Subgroups — C₂×C₇×C₂₃×C₃₂ #72188939

Order 10304.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×C₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₇×C₂₃×C₃₂	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₇×C₈×C₂₃	C₄	1	1	yes	characteristic
4	C₇×C₁₆×C₂₃	C₄	1	1	yes	characteristic
5	C₇×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
6	C₂×C₂₃×C₃₂	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₄×C₇×C₂₃	C₈	1	1	yes	characteristic
8	C₇×C₈×C₂₃	C₈	1	1	yes	characteristic
9	C₇×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
10	C₂×C₁₆×C₂₃	C₂×C₇	1	1	yes	characteristic
11	C₂₃×C₃₂	C₂×C₇	1	2	yes	direct product factor
12	C₂²×C₇×C₂₃	C₁₆	1	1	yes	characteristic
13	C₄×C₇×C₂₃	C₁₆	1	1	yes	characteristic
14	C₄×C₇×C₂₃	C₂×C₈	1	1	yes	characteristic
15	C₂×C₇×C₃₂	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂×C₈×C₂₃	C₄×C₇	1	1	yes	characteristic
17	C₁₆×C₂₃	C₄×C₇	1	1	yes	characteristic
18	C₁₆×C₂₃	C₂²×C₇	1	1	yes	characteristic
19	C₂×C₇×C₂₃	C₃₂	1	2	yes	direct product factor
20	C₂×C₇×C₂₃	C₂×C₁₆	1	1	yes	characteristic
21	C₂×C₇×C₁₆	C₂×C₂₃	1	1	yes	characteristic
22	C₇×C₃₂	C₂×C₂₃	1	2	yes	direct product factor
23	C₂×C₄×C₂₃	C₇×C₈	1	1	yes	characteristic
24	C₈×C₂₃	C₇×C₈	1	1	yes	characteristic
25	C₈×C₂₃	C₂×C₄×C₇	1	1	yes	characteristic
26	C₇×C₂₃	C₂×C₃₂	1	1	yes	characteristic, direct product factor, Hall
27	C₂×C₇×C₈	C₄×C₂₃	1	1	yes	characteristic
28	C₇×C₁₆	C₄×C₂₃	1	1	yes	characteristic
29	C₇×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
30	C₂²×C₂₃	C₇×C₁₆	1	1	yes	characteristic
31	C₄×C₂₃	C₇×C₁₆	1	1	yes	characteristic
32	C₄×C₂₃	C₂×C₇×C₈	1	1	yes	characteristic
33	C₂×C₃₂	C₇×C₂₃	1	1	yes	characteristic, direct product factor, Sylow
34	C₂×C₄×C₇	C₈×C₂₃	1	1	yes	characteristic
35	C₇×C₈	C₈×C₂₃	1	1	yes	characteristic
36	C₇×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
37	C₂×C₂₃	C₇×C₃₂	1	2	yes	direct product factor
38	C₂×C₂₃	C₂×C₇×C₁₆	1	1	yes	characteristic
39	C₂×C₁₆	C₂×C₇×C₂₃	1	1	yes	characteristic
40	C₃₂	C₂×C₇×C₂₃	1	2	yes	direct product factor
41	C₂²×C₇	C₁₆×C₂₃	1	1	yes	characteristic
42	C₄×C₇	C₁₆×C₂₃	1	1	yes	characteristic
43	C₄×C₇	C₂×C₈×C₂₃	1	1	yes	characteristic
44	C₂₃	C₂×C₇×C₃₂	1	1	yes	characteristic, direct product factor, Sylow
45	C₂×C₈	C₄×C₇×C₂₃	1	1	yes	characteristic
46	C₁₆	C₄×C₇×C₂₃	1	1	yes	characteristic
47	C₁₆	C₂²×C₇×C₂₃	1	1	yes	characteristic
48	C₂×C₇	C₂₃×C₃₂	1	2	yes	direct product factor
49	C₂×C₇	C₂×C₁₆×C₂₃	1	1	yes	characteristic
50	C₂×C₄	C₇×C₈×C₂₃	1	1	yes	characteristic
51	C₈	C₇×C₈×C₂₃	1	1	yes	characteristic
52	C₈	C₂×C₄×C₇×C₂₃	1	1	yes	characteristic
53	C₇	C₂×C₂₃×C₃₂	1	1	yes	characteristic, direct product factor, Sylow
54	C₂²	C₇×C₁₆×C₂₃	1	1	yes	characteristic
55	C₄	C₇×C₁₆×C₂₃	1	1	yes	characteristic
56	C₄	C₂×C₇×C₈×C₂₃	1	1	yes	characteristic
57	C₂	C₇×C₂₃×C₃₂	1	2	yes	direct product factor
58	C₂	C₂×C₇×C₁₆×C₂₃	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₁₆	C₂²×C₇×C₂₃
socle, prime radical	C₂²×C₇×C₂₃	C₁₆
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×C₂₃×C₃₂	C₂	direct (N × H)
6. C₂×C₂₃×C₃₂	C₇	direct (N × H)
11. C₂₃×C₃₂	C₂×C₇	direct (N × H)
15. C₂×C₇×C₃₂	C₂₃	direct (N × H)
19. C₂×C₇×C₂₃	C₃₂	direct (N × H)
22. C₇×C₃₂	C₂×C₂₃	direct (N × H)
26. C₇×C₂₃	C₂×C₃₂	direct (N × H)
33. C₂×C₃₂	C₇×C₂₃	direct (N × H)
37. C₂×C₂₃	C₇×C₃₂	direct (N × H)
40. C₃₂	C₂×C₇×C₂₃	direct (N × H)
44. C₂₃	C₂×C₇×C₃₂	direct (N × H)
48. C₂×C₇	C₂₃×C₃₂	direct (N × H)
53. C₇	C₂×C₂₃×C₃₂	direct (N × H)
57. C₂	C₇×C₂₃×C₃₂	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×C₇×C₁₆×C₂₃	2
C₇×C₂₃×C₃₂	2
C₂×C₂₃×C₃₂	7
C₂×C₇×C₃₂	23

Subgroups — C2×C7×C23×C32 #72188939

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×C₂₃×C₃₂ #72188939