Subgroups — C₂×C₃×C₄×#3861 #72245814

Order 3528.

480 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×#3861	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×#3861	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃×C₄×C₇²	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×#3861	C₃	1	3	yes	maximal, direct product factor
5	C₂×C₃×#3861	C₄	1	2	yes	direct product factor
6	C₂×C₃×#3861	C₂²	1	1	yes	characteristic
7	C₂²×C₃×C₇²	C₂×C₃	1	1	yes	characteristic
8	C₂²×#3861	C₂×C₃	1	3	yes	—
9	C₃×C₄×C₇²	C₂×C₃	1	2	yes	complemented
10	C₄×#3861	C₂×C₃	1	6	yes	direct product factor
11	C₂×C₃×C₄×ΓL(1,𝔽₈)	—	7	8	no	maximal
12	C₃×#3861	C₂×C₄	1	1	yes	characteristic, direct product factor, Hall
13	C₂×C₄×C₇²	C₃²	1	1	yes	characteristic, complemented, Hall
14	C₂×C₃×C₇²	C₃×C₄	1	2	yes	complemented
15	C₂×C₃×C₇²	C₂²×C₃	1	1	yes	characteristic
16	C₂×#3861	C₃×C₄	1	6	yes	direct product factor
17	C₂×#3861	C₂²×C₃	1	3	yes	—
18	C₂²×C₃×ΓL(1,𝔽₈)	—	7	8	no	—
19	C₃×C₄×ΓL(1,𝔽₈)	—	7	16	no	—
20	C₂²×C₇²	C₂×C₃²	1	1	yes	characteristic
21	C₄×C₇²	C₂×C₃²	1	2	yes	complemented
22	C₂×C₃×C₄×C₇	ΓL(1,𝔽₈)	1	8	yes	complemented
23	C₂×C₄×ΓL(1,𝔽₈)	—	7	24	no	—
24	C₃×C₇²	C₂×C₃×C₄	1	1	yes	characteristic, complemented
25	#3861	C₂×C₃×C₄	1	3	yes	direct product factor
26	C₂×C₃×ΓL(1,𝔽₈)	—	7	16	no	—
27	C₂×C₃×ΓL(1,𝔽₈)	—	7	8	no	—
28	C₂×C₇²	C₃²×C₄	1	2	yes	complemented
29	C₂×C₇²	C₂²×C₃²	1	1	yes	characteristic
30	C₂²×C₃×C₇	C₂×ΓL(1,𝔽₈)	1	8	yes	—
31	C₂²×ΓL(1,𝔽₈)	—	7	24	no	—
32	C₃×C₄×C₇	C₂×ΓL(1,𝔽₈)	1	16	yes	complemented
33	C₄×ΓL(1,𝔽₈)	—	7	48	no	—
34	C₂×C₃²×C₄	—	49	1	no	Hall
35	C₃×ΓL(1,𝔽₈)	—	7	8	no	—
36	C₂×C₄×C₇	C₃×ΓL(1,𝔽₈)	1	8	yes	complemented
37	C₇²	C₂×C₃²×C₄	1	1	yes	characteristic, complemented, Sylow
38	C₂×C₃×C₇	C₄×ΓL(1,𝔽₈)	1	16	yes	complemented
39	C₂×C₃×C₇	C₂²×ΓL(1,𝔽₈)	1	8	yes	—
40	C₂×ΓL(1,𝔽₈)	—	7	48	no	—
41	C₂×ΓL(1,𝔽₈)	—	7	24	no	—
42	C₂²×C₃²	—	49	1	no	—
43	C₃²×C₄	—	49	2	no	—
44	C₂²×C₇	C₂×C₃×ΓL(1,𝔽₈)	1	8	yes	—
45	C₄×C₇	C₂×C₃×ΓL(1,𝔽₈)	1	16	yes	complemented
46	C₂×C₃×C₄	—	49	3	no	—
47	C₂×C₃×C₄	#3861	1	1	yes	characteristic, direct product factor
48	C₃×C₇	C₂×C₄×ΓL(1,𝔽₈)	1	8	yes	complemented
49	ΓL(1,𝔽₈)	—	7	24	no	—
50	C₂×C₃²	—	49	2	no	—
51	C₂×C₃²	—	49	1	no	—
52	C₂×C₇	C₃×C₄×ΓL(1,𝔽₈)	1	16	yes	complemented
53	C₂×C₇	C₂²×C₃×ΓL(1,𝔽₈)	1	8	yes	—
54	C₂²×C₃	—	49	3	no	—
55	C₂²×C₃	C₂×#3861	1	1	yes	characteristic
56	C₃×C₄	—	49	6	no	—
57	C₃×C₄	C₂×#3861	1	2	yes	direct product factor
58	C₃²	—	49	1	no	Sylow
59	C₂×C₄	C₃×#3861	1	1	yes	characteristic, direct product factor, Sylow
60	C₇	C₂×C₃×C₄×ΓL(1,𝔽₈)	1	8	yes	complemented
61	C₂×C₃	—	49	6	no	—
62	C₂×C₃	—	49	3	no	—
63	C₂×C₃	C₄×#3861	1	2	yes	direct product factor
64	C₂×C₃	C₂²×#3861	1	1	yes	characteristic
65	C₂²	C₂×C₃×#3861	1	1	yes	characteristic
66	C₄	C₂×C₃×#3861	1	2	yes	direct product factor
67	C₃	—	49	3	no	—
68	C₃	C₂×C₄×#3861	1	1	yes	characteristic, direct product factor
69	C₂	C₃×C₄×#3861	1	2	yes	direct product factor
70	C₂	C₂²×C₃×#3861	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃×C₄	#3861
commutator, hypocenter	C₇²	C₂×C₃²×C₄
Frattini	C₂	C₂²×C₃×#3861
Fitting	C₂×C₃×C₄×C₇²	C₃
socle	C₂²×C₃×C₇²	C₂×C₃
prime radical	C₂²×C₃×#3861	C₂
cosocle	C₂×C₇²	C₂²×C₃²
LCS	G ▷ C₇²
UCS	C₁ ◁ C₂×C₃×C₄

Complements

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

N	H ≅ G/N	type
2. C₃×C₄×#3861	C₂	direct (N × H)
3. C₂×C₃×C₄×C₇²	C₃	semidirect
4. C₂×C₄×#3861	C₃	direct (N × H)
5. C₂×C₃×#3861	C₄	direct (N × H)
9. C₃×C₄×C₇²	C₂×C₃	semidirect
10. C₄×#3861	C₂×C₃	direct (N × H)
12. C₃×#3861	C₂×C₄	direct (N × H)
13. C₂×C₄×C₇²	C₃²	semidirect
14. C₂×C₃×C₇²	C₃×C₄	semidirect
16. C₂×#3861	C₃×C₄	direct (N × H)
21. C₄×C₇²	C₂×C₃²	semidirect
22. C₂×C₃×C₄×C₇	ΓL(1,𝔽₈)	semidirect
24. C₃×C₇²	C₂×C₃×C₄	semidirect
25. #3861	C₂×C₃×C₄	direct (N × H)
28. C₂×C₇²	C₃²×C₄	semidirect
32. C₃×C₄×C₇	C₂×ΓL(1,𝔽₈)	semidirect
36. C₂×C₄×C₇	C₃×ΓL(1,𝔽₈)	semidirect
37. C₇²	C₂×C₃²×C₄	semidirect
38. C₂×C₃×C₇	C₄×ΓL(1,𝔽₈)	semidirect
45. C₄×C₇	C₂×C₃×ΓL(1,𝔽₈)	semidirect
47. C₂×C₃×C₄	#3861	direct (N × H)
48. C₃×C₇	C₂×C₄×ΓL(1,𝔽₈)	semidirect
52. C₂×C₇	C₃×C₄×ΓL(1,𝔽₈)	semidirect
57. C₃×C₄	C₂×#3861	direct (N × H)
59. C₂×C₄	C₃×#3861	direct (N × H)
60. C₇	C₂×C₃×C₄×ΓL(1,𝔽₈)	semidirect
63. C₂×C₃	C₄×#3861	direct (N × H)
66. C₄	C₂×C₃×#3861	direct (N × H)
68. C₃	C₂×C₄×#3861	direct (N × H)
69. C₂	C₃×C₄×#3861	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×#3861	2
C₃×C₄×#3861	2
C₂×C₃×C₄×C₇²	3
C₂×C₄×#3861	3
C₂×C₃×C₄×ΓL(1,𝔽₈)	7

Subgroups — C2×C3×C4×#3861 #72245814

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₄×#3861 #72245814