Subgroups — C₃²×C₄×C₅×Q₈ #13088228

Order 1440.

420 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₅×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₅×C₄⋊C₄	C₂	1	3	yes	maximal
3	C₃²×C₄²×C₅	C₂	1	3	yes	maximal
4	C₃×C₄×C₅×Q₈	C₃	1	4	yes	maximal, direct product factor
5	C₃²×C₅×Q₈	C₄	1	4	yes	direct product factor
6	C₂×C₃²×C₄×C₅	C₂²	1	3	yes	—
7	C₂×C₃²×C₄×C₅	C₂²	1	3	yes	—
8	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
9	C₃²×C₄×Q₈	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂×C₃×C₅×Q₈	C₂×C₃	1	4	yes	—
11	C₃×C₅×C₄⋊C₄	C₂×C₃	1	12	yes	—
12	C₃×C₄²×C₅	C₂×C₃	1	12	yes	—
13	C₂²×C₃²×C₅	C₂³	1	1	yes	characteristic
14	C₃²×C₄×C₅	—	2	3	no	—
15	C₃²×C₄×C₅	C₂×C₄	1	6	yes	—
16	C₃²×C₄×C₅	Q₈	1	2	yes	direct product factor
17	C₄×C₅×Q₈	C₃²	1	1	yes	characteristic, direct product factor, Hall
18	C₂×C₃²×Q₈	C₂×C₅	1	1	yes	characteristic
19	C₃²×C₄⋊C₄	C₂×C₅	1	3	yes	—
20	C₃²×C₄²	C₂×C₅	1	3	yes	—
21	C₃×C₅×Q₈	C₃×C₄	1	16	yes	direct product factor
22	C₂×C₃×C₄×C₅	C₂²×C₃	1	12	yes	—
23	C₂×C₃×C₄×C₅	C₂²×C₃	1	12	yes	—
24	C₂×C₃×C₄×C₅	C₂²×C₃	1	4	yes	—
25	C₃×C₄×Q₈	C₃×C₅	1	4	yes	direct product factor
26	C₂×C₃²×C₅	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₃²×C₅	C₂×Q₈	1	1	yes	characteristic
28	C₂×C₃²×C₅	#41	1	1	yes	characteristic
29	C₂×C₅×Q₈	C₂×C₃²	1	1	yes	characteristic
30	C₅×C₄⋊C₄	C₂×C₃²	1	3	yes	—
31	C₄²×C₅	C₂×C₃²	1	3	yes	—
32	C₃²×Q₈	C₄×C₅	1	4	yes	direct product factor
33	C₂×C₃²×C₄	C₂²×C₅	1	3	yes	—
34	C₂×C₃²×C₄	C₂²×C₅	1	3	yes	—
35	C₂×C₃²×C₄	C₂²×C₅	1	1	yes	characteristic
36	C₂²×C₃×C₅	C₂³×C₃	1	4	yes	—
37	C₃×C₄×C₅	—	2	12	no	—
38	C₃×C₄×C₅	C₂×C₃×C₄	1	24	yes	—
39	C₃×C₄×C₅	C₃×Q₈	1	8	yes	direct product factor
40	C₂×C₃×Q₈	C₂×C₃×C₅	1	4	yes	—
41	C₃×C₄⋊C₄	C₂×C₃×C₅	1	12	yes	—
42	C₃×C₄²	C₂×C₃×C₅	1	12	yes	—
43	C₃²×C₅	C₄×Q₈	1	1	yes	characteristic, direct product factor, Hall
44	C₅×Q₈	C₃²×C₄	1	4	yes	direct product factor
45	C₂×C₄×C₅	C₂²×C₃²	1	3	yes	—
46	C₂×C₄×C₅	C₂²×C₃²	1	3	yes	—
47	C₂×C₄×C₅	C₂²×C₃²	1	1	yes	characteristic
48	C₂²×C₃²	C₂³×C₅	1	1	yes	characteristic
49	C₃²×C₄	—	2	3	no	—
50	C₃²×C₄	C₂×C₄×C₅	1	6	yes	—
51	C₃²×C₄	C₅×Q₈	1	2	yes	direct product factor
52	C₄×Q₈	C₃²×C₅	1	1	yes	characteristic, direct product factor, Sylow
53	C₂×C₃×C₅	C₂²×C₃×C₄	1	4	yes	—
54	C₂×C₃×C₅	C₂×C₃×Q₈	1	4	yes	—
55	C₂×C₃×C₅	C₃×#41	1	4	yes	—
56	C₃×Q₈	C₃×C₄×C₅	1	16	yes	direct product factor
57	C₂×C₃×C₄	C₂²×C₃×C₅	1	12	yes	—
58	C₂×C₃×C₄	C₂²×C₃×C₅	1	12	yes	—
59	C₂×C₃×C₄	C₂²×C₃×C₅	1	4	yes	—
60	C₂²×C₅	C₂³×C₃²	1	1	yes	characteristic
61	C₄×C₅	—	2	3	no	—
62	C₄×C₅	C₂×C₃²×C₄	1	6	yes	—
63	C₄×C₅	C₃²×Q₈	1	2	yes	direct product factor
64	C₂×C₃²	C₂²×C₄×C₅	1	1	yes	characteristic
65	C₂×C₃²	C₂×C₅×Q₈	1	1	yes	characteristic
66	C₂×C₃²	C₅×#41	1	1	yes	characteristic
67	C₂×Q₈	C₂×C₃²×C₅	1	1	yes	characteristic
68	C₄⋊C₄	C₂×C₃²×C₅	1	3	yes	—
69	C₄²	C₂×C₃²×C₅	1	3	yes	—
70	C₃×C₅	C₃×C₄×Q₈	1	4	yes	direct product factor
71	C₂²×C₃	C₂³×C₃×C₅	1	4	yes	—
72	C₃×C₄	—	2	12	no	—
73	C₃×C₄	C₂×C₃×C₄×C₅	1	24	yes	—
74	C₃×C₄	C₃×C₅×Q₈	1	8	yes	direct product factor
75	C₂×C₅	C₂²×C₃²×C₄	1	1	yes	characteristic
76	C₂×C₅	C₂×C₃²×Q₈	1	1	yes	characteristic
77	C₂×C₅	C₃²×#41	1	1	yes	characteristic
78	C₃²	C₄×C₅×Q₈	1	1	yes	characteristic, direct product factor, Sylow
79	Q₈	C₃²×C₄×C₅	1	4	yes	direct product factor
80	C₂×C₄	C₂²×C₃²×C₅	1	3	yes	—
81	C₂×C₄	C₂²×C₃²×C₅	1	3	yes	—
82	C₂×C₄	C₂²×C₃²×C₅	1	1	yes	characteristic
83	C₂×C₃	C₂²×C₃×C₄×C₅	1	4	yes	—
84	C₂×C₃	C₂×C₃×C₅×Q₈	1	4	yes	—
85	C₂×C₃	C₃×C₅×#41	1	4	yes	—
86	C₅	C₃²×C₄×Q₈	1	1	yes	characteristic, direct product factor, Sylow
87	C₂²	C₂³×C₃²×C₅	1	1	yes	characteristic
88	C₄	—	2	3	no	—
89	C₄	C₂×C₃²×C₄×C₅	1	6	yes	—
90	C₄	C₃²×C₅×Q₈	1	2	yes	direct product factor
91	C₃	C₃×C₄×C₅×Q₈	1	4	yes	direct product factor
92	C₂	C₂²×C₃²×C₄×C₅	1	1	yes	characteristic
93	C₂	C₂×C₃²×C₅×Q₈	1	1	yes	characteristic
94	C₂	C₃²×C₅×#41	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃²×C₄×C₅	C₂²
commutator	C₂	C₂²×C₃²×C₄×C₅
Frattini, cosocle	C₂²	C₂³×C₃²×C₅
socle, prime radical	C₂²×C₃²×C₅	C₂³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃²×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃×C₄×C₅×Q₈	C₃	direct (N × H)
5. C₃²×C₅×Q₈	C₄	direct (N × H)
9. C₃²×C₄×Q₈	C₅	direct (N × H)
16. C₃²×C₄×C₅	Q₈	direct (N × H)
17. C₄×C₅×Q₈	C₃²	direct (N × H)
21. C₃×C₅×Q₈	C₃×C₄	direct (N × H)
25. C₃×C₄×Q₈	C₃×C₅	direct (N × H)
32. C₃²×Q₈	C₄×C₅	direct (N × H)
39. C₃×C₄×C₅	C₃×Q₈	direct (N × H)
43. C₃²×C₅	C₄×Q₈	direct (N × H)
44. C₅×Q₈	C₃²×C₄	direct (N × H)
51. C₃²×C₄	C₅×Q₈	direct (N × H)
52. C₄×Q₈	C₃²×C₅	direct (N × H)
56. C₃×Q₈	C₃×C₄×C₅	direct (N × H)
63. C₄×C₅	C₃²×Q₈	direct (N × H)
70. C₃×C₅	C₃×C₄×Q₈	direct (N × H)
74. C₃×C₄	C₃×C₅×Q₈	direct (N × H)
78. C₃²	C₄×C₅×Q₈	direct (N × H)
79. Q₈	C₃²×C₄×C₅	direct (N × H)
86. C₅	C₃²×C₄×Q₈	direct (N × H)
90. C₄	C₃²×C₅×Q₈	direct (N × H)
91. C₃	C₃×C₄×C₅×Q₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃²×C₅×Q₈	2
C₃²×C₅×C₄⋊C₄	2
C₃²×C₄²×C₅	2
C₃×C₄×C₅×Q₈	3
C₃²×C₄×Q₈	5

Subgroups — C32×C4×C5×Q8 #13088228

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₄×C₅×Q₈ #13088228