Subgroups — Q₈×#14582115 #36342420

Order 21568.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#14582115	C₂	1	3	yes	maximal
2	C₄×#14582115	C₂	1	3	yes	maximal
3	Q₈×#13062438	C₂	1	1	yes	maximal, characteristic
4	C₂×#14582115	C₂²	1	3	yes	—
5	C₂×#14582115	C₂²	1	1	yes	characteristic
6	C₄⋊#13062438	C₄	1	3	yes	—
7	C₄×#13062438	C₂²	1	3	yes	—
8	Q₈×D₃₃₇	C₄	1	1	yes	characteristic
9	#14582115	—	2	3	no	—
10	#14582115	Q₈	1	2	yes	direct product factor
11	C₂×#13062438	C₂×C₄	1	3	yes	—
12	C₂×#13062438	C₂³	1	1	yes	characteristic
13	Dic₆₇₄	C₈	1	3	yes	complemented
14	C₄×D₃₃₇	C₂×C₄	1	3	yes	—
15	Q₈×C₃₃₇	C₈	1	1	yes	characteristic, complemented
16	C₂×D₃₃₇	C₂²×C₄	1	1	yes	characteristic
17	#13062438	—	2	3	no	—
18	#13062438	C₂×Q₈	1	1	yes	characteristic
19	#13062438	#41	1	1	yes	characteristic
20	C₄×C₃₃₇	C₂×C₈	1	3	yes	—
21	Dic₃₃₇	C₂×C₈	1	3	yes	—
22	C₂×C₃₃₇	C₂²×C₈	1	1	yes	characteristic
23	D₃₃₇	C₄×Q₈	1	1	yes	characteristic
24	D₃₃₇	#131	1	1	yes	characteristic
25	C₃₃₇	C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
26	C₈×Q₈	—	337	1	no	maximal, Sylow
27	C₄×Q₈	—	337	1	no	—
28	C₄⋊C₈	—	337	3	no	—
29	C₄×C₈	—	337	3	no	—
30	C₂×Q₈	—	337	1	no	—
31	C₂×C₈	—	337	1	no	—
32	C₂×C₈	—	337	3	no	—
33	C₄⋊C₄	—	337	3	no	—
34	C₄²	—	337	3	no	—
35	Q₈	—	337	3	no	—
36	Q₈	#14582115	1	1	yes	characteristic, direct product factor
37	C₂×C₄	—	337	1	no	—
38	C₂×C₄	—	337	3	no	—
39	C₂×C₄	—	337	3	no	—
40	C₈	—	337	2	no	—
41	C₈	—	674	3	no	—
42	C₂²	—	337	1	no	—
43	C₄	—	337	1	no	—
44	C₄	—	337	1	no	—
45	C₄	—	337	3	no	—
46	C₄	—	674	3	no	—
47	C₄	C₂×#14582115	1	3	yes	—
48	C₂	—	337	1	no	—
49	C₂	—	337	1	no	—
50	C₂	C₂²×#14582115	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂²×#14582115
commutator, socle	C₂×C₃₃₇	C₂²×C₈
Fitting	Q₈×C₃₃₇	C₈
prime radical	C₂×D₃₃₇	C₂²×C₄
cosocle	C₂×#13062438	C₂³
hypercenter	Q₈	#14582115
hypocenter	C₃₃₇	C₈×Q₈
LCS	G ▷ C₂×C₃₃₇ ▷ C₃₃₇
UCS	C₁ ◁ C₂ ◁ Q₈

Complements

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

N	H ≅ G/N	type
10. #14582115	Q₈	direct (N × H)
13. Dic₆₇₄	C₈	semidirect
15. Q₈×C₃₃₇	C₈	semidirect
25. C₃₃₇	C₈×Q₈	semidirect
36. Q₈	#14582115	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⋊#14582115	2
C₄×#14582115	2
Q₈×#13062438	2
C₈×Q₈	337

Subgroups — Q8×#14582115 #36342420

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#14582115 #36342420