Subgroups — C₂×ΓL(1,𝔽₈)×C₂₅×D₁₃ #41031245

Order 27300.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	ΓL(1,𝔽₈)×C₂₅×D₁₃	C₂	1	2	yes	maximal, direct product factor
2	C₂×C₁₃×ΓL(1,𝔽₈)×C₂₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×C₂₅×D₁₃	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₁₃×ΓL(1,𝔽₈)×C₂₅	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×C₅×ΓL(1,𝔽₈)×D₁₃	C₅	1	1	yes	maximal, characteristic
6	C₇×C₂₅×D₁₃	C₂×C₃	1	2	yes	complemented
7	C₂×C₇×C₁₃×C₂₅	C₂×C₃	1	1	yes	characteristic, complemented
8	C₂×C₃×C₂₅×D₁₃	—	7	1	no	maximal, Hall
9	C₅×ΓL(1,𝔽₈)×D₁₃	C₂×C₅	1	2	yes	—
10	C₂×C₅×C₁₃×ΓL(1,𝔽₈)	C₂×C₅	1	1	yes	characteristic
11	C₇×C₁₃×C₂₅	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
12	C₂²×ΓL(1,𝔽₈)×C₂₅	—	13	1	no	maximal, Hall
13	C₂×C₃×C₁₃×C₂₅	—	7	1	no	—
14	C₃×C₂₅×D₁₃	—	7	2	no	—
15	C₂×C₅×C₇×D₁₃	C₃×C₅	1	1	yes	characteristic
16	C₅×C₁₃×ΓL(1,𝔽₈)	C₂²×C₅	1	1	yes	characteristic
17	C₂×C₂₅×D₁₃	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
18	C₂×ΓL(1,𝔽₈)×D₁₃	C₂₅	1	1	yes	characteristic, direct product factor, Hall
19	C₂×ΓL(1,𝔽₈)×C₂₅	—	13	2	no	—
20	C₂×ΓL(1,𝔽₈)×C₂₅	D₁₃	1	1	yes	characteristic, direct product factor
21	C₃×C₁₃×C₂₅	—	7	1	no	Hall
22	C₂×C₅×C₇×C₁₃	C₂×C₃×C₅	1	1	yes	characteristic
23	C₅×C₇×D₁₃	C₂×C₃×C₅	1	2	yes	—
24	C₂×C₃×C₅×D₁₃	—	7	1	no	—
25	C₂²×C₇×C₂₅	—	13	1	no	Hall
26	C₂×C₁₃×C₂₅	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
27	C₂₅×D₁₃	C₂×ΓL(1,𝔽₈)	1	2	yes	direct product factor
28	C₂×C₁₃×ΓL(1,𝔽₈)	C₂×C₂₅	1	1	yes	characteristic, complemented
29	ΓL(1,𝔽₈)×D₁₃	C₂×C₂₅	1	2	yes	direct product factor
30	ΓL(1,𝔽₈)×C₂₅	C₂×D₁₃	1	1	yes	characteristic, direct product factor, Hall
31	C₅×C₇×C₁₃	C₂²×C₃×C₅	1	1	yes	characteristic
32	C₂²×C₅×ΓL(1,𝔽₈)	—	13	1	no	—
33	C₂×C₃×C₅×C₁₃	—	7	1	no	—
34	C₃×C₅×D₁₃	—	7	2	no	—
35	C₂×C₇×D₁₃	C₃×C₂₅	1	1	yes	characteristic, complemented, Hall
36	C₂×C₇×C₂₅	—	13	2	no	—
37	C₂×C₇×C₂₅	C₃×D₁₃	1	1	yes	characteristic, complemented
38	C₁₃×C₂₅	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
39	C₂²×C₃×C₂₅	—	91	1	no	Hall
40	C₁₃×ΓL(1,𝔽₈)	C₂²×C₂₅	1	1	yes	characteristic, complemented, Hall
41	C₂×C₅×D₁₃	C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic
42	C₂×C₅×ΓL(1,𝔽₈)	—	13	2	no	—
43	C₂×C₅×ΓL(1,𝔽₈)	C₅×D₁₃	1	1	yes	characteristic
44	C₃×C₅×C₁₃	—	7	1	no	—
45	C₂×C₇×C₁₃	C₂×C₃×C₂₅	1	1	yes	characteristic, complemented
46	C₇×D₁₃	C₂×C₃×C₂₅	1	2	yes	complemented
47	C₇×C₂₅	C₂×C₃×D₁₃	1	1	yes	characteristic, complemented, Hall
48	C₂×C₃×D₁₃	—	7	1	no	Hall
49	C₂×C₃×C₂₅	—	7	1	no	—
50	C₂×C₃×C₂₅	—	91	2	no	—
51	C₂²×C₅×C₇	—	13	1	no	—
52	C₂×C₅×C₁₃	C₂×C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic
53	C₅×D₁₃	C₂×C₅×ΓL(1,𝔽₈)	1	2	yes	—
54	C₅×ΓL(1,𝔽₈)	C₂×C₅×D₁₃	1	1	yes	characteristic
55	C₂²×C₂₅	—	13	1	no	Hall
56	C₇×C₁₃	C₂²×C₃×C₂₅	1	1	yes	characteristic, complemented, Hall
57	C₂²×ΓL(1,𝔽₈)	—	13	1	no	Hall
58	C₂×C₃×C₁₃	—	7	1	no	—
59	C₃×D₁₃	—	7	2	no	—
60	C₃×C₂₅	—	7	1	no	Hall
61	C₂×C₅×C₇	—	13	2	no	—
62	C₂×C₅×C₇	C₃×C₅×D₁₃	1	1	yes	characteristic
63	C₅×C₁₃	C₂²×C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic
64	C₂²×C₃×C₅	—	91	1	no	—
65	C₂×D₁₃	ΓL(1,𝔽₈)×C₂₅	1	1	yes	characteristic, direct product factor, Hall
66	C₂×C₂₅	—	13	2	no	—
67	C₂×C₂₅	ΓL(1,𝔽₈)×D₁₃	1	1	yes	characteristic, direct product factor
68	C₂×ΓL(1,𝔽₈)	—	13	2	no	—
69	C₂×ΓL(1,𝔽₈)	C₂₅×D₁₃	1	1	yes	characteristic, direct product factor
70	C₃×C₁₃	—	7	1	no	Hall
71	C₅×C₇	C₂×C₃×C₅×D₁₃	1	1	yes	characteristic
72	C₂×C₃×C₅	—	7	1	no	—
73	C₂×C₃×C₅	—	91	2	no	—
74	C₂²×C₇	—	13	1	no	Hall
75	C₂×C₁₃	C₂×ΓL(1,𝔽₈)×C₂₅	1	1	yes	characteristic, complemented
76	D₁₃	C₂×ΓL(1,𝔽₈)×C₂₅	1	2	yes	direct product factor
77	C₂₅	C₂×ΓL(1,𝔽₈)×D₁₃	1	1	yes	characteristic, direct product factor, Sylow
78	ΓL(1,𝔽₈)	C₂×C₂₅×D₁₃	1	1	yes	characteristic, direct product factor, Hall
79	C₂²×C₅	—	13	1	no	—
80	C₃×C₅	—	7	1	no	—
81	C₂×C₇	—	13	2	no	—
82	C₂×C₇	C₃×C₂₅×D₁₃	1	1	yes	characteristic, complemented
83	C₁₃	C₂²×ΓL(1,𝔽₈)×C₂₅	1	1	yes	characteristic, complemented, Sylow
84	C₂²×C₃	—	91	1	no	Hall
85	C₂×C₅	—	13	2	no	—
86	C₂×C₅	C₅×ΓL(1,𝔽₈)×D₁₃	1	1	yes	characteristic
87	C₇	C₂×C₃×C₂₅×D₁₃	1	1	yes	characteristic, complemented, Sylow
88	C₂×C₃	—	7	1	no	—
89	C₂×C₃	—	91	2	no	—
90	C₅	C₂×C₅×ΓL(1,𝔽₈)×D₁₃	1	1	yes	characteristic
91	C₂²	—	13	1	no	Sylow
92	C₃	—	7	1	no	Sylow
93	C₂	—	13	2	no	—
94	C₂	ΓL(1,𝔽₈)×C₂₅×D₁₃	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₂₅	ΓL(1,𝔽₈)×D₁₃
commutator, hypocenter	C₇×C₁₃	C₂²×C₃×C₂₅
Frattini	C₅	C₂×C₅×ΓL(1,𝔽₈)×D₁₃
Fitting	C₂×C₇×C₁₃×C₂₅	C₂×C₃
socle	C₂×C₅×C₇×C₁₃	C₂×C₃×C₅
prime radical	C₂×C₅×ΓL(1,𝔽₈)×D₁₃	C₅
cosocle	C₅×C₇×C₁₃	C₂²×C₃×C₅
LCS	G ▷ C₇×C₁₃
UCS	C₁ ◁ C₂×C₂₅

Complements

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

N	H ≅ G/N	type
1. ΓL(1,𝔽₈)×C₂₅×D₁₃	C₂	direct (N × H)
2. C₂×C₁₃×ΓL(1,𝔽₈)×C₂₅	C₂	semidirect
3. C₂×C₇×C₂₅×D₁₃	C₃	semidirect
4. C₁₃×ΓL(1,𝔽₈)×C₂₅	C₂²	semidirect
6. C₇×C₂₅×D₁₃	C₂×C₃	semidirect
7. C₂×C₇×C₁₃×C₂₅	C₂×C₃	semidirect
11. C₇×C₁₃×C₂₅	C₂²×C₃	semidirect
17. C₂×C₂₅×D₁₃	ΓL(1,𝔽₈)	direct (N × H)
18. C₂×ΓL(1,𝔽₈)×D₁₃	C₂₅	direct (N × H)
20. C₂×ΓL(1,𝔽₈)×C₂₅	D₁₃	direct (N × H)
26. C₂×C₁₃×C₂₅	C₂×ΓL(1,𝔽₈)	semidirect
27. C₂₅×D₁₃	C₂×ΓL(1,𝔽₈)	direct (N × H)
28. C₂×C₁₃×ΓL(1,𝔽₈)	C₂×C₂₅	semidirect
29. ΓL(1,𝔽₈)×D₁₃	C₂×C₂₅	direct (N × H)
30. ΓL(1,𝔽₈)×C₂₅	C₂×D₁₃	direct (N × H)
35. C₂×C₇×D₁₃	C₃×C₂₅	semidirect
37. C₂×C₇×C₂₅	C₃×D₁₃	semidirect
38. C₁₃×C₂₅	C₂²×ΓL(1,𝔽₈)	semidirect
40. C₁₃×ΓL(1,𝔽₈)	C₂²×C₂₅	semidirect
45. C₂×C₇×C₁₃	C₂×C₃×C₂₅	semidirect
46. C₇×D₁₃	C₂×C₃×C₂₅	semidirect
47. C₇×C₂₅	C₂×C₃×D₁₃	semidirect
56. C₇×C₁₃	C₂²×C₃×C₂₅	semidirect
65. C₂×D₁₃	ΓL(1,𝔽₈)×C₂₅	direct (N × H)
67. C₂×C₂₅	ΓL(1,𝔽₈)×D₁₃	direct (N × H)
69. C₂×ΓL(1,𝔽₈)	C₂₅×D₁₃	direct (N × H)
75. C₂×C₁₃	C₂×ΓL(1,𝔽₈)×C₂₅	semidirect
76. D₁₃	C₂×ΓL(1,𝔽₈)×C₂₅	direct (N × H)
77. C₂₅	C₂×ΓL(1,𝔽₈)×D₁₃	direct (N × H)
78. ΓL(1,𝔽₈)	C₂×C₂₅×D₁₃	direct (N × H)
82. C₂×C₇	C₃×C₂₅×D₁₃	semidirect
83. C₁₃	C₂²×ΓL(1,𝔽₈)×C₂₅	semidirect
87. C₇	C₂×C₃×C₂₅×D₁₃	semidirect
94. C₂	ΓL(1,𝔽₈)×C₂₅×D₁₃	direct (N × H)

Maximal subgroups

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

subgroup	index
ΓL(1,𝔽₈)×C₂₅×D₁₃	2
C₂×C₁₃×ΓL(1,𝔽₈)×C₂₅	2
C₂×C₇×C₂₅×D₁₃	3
C₂×C₅×ΓL(1,𝔽₈)×D₁₃	5
C₂×C₃×C₂₅×D₁₃	7
C₂²×ΓL(1,𝔽₈)×C₂₅	13

Subgroups — C2×ΓL(1,𝔽8)×C25×D13 #41031245

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×ΓL(1,𝔽₈)×C₂₅×D₁₃ #41031245