Subgroups — C₂×C₇²×C₈×C₃₁ #72201596

Order 24304.

220 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇²×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₇²×C₈×C₃₁	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₇×C₈×C₃₁	C₇	1	8	yes	maximal, direct product factor
4	C₂²×C₇²×C₃₁	C₄	1	1	yes	characteristic
5	C₄×C₇²×C₃₁	C₄	1	1	yes	characteristic
6	C₄×C₇²×C₃₁	C₂²	1	1	yes	characteristic
7	C₂×C₇²×C₃₁	C₈	1	2	yes	direct product factor
8	C₂×C₇²×C₃₁	C₂×C₄	1	1	yes	characteristic
9	C₂×C₄×C₇×C₃₁	C₂×C₇	1	8	yes	—
10	C₇×C₈×C₃₁	C₂×C₇	1	16	yes	direct product factor
11	C₇²×C₃₁	C₂×C₈	1	1	yes	characteristic, direct product factor, Hall
12	C₂²×C₇×C₃₁	C₄×C₇	1	8	yes	—
13	C₄×C₇×C₃₁	C₄×C₇	1	8	yes	—
14	C₄×C₇×C₃₁	C₂²×C₇	1	8	yes	—
15	C₂×C₇²×C₈	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂×C₈×C₃₁	C₇²	1	1	yes	characteristic, direct product factor, Hall
17	C₂×C₇×C₃₁	C₇×C₈	1	16	yes	direct product factor
18	C₂×C₇×C₃₁	C₂×C₄×C₇	1	8	yes	—
19	C₂×C₄×C₇²	C₂×C₃₁	1	1	yes	characteristic
20	C₇²×C₈	C₂×C₃₁	1	2	yes	direct product factor
21	C₂×C₄×C₃₁	C₂×C₇²	1	1	yes	characteristic
22	C₈×C₃₁	C₂×C₇²	1	2	yes	direct product factor
23	C₇×C₃₁	C₂×C₇×C₈	1	8	yes	direct product factor
24	C₂²×C₇²	C₄×C₃₁	1	1	yes	characteristic
25	C₄×C₇²	C₄×C₃₁	1	1	yes	characteristic
26	C₄×C₇²	C₂²×C₃₁	1	1	yes	characteristic
27	C₂²×C₃₁	C₄×C₇²	1	1	yes	characteristic
28	C₄×C₃₁	C₄×C₇²	1	1	yes	characteristic
29	C₄×C₃₁	C₂²×C₇²	1	1	yes	characteristic
30	C₂×C₇×C₈	C₇×C₃₁	1	8	yes	direct product factor
31	C₂×C₇²	C₈×C₃₁	1	2	yes	direct product factor
32	C₂×C₇²	C₂×C₄×C₃₁	1	1	yes	characteristic
33	C₂×C₃₁	C₇²×C₈	1	2	yes	direct product factor
34	C₂×C₃₁	C₂×C₄×C₇²	1	1	yes	characteristic
35	C₂×C₄×C₇	C₂×C₇×C₃₁	1	8	yes	—
36	C₇×C₈	C₂×C₇×C₃₁	1	16	yes	direct product factor
37	C₇²	C₂×C₈×C₃₁	1	1	yes	characteristic, direct product factor, Sylow
38	C₃₁	C₂×C₇²×C₈	1	1	yes	characteristic, direct product factor, Sylow
39	C₂²×C₇	C₄×C₇×C₃₁	1	8	yes	—
40	C₄×C₇	C₄×C₇×C₃₁	1	8	yes	—
41	C₄×C₇	C₂²×C₇×C₃₁	1	8	yes	—
42	C₂×C₈	C₇²×C₃₁	1	1	yes	characteristic, direct product factor, Sylow
43	C₂×C₇	C₇×C₈×C₃₁	1	16	yes	direct product factor
44	C₂×C₇	C₂×C₄×C₇×C₃₁	1	8	yes	—
45	C₂×C₄	C₂×C₇²×C₃₁	1	1	yes	characteristic
46	C₈	C₂×C₇²×C₃₁	1	2	yes	direct product factor
47	C₇	C₂×C₇×C₈×C₃₁	1	8	yes	direct product factor
48	C₂²	C₄×C₇²×C₃₁	1	1	yes	characteristic
49	C₄	C₄×C₇²×C₃₁	1	1	yes	characteristic
50	C₄	C₂²×C₇²×C₃₁	1	1	yes	characteristic
51	C₂	C₇²×C₈×C₃₁	1	2	yes	direct product factor
52	C₂	C₂×C₄×C₇²×C₃₁	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄	C₂²×C₇²×C₃₁
socle, prime radical	C₂²×C₇²×C₃₁	C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇²×C₈×C₃₁	C₂	direct (N × H)
3. C₂×C₇×C₈×C₃₁	C₇	direct (N × H)
7. C₂×C₇²×C₃₁	C₈	direct (N × H)
10. C₇×C₈×C₃₁	C₂×C₇	direct (N × H)
11. C₇²×C₃₁	C₂×C₈	direct (N × H)
15. C₂×C₇²×C₈	C₃₁	direct (N × H)
16. C₂×C₈×C₃₁	C₇²	direct (N × H)
17. C₂×C₇×C₃₁	C₇×C₈	direct (N × H)
20. C₇²×C₈	C₂×C₃₁	direct (N × H)
22. C₈×C₃₁	C₂×C₇²	direct (N × H)
23. C₇×C₃₁	C₂×C₇×C₈	direct (N × H)
30. C₂×C₇×C₈	C₇×C₃₁	direct (N × H)
31. C₂×C₇²	C₈×C₃₁	direct (N × H)
33. C₂×C₃₁	C₇²×C₈	direct (N × H)
36. C₇×C₈	C₂×C₇×C₃₁	direct (N × H)
37. C₇²	C₂×C₈×C₃₁	direct (N × H)
38. C₃₁	C₂×C₇²×C₈	direct (N × H)
42. C₂×C₈	C₇²×C₃₁	direct (N × H)
43. C₂×C₇	C₇×C₈×C₃₁	direct (N × H)
46. C₈	C₂×C₇²×C₃₁	direct (N × H)
47. C₇	C₂×C₇×C₈×C₃₁	direct (N × H)
51. C₂	C₇²×C₈×C₃₁	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×C₄×C₇²×C₃₁	2
C₇²×C₈×C₃₁	2
C₂×C₇×C₈×C₃₁	7
C₂×C₇²×C₈	31

Subgroups — C2×C72×C8×C31 #72201596

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇²×C₈×C₃₁ #72201596