Subgroups — C₂×C₇×ΓL(1,𝔽₂₅) #10621297

Order 672.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₃×C₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×ΓL(1,𝔽₂₅)	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₇×M₁₆	—	3	1	no	maximal, Hall
6	C₂²×D₃×C₇	C₄	1	1	yes	characteristic
7	C₂×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
8	C₂×C₇×Dic₃	C₄	1	1	yes	characteristic
9	C₄×D₃×C₇	C₄	1	2	yes	—
10	C₄×D₃×C₇	C₂²	1	2	yes	—
11	C₃×C₇×C₈	C₂²	1	2	yes	complemented
12	C₇×C₃⋊C₈	C₂²	1	2	yes	complemented
13	C₂²×C₄×C₇	—	3	1	no	—
14	C₇×M₁₆	—	3	4	no	—
15	C₂×C₇×C₈	—	3	1	no	—
16	C₂×C₇×C₈	D₃	1	1	yes	characteristic, complemented
17	C₂×ΓL(1,𝔽₂₅)	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂²×C₃×C₇	C₂×C₄	1	1	yes	characteristic
19	C₂×D₃×C₇	—	2	2	no	—
20	C₂×D₃×C₇	C₂×C₄	1	2	yes	—
21	C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
22	C₃×C₄×C₇	C₂³	1	1	yes	characteristic
23	C₇×Dic₃	C₂×C₄	1	2	yes	—
24	C₂³×C₇	—	3	1	no	—
25	C₂×C₄×C₇	—	3	1	no	—
26	C₂×C₄×C₇	—	3	2	no	—
27	C₂×C₄×C₇	—	3	2	no	—
28	C₂×C₄×C₇	C₂×D₃	1	1	yes	characteristic
29	C₇×C₈	—	3	2	no	—
30	C₇×C₈	C₂×D₃	1	2	yes	complemented
31	C₂×C₄×D₃	C₂×C₇	1	1	yes	characteristic
32	C₂×C₃×C₈	C₂×C₇	1	1	yes	characteristic, complemented
33	C₂×C₃⋊C₈	C₂×C₇	1	1	yes	characteristic, complemented
34	ΓL(1,𝔽₂₅)	C₂×C₇	1	4	yes	direct product factor
35	C₂×C₃×C₇	M₁₆	1	2	yes	complemented
36	C₂×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
37	D₃×C₇	—	2	2	no	—
38	C₂×M₁₆	—	3	1	no	Sylow
39	C₂²×C₇	—	3	2	no	—
40	C₂²×C₇	—	6	2	no	—
41	C₂²×C₇	C₄×D₃	1	1	yes	characteristic
42	C₄×C₇	—	3	2	no	—
43	C₄×C₇	C₄×D₃	1	1	yes	characteristic
44	C₄×C₇	C₂²×D₃	1	1	yes	characteristic
45	C₂²×D₃	C₄×C₇	1	1	yes	characteristic
46	C₂×C₃×C₄	C₂²×C₇	1	1	yes	characteristic
47	C₂×Dic₃	C₄×C₇	1	1	yes	characteristic
48	C₄×D₃	C₄×C₇	1	2	yes	—
49	C₄×D₃	C₂²×C₇	1	2	yes	—
50	C₃×C₈	C₂²×C₇	1	2	yes	complemented
51	C₃⋊C₈	C₂²×C₇	1	2	yes	complemented
52	C₃×C₇	C₂×M₁₆	1	1	yes	characteristic, complemented, Hall
53	C₂²×C₄	—	3	1	no	—
54	M₁₆	—	3	4	no	—
55	C₂×C₈	—	3	1	no	—
56	C₂×C₈	D₃×C₇	1	1	yes	characteristic, complemented
57	C₂×C₇	—	6	2	no	—
58	C₂×C₇	ΓL(1,𝔽₂₅)	1	2	yes	direct product factor
59	C₂×C₇	C₂×C₄×D₃	1	1	yes	characteristic
60	C₂²×C₃	C₂×C₄×C₇	1	1	yes	characteristic
61	C₂×D₃	—	2	2	no	—
62	C₂×D₃	C₂×C₄×C₇	1	2	yes	—
63	C₃×C₄	C₂×C₄×C₇	1	1	yes	characteristic
64	C₃×C₄	C₂³×C₇	1	1	yes	characteristic
65	Dic₃	C₂×C₄×C₇	1	2	yes	—
66	C₂³	—	3	1	no	—
67	C₂×C₄	—	3	1	no	—
68	C₂×C₄	—	3	2	no	—
69	C₂×C₄	—	3	2	no	—
70	C₂×C₄	C₂×D₃×C₇	1	1	yes	characteristic
71	C₈	—	3	2	no	—
72	C₈	C₂×D₃×C₇	1	2	yes	complemented
73	C₇	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic, direct product factor, Sylow
74	C₂×C₃	C₇×M₁₆	1	2	yes	complemented
75	C₂×C₃	C₂²×C₄×C₇	1	1	yes	characteristic
76	D₃	—	2	2	no	—
77	C₂²	—	3	2	no	—
78	C₂²	—	6	2	no	—
79	C₂²	C₄×D₃×C₇	1	1	yes	characteristic
80	C₄	—	3	2	no	—
81	C₄	C₄×D₃×C₇	1	1	yes	characteristic
82	C₄	C₂²×D₃×C₇	1	1	yes	characteristic
83	C₃	C₂×C₇×M₁₆	1	1	yes	characteristic, complemented, Sylow
84	C₂	—	6	2	no	—
85	C₂	C₇×ΓL(1,𝔽₂₅)	1	2	yes	direct product factor
86	C₂	C₂×C₄×D₃×C₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	C₂×D₃
commutator	C₂×C₃	C₂²×C₄×C₇
Frattini	C₄	C₂²×D₃×C₇
Fitting	C₂×C₃×C₇×C₈	C₂
socle	C₂²×C₃×C₇	C₂×C₄
prime radical	C₂²×D₃×C₇	C₄
cosocle	C₃×C₄	C₂³×C₇
hypercenter	C₂×C₇×C₈	D₃
hypocenter	C₃	C₂×C₇×M₁₆
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₇×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₇×C₈	C₂	semidirect
3. C₂×C₇×C₃⋊C₈	C₂	semidirect
4. C₇×ΓL(1,𝔽₂₅)	C₂	direct (N × H)
11. C₃×C₇×C₈	C₂²	semidirect
12. C₇×C₃⋊C₈	C₂²	semidirect
16. C₂×C₇×C₈	D₃	semidirect
17. C₂×ΓL(1,𝔽₂₅)	C₇	direct (N × H)
30. C₇×C₈	C₂×D₃	semidirect
32. C₂×C₃×C₈	C₂×C₇	semidirect
33. C₂×C₃⋊C₈	C₂×C₇	semidirect
34. ΓL(1,𝔽₂₅)	C₂×C₇	direct (N × H)
35. C₂×C₃×C₇	M₁₆	semidirect
50. C₃×C₈	C₂²×C₇	semidirect
51. C₃⋊C₈	C₂²×C₇	semidirect
52. C₃×C₇	C₂×M₁₆	semidirect
56. C₂×C₈	D₃×C₇	semidirect
58. C₂×C₇	ΓL(1,𝔽₂₅)	direct (N × H)
72. C₈	C₂×D₃×C₇	semidirect
73. C₇	C₂×ΓL(1,𝔽₂₅)	direct (N × H)
74. C₂×C₃	C₇×M₁₆	semidirect
83. C₃	C₂×C₇×M₁₆	semidirect
85. C₂	C₇×ΓL(1,𝔽₂₅)	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×D₃×C₇	2
C₂×C₃×C₇×C₈	2
C₂×C₇×C₃⋊C₈	2
C₇×ΓL(1,𝔽₂₅)	2
C₂×C₇×M₁₆	3
C₂×ΓL(1,𝔽₂₅)	7

Subgroups — C2×C7×ΓL(1,𝔽25) #10621297

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×ΓL(1,𝔽₂₅) #10621297