Subgroups — C₃×Q₈×#170 #11752319

Order 960.

204 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×Q₈×Dic₅	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄⋊#170	C₂	1	3	yes	maximal
3	C₃×C₄×#170	C₂	1	3	yes	maximal
4	Q₈×#170	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂×C₃×C₅×Q₈	C₄	1	1	yes	characteristic
6	C₂×C₃×#170	C₂²	1	3	yes	—
7	C₂×C₃×#170	C₂²	1	1	yes	characteristic
8	C₃×C₄⋊Dic₅	C₄	1	3	yes	—
9	C₃×C₄×Dic₅	C₂²	1	3	yes	—
10	C₃×C₈×Q₈	—	5	1	no	maximal, Hall
11	Q₈×Dic₅	C₂×C₃	1	1	yes	characteristic
12	C₄⋊#170	C₂×C₃	1	3	yes	—
13	C₄×#170	C₂×C₃	1	3	yes	—
14	C₃×C₅×Q₈	C₈	1	4	yes	complemented
15	C₂×C₃×C₄×C₅	C₂×C₄	1	3	yes	—
16	C₂×C₃×Dic₅	C₂×C₄	1	3	yes	—
17	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
18	C₃×#170	—	2	3	no	—
19	C₃×#170	Q₈	1	2	yes	direct product factor
20	C₃×C₄×Q₈	—	5	1	no	—
21	C₃×C₄⋊C₈	—	5	3	no	—
22	C₃×C₄×C₈	—	5	3	no	—
23	C₂×C₅×Q₈	C₃×C₄	1	1	yes	characteristic
24	C₂×#170	C₂²×C₃	1	3	yes	—
25	C₂×#170	C₂²×C₃	1	1	yes	characteristic
26	C₄⋊Dic₅	C₃×C₄	1	3	yes	—
27	C₄×Dic₅	C₂²×C₃	1	3	yes	—
28	C₈×Q₈	—	5	1	no	Sylow
29	C₂²×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
30	C₃×C₄×C₅	C₂×C₈	1	6	yes	—
31	C₃×Dic₅	—	2	3	no	—
32	C₃×Dic₅	C₂×Q₈	1	1	yes	characteristic
33	C₃×Dic₅	#41	1	1	yes	characteristic
34	C₂×C₃×Q₈	AGL(1,𝔽₅)	1	1	yes	characteristic
35	C₂×C₃×C₈	—	5	1	no	—
36	C₂×C₃×C₈	—	5	3	no	—
37	C₃×C₄⋊C₄	—	5	3	no	—
38	C₃×C₄²	—	5	3	no	—
39	C₅×Q₈	C₃×C₈	1	4	yes	complemented
40	C₂×C₄×C₅	C₂×C₃×C₄	1	3	yes	—
41	C₂×Dic₅	C₂×C₃×C₄	1	3	yes	—
42	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
43	#170	—	2	3	no	—
44	#170	C₃×Q₈	1	2	yes	direct product factor
45	C₄×Q₈	—	5	1	no	—
46	C₄⋊C₈	—	5	3	no	—
47	C₄×C₈	—	5	3	no	—
48	C₂×C₃×C₅	C₄×Q₈	1	1	yes	characteristic
49	C₂×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
50	C₂×C₃×C₅	#131	1	1	yes	characteristic
51	C₃×Q₈	#170	1	4	yes	direct product factor
52	C₂×C₃×C₄	—	5	1	no	—
53	C₂×C₃×C₄	—	5	3	no	—
54	C₂×C₃×C₄	C₂×AGL(1,𝔽₅)	1	3	yes	—
55	C₃×C₈	—	5	2	no	—
56	C₃×C₈	—	10	3	no	—
57	C₂²×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
58	C₄×C₅	C₂×C₃×C₈	1	6	yes	—
59	Dic₅	—	2	3	no	—
60	Dic₅	C₂×C₃×Q₈	1	1	yes	characteristic
61	Dic₅	C₃×#41	1	1	yes	characteristic
62	C₂×Q₈	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
63	C₂×C₈	—	5	1	no	—
64	C₂×C₈	—	5	3	no	—
65	C₄⋊C₄	—	5	3	no	—
66	C₄²	—	5	3	no	—
67	C₃×C₅	C₈×Q₈	1	1	yes	characteristic, complemented, Hall
68	C₂²×C₃	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
69	C₃×C₄	—	5	1	no	—
70	C₃×C₄	—	5	1	no	—
71	C₃×C₄	—	10	3	no	—
72	C₃×C₄	C₂×#170	1	6	yes	—
73	C₂×C₅	C₃×C₄×Q₈	1	1	yes	characteristic
74	C₂×C₅	C₂²×C₃×C₈	1	1	yes	characteristic
75	C₂×C₅	C₃×#131	1	1	yes	characteristic
76	Q₈	C₃×#170	1	4	yes	direct product factor
77	C₂×C₄	—	5	1	no	—
78	C₂×C₄	—	5	3	no	—
79	C₂×C₄	C₂×C₃×AGL(1,𝔽₅)	1	3	yes	—
80	C₈	—	5	2	no	—
81	C₈	—	10	3	no	—
82	C₂×C₃	#4133	1	1	yes	characteristic
83	C₂×C₃	Q₈×AGL(1,𝔽₅)	1	1	yes	characteristic
84	C₂×C₃	C₂²×#170	1	1	yes	characteristic
85	C₅	C₃×C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
86	C₂²	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
87	C₄	—	5	1	no	—
88	C₄	—	5	1	no	—
89	C₄	—	10	3	no	—
90	C₄	C₂×C₃×#170	1	6	yes	—
91	C₃	Q₈×#170	1	1	yes	characteristic, direct product factor, Sylow
92	C₂	C₃×#4133	1	1	yes	characteristic
93	C₂	C₃×Q₈×AGL(1,𝔽₅)	1	1	yes	characteristic
94	C₂	C₂²×C₃×#170	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂²×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂²×C₃×C₈
Frattini	C₂²	C₂²×C₃×AGL(1,𝔽₅)
Fitting	C₂×C₃×C₅×Q₈	C₄
socle, prime radical	C₂²×C₃×C₅	C₂²×C₄
cosocle	C₂×Dic₅	C₂³×C₃
hypercenter	C₂×C₃×Q₈	AGL(1,𝔽₅)
hypocenter	C₅	C₃×C₈×Q₈
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×Q₈

Complements

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

N	H ≅ G/N	type
4. Q₈×#170	C₃	direct (N × H)
14. C₃×C₅×Q₈	C₈	semidirect
19. C₃×#170	Q₈	direct (N × H)
39. C₅×Q₈	C₃×C₈	semidirect
44. #170	C₃×Q₈	direct (N × H)
51. C₃×Q₈	#170	direct (N × H)
67. C₃×C₅	C₈×Q₈	semidirect
76. Q₈	C₃×#170	direct (N × H)
85. C₅	C₃×C₈×Q₈	semidirect
91. C₃	Q₈×#170	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×Q₈×Dic₅	2
C₃×C₄⋊#170	2
C₃×C₄×#170	2
Q₈×#170	3
C₃×C₈×Q₈	5

Subgroups — C3×Q8×#170 #11752319

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×Q₈×#170 #11752319