Subgroups — Q₈×#15668452 #37550134

Order 23168.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#15668452	C₂	1	3	yes	maximal
2	C₄×#15668452	C₂	1	3	yes	maximal
3	Q₈×#13089678	C₂	1	1	yes	maximal, characteristic
4	C₂×#15668452	C₂²	1	3	yes	—
5	C₂×#15668452	C₂²	1	1	yes	characteristic
6	C₄⋊#13089678	C₄	1	3	yes	—
7	C₄×#13089678	C₂²	1	3	yes	—
8	C₄×Q₈×C₁₈₁	C₄	1	1	yes	characteristic
9	#15668452	—	2	3	no	—
10	#15668452	Q₈	1	2	yes	direct product factor
11	C₂×#13089678	C₂×C₄	1	3	yes	—
12	C₂×#13089678	C₂³	1	1	yes	characteristic
13	C₂×Q₈×C₁₈₁	C₈	1	1	yes	characteristic
14	C₄⋊C₄×C₁₈₁	C₈	1	3	yes	—
15	C₄²×C₁₈₁	C₂×C₄	1	3	yes	—
16	#13089678	—	2	3	no	—
17	#13089678	C₂×Q₈	1	1	yes	characteristic
18	#13089678	#41	1	1	yes	characteristic
19	Q₈×C₁₈₁	C₁₆	1	4	yes	complemented
20	C₂×C₄×C₁₈₁	C₂×C₈	1	3	yes	—
21	C₂×C₄×C₁₈₁	C₂×C₈	1	3	yes	—
22	C₂×C₄×C₁₈₁	C₂²×C₄	1	1	yes	characteristic
23	C₂²×C₁₈₁	C₂²×C₈	1	1	yes	characteristic
24	C₄×C₁₈₁	—	2	3	no	—
25	C₄×C₁₈₁	C₂×C₁₆	1	6	yes	—
26	C₄×C₁₈₁	C₄×Q₈	1	1	yes	characteristic
27	C₄×C₁₈₁	#131	1	1	yes	characteristic
28	C₂×C₁₈₁	C₈×Q₈	1	1	yes	characteristic
29	C₂×C₁₈₁	C₂²×C₁₆	1	1	yes	characteristic
30	C₂×C₁₈₁	#504	1	1	yes	characteristic
31	C₁₈₁	Q₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
32	Q₈×C₁₆	—	181	1	no	maximal, Sylow
33	C₈×Q₈	—	181	1	no	—
34	C₄⋊C₁₆	—	181	3	no	—
35	C₄×C₁₆	—	181	3	no	—
36	C₄×Q₈	#10624382	1	1	yes	characteristic
37	C₂×C₁₆	—	181	3	no	—
38	C₂×C₁₆	—	181	1	no	—
39	C₄⋊C₈	—	181	3	no	—
40	C₄×C₈	—	181	3	no	—
41	C₂×Q₈	#13089685	1	1	yes	characteristic
42	C₂×C₈	—	181	3	no	—
43	C₂×C₈	—	181	1	no	—
44	C₄⋊C₄	#13089685	1	3	yes	—
45	C₄²	C₂×#10624382	1	3	yes	—
46	C₁₆	—	362	3	no	—
47	C₁₆	—	181	2	no	—
48	Q₈	#15668452	1	4	yes	direct product factor
49	C₂×C₄	C₂×#13089685	1	3	yes	—
50	C₂×C₄	C₂×#13089685	1	3	yes	—
51	C₂×C₄	C₂²×#10624382	1	1	yes	characteristic
52	C₈	—	362	3	no	—
53	C₈	—	181	1	no	—
54	C₈	—	181	1	no	—
55	C₂²	C₂²×#13089685	1	1	yes	characteristic
56	C₄	—	2	3	no	—
57	C₄	C₂×#15668452	1	6	yes	—
58	C₄	Q₈×#10624382	1	1	yes	characteristic
59	C₄	#19153904	1	1	yes	characteristic
60	C₂	Q₈×#13089685	1	1	yes	characteristic
61	C₂	C₂²×#15668452	1	1	yes	characteristic
62	C₂	#26042337	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#10624382
commutator	C₂×C₁₈₁	C₂²×C₁₆
Fitting	C₄×Q₈×C₁₈₁	C₄
socle, prime radical	C₂²×C₁₈₁	C₂²×C₈
cosocle	C₂×#13089678	C₂³
hypercenter	C₄×Q₈	#10624382
hypocenter	C₁₈₁	Q₈×C₁₆
LCS	G ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
10. #15668452	Q₈	direct (N × H)
19. Q₈×C₁₈₁	C₁₆	semidirect
31. C₁₈₁	Q₈×C₁₆	semidirect
48. Q₈	#15668452	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⋊#15668452	2
C₄×#15668452	2
Q₈×#13089678	2
Q₈×C₁₆	181

Subgroups — Q8×#15668452 #37550134

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#15668452 #37550134