Subgroups — Q₈×#11719910 #20291128

Order 6784.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#11719910	C₂	1	3	yes	maximal
2	C₄×#11719910	C₂	1	3	yes	maximal
3	Q₈×#88322	C₂	1	1	yes	maximal, characteristic
4	C₂×#11719910	C₂²	1	3	yes	—
5	C₂×#11719910	C₂²	1	1	yes	characteristic
6	C₄⋊#88322	C₄	1	3	yes	—
7	C₄×#88322	C₂²	1	3	yes	—
8	C₄×Q₈×C₅₃	C₄	1	1	yes	characteristic
9	C₂×Q₈×C₅₃	C₈	1	1	yes	characteristic
10	C₄⋊C₄×C₅₃	C₈	1	3	yes	—
11	C₄²×C₅₃	C₂×C₄	1	3	yes	—
12	C₂×#88322	C₂×C₄	1	3	yes	—
13	C₂×#88322	C₂³	1	1	yes	characteristic
14	#11719910	—	2	3	no	—
15	#11719910	Q₈	1	2	yes	direct product factor
16	Q₈×C₅₃	C₁₆	1	4	yes	complemented
17	C₂×C₄×C₅₃	C₂×C₈	1	3	yes	—
18	C₂×C₄×C₅₃	C₂×C₈	1	3	yes	—
19	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
20	#88322	—	2	3	no	—
21	#88322	C₂×Q₈	1	1	yes	characteristic
22	#88322	#41	1	1	yes	characteristic
23	C₂²×C₅₃	C₂²×C₈	1	1	yes	characteristic
24	C₄×C₅₃	—	2	3	no	—
25	C₄×C₅₃	C₂×C₁₆	1	6	yes	—
26	C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
27	C₄×C₅₃	#131	1	1	yes	characteristic
28	Q₈×C₁₆	—	53	1	no	maximal, Sylow
29	C₂×C₅₃	C₈×Q₈	1	1	yes	characteristic
30	C₂×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
31	C₂×C₅₃	#504	1	1	yes	characteristic
32	C₈×Q₈	—	53	1	no	—
33	C₄⋊C₁₆	—	53	3	no	—
34	C₄×C₁₆	—	53	3	no	—
35	C₅₃	Q₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
36	C₄×Q₈	#6157	1	1	yes	characteristic
37	C₂×C₁₆	—	53	1	no	—
38	C₂×C₁₆	—	53	3	no	—
39	C₄⋊C₈	—	53	3	no	—
40	C₄×C₈	—	53	3	no	—
41	C₂×Q₈	#88324	1	1	yes	characteristic
42	C₂×C₈	—	53	1	no	—
43	C₂×C₈	—	53	3	no	—
44	C₄⋊C₄	#88324	1	3	yes	—
45	C₄²	C₂×#6157	1	3	yes	—
46	C₁₆	—	53	2	no	—
47	C₁₆	—	106	3	no	—
48	Q₈	#11719910	1	4	yes	direct product factor
49	C₂×C₄	C₂×#88324	1	3	yes	—
50	C₂×C₄	C₂×#88324	1	3	yes	—
51	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
52	C₈	—	53	1	no	—
53	C₈	—	53	1	no	—
54	C₈	—	106	3	no	—
55	C₂²	C₂²×#88324	1	1	yes	characteristic
56	C₄	—	2	3	no	—
57	C₄	C₂×#11719910	1	6	yes	—
58	C₄	Q₈×#6157	1	1	yes	characteristic
59	C₄	#13133305	1	1	yes	characteristic
60	C₂	Q₈×#88324	1	1	yes	characteristic
61	C₂	C₂²×#11719910	1	1	yes	characteristic
62	C₂	#16811681	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#6157
commutator	C₂×C₅₃	C₂²×C₁₆
Fitting	C₄×Q₈×C₅₃	C₄
socle	C₂²×C₅₃	C₂²×C₈
hypercenter	C₄×Q₈	#6157
hypocenter	C₅₃	Q₈×C₁₆
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
15. #11719910	Q₈	direct (N × H)
16. Q₈×C₅₃	C₁₆	semidirect
35. C₅₃	Q₈×C₁₆	semidirect
48. Q₈	#11719910	direct (N × H)

Subgroups — Q8×#11719910 #20291128

Conjugacy classes of subgroups

Complements

Subgroups — Q₈×#11719910 #20291128