Subgroups — #576×C₁₆₇ #24922977

Order 10688.

268 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	H(5,𝔽₂)×C₁₆₇	C₂	1	2	yes	maximal, complemented
2	AGL(1,ℤ/8)×C₁₆₇	C₂	1	6	yes	maximal, complemented
3	#135×C₁₆₇	C₂	1	3	yes	maximal, complemented
4	C₂×D₈×C₁₆₇	C₂	1	3	yes	maximal, complemented
5	#131×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#41×C₁₆₇	—	2	2	no	—
7	#41×C₁₆₇	C₂²	1	1	yes	characteristic
8	#41×C₁₆₇	C₂²	1	6	yes	complemented
9	C₂×D₄×C₁₆₇	—	2	6	no	—
10	C₂×D₄×C₁₆₇	C₂²	1	6	yes	complemented
11	Q₁₆×C₁₆₇	C₂²	1	1	yes	characteristic, complemented
12	SD₁₆×C₁₆₇	C₂²	1	6	yes	complemented
13	D₈×C₁₆₇	C₂²	1	3	yes	complemented
14	D₈×C₁₆₇	C₂²	1	6	yes	complemented
15	M₁₆×C₁₆₇	C₂²	1	3	yes	complemented
16	C₂×C₈×C₁₆₇	C₂²	1	3	yes	complemented
17	C₂³×C₁₆₇	—	2	6	no	—
18	Q₈×C₁₆₇	D₄	1	1	yes	characteristic
19	Q₈×C₁₆₇	C₂³	1	2	yes	—
20	D₄×C₁₆₇	—	2	6	no	—
21	D₄×C₁₆₇	—	2	6	no	—
22	D₄×C₁₆₇	D₄	1	3	yes	—
23	D₄×C₁₆₇	C₂³	1	6	yes	—
24	C₂×C₄×C₁₆₇	—	2	6	no	—
25	C₂×C₄×C₁₆₇	C₂³	1	3	yes	—
26	C₈×C₁₆₇	C₂³	1	1	yes	characteristic
27	C₈×C₁₆₇	C₂³	1	3	yes	—
28	C₂²×C₁₆₇	—	2	6	no	—
29	C₂²×C₁₆₇	—	8	6	no	—
30	C₂²×C₁₆₇	C₂×D₄	1	3	yes	—
31	C₄×C₁₆₇	—	2	2	no	—
32	C₄×C₁₆₇	C₂×D₄	1	3	yes	—
33	C₄×C₁₆₇	C₂⁴	1	1	yes	characteristic
34	C₂×C₁₆₇	—	2	3	no	—
35	C₂×C₁₆₇	—	4	6	no	—
36	C₂×C₁₆₇	C₂²×D₄	1	1	yes	characteristic
37	C₁₆₇	#576	1	1	yes	characteristic, direct product factor, Sylow
38	#576	C₁₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
39	H(5,𝔽₂)	C₂×C₁₆₇	1	2	yes	complemented
40	AGL(1,ℤ/8)	C₂×C₁₆₇	1	6	yes	complemented
41	#135	C₂×C₁₆₇	1	3	yes	complemented
42	C₂×D₈	C₂×C₁₆₇	1	3	yes	complemented
43	#131	C₂×C₁₆₇	1	1	yes	characteristic, complemented
44	#41	—	2	2	no	—
45	#41	C₂²×C₁₆₇	1	1	yes	characteristic
46	#41	C₂²×C₁₆₇	1	6	yes	complemented
47	C₂×D₄	—	2	6	no	—
48	C₂×D₄	C₂²×C₁₆₇	1	6	yes	complemented
49	Q₁₆	C₂²×C₁₆₇	1	1	yes	characteristic, complemented
50	SD₁₆	C₂²×C₁₆₇	1	6	yes	complemented
51	D₈	C₂²×C₁₆₇	1	3	yes	complemented
52	D₈	C₂²×C₁₆₇	1	6	yes	complemented
53	M₁₆	C₂²×C₁₆₇	1	3	yes	complemented
54	C₂×C₈	C₂²×C₁₆₇	1	3	yes	complemented
55	C₂³	—	2	6	no	—
56	Q₈	D₄×C₁₆₇	1	1	yes	characteristic
57	Q₈	C₂³×C₁₆₇	1	2	yes	—
58	D₄	—	2	6	no	—
59	D₄	—	2	6	no	—
60	D₄	D₄×C₁₆₇	1	3	yes	—
61	D₄	C₂³×C₁₆₇	1	6	yes	—
62	C₂×C₄	—	2	6	no	—
63	C₂×C₄	C₂³×C₁₆₇	1	3	yes	—
64	C₈	C₂³×C₁₆₇	1	1	yes	characteristic
65	C₈	C₂³×C₁₆₇	1	3	yes	—
66	C₂²	—	2	6	no	—
67	C₂²	—	8	6	no	—
68	C₂²	C₂×D₄×C₁₆₇	1	3	yes	—
69	C₄	—	2	2	no	—
70	C₄	C₂×D₄×C₁₆₇	1	3	yes	—
71	C₄	C₂⁴×C₁₆₇	1	1	yes	characteristic
72	C₂	—	2	3	no	—
73	C₂	—	4	6	no	—
74	C₂	C₂²×D₄×C₁₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₄	C₂⁴×C₁₆₇
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆₇ ◁ #41×C₁₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. H(5,𝔽₂)×C₁₆₇	C₂	semidirect
2. AGL(1,ℤ/8)×C₁₆₇	C₂	semidirect
3. #135×C₁₆₇	C₂	semidirect
4. C₂×D₈×C₁₆₇	C₂	semidirect
5. #131×C₁₆₇	C₂	semidirect
8. #41×C₁₆₇	C₂²	semidirect
10. C₂×D₄×C₁₆₇	C₂²	semidirect
11. Q₁₆×C₁₆₇	C₂²	semidirect
12. SD₁₆×C₁₆₇	C₂²	semidirect
13. D₈×C₁₆₇	C₂²	semidirect
14. D₈×C₁₆₇	C₂²	semidirect
15. M₁₆×C₁₆₇	C₂²	semidirect
16. C₂×C₈×C₁₆₇	C₂²	semidirect
37. C₁₆₇	#576	direct (N × H)
38. #576	C₁₆₇	direct (N × H)
39. H(5,𝔽₂)	C₂×C₁₆₇	semidirect
40. AGL(1,ℤ/8)	C₂×C₁₆₇	semidirect
41. #135	C₂×C₁₆₇	semidirect
42. C₂×D₈	C₂×C₁₆₇	semidirect
43. #131	C₂×C₁₆₇	semidirect
46. #41	C₂²×C₁₆₇	semidirect
48. C₂×D₄	C₂²×C₁₆₇	semidirect
49. Q₁₆	C₂²×C₁₆₇	semidirect
50. SD₁₆	C₂²×C₁₆₇	semidirect
51. D₈	C₂²×C₁₆₇	semidirect
52. D₈	C₂²×C₁₆₇	semidirect
53. M₁₆	C₂²×C₁₆₇	semidirect
54. C₂×C₈	C₂²×C₁₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
H(5,𝔽₂)×C₁₆₇	2
AGL(1,ℤ/8)×C₁₆₇	2
#135×C₁₆₇	2
C₂×D₈×C₁₆₇	2
#131×C₁₆₇	2
#576	167

Subgroups — #576×C167 #24922977

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #576×C₁₆₇ #24922977