Subgroups — C₅⋊(ΓL(1,𝔽₈)⋊(D₃₇⋊C₃)) #37552049

Order 23310.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×ΓL(1,𝔽₈)×C₃₇⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₅⋊#13095538	C₃	1	1	yes	maximal, characteristic, complemented
3	C₅⋊#13095537	C₃	1	1	yes	maximal, characteristic, complemented
4	C₅⋊(ΓL(1,𝔽₈)⋊D₃₇)	C₃	1	1	yes	maximal, characteristic, complemented
5	C₅⋊((C₃₇⋊C₃)⋊D₇)	C₃	1	1	yes	maximal, characteristic, complemented
6	ΓL(1,𝔽₈)⋊(D₃₇⋊C₃)	—	5	1	no	maximal, Hall
7	C₅×#11715912	C₂×C₃	1	1	yes	characteristic, complemented
8	C₅×#11715913	C₂×C₃	1	1	yes	characteristic, complemented
9	C₅×ΓL(1,𝔽₈)×C₃₇	C₂×C₃	1	1	yes	characteristic, complemented
10	C₅×C₇×C₃₇⋊C₃	C₂×C₃	1	1	yes	characteristic, complemented
11	C₃×C₅⋊(D₃₇⋊C₃)	—	7	1	no	maximal, Hall
12	D₁₂₉₅	C₃²	1	1	yes	characteristic, complemented, Hall
13	ΓL(1,𝔽₈)×C₃₇⋊C₃	D₅	1	1	yes	characteristic, complemented, Hall
14	C₃×C₅×C₃₇⋊C₃	—	7	1	no	Hall
15	#13095538	—	5	1	no	—
16	#13095537	—	5	1	no	—
17	ΓL(1,𝔽₈)⋊D₃₇	—	5	1	no	—
18	(C₃₇⋊C₃)⋊D₇	—	5	1	no	—
19	C₅×C₇×C₃₇	C₂×C₃²	1	1	yes	characteristic, complemented, Hall
20	C₃×D₁₈₅	—	7	1	no	—
21	C₅⋊(D₃₇⋊C₃)	—	7	1	no	—
22	C₅⋊(D₃₇⋊C₃)	—	7	1	no	—
23	C₅⋊(D₃₇⋊C₃)	—	7	1	no	—
24	#11715913	C₃×D₅	1	1	yes	characteristic, complemented
25	#11715912	C₃×D₅	1	1	yes	characteristic, complemented
26	C₇×C₃₇⋊C₃	C₃×D₅	1	1	yes	characteristic, complemented
27	ΓL(1,𝔽₈)×C₃₇	C₃×D₅	1	1	yes	characteristic, complemented
28	C₃×D₃₇⋊C₃	—	35	1	no	Hall
29	C₃×C₅⋊AGL(1,𝔽₇)	—	37	1	no	maximal, Hall
30	C₃×C₅×C₃₇	—	7	1	no	—
31	C₅×C₃₇⋊C₃	—	7	1	no	—
32	C₅×C₃₇⋊C₃	—	7	1	no	—
33	C₅×C₃₇⋊C₃	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
34	D₂₅₉	—	5	1	no	Hall
35	D₁₈₅	—	7	1	no	Hall
36	C₃×C₃₇⋊C₃	—	7	1	no	Hall
37	C₃×C₅×ΓL(1,𝔽₈)	—	37	1	no	Hall
38	C₇×C₃₇	C₃²×D₅	1	1	yes	characteristic, complemented, Hall
39	C₃×D₃₇	—	35	1	no	—
40	D₃₇⋊C₃	—	35	1	no	—
41	D₃₇⋊C₃	—	35	1	no	—
42	D₃₇⋊C₃	—	105	1	no	—
43	C₃×D₃₅	—	37	1	no	—
44	C₅⋊AGL(1,𝔽₇)	—	37	1	no	—
45	C₅⋊AGL(1,𝔽₇)	—	37	1	no	—
46	C₅⋊AGL(1,𝔽₇)	—	37	1	no	—
47	C₅×C₃₇	C₃×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, Hall
48	C₃×AGL(1,𝔽₇)	—	185	1	no	Hall
49	C₃×C₃₇	—	7	1	no	—
50	C₃₇⋊C₃	—	7	1	no	—
51	C₃₇⋊C₃	—	7	1	no	—
52	C₃₇⋊C₃	C₅⋊AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
53	C₃×C₅×C₇	—	37	1	no	—
54	C₅×ΓL(1,𝔽₈)	—	37	1	no	—
55	C₅×ΓL(1,𝔽₈)	—	37	1	no	—
56	C₅×ΓL(1,𝔽₈)	D₃₇⋊C₃	1	1	yes	characteristic, complemented
57	C₃²×D₅	—	259	1	no	Hall
58	D₃₇	—	105	1	no	Hall
59	D₃₅	—	37	1	no	Hall
60	C₃×ΓL(1,𝔽₈)	—	37	1	no	Hall
61	C₃²×C₅	—	259	1	no	Hall
62	C₃×D₇	—	185	1	no	—
63	AGL(1,𝔽₇)	—	555	1	no	—
64	AGL(1,𝔽₇)	—	185	1	no	—
65	AGL(1,𝔽₇)	—	185	1	no	—
66	C₃₇	C₃×C₅⋊AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, Sylow
67	C₅×C₇	C₃×D₃₇⋊C₃	1	1	yes	characteristic, complemented, Hall
68	C₃×D₅	—	259	1	no	—
69	C₃×D₅	—	259	1	no	—
70	C₃×D₅	—	777	1	no	—
71	C₃×D₅	—	777	1	no	—
72	C₃×C₇	—	37	1	no	—
73	ΓL(1,𝔽₈)	—	37	1	no	—
74	ΓL(1,𝔽₈)	—	37	1	no	—
75	ΓL(1,𝔽₈)	C₅⋊(D₃₇⋊C₃)	1	1	yes	characteristic, complemented
76	C₂×C₃²	—	1295	1	no	Hall
77	C₃×C₅	—	259	1	no	—
78	C₃×C₅	—	259	1	no	—
79	C₃×C₅	—	7	1	no	—
80	C₃×C₅	—	37	1	no	—
81	D₇	—	555	1	no	Hall
82	D₅	—	777	1	no	Hall
83	C₃²	—	259	1	no	Sylow
84	C₇	C₃×C₅⋊(D₃₇⋊C₃)	1	1	yes	characteristic, complemented, Sylow
85	C₂×C₃	—	1295	1	no	—
86	C₂×C₃	—	1295	1	no	—
87	C₂×C₃	—	3885	1	no	—
88	C₂×C₃	—	3885	1	no	—
89	C₅	ΓL(1,𝔽₈)⋊(D₃₇⋊C₃)	1	1	yes	characteristic, complemented, Sylow
90	C₃	—	259	1	no	—
91	C₃	—	259	1	no	—
92	C₃	—	7	1	no	—
93	C₃	—	37	1	no	—
94	C₂	—	11655	1	no	Sylow

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₅×C₇×C₃₇	C₂×C₃²
LCS	G ▷ C₅×C₇×C₃₇
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₅×ΓL(1,𝔽₈)×C₃₇⋊C₃	C₂	semidirect
2. C₅⋊#13095538	C₃	semidirect
3. C₅⋊#13095537	C₃	semidirect
4. C₅⋊(ΓL(1,𝔽₈)⋊D₃₇)	C₃	semidirect
5. C₅⋊((C₃₇⋊C₃)⋊D₇)	C₃	semidirect
7. C₅×#11715912	C₂×C₃	semidirect
8. C₅×#11715913	C₂×C₃	semidirect
9. C₅×ΓL(1,𝔽₈)×C₃₇	C₂×C₃	semidirect
10. C₅×C₇×C₃₇⋊C₃	C₂×C₃	semidirect
12. D₁₂₉₅	C₃²	semidirect
13. ΓL(1,𝔽₈)×C₃₇⋊C₃	D₅	semidirect
19. C₅×C₇×C₃₇	C₂×C₃²	semidirect
24. #11715913	C₃×D₅	semidirect
25. #11715912	C₃×D₅	semidirect
26. C₇×C₃₇⋊C₃	C₃×D₅	semidirect
27. ΓL(1,𝔽₈)×C₃₇	C₃×D₅	semidirect
33. C₅×C₃₇⋊C₃	AGL(1,𝔽₇)	semidirect
38. C₇×C₃₇	C₃²×D₅	semidirect
47. C₅×C₃₇	C₃×AGL(1,𝔽₇)	semidirect
52. C₃₇⋊C₃	C₅⋊AGL(1,𝔽₇)	semidirect
56. C₅×ΓL(1,𝔽₈)	D₃₇⋊C₃	semidirect
66. C₃₇	C₃×C₅⋊AGL(1,𝔽₇)	semidirect
67. C₅×C₇	C₃×D₃₇⋊C₃	semidirect
75. ΓL(1,𝔽₈)	C₅⋊(D₃₇⋊C₃)	semidirect
84. C₇	C₃×C₅⋊(D₃₇⋊C₃)	semidirect
89. C₅	ΓL(1,𝔽₈)⋊(D₃₇⋊C₃)	semidirect

Maximal subgroups

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

subgroup	index
C₅×ΓL(1,𝔽₈)×C₃₇⋊C₃	2
C₅⋊#13095538	3
C₅⋊#13095537	3
C₅⋊(ΓL(1,𝔽₈)⋊D₃₇)	3
C₅⋊((C₃₇⋊C₃)⋊D₇)	3
ΓL(1,𝔽₈)⋊(D₃₇⋊C₃)	5
C₃×C₅⋊(D₃₇⋊C₃)	7
C₃×C₅⋊AGL(1,𝔽₇)	37

Subgroups — C5⋊(ΓL(1,𝔽8)⋊(D37⋊C3)) #37552049

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅⋊(ΓL(1,𝔽₈)⋊(D₃₇⋊C₃)) #37552049