Subgroups — C₂⁴×#64423 #72260008

Order 4704.

3394 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×#3861	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×#64423	C₂	1	30	yes	maximal, direct product factor
3	C₂⁴×C₇×D₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₂³×#3861	C₂²	1	15	yes	complemented
5	C₂²×#64423	C₂²	1	140	yes	direct product factor
6	C₂⁴×C₇²	C₂×C₃	1	1	yes	characteristic, complemented
7	C₂³×C₇×D₇	C₂×C₃	1	30	yes	complemented
8	C₂⁵×ΓL(1,𝔽₈)	—	7	1	no	maximal
9	C₂⁴×AGL(1,𝔽₇)	—	7	1	no	maximal
10	C₂²×#3861	C₂³	1	35	yes	complemented
11	C₂×#64423	C₂³	1	120	yes	direct product factor
12	C₂³×C₇²	C₂²×C₃	1	15	yes	complemented
13	C₂²×C₇×D₇	C₂²×C₃	1	140	yes	complemented
14	C₂⁴×ΓL(1,𝔽₈)	—	7	30	no	—
15	C₂⁴×ΓL(1,𝔽₈)	—	7	1	no	—
16	C₂⁴×ΓL(1,𝔽₈)	—	7	1	no	—
17	C₂⁴×ΓL(1,𝔽₈)	—	14	3	no	—
18	C₂³×AGL(1,𝔽₇)	—	7	30	no	—
19	C₂×#3861	C₂⁴	1	15	yes	complemented
20	#64423	C₂⁴	1	16	yes	direct product factor
21	C₂⁵×C₇	—	7	1	no	—
22	C₂⁴×D₇	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
23	C₂²×C₇²	C₂³×C₃	1	35	yes	complemented
24	C₂×C₇×D₇	C₂³×C₃	1	120	yes	complemented
25	C₂³×ΓL(1,𝔽₈)	—	7	140	no	—
26	C₂³×ΓL(1,𝔽₈)	—	7	15	no	—
27	C₂³×ΓL(1,𝔽₈)	—	7	15	no	—
28	C₂³×ΓL(1,𝔽₈)	—	14	45	no	—
29	C₂²×AGL(1,𝔽₇)	—	7	140	no	—
30	#3861	C₂⁵	1	1	yes	characteristic, complemented, Hall
31	C₂⁴×C₇	—	2	3	no	—
32	C₂⁴×C₇	—	7	30	no	—
33	C₂⁴×C₇	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
34	C₂⁴×C₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
35	C₂³×D₇	C₂×ΓL(1,𝔽₈)	1	30	yes	complemented
36	C₂×C₇²	C₂⁴×C₃	1	15	yes	complemented
37	C₇×D₇	C₂⁴×C₃	1	16	yes	complemented
38	C₂⁵×C₃	—	49	1	no	Hall
39	C₂²×ΓL(1,𝔽₈)	—	7	120	no	—
40	C₂²×ΓL(1,𝔽₈)	—	7	35	no	—
41	C₂²×ΓL(1,𝔽₈)	—	7	35	no	—
42	C₂²×ΓL(1,𝔽₈)	—	14	105	no	—
43	C₂×AGL(1,𝔽₇)	—	7	120	no	—
44	C₂³×C₇	—	2	45	no	—
45	C₂³×C₇	—	7	140	no	—
46	C₂³×C₇	C₂×AGL(1,𝔽₇)	1	15	yes	complemented
47	C₂³×C₇	C₂²×ΓL(1,𝔽₈)	1	15	yes	complemented
48	C₂²×D₇	C₂²×ΓL(1,𝔽₈)	1	140	yes	complemented
49	C₇²	C₂⁵×C₃	1	1	yes	characteristic, complemented, Sylow
50	C₂⁴×C₃	—	49	30	no	—
51	C₂⁴×C₃	—	49	1	no	—
52	C₂×ΓL(1,𝔽₈)	—	7	16	no	—
53	C₂×ΓL(1,𝔽₈)	—	7	15	no	—
54	C₂×ΓL(1,𝔽₈)	—	7	15	no	—
55	C₂×ΓL(1,𝔽₈)	—	14	45	no	—
56	AGL(1,𝔽₇)	—	7	16	no	—
57	C₂⁵	—	7	1	no	Sylow
58	C₂²×C₇	—	2	105	no	—
59	C₂²×C₇	—	7	120	no	—
60	C₂²×C₇	C₂²×AGL(1,𝔽₇)	1	35	yes	complemented
61	C₂²×C₇	C₂³×ΓL(1,𝔽₈)	1	35	yes	complemented
62	C₂×D₇	C₂³×ΓL(1,𝔽₈)	1	120	yes	complemented
63	C₂³×C₃	—	49	140	no	—
64	C₂³×C₃	—	49	15	no	—
65	ΓL(1,𝔽₈)	—	7	1	no	—
66	ΓL(1,𝔽₈)	—	7	1	no	—
67	ΓL(1,𝔽₈)	—	14	3	no	—
68	C₂⁴	—	7	30	no	—
69	C₂⁴	#64423	1	1	yes	characteristic, direct product factor
70	C₂×C₇	—	2	45	no	—
71	C₂×C₇	—	7	16	no	—
72	C₂×C₇	C₂³×AGL(1,𝔽₇)	1	15	yes	complemented
73	C₂×C₇	C₂⁴×ΓL(1,𝔽₈)	1	15	yes	complemented
74	D₇	C₂⁴×ΓL(1,𝔽₈)	1	16	yes	complemented
75	C₂²×C₃	—	49	120	no	—
76	C₂²×C₃	—	49	35	no	—
77	C₂³	—	7	140	no	—
78	C₂³	C₂×#64423	1	15	yes	direct product factor
79	C₇	—	2	3	no	—
80	C₇	C₂⁴×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
81	C₇	C₂⁵×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
82	C₂×C₃	—	49	16	no	—
83	C₂×C₃	—	49	15	no	—
84	C₂²	—	7	120	no	—
85	C₂²	C₂²×#64423	1	35	yes	direct product factor
86	C₃	—	49	1	no	Sylow
87	C₂	—	7	16	no	—
88	C₂	C₂³×#64423	1	15	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂⁴	#64423
commutator, cosocle, hypocenter	C₇²	C₂⁵×C₃
Frattini	C₁	G
Fitting, socle	C₂⁴×C₇²	C₂×C₃
LCS	G ▷ C₇²
UCS	C₁ ◁ C₂⁴

Complements

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

N	H ≅ G/N	type
1. C₂⁴×#3861	C₂	semidirect
2. C₂³×#64423	C₂	direct (N × H)
3. C₂⁴×C₇×D₇	C₃	semidirect
4. C₂³×#3861	C₂²	semidirect
5. C₂²×#64423	C₂²	direct (N × H)
6. C₂⁴×C₇²	C₂×C₃	semidirect
7. C₂³×C₇×D₇	C₂×C₃	semidirect
10. C₂²×#3861	C₂³	semidirect
11. C₂×#64423	C₂³	direct (N × H)
12. C₂³×C₇²	C₂²×C₃	semidirect
13. C₂²×C₇×D₇	C₂²×C₃	semidirect
19. C₂×#3861	C₂⁴	semidirect
20. #64423	C₂⁴	direct (N × H)
22. C₂⁴×D₇	ΓL(1,𝔽₈)	semidirect
23. C₂²×C₇²	C₂³×C₃	semidirect
24. C₂×C₇×D₇	C₂³×C₃	semidirect
30. #3861	C₂⁵	semidirect
33. C₂⁴×C₇	AGL(1,𝔽₇)	semidirect
34. C₂⁴×C₇	C₂×ΓL(1,𝔽₈)	semidirect
35. C₂³×D₇	C₂×ΓL(1,𝔽₈)	semidirect
36. C₂×C₇²	C₂⁴×C₃	semidirect
37. C₇×D₇	C₂⁴×C₃	semidirect
46. C₂³×C₇	C₂×AGL(1,𝔽₇)	semidirect
47. C₂³×C₇	C₂²×ΓL(1,𝔽₈)	semidirect
48. C₂²×D₇	C₂²×ΓL(1,𝔽₈)	semidirect
49. C₇²	C₂⁵×C₃	semidirect
60. C₂²×C₇	C₂²×AGL(1,𝔽₇)	semidirect
61. C₂²×C₇	C₂³×ΓL(1,𝔽₈)	semidirect
62. C₂×D₇	C₂³×ΓL(1,𝔽₈)	semidirect
69. C₂⁴	#64423	direct (N × H)
72. C₂×C₇	C₂³×AGL(1,𝔽₇)	semidirect
73. C₂×C₇	C₂⁴×ΓL(1,𝔽₈)	semidirect
74. D₇	C₂⁴×ΓL(1,𝔽₈)	semidirect
78. C₂³	C₂×#64423	direct (N × H)
80. C₇	C₂⁴×AGL(1,𝔽₇)	semidirect
81. C₇	C₂⁵×ΓL(1,𝔽₈)	semidirect
85. C₂²	C₂²×#64423	direct (N × H)
88. C₂	C₂³×#64423	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×#3861	2
C₂³×#64423	2
C₂⁴×C₇×D₇	3
C₂⁵×ΓL(1,𝔽₈)	7
C₂⁴×AGL(1,𝔽₇)	7

Subgroups — C24×#64423 #72260008

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×#64423 #72260008