Subgroups — C₂³×C₃₂×C₆₇ #30713222

Order 17152.

542 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃₂×C₆₇	C₂	1	14	yes	maximal, direct product factor
3	C₂³×C₈×C₆₇	C₄	1	1	yes	characteristic
4	C₂²×C₁₆×C₆₇	C₄	1	7	yes	—
5	C₂²×C₁₆×C₆₇	C₂²	1	7	yes	—
6	C₂×C₃₂×C₆₇	C₂²	1	28	yes	direct product factor
7	C₂³×C₄×C₆₇	C₈	1	1	yes	characteristic
8	C₂²×C₈×C₆₇	C₈	1	7	yes	—
9	C₂²×C₈×C₆₇	C₂×C₄	1	7	yes	—
10	C₂×C₁₆×C₆₇	C₂×C₄	1	21	yes	—
11	C₂×C₁₆×C₆₇	C₂³	1	7	yes	—
12	C₃₂×C₆₇	C₂³	1	8	yes	direct product factor
13	C₂⁴×C₆₇	C₁₆	1	1	yes	characteristic
14	C₂²×C₄×C₆₇	C₁₆	1	7	yes	—
15	C₂²×C₄×C₆₇	C₂×C₈	1	7	yes	—
16	C₂×C₈×C₆₇	C₂×C₈	1	21	yes	—
17	C₂×C₈×C₆₇	C₂²×C₄	1	7	yes	—
18	C₁₆×C₆₇	C₂²×C₄	1	7	yes	—
19	C₁₆×C₆₇	C₂⁴	1	1	yes	characteristic
20	C₂³×C₆₇	C₃₂	1	8	yes	direct product factor
21	C₂³×C₆₇	C₂×C₁₆	1	7	yes	—
22	C₂×C₄×C₆₇	C₂×C₁₆	1	21	yes	—
23	C₂×C₄×C₆₇	C₂²×C₈	1	7	yes	—
24	C₈×C₆₇	C₂²×C₈	1	7	yes	—
25	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
26	C₂²×C₆₇	C₂×C₃₂	1	28	yes	direct product factor
27	C₂²×C₆₇	C₂²×C₁₆	1	7	yes	—
28	C₄×C₆₇	C₂²×C₁₆	1	7	yes	—
29	C₄×C₆₇	C₂³×C₈	1	1	yes	characteristic
30	C₂³×C₃₂	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
31	C₂×C₆₇	C₂²×C₃₂	1	14	yes	direct product factor
32	C₂×C₆₇	C₂³×C₁₆	1	1	yes	characteristic
33	C₂³×C₁₆	C₂×C₆₇	1	1	yes	characteristic
34	C₂²×C₃₂	C₂×C₆₇	1	14	yes	direct product factor
35	C₆₇	C₂³×C₃₂	1	1	yes	characteristic, direct product factor, Sylow
36	C₂³×C₈	C₄×C₆₇	1	1	yes	characteristic
37	C₂²×C₁₆	C₄×C₆₇	1	7	yes	—
38	C₂²×C₁₆	C₂²×C₆₇	1	7	yes	—
39	C₂×C₃₂	C₂²×C₆₇	1	28	yes	direct product factor
40	C₂³×C₄	C₈×C₆₇	1	1	yes	characteristic
41	C₂²×C₈	C₈×C₆₇	1	7	yes	—
42	C₂²×C₈	C₂×C₄×C₆₇	1	7	yes	—
43	C₂×C₁₆	C₂×C₄×C₆₇	1	21	yes	—
44	C₂×C₁₆	C₂³×C₆₇	1	7	yes	—
45	C₃₂	C₂³×C₆₇	1	8	yes	direct product factor
46	C₂⁴	C₁₆×C₆₇	1	1	yes	characteristic
47	C₂²×C₄	C₁₆×C₆₇	1	7	yes	—
48	C₂²×C₄	C₂×C₈×C₆₇	1	7	yes	—
49	C₂×C₈	C₂×C₈×C₆₇	1	21	yes	—
50	C₂×C₈	C₂²×C₄×C₆₇	1	7	yes	—
51	C₁₆	C₂²×C₄×C₆₇	1	7	yes	—
52	C₁₆	C₂⁴×C₆₇	1	1	yes	characteristic
53	C₂³	C₃₂×C₆₇	1	8	yes	direct product factor
54	C₂³	C₂×C₁₆×C₆₇	1	7	yes	—
55	C₂×C₄	C₂×C₁₆×C₆₇	1	21	yes	—
56	C₂×C₄	C₂²×C₈×C₆₇	1	7	yes	—
57	C₈	C₂²×C₈×C₆₇	1	7	yes	—
58	C₈	C₂³×C₄×C₆₇	1	1	yes	characteristic
59	C₂²	C₂×C₃₂×C₆₇	1	28	yes	direct product factor
60	C₂²	C₂²×C₁₆×C₆₇	1	7	yes	—
61	C₄	C₂²×C₁₆×C₆₇	1	7	yes	—
62	C₄	C₂³×C₈×C₆₇	1	1	yes	characteristic
63	C₂	C₂²×C₃₂×C₆₇	1	14	yes	direct product factor
64	C₂	C₂³×C₁₆×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₁₆	C₂⁴×C₆₇
socle, prime radical	C₂⁴×C₆₇	C₁₆
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃₂×C₆₇	C₂	direct (N × H)
6. C₂×C₃₂×C₆₇	C₂²	direct (N × H)
12. C₃₂×C₆₇	C₂³	direct (N × H)
20. C₂³×C₆₇	C₃₂	direct (N × H)
26. C₂²×C₆₇	C₂×C₃₂	direct (N × H)
30. C₂³×C₃₂	C₆₇	direct (N × H)
31. C₂×C₆₇	C₂²×C₃₂	direct (N × H)
34. C₂²×C₃₂	C₂×C₆₇	direct (N × H)
35. C₆₇	C₂³×C₃₂	direct (N × H)
39. C₂×C₃₂	C₂²×C₆₇	direct (N × H)
45. C₃₂	C₂³×C₆₇	direct (N × H)
53. C₂³	C₃₂×C₆₇	direct (N × H)
59. C₂²	C₂×C₃₂×C₆₇	direct (N × H)
63. C₂	C₂²×C₃₂×C₆₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₁₆×C₆₇	2
C₂²×C₃₂×C₆₇	2
C₂³×C₃₂	67

Subgroups — C23×C32×C67 #30713222

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₃₂×C₆₇ #30713222