Subgroups — C₂×ΓL(1,𝔽₈)×D₁₁×C₂₅ #37528261

Order 23100.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	ΓL(1,𝔽₈)×D₁₁×C₂₅	C₂	1	2	yes	maximal, direct product factor
2	C₂×C₁₁×ΓL(1,𝔽₈)×C₂₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×D₁₁×C₂₅	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₁₁×ΓL(1,𝔽₈)×C₂₅	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×C₅×ΓL(1,𝔽₈)×D₁₁	C₅	1	1	yes	maximal, characteristic
6	C₇×D₁₁×C₂₅	C₂×C₃	1	2	yes	complemented
7	C₂×C₇×C₁₁×C₂₅	C₂×C₃	1	1	yes	characteristic, complemented
8	C₂×C₃×D₁₁×C₂₅	—	7	1	no	maximal, Hall
9	C₅×ΓL(1,𝔽₈)×D₁₁	C₂×C₅	1	2	yes	—
10	C₂×C₅×C₁₁×ΓL(1,𝔽₈)	C₂×C₅	1	1	yes	characteristic
11	C₂²×ΓL(1,𝔽₈)×C₂₅	—	11	1	no	maximal, Hall
12	C₇×C₁₁×C₂₅	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
13	C₂×C₃×C₁₁×C₂₅	—	7	1	no	—
14	C₃×D₁₁×C₂₅	—	7	2	no	—
15	C₂×C₅×C₇×D₁₁	C₃×C₅	1	1	yes	characteristic
16	C₅×C₁₁×ΓL(1,𝔽₈)	C₂²×C₅	1	1	yes	characteristic
17	C₂×D₁₁×C₂₅	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
18	C₂×ΓL(1,𝔽₈)×C₂₅	—	11	2	no	—
19	C₂×ΓL(1,𝔽₈)×C₂₅	D₁₁	1	1	yes	characteristic, direct product factor
20	C₂×ΓL(1,𝔽₈)×D₁₁	C₂₅	1	1	yes	characteristic, direct product factor, Hall
21	C₃×C₁₁×C₂₅	—	7	1	no	Hall
22	C₂×C₅×C₇×C₁₁	C₂×C₃×C₅	1	1	yes	characteristic
23	C₅×C₇×D₁₁	C₂×C₃×C₅	1	2	yes	—
24	C₂²×C₇×C₂₅	—	11	1	no	Hall
25	C₂×C₃×C₅×D₁₁	—	7	1	no	—
26	C₂×C₁₁×C₂₅	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
27	D₁₁×C₂₅	C₂×ΓL(1,𝔽₈)	1	2	yes	direct product factor
28	ΓL(1,𝔽₈)×C₂₅	C₂×D₁₁	1	1	yes	characteristic, direct product factor, Hall
29	C₂×C₁₁×ΓL(1,𝔽₈)	C₂×C₂₅	1	1	yes	characteristic, complemented
30	ΓL(1,𝔽₈)×D₁₁	C₂×C₂₅	1	2	yes	direct product factor
31	C₂²×C₅×ΓL(1,𝔽₈)	—	11	1	no	—
32	C₅×C₇×C₁₁	C₂²×C₃×C₅	1	1	yes	characteristic
33	C₂×C₇×C₂₅	—	11	2	no	—
34	C₂×C₇×C₂₅	C₃×D₁₁	1	1	yes	characteristic, complemented
35	C₂×C₃×C₅×C₁₁	—	7	1	no	—
36	C₃×C₅×D₁₁	—	7	2	no	—
37	C₂×C₇×D₁₁	C₃×C₂₅	1	1	yes	characteristic, complemented, Hall
38	C₂²×C₃×C₂₅	—	77	1	no	Hall
39	C₁₁×C₂₅	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
40	C₁₁×ΓL(1,𝔽₈)	C₂²×C₂₅	1	1	yes	characteristic, complemented, Hall
41	C₂×C₅×D₁₁	C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic
42	C₂×C₅×ΓL(1,𝔽₈)	—	11	2	no	—
43	C₂×C₅×ΓL(1,𝔽₈)	C₅×D₁₁	1	1	yes	characteristic
44	C₇×C₂₅	C₂×C₃×D₁₁	1	1	yes	characteristic, complemented, Hall
45	C₃×C₅×C₁₁	—	7	1	no	—
46	C₂×C₇×C₁₁	C₂×C₃×C₂₅	1	1	yes	characteristic, complemented
47	C₇×D₁₁	C₂×C₃×C₂₅	1	2	yes	complemented
48	C₂×C₃×C₂₅	—	7	1	no	—
49	C₂×C₃×C₂₅	—	77	2	no	—
50	C₂²×C₅×C₇	—	11	1	no	—
51	C₂×C₃×D₁₁	—	7	1	no	Hall
52	C₂×C₅×C₁₁	C₂×C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic
53	C₅×D₁₁	C₂×C₅×ΓL(1,𝔽₈)	1	2	yes	—
54	C₅×ΓL(1,𝔽₈)	C₂×C₅×D₁₁	1	1	yes	characteristic
55	C₂²×C₂₅	—	11	1	no	Hall
56	C₂²×ΓL(1,𝔽₈)	—	11	1	no	Hall
57	C₇×C₁₁	C₂²×C₃×C₂₅	1	1	yes	characteristic, complemented, Hall
58	C₃×C₂₅	—	7	1	no	Hall
59	C₂×C₅×C₇	—	11	2	no	—
60	C₂×C₅×C₇	C₃×C₅×D₁₁	1	1	yes	characteristic
61	C₂×C₃×C₁₁	—	7	1	no	—
62	C₃×D₁₁	—	7	2	no	—
63	C₂²×C₃×C₅	—	77	1	no	—
64	C₅×C₁₁	C₂²×C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic
65	C₂×C₂₅	—	11	2	no	—
66	C₂×C₂₅	ΓL(1,𝔽₈)×D₁₁	1	1	yes	characteristic, direct product factor
67	C₂×D₁₁	ΓL(1,𝔽₈)×C₂₅	1	1	yes	characteristic, direct product factor, Hall
68	C₂×ΓL(1,𝔽₈)	—	11	2	no	—
69	C₂×ΓL(1,𝔽₈)	D₁₁×C₂₅	1	1	yes	characteristic, direct product factor
70	C₅×C₇	C₂×C₃×C₅×D₁₁	1	1	yes	characteristic
71	C₃×C₁₁	—	7	1	no	Hall
72	C₂×C₃×C₅	—	7	1	no	—
73	C₂×C₃×C₅	—	77	2	no	—
74	C₂²×C₇	—	11	1	no	Hall
75	C₂₅	C₂×ΓL(1,𝔽₈)×D₁₁	1	1	yes	characteristic, direct product factor, Sylow
76	C₂×C₁₁	C₂×ΓL(1,𝔽₈)×C₂₅	1	1	yes	characteristic, complemented
77	D₁₁	C₂×ΓL(1,𝔽₈)×C₂₅	1	2	yes	direct product factor
78	ΓL(1,𝔽₈)	C₂×D₁₁×C₂₅	1	1	yes	characteristic, direct product factor, Hall
79	C₂²×C₅	—	11	1	no	—
80	C₃×C₅	—	7	1	no	—
81	C₂×C₇	—	11	2	no	—
82	C₂×C₇	C₃×D₁₁×C₂₅	1	1	yes	characteristic, complemented
83	C₂²×C₃	—	77	1	no	Hall
84	C₁₁	C₂²×ΓL(1,𝔽₈)×C₂₅	1	1	yes	characteristic, complemented, Sylow
85	C₂×C₅	—	11	2	no	—
86	C₂×C₅	C₅×ΓL(1,𝔽₈)×D₁₁	1	1	yes	characteristic
87	C₇	C₂×C₃×D₁₁×C₂₅	1	1	yes	characteristic, complemented, Sylow
88	C₂×C₃	—	7	1	no	—
89	C₂×C₃	—	77	2	no	—
90	C₅	C₂×C₅×ΓL(1,𝔽₈)×D₁₁	1	1	yes	characteristic
91	C₂²	—	11	1	no	Sylow
92	C₃	—	7	1	no	Sylow
93	C₂	—	11	2	no	—
94	C₂	ΓL(1,𝔽₈)×D₁₁×C₂₅	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₂₅	ΓL(1,𝔽₈)×D₁₁
commutator, hypocenter	C₇×C₁₁	C₂²×C₃×C₂₅
Frattini	C₅	C₂×C₅×ΓL(1,𝔽₈)×D₁₁
Fitting	C₂×C₇×C₁₁×C₂₅	C₂×C₃
socle	C₂×C₅×C₇×C₁₁	C₂×C₃×C₅
prime radical	C₂×C₅×ΓL(1,𝔽₈)×D₁₁	C₅
cosocle	C₅×C₇×C₁₁	C₂²×C₃×C₅
LCS	G ▷ C₇×C₁₁
UCS	C₁ ◁ C₂×C₂₅

Complements

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

N	H ≅ G/N	type
1. ΓL(1,𝔽₈)×D₁₁×C₂₅	C₂	direct (N × H)
2. C₂×C₁₁×ΓL(1,𝔽₈)×C₂₅	C₂	semidirect
3. C₂×C₇×D₁₁×C₂₅	C₃	semidirect
4. C₁₁×ΓL(1,𝔽₈)×C₂₅	C₂²	semidirect
6. C₇×D₁₁×C₂₅	C₂×C₃	semidirect
7. C₂×C₇×C₁₁×C₂₅	C₂×C₃	semidirect
12. C₇×C₁₁×C₂₅	C₂²×C₃	semidirect
17. C₂×D₁₁×C₂₅	ΓL(1,𝔽₈)	direct (N × H)
19. C₂×ΓL(1,𝔽₈)×C₂₅	D₁₁	direct (N × H)
20. C₂×ΓL(1,𝔽₈)×D₁₁	C₂₅	direct (N × H)
26. C₂×C₁₁×C₂₅	C₂×ΓL(1,𝔽₈)	semidirect
27. D₁₁×C₂₅	C₂×ΓL(1,𝔽₈)	direct (N × H)
28. ΓL(1,𝔽₈)×C₂₅	C₂×D₁₁	direct (N × H)
29. C₂×C₁₁×ΓL(1,𝔽₈)	C₂×C₂₅	semidirect
30. ΓL(1,𝔽₈)×D₁₁	C₂×C₂₅	direct (N × H)
34. C₂×C₇×C₂₅	C₃×D₁₁	semidirect
37. C₂×C₇×D₁₁	C₃×C₂₅	semidirect
39. C₁₁×C₂₅	C₂²×ΓL(1,𝔽₈)	semidirect
40. C₁₁×ΓL(1,𝔽₈)	C₂²×C₂₅	semidirect
44. C₇×C₂₅	C₂×C₃×D₁₁	semidirect
46. C₂×C₇×C₁₁	C₂×C₃×C₂₅	semidirect
47. C₇×D₁₁	C₂×C₃×C₂₅	semidirect
57. C₇×C₁₁	C₂²×C₃×C₂₅	semidirect
66. C₂×C₂₅	ΓL(1,𝔽₈)×D₁₁	direct (N × H)
67. C₂×D₁₁	ΓL(1,𝔽₈)×C₂₅	direct (N × H)
69. C₂×ΓL(1,𝔽₈)	D₁₁×C₂₅	direct (N × H)
75. C₂₅	C₂×ΓL(1,𝔽₈)×D₁₁	direct (N × H)
76. C₂×C₁₁	C₂×ΓL(1,𝔽₈)×C₂₅	semidirect
77. D₁₁	C₂×ΓL(1,𝔽₈)×C₂₅	direct (N × H)
78. ΓL(1,𝔽₈)	C₂×D₁₁×C₂₅	direct (N × H)
82. C₂×C₇	C₃×D₁₁×C₂₅	semidirect
84. C₁₁	C₂²×ΓL(1,𝔽₈)×C₂₅	semidirect
87. C₇	C₂×C₃×D₁₁×C₂₅	semidirect
94. C₂	ΓL(1,𝔽₈)×D₁₁×C₂₅	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)×D₁₁×C₂₅	2
C₂×C₁₁×ΓL(1,𝔽₈)×C₂₅	2
C₂×C₇×D₁₁×C₂₅	3
C₂×C₅×ΓL(1,𝔽₈)×D₁₁	5
C₂×C₃×D₁₁×C₂₅	7
C₂²×ΓL(1,𝔽₈)×C₂₅	11

Subgroups — C2×ΓL(1,𝔽8)×D11×C25 #37528261

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×ΓL(1,𝔽₈)×D₁₁×C₂₅ #37528261