Subgroups — C₃×C₇×(C₅⋊D₅)⋊ΓL(1,𝔽₈) #36347948

Order 22050.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇×C₇⋊(C₅²⋊C₃)	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₃×C₇²×C₅⋊D₅	C₃	1	1	yes	maximal, characteristic, complemented
3	C₇×(C₅⋊D₅)⋊ΓL(1,𝔽₈)	C₃	1	3	yes	maximal, direct product factor
4	C₇×C₇⋊(C₅²⋊C₃)	C₂×C₃	1	3	yes	complemented
5	C₃×C₅²×C₇²	C₂×C₃	1	1	yes	characteristic, complemented
6	C₃×(C₅⋊D₅)⋊ΓL(1,𝔽₈)	C₇	1	1	yes	maximal, characteristic, direct product factor
7	C₃×C₇×(C₅⋊D₅)⋊C₃	—	7	1	no	maximal
8	C₇²×C₅⋊D₅	C₃²	1	1	yes	characteristic, complemented, Hall
9	C₃×C₇⋊(C₅²⋊C₃)	C₂×C₇	1	1	yes	characteristic, complemented
10	C₃×C₇×C₅²⋊C₃	—	7	1	no	—
11	C₃×C₇²×D₅	—	15	2	no	—
12	C₅²×C₇²	C₂×C₃²	1	1	yes	characteristic, complemented, Hall
13	C₃×C₇×C₅⋊D₅	—	3	2	no	—
14	C₃×C₇×C₅⋊D₅	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
15	C₃×C₇×C₅⋊D₅	C₃×C₇	1	1	yes	characteristic, complemented
16	(C₅⋊D₅)⋊ΓL(1,𝔽₈)	C₃×C₇	1	3	yes	direct product factor
17	C₇×(C₅⋊D₅)⋊C₃	—	7	3	no	—
18	C₂×C₃×C₇×ΓL(1,𝔽₈)	—	25	1	no	maximal, Hall
19	C₃×C₅×C₇²	—	3	2	no	—
20	C₃×C₅²×C₇	—	3	2	no	—
21	C₃×C₅²×C₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
22	C₃×C₅²×C₇	C₂×C₃×C₇	1	1	yes	characteristic, complemented
23	C₇⋊(C₅²⋊C₃)	C₂×C₃×C₇	1	3	yes	complemented
24	C₇×C₅²⋊C₃	—	7	3	no	—
25	C₇²×D₅	—	15	2	no	—
26	C₃×(C₅⋊D₅)⋊C₃	—	7	1	no	Hall
27	C₃×C₇×ΓL(1,𝔽₈)	—	25	1	no	Hall
28	C₇×C₅⋊D₅	—	3	2	no	—
29	C₇×C₅⋊D₅	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
30	C₇×C₅⋊D₅	C₃²×C₇	1	1	yes	characteristic, complemented
31	C₂×C₃×C₇²	—	75	1	no	—
32	C₂×C₇×ΓL(1,𝔽₈)	—	25	3	no	—
33	C₅×C₇²	—	3	2	no	—
34	C₃×C₅²⋊C₃	—	7	1	no	Hall
35	C₃×C₇×D₅	—	15	2	no	—
36	C₃×C₇×D₅	—	15	12	no	—
37	C₃×C₇×D₅	—	15	2	no	—
38	C₅²×C₇	—	3	2	no	—
39	C₅²×C₇	C₂×C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
40	C₅²×C₇	C₂×C₃²×C₇	1	1	yes	characteristic, complemented
41	C₃×C₅⋊D₅	C₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
42	(C₅⋊D₅)⋊C₃	—	7	3	no	—
43	C₃×C₇²	(C₅⋊D₅)⋊C₃	1	1	yes	characteristic, complemented
44	C₇×ΓL(1,𝔽₈)	—	25	3	no	—
45	C₂×C₃²×C₇	—	175	1	no	—
46	C₂×C₃×ΓL(1,𝔽₈)	—	25	1	no	—
47	C₃×C₅×C₇	—	3	12	no	—
48	C₃×C₅×C₇	—	3	2	no	—
49	C₃×C₅×C₇	—	3	2	no	—
50	C₂×C₇²	—	75	1	no	Hall
51	C₃×C₅²	C₂×C₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
52	C₅²⋊C₃	—	7	3	no	—
53	C₇×D₅	—	15	2	no	—
54	C₇×D₅	—	15	12	no	—
55	C₇×D₅	—	15	2	no	—
56	C₃²×C₇	—	175	1	no	—
57	C₃×ΓL(1,𝔽₈)	—	25	1	no	—
58	C₅⋊D₅	C₃×C₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
59	C₇²	C₃×(C₅⋊D₅)⋊C₃	1	1	yes	characteristic, complemented, Sylow
60	C₂×C₃×C₇	—	25	1	no	—
61	C₂×C₃×C₇	—	75	2	no	—
62	C₂×C₃×C₇	—	75	1	no	—
63	C₂×C₃×C₇	—	175	3	no	—
64	C₂×ΓL(1,𝔽₈)	—	25	3	no	—
65	C₅×C₇	—	3	12	no	—
66	C₅×C₇	—	3	2	no	—
67	C₅×C₇	—	3	2	no	—
68	C₃×D₅	—	15	2	no	—
69	C₅²	C₂×C₃×C₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
70	C₃×C₇	—	3	2	no	—
71	C₃×C₇	—	175	3	no	—
72	C₃×C₇	C₇×(C₅⋊D₅)⋊C₃	1	1	yes	characteristic, complemented
73	C₃×C₇	(C₅⋊D₅)⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
74	ΓL(1,𝔽₈)	—	25	3	no	—
75	C₂×C₃²	—	175	1	no	Hall
76	C₃×C₅	—	3	2	no	—
77	C₂×C₇	—	25	1	no	—
78	C₂×C₇	—	75	2	no	—
79	C₂×C₇	—	75	1	no	—
80	D₅	—	15	2	no	—
81	C₃²	—	175	1	no	Sylow
82	C₇	—	3	2	no	—
83	C₇	C₃×C₇×(C₅⋊D₅)⋊C₃	1	1	yes	characteristic, complemented
84	C₇	C₃×(C₅⋊D₅)⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
85	C₂×C₃	—	75	1	no	—
86	C₂×C₃	—	175	3	no	—
87	C₅	—	3	2	no	—
88	C₃	—	175	3	no	—
89	C₃	C₇×(C₅⋊D₅)⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
90	C₂	—	75	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃×C₇	(C₅⋊D₅)⋊ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₅²×C₇	C₂×C₃²×C₇
Frattini	C₁	G
Fitting, socle	C₃×C₅²×C₇²	C₂×C₃
LCS	G ▷ C₅²×C₇
UCS	C₁ ◁ C₃×C₇

Complements

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

N	H ≅ G/N	type
1. C₃×C₇×C₇⋊(C₅²⋊C₃)	C₂	semidirect
2. C₃×C₇²×C₅⋊D₅	C₃	semidirect
3. C₇×(C₅⋊D₅)⋊ΓL(1,𝔽₈)	C₃	direct (N × H)
4. C₇×C₇⋊(C₅²⋊C₃)	C₂×C₃	semidirect
5. C₃×C₅²×C₇²	C₂×C₃	semidirect
6. C₃×(C₅⋊D₅)⋊ΓL(1,𝔽₈)	C₇	direct (N × H)
8. C₇²×C₅⋊D₅	C₃²	semidirect
9. C₃×C₇⋊(C₅²⋊C₃)	C₂×C₇	semidirect
12. C₅²×C₇²	C₂×C₃²	semidirect
14. C₃×C₇×C₅⋊D₅	ΓL(1,𝔽₈)	semidirect
15. C₃×C₇×C₅⋊D₅	C₃×C₇	semidirect
16. (C₅⋊D₅)⋊ΓL(1,𝔽₈)	C₃×C₇	direct (N × H)
21. C₃×C₅²×C₇	C₂×ΓL(1,𝔽₈)	semidirect
22. C₃×C₅²×C₇	C₂×C₃×C₇	semidirect
23. C₇⋊(C₅²⋊C₃)	C₂×C₃×C₇	semidirect
29. C₇×C₅⋊D₅	C₃×ΓL(1,𝔽₈)	semidirect
30. C₇×C₅⋊D₅	C₃²×C₇	semidirect
39. C₅²×C₇	C₂×C₃×ΓL(1,𝔽₈)	semidirect
40. C₅²×C₇	C₂×C₃²×C₇	semidirect
41. C₃×C₅⋊D₅	C₇×ΓL(1,𝔽₈)	semidirect
43. C₃×C₇²	(C₅⋊D₅)⋊C₃	semidirect
51. C₃×C₅²	C₂×C₇×ΓL(1,𝔽₈)	semidirect
58. C₅⋊D₅	C₃×C₇×ΓL(1,𝔽₈)	semidirect
59. C₇²	C₃×(C₅⋊D₅)⋊C₃	semidirect
69. C₅²	C₂×C₃×C₇×ΓL(1,𝔽₈)	semidirect
72. C₃×C₇	C₇×(C₅⋊D₅)⋊C₃	semidirect
73. C₃×C₇	(C₅⋊D₅)⋊ΓL(1,𝔽₈)	direct (N × H)
83. C₇	C₃×C₇×(C₅⋊D₅)⋊C₃	semidirect
84. C₇	C₃×(C₅⋊D₅)⋊ΓL(1,𝔽₈)	direct (N × H)
89. C₃	C₇×(C₅⋊D₅)⋊ΓL(1,𝔽₈)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₇×C₇⋊(C₅²⋊C₃)	2
C₃×C₇²×C₅⋊D₅	3
C₇×(C₅⋊D₅)⋊ΓL(1,𝔽₈)	3
C₃×(C₅⋊D₅)⋊ΓL(1,𝔽₈)	7
C₃×C₇×(C₅⋊D₅)⋊C₃	7
C₂×C₃×C₇×ΓL(1,𝔽₈)	25

Subgroups — C3×C7×(C5⋊D5)⋊ΓL(1,𝔽8) #36347948

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₇×(C₅⋊D₅)⋊ΓL(1,𝔽₈) #36347948