Subgroups — C₂²×C₄³×C₆₇ #30713236

Order 17152.

8094 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄²×C₆₇	C₂	1	7	yes	maximal
2	C₂×C₄³×C₆₇	C₂	1	24	yes	maximal, direct product factor
3	C₂⁴×C₄×C₆₇	C₂²	1	7	yes	—
4	C₂²×C₄²×C₆₇	C₄	1	112	yes	direct product factor
5	C₂²×C₄²×C₆₇	C₂²	1	84	yes	—
6	C₄³×C₆₇	C₂²	1	64	yes	direct product factor
7	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
8	C₂³×C₄×C₆₇	C₂×C₄	1	168	yes	—
9	C₂³×C₄×C₆₇	C₂³	1	42	yes	—
10	C₂×C₄²×C₆₇	C₂×C₄	1	672	yes	direct product factor
11	C₂×C₄²×C₆₇	C₂³	1	112	yes	—
12	C₂⁴×C₆₇	C₂²×C₄	1	28	yes	—
13	C₂⁴×C₆₇	C₂⁴	1	3	yes	—
14	C₂²×C₄×C₆₇	C₄²	1	448	yes	direct product factor
15	C₂²×C₄×C₆₇	C₂²×C₄	1	504	yes	—
16	C₂²×C₄×C₆₇	C₂⁴	1	28	yes	—
17	C₄²×C₆₇	C₂²×C₄	1	448	yes	direct product factor
18	C₂³×C₆₇	C₂×C₄²	1	112	yes	—
19	C₂³×C₆₇	C₂³×C₄	1	42	yes	—
20	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
21	C₂×C₄×C₆₇	C₂×C₄²	1	672	yes	direct product factor
22	C₂×C₄×C₆₇	C₂³×C₄	1	168	yes	—
23	C₂²×C₆₇	C₄³	1	64	yes	direct product factor
24	C₂²×C₆₇	C₂²×C₄²	1	84	yes	—
25	C₂²×C₆₇	C₂⁴×C₄	1	7	yes	—
26	C₄×C₆₇	C₂²×C₄²	1	112	yes	direct product factor
27	C₂²×C₄³	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂×C₆₇	C₂×C₄³	1	24	yes	direct product factor
29	C₂×C₆₇	C₂³×C₄²	1	7	yes	—
30	C₂³×C₄²	C₂×C₆₇	1	7	yes	—
31	C₂×C₄³	C₂×C₆₇	1	24	yes	direct product factor
32	C₆₇	C₂²×C₄³	1	1	yes	characteristic, direct product factor, Sylow
33	C₂⁴×C₄	C₂²×C₆₇	1	7	yes	—
34	C₂²×C₄²	C₄×C₆₇	1	112	yes	direct product factor
35	C₂²×C₄²	C₂²×C₆₇	1	84	yes	—
36	C₄³	C₂²×C₆₇	1	64	yes	direct product factor
37	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
38	C₂³×C₄	C₂×C₄×C₆₇	1	168	yes	—
39	C₂³×C₄	C₂³×C₆₇	1	42	yes	—
40	C₂×C₄²	C₂×C₄×C₆₇	1	672	yes	direct product factor
41	C₂×C₄²	C₂³×C₆₇	1	112	yes	—
42	C₂⁴	C₂²×C₄×C₆₇	1	28	yes	—
43	C₂⁴	C₂⁴×C₆₇	1	3	yes	—
44	C₂²×C₄	C₄²×C₆₇	1	448	yes	direct product factor
45	C₂²×C₄	C₂²×C₄×C₆₇	1	504	yes	—
46	C₂²×C₄	C₂⁴×C₆₇	1	28	yes	—
47	C₄²	C₂²×C₄×C₆₇	1	448	yes	direct product factor
48	C₂³	C₂×C₄²×C₆₇	1	112	yes	—
49	C₂³	C₂³×C₄×C₆₇	1	42	yes	—
50	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
51	C₂×C₄	C₂×C₄²×C₆₇	1	672	yes	direct product factor
52	C₂×C₄	C₂³×C₄×C₆₇	1	168	yes	—
53	C₂²	C₄³×C₆₇	1	64	yes	direct product factor
54	C₂²	C₂²×C₄²×C₆₇	1	84	yes	—
55	C₂²	C₂⁴×C₄×C₆₇	1	7	yes	—
56	C₄	C₂²×C₄²×C₆₇	1	112	yes	direct product factor
57	C₂	C₂×C₄³×C₆₇	1	24	yes	direct product factor
58	C₂	C₂³×C₄²×C₆₇	1	7	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂³	C₂⁵×C₆₇
socle, prime radical	C₂⁵×C₆₇	C₂³
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄³×C₆₇	C₂	direct (N × H)
4. C₂²×C₄²×C₆₇	C₄	direct (N × H)
6. C₄³×C₆₇	C₂²	direct (N × H)
10. C₂×C₄²×C₆₇	C₂×C₄	direct (N × H)
14. C₂²×C₄×C₆₇	C₄²	direct (N × H)
17. C₄²×C₆₇	C₂²×C₄	direct (N × H)
21. C₂×C₄×C₆₇	C₂×C₄²	direct (N × H)
23. C₂²×C₆₇	C₄³	direct (N × H)
26. C₄×C₆₇	C₂²×C₄²	direct (N × H)
27. C₂²×C₄³	C₆₇	direct (N × H)
28. C₂×C₆₇	C₂×C₄³	direct (N × H)
31. C₂×C₄³	C₂×C₆₇	direct (N × H)
32. C₆₇	C₂²×C₄³	direct (N × H)
34. C₂²×C₄²	C₄×C₆₇	direct (N × H)
36. C₄³	C₂²×C₆₇	direct (N × H)
40. C₂×C₄²	C₂×C₄×C₆₇	direct (N × H)
44. C₂²×C₄	C₄²×C₆₇	direct (N × H)
47. C₄²	C₂²×C₄×C₆₇	direct (N × H)
51. C₂×C₄	C₂×C₄²×C₆₇	direct (N × H)
53. C₂²	C₄³×C₆₇	direct (N × H)
56. C₄	C₂²×C₄²×C₆₇	direct (N × H)
57. C₂	C₂×C₄³×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×C₄²×C₆₇	2
C₂×C₄³×C₆₇	2
C₂²×C₄³	67

Subgroups — C22×C43×C67 #30713236

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄³×C₆₇ #30713236