Subgroups — Q₈×C₃₂×C₅₃ #27221171

Order 13568.

166 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×C₄⋊C₃₂	C₂	1	3	yes	maximal
3	C₄×C₃₂×C₅₃	C₂	1	3	yes	maximal
4	C₈×Q₈×C₅₃	C₄	1	1	yes	characteristic
5	C₂×C₃₂×C₅₃	C₂²	1	3	yes	—
6	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
7	C₅₃×C₄⋊C₁₆	C₄	1	3	yes	—
8	C₄×C₁₆×C₅₃	C₂²	1	3	yes	—
9	C₄×Q₈×C₅₃	C₈	1	1	yes	characteristic
10	C₂×C₁₆×C₅₃	C₂×C₄	1	3	yes	—
11	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
12	C₄⋊C₈×C₅₃	C₈	1	3	yes	—
13	C₄×C₈×C₅₃	C₂×C₄	1	3	yes	—
14	C₃₂×C₅₃	—	2	3	no	—
15	C₃₂×C₅₃	Q₈	1	2	yes	direct product factor
16	C₂×Q₈×C₅₃	C₁₆	1	1	yes	characteristic
17	C₂×C₈×C₅₃	C₂×C₈	1	3	yes	—
18	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
19	C₄⋊C₄×C₅₃	C₁₆	1	3	yes	—
20	C₄²×C₅₃	C₂×C₈	1	3	yes	—
21	C₁₆×C₅₃	—	2	3	no	—
22	C₁₆×C₅₃	C₂×Q₈	1	1	yes	characteristic
23	C₁₆×C₅₃	#41	1	1	yes	characteristic
24	Q₈×C₅₃	C₃₂	1	4	yes	direct product factor
25	C₂×C₄×C₅₃	C₂×C₁₆	1	3	yes	—
26	C₂×C₄×C₅₃	C₂×C₁₆	1	3	yes	—
27	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
28	C₈×C₅₃	—	2	3	no	—
29	C₈×C₅₃	C₄×Q₈	1	1	yes	characteristic
30	C₈×C₅₃	#131	1	1	yes	characteristic
31	Q₈×C₃₂	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
33	C₄×C₅₃	—	2	3	no	—
34	C₄×C₅₃	C₂×C₃₂	1	6	yes	—
35	C₄×C₅₃	C₈×Q₈	1	1	yes	characteristic
36	C₄×C₅₃	#504	1	1	yes	characteristic
37	Q₈×C₁₆	C₂×C₅₃	1	1	yes	characteristic
38	C₄⋊C₃₂	C₂×C₅₃	1	3	yes	—
39	C₄×C₃₂	C₂×C₅₃	1	3	yes	—
40	C₂×C₅₃	Q₈×C₁₆	1	1	yes	characteristic
41	C₂×C₅₃	C₂²×C₃₂	1	1	yes	characteristic
42	C₂×C₅₃	#2258	1	1	yes	characteristic
43	C₈×Q₈	C₄×C₅₃	1	1	yes	characteristic
44	C₂×C₃₂	C₂²×C₅₃	1	3	yes	—
45	C₂×C₃₂	C₂²×C₅₃	1	1	yes	characteristic
46	C₄⋊C₁₆	C₄×C₅₃	1	3	yes	—
47	C₄×C₁₆	C₂²×C₅₃	1	3	yes	—
48	C₅₃	Q₈×C₃₂	1	1	yes	characteristic, direct product factor, Sylow
49	C₄×Q₈	C₈×C₅₃	1	1	yes	characteristic
50	C₂×C₁₆	C₂×C₄×C₅₃	1	3	yes	—
51	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
52	C₄⋊C₈	C₈×C₅₃	1	3	yes	—
53	C₄×C₈	C₂×C₄×C₅₃	1	3	yes	—
54	C₃₂	—	2	3	no	—
55	C₃₂	Q₈×C₅₃	1	2	yes	direct product factor
56	C₂×Q₈	C₁₆×C₅₃	1	1	yes	characteristic
57	C₂×C₈	C₂×C₈×C₅₃	1	3	yes	—
58	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
59	C₄⋊C₄	C₁₆×C₅₃	1	3	yes	—
60	C₄²	C₂×C₈×C₅₃	1	3	yes	—
61	C₁₆	—	2	3	no	—
62	C₁₆	C₂×Q₈×C₅₃	1	1	yes	characteristic
63	C₁₆	#41×C₅₃	1	1	yes	characteristic
64	Q₈	C₃₂×C₅₃	1	4	yes	direct product factor
65	C₂×C₄	C₂×C₁₆×C₅₃	1	3	yes	—
66	C₂×C₄	C₂×C₁₆×C₅₃	1	3	yes	—
67	C₂×C₄	C₂²×C₈×C₅₃	1	1	yes	characteristic
68	C₈	—	2	3	no	—
69	C₈	C₄×Q₈×C₅₃	1	1	yes	characteristic
70	C₈	#131×C₅₃	1	1	yes	characteristic
71	C₂²	C₂²×C₁₆×C₅₃	1	1	yes	characteristic
72	C₄	—	2	3	no	—
73	C₄	C₂×C₃₂×C₅₃	1	6	yes	—
74	C₄	C₈×Q₈×C₅₃	1	1	yes	characteristic
75	C₄	C₅₃×#504	1	1	yes	characteristic
76	C₂	Q₈×C₁₆×C₅₃	1	1	yes	characteristic
77	C₂	C₂²×C₃₂×C₅₃	1	1	yes	characteristic
78	C₂	C₅₃×#2258	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃₂×C₅₃	C₂²
commutator	C₂	C₂²×C₃₂×C₅₃
Frattini, cosocle	C₂×C₁₆	C₂³×C₅₃
socle, prime radical	C₂²×C₅₃	C₂²×C₁₆
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₂×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
15. C₃₂×C₅₃	Q₈	direct (N × H)
24. Q₈×C₅₃	C₃₂	direct (N × H)
31. Q₈×C₃₂	C₅₃	direct (N × H)
48. C₅₃	Q₈×C₃₂	direct (N × H)
55. C₃₂	Q₈×C₅₃	direct (N × H)
64. Q₈	C₃₂×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
Q₈×C₁₆×C₅₃	2
C₅₃×C₄⋊C₃₂	2
C₄×C₃₂×C₅₃	2
Q₈×C₃₂	53

Subgroups — Q8×C32×C53 #27221171

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₃₂×C₅₃ #27221171