Subgroups — C₂³×C₇×#127 #13239880

Order 1792.

24030 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₇×D₄	C₂	1	6	yes	maximal, complemented
2	C₂²×C₇×#127	C₂	1	56	yes	maximal, direct product factor
3	C₂³×C₄²×C₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁶×C₇	C₂²	1	4	yes	—
5	C₂³×C₇×D₄	—	2	96	no	—
6	C₂³×C₇×D₄	C₂²	1	168	yes	complemented
7	C₂⁴×C₄×C₇	C₂²	1	3	yes	—
8	C₂×C₇×#127	C₂²	1	448	yes	direct product factor
9	C₂²×C₄²×C₇	C₂²	1	28	yes	complemented
10	C₂³×#127	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂⁵×C₇	—	2	96	no	—
12	C₂⁵×C₇	C₂³	1	56	yes	—
13	C₂⁵×C₇	C₂³	1	1	yes	characteristic
14	C₂²×C₇×D₄	—	2	1344	no	—
15	C₂²×C₇×D₄	C₂³	1	672	yes	complemented
16	C₂³×C₄×C₇	D₄	1	48	yes	complemented
17	C₂³×C₄×C₇	C₂³	1	42	yes	—
18	C₇×#127	C₂³	1	512	yes	direct product factor
19	C₂×C₄²×C₇	C₂³	1	112	yes	complemented
20	C₂⁴×D₄	C₂×C₇	1	6	yes	complemented
21	C₂²×#127	C₂×C₇	1	56	yes	direct product factor
22	C₂³×C₄²	C₂×C₇	1	1	yes	characteristic, complemented
23	C₂⁴×C₇	—	2	672	no	—
24	C₂⁴×C₇	—	4	256	no	—
25	C₂⁴×C₇	C₂×D₄	1	24	yes	—
26	C₂⁴×C₇	C₂⁴	1	112	yes	—
27	C₂⁴×C₇	C₂⁴	1	7	yes	—
28	C₂×C₇×D₄	—	2	2688	no	—
29	C₂×C₇×D₄	C₂⁴	1	384	yes	complemented
30	C₂²×C₄×C₇	C₂×D₄	1	336	yes	complemented
31	C₂²×C₄×C₇	C₂⁴	1	84	yes	—
32	C₄²×C₇	C₂⁴	1	64	yes	complemented
33	C₂⁶	C₂²×C₇	1	4	yes	—
34	C₂³×D₄	—	2	96	no	—
35	C₂³×D₄	C₂²×C₇	1	168	yes	complemented
36	C₂⁴×C₄	C₂²×C₇	1	3	yes	—
37	C₂×#127	C₂²×C₇	1	448	yes	direct product factor
38	C₂²×C₄²	C₂²×C₇	1	28	yes	complemented
39	C₂³×C₇	—	2	672	no	—
40	C₂³×C₇	—	4	896	no	—
41	C₂³×C₇	#127	1	64	yes	direct product factor
42	C₂³×C₇	C₂²×D₄	1	84	yes	—
43	C₂³×C₇	C₂⁵	1	32	yes	—
44	C₂³×C₇	C₂⁵	1	7	yes	—
45	C₇×D₄	—	2	768	no	—
46	C₂×C₄×C₇	C₂²×D₄	1	336	yes	complemented
47	C₂×C₄×C₇	C₂⁵	1	24	yes	—
48	C₂⁵	—	2	96	no	—
49	C₂⁵	C₂³×C₇	1	56	yes	—
50	C₂⁵	C₂³×C₇	1	1	yes	characteristic
51	C₂²×D₄	—	2	1344	no	—
52	C₂²×D₄	C₂³×C₇	1	672	yes	complemented
53	C₂³×C₄	C₇×D₄	1	48	yes	complemented
54	C₂³×C₄	C₂³×C₇	1	42	yes	—
55	#127	C₂³×C₇	1	512	yes	direct product factor
56	C₂×C₄²	C₂³×C₇	1	112	yes	complemented
57	C₂²×C₇	—	2	96	no	—
58	C₂²×C₇	—	4	448	no	—
59	C₂²×C₇	C₂×#127	1	112	yes	direct product factor
60	C₂²×C₇	C₂³×D₄	1	42	yes	—
61	C₂²×C₇	C₂⁶	1	1	yes	characteristic
62	C₄×C₇	C₂³×D₄	1	48	yes	complemented
63	C₂⁴	—	2	672	no	—
64	C₂⁴	—	4	256	no	—
65	C₂⁴	C₂×C₇×D₄	1	24	yes	—
66	C₂⁴	C₂⁴×C₇	1	112	yes	—
67	C₂⁴	C₂⁴×C₇	1	7	yes	—
68	C₂×D₄	—	2	2688	no	—
69	C₂×D₄	C₂⁴×C₇	1	384	yes	complemented
70	C₂²×C₄	C₂×C₇×D₄	1	336	yes	complemented
71	C₂²×C₄	C₂⁴×C₇	1	84	yes	—
72	C₄²	C₂⁴×C₇	1	64	yes	complemented
73	C₂×C₇	—	4	32	no	—
74	C₂×C₇	C₂²×#127	1	28	yes	direct product factor
75	C₂×C₇	C₂⁴×D₄	1	3	yes	—
76	C₂³	—	2	672	no	—
77	C₂³	—	4	896	no	—
78	C₂³	C₇×#127	1	64	yes	direct product factor
79	C₂³	C₂²×C₇×D₄	1	84	yes	—
80	C₂³	C₂⁵×C₇	1	32	yes	—
81	C₂³	C₂⁵×C₇	1	7	yes	—
82	D₄	—	2	768	no	—
83	C₂×C₄	C₂²×C₇×D₄	1	336	yes	complemented
84	C₂×C₄	C₂⁵×C₇	1	24	yes	—
85	C₇	C₂³×#127	1	1	yes	characteristic, direct product factor, Sylow
86	C₂²	—	2	96	no	—
87	C₂²	—	4	448	no	—
88	C₂²	C₂×C₇×#127	1	112	yes	direct product factor
89	C₂²	C₂³×C₇×D₄	1	42	yes	—
90	C₂²	C₂⁶×C₇	1	1	yes	characteristic
91	C₄	C₂³×C₇×D₄	1	48	yes	complemented
92	C₂	—	4	32	no	—
93	C₂	C₂²×C₇×#127	1	28	yes	direct product factor
94	C₂	C₂⁴×C₇×D₄	1	3	yes	—

	subgroup	G/H
center, socle	C₂⁵×C₇	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁶×C₇
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂⁴×C₇×D₄	C₂	semidirect
2. C₂²×C₇×#127	C₂	direct (N × H)
3. C₂³×C₄²×C₇	C₂	semidirect
6. C₂³×C₇×D₄	C₂²	semidirect
8. C₂×C₇×#127	C₂²	direct (N × H)
9. C₂²×C₄²×C₇	C₂²	semidirect
10. C₂³×#127	C₇	direct (N × H)
15. C₂²×C₇×D₄	C₂³	semidirect
16. C₂³×C₄×C₇	D₄	semidirect
18. C₇×#127	C₂³	direct (N × H)
19. C₂×C₄²×C₇	C₂³	semidirect
20. C₂⁴×D₄	C₂×C₇	semidirect
21. C₂²×#127	C₂×C₇	direct (N × H)
22. C₂³×C₄²	C₂×C₇	semidirect
29. C₂×C₇×D₄	C₂⁴	semidirect
30. C₂²×C₄×C₇	C₂×D₄	semidirect
32. C₄²×C₇	C₂⁴	semidirect
35. C₂³×D₄	C₂²×C₇	semidirect
37. C₂×#127	C₂²×C₇	direct (N × H)
38. C₂²×C₄²	C₂²×C₇	semidirect
41. C₂³×C₇	#127	direct (N × H)
46. C₂×C₄×C₇	C₂²×D₄	semidirect
52. C₂²×D₄	C₂³×C₇	semidirect
53. C₂³×C₄	C₇×D₄	semidirect
55. #127	C₂³×C₇	direct (N × H)
56. C₂×C₄²	C₂³×C₇	semidirect
59. C₂²×C₇	C₂×#127	direct (N × H)
62. C₄×C₇	C₂³×D₄	semidirect
69. C₂×D₄	C₂⁴×C₇	semidirect
70. C₂²×C₄	C₂×C₇×D₄	semidirect
72. C₄²	C₂⁴×C₇	semidirect
74. C₂×C₇	C₂²×#127	direct (N × H)
78. C₂³	C₇×#127	direct (N × H)
83. C₂×C₄	C₂²×C₇×D₄	semidirect
85. C₇	C₂³×#127	direct (N × H)
88. C₂²	C₂×C₇×#127	direct (N × H)
91. C₄	C₂³×C₇×D₄	semidirect
93. C₂	C₂²×C₇×#127	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×C₇×D₄	2
C₂²×C₇×#127	2
C₂³×C₄²×C₇	2
C₂³×#127	7

Subgroups — C23×C7×#127 #13239880

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₇×#127 #13239880