Subgroups — C₂×C₇²×C₁₂₈ #72190883

Order 12544.

230 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇²×C₆₄	C₂	1	1	yes	maximal, characteristic
2	C₇²×C₁₂₈	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₇²×C₃₂	C₄	1	1	yes	characteristic
4	C₇²×C₆₄	C₄	1	1	yes	characteristic
5	C₇²×C₆₄	C₂²	1	1	yes	characteristic
6	C₂×C₇×C₁₂₈	C₇	1	8	yes	maximal, direct product factor
7	C₂×C₇²×C₁₆	C₈	1	1	yes	characteristic
8	C₇²×C₃₂	C₈	1	1	yes	characteristic
9	C₇²×C₃₂	C₂×C₄	1	1	yes	characteristic
10	C₂×C₇×C₆₄	C₂×C₇	1	8	yes	—
11	C₇×C₁₂₈	C₂×C₇	1	16	yes	direct product factor
12	C₂×C₇²×C₈	C₁₆	1	1	yes	characteristic
13	C₇²×C₁₆	C₁₆	1	1	yes	characteristic
14	C₇²×C₁₆	C₂×C₈	1	1	yes	characteristic
15	C₂×C₇×C₃₂	C₄×C₇	1	8	yes	—
16	C₇×C₆₄	C₄×C₇	1	8	yes	—
17	C₇×C₆₄	C₂²×C₇	1	8	yes	—
18	C₂×C₄×C₇²	C₃₂	1	1	yes	characteristic
19	C₇²×C₈	C₃₂	1	1	yes	characteristic
20	C₇²×C₈	C₂×C₁₆	1	1	yes	characteristic
21	C₂×C₁₂₈	C₇²	1	1	yes	characteristic, direct product factor, Sylow
22	C₂×C₇×C₁₆	C₇×C₈	1	8	yes	—
23	C₇×C₃₂	C₇×C₈	1	8	yes	—
24	C₇×C₃₂	C₂×C₄×C₇	1	8	yes	—
25	C₂²×C₇²	C₆₄	1	1	yes	characteristic
26	C₄×C₇²	C₆₄	1	1	yes	characteristic
27	C₄×C₇²	C₂×C₃₂	1	1	yes	characteristic
28	C₂×C₆₄	C₂×C₇²	1	1	yes	characteristic
29	C₁₂₈	C₂×C₇²	1	2	yes	direct product factor
30	C₂×C₇×C₈	C₇×C₁₆	1	8	yes	—
31	C₇×C₁₆	C₇×C₁₆	1	8	yes	—
32	C₇×C₁₆	C₂×C₇×C₈	1	8	yes	—
33	C₂×C₇²	C₁₂₈	1	2	yes	direct product factor
34	C₂×C₇²	C₂×C₆₄	1	1	yes	characteristic
35	C₂×C₃₂	C₄×C₇²	1	1	yes	characteristic
36	C₆₄	C₄×C₇²	1	1	yes	characteristic
37	C₆₄	C₂²×C₇²	1	1	yes	characteristic
38	C₂×C₄×C₇	C₇×C₃₂	1	8	yes	—
39	C₇×C₈	C₇×C₃₂	1	8	yes	—
40	C₇×C₈	C₂×C₇×C₁₆	1	8	yes	—
41	C₇²	C₂×C₁₂₈	1	1	yes	characteristic, direct product factor, Sylow
42	C₂×C₁₆	C₇²×C₈	1	1	yes	characteristic
43	C₃₂	C₇²×C₈	1	1	yes	characteristic
44	C₃₂	C₂×C₄×C₇²	1	1	yes	characteristic
45	C₂²×C₇	C₇×C₆₄	1	8	yes	—
46	C₄×C₇	C₇×C₆₄	1	8	yes	—
47	C₄×C₇	C₂×C₇×C₃₂	1	8	yes	—
48	C₂×C₈	C₇²×C₁₆	1	1	yes	characteristic
49	C₁₆	C₇²×C₁₆	1	1	yes	characteristic
50	C₁₆	C₂×C₇²×C₈	1	1	yes	characteristic
51	C₂×C₇	C₇×C₁₂₈	1	16	yes	direct product factor
52	C₂×C₇	C₂×C₇×C₆₄	1	8	yes	—
53	C₂×C₄	C₇²×C₃₂	1	1	yes	characteristic
54	C₈	C₇²×C₃₂	1	1	yes	characteristic
55	C₈	C₂×C₇²×C₁₆	1	1	yes	characteristic
56	C₇	C₂×C₇×C₁₂₈	1	8	yes	direct product factor
57	C₂²	C₇²×C₆₄	1	1	yes	characteristic
58	C₄	C₇²×C₆₄	1	1	yes	characteristic
59	C₄	C₂×C₇²×C₃₂	1	1	yes	characteristic
60	C₂	C₇²×C₁₂₈	1	2	yes	direct product factor
61	C₂	C₂×C₇²×C₆₄	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₆₄	C₂²×C₇²
socle, prime radical	C₂²×C₇²	C₆₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇²×C₁₂₈	C₂	direct (N × H)
6. C₂×C₇×C₁₂₈	C₇	direct (N × H)
11. C₇×C₁₂₈	C₂×C₇	direct (N × H)
21. C₂×C₁₂₈	C₇²	direct (N × H)
29. C₁₂₈	C₂×C₇²	direct (N × H)
33. C₂×C₇²	C₁₂₈	direct (N × H)
41. C₇²	C₂×C₁₂₈	direct (N × H)
51. C₂×C₇	C₇×C₁₂₈	direct (N × H)
56. C₇	C₂×C₇×C₁₂₈	direct (N × H)
60. C₂	C₇²×C₁₂₈	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×C₇²×C₆₄	2
C₇²×C₁₂₈	2
C₂×C₇×C₁₂₈	7

Subgroups — C2×C72×C128 #72190883

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇²×C₁₂₈ #72190883