Subgroups — Q₈×#13133123 #27304723

Order 13568.

166 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#13133123	C₂	1	3	yes	maximal
2	C₄×#13133123	C₂	1	3	yes	maximal
3	Q₈×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133123	C₂²	1	3	yes	—
5	C₂×#13133123	C₂²	1	1	yes	characteristic
6	C₈×Q₈×C₅₃	C₄	1	1	yes	characteristic
7	C₅₃×C₄⋊C₁₆	C₄	1	3	yes	—
8	C₄×C₁₆×C₅₃	C₂²	1	3	yes	—
9	#13133123	—	2	3	no	—
10	#13133123	Q₈	1	2	yes	direct product factor
11	C₄×Q₈×C₅₃	C₈	1	1	yes	characteristic
12	C₂×C₁₆×C₅₃	C₂×C₄	1	3	yes	—
13	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
14	C₄⋊C₈×C₅₃	C₈	1	3	yes	—
15	C₄×C₈×C₅₃	C₂×C₄	1	3	yes	—
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	complemented
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	—	53	1	no	maximal, 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	D₅₃	1	1	yes	characteristic
38	C₄⋊C₃₂	—	53	3	no	—
39	C₄×C₃₂	—	53	3	no	—
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	Dic₅₃	1	1	yes	characteristic
44	C₂×C₃₂	—	53	1	no	—
45	C₂×C₃₂	—	53	3	no	—
46	C₄⋊C₁₆	Dic₅₃	1	3	yes	—
47	C₄×C₁₆	C₂×D₅₃	1	3	yes	—
48	C₅₃	Q₈×C₃₂	1	1	yes	characteristic, complemented, Sylow
49	C₄×Q₈	#88322	1	1	yes	characteristic
50	C₂×C₁₆	C₂×Dic₅₃	1	3	yes	—
51	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
52	C₄⋊C₈	#88322	1	3	yes	—
53	C₄×C₈	C₂×Dic₅₃	1	3	yes	—
54	C₃₂	—	53	2	no	—
55	C₃₂	—	106	3	no	—
56	C₂×Q₈	#11719908	1	1	yes	characteristic
57	C₂×C₈	C₂×#88322	1	3	yes	—
58	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
59	C₄⋊C₄	#11719908	1	3	yes	—
60	C₄²	C₂×#88322	1	3	yes	—
61	C₁₆	—	2	3	no	—
62	C₁₆	Q₈×D₅₃	1	1	yes	characteristic
63	C₁₆	#11719949	1	1	yes	characteristic
64	Q₈	#13133123	1	4	yes	direct product factor
65	C₂×C₄	C₂×#11719908	1	3	yes	—
66	C₂×C₄	C₂×#11719908	1	3	yes	—
67	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
68	C₈	—	2	3	no	—
69	C₈	Q₈×Dic₅₃	1	1	yes	characteristic
70	C₈	#13133224	1	1	yes	characteristic
71	C₂²	C₂²×#11719908	1	1	yes	characteristic
72	C₄	—	2	3	no	—
73	C₄	C₂×#13133123	1	6	yes	—
74	C₄	Q₈×#88322	1	1	yes	characteristic
75	C₄	#16811040	1	1	yes	characteristic
76	C₂	Q₈×#11719908	1	1	yes	characteristic
77	C₂	C₂²×#13133123	1	1	yes	characteristic
78	C₂	#20277009	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₅₃
commutator	C₂×C₅₃	C₂²×C₃₂
Fitting	Q₈×C₁₆×C₅₃	C₂
socle, prime radical	C₂²×C₅₃	C₂²×C₁₆
cosocle	C₂×C₁₆×C₅₃	C₂³
hypercenter	Q₈×C₁₆	D₅₃
hypocenter	C₅₃	Q₈×C₃₂
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₁₆ ◁ Q₈×C₁₆

Complements

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

N	H ≅ G/N	type
10. #13133123	Q₈	direct (N × H)
24. Q₈×C₅₃	C₃₂	semidirect
48. C₅₃	Q₈×C₃₂	semidirect
64. Q₈	#13133123	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⋊#13133123	2
C₄×#13133123	2
Q₈×C₁₆×C₅₃	2
Q₈×C₃₂	53

Subgroups — Q8×#13133123 #27304723

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#13133123 #27304723