	complemented
94	C₂³	C₄×C₈×C₅₃	1	1	yes	characteristic
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	C₄×C₈×C₅₃	1	2	yes	—
97	C₂×C₄	C₄×C₈×C₅₃	1	1	yes	characteristic
98	C₂×C₄	C₄×C₈×C₅₃	1	4	yes	—
99	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
100	C₂×C₄	C₂²×C₈×C₅₃	1	2	yes	—
101	C₈	—	2	4	no	—
102	C₈	C₄×C₈×C₅₃	1	4	yes	—
103	C₈	M₃₂×C₅₃	1	8	yes	direct product factor
104	C₈	C₂²×C₈×C₅₃	1	4	yes	—
105	C₂²	—	2	2	no	—
106	C₂²	C₈²×C₅₃	1	2	yes	—
107	C₂²	C₂×C₄×C₈×C₅₃	1	1	yes	characteristic
108	C₄	—	2	2	no	—
109	C₄	C₈²×C₅₃	1	2	yes	—
110	C₄	C₂×C₄×C₈×C₅₃	1	2	yes	—
111	C₄	C₂×M₃₂×C₅₃	1	4	yes	—
112	C₂	—	2	2	no	—
113	C₂	C₂×C₈²×C₅₃	1	1	yes	characteristic
114	C₂	C₄×M₃₂×C₅₃	1	2	yes	—

	subgroup	G/H
center	C₈²×C₅₃	C₂²
commutator	C₂	C₂×C₈²×C₅₃
Frattini, cosocle	C₄×C₈	C₂³×C₅₃
socle	C₂²×C₅₃	C₂×C₄×C₈
prime radical	C₂³×C₅₃	C₄×C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1311	C₂	semidirect
4. C₈×C₁₆×C₅₃	C₂	semidirect
16. M₃₂×C₅₃	C₈	direct (N × H)
35. C₁₆×C₅₃	C₂×C₈	semidirect
45. C₈×C₅₃	M₃₂	direct (N × H)
47. C₈×M₃₂	C₅₃	direct (N × H)
57. #1311	C₂×C₅₃	semidirect
58. C₈×C₁₆	C₂×C₅₃	semidirect
70. C₅₃	C₈×M₃₂	direct (N × H)
74. M₃₂	C₈×C₅₃	direct (N × H)
93. C₁₆	C₂×C₈×C₅₃	semidirect
103. C₈	M₃₂×C₅₃	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×M₃₂×C₅₃	2
C₂×C₈²×C₅₃	2
C₅₃×#1311	2
C₈×C₁₆×C₅₃	2
C₈×M₃₂	53

Subgroups — C8×M32×C53 #27218910

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×M₃₂×C₅₃ #27218910