Subgroups — C₂×C₅×C₇×SD₁₆ #11765303

Order 1120.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₇×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×C₇×D₄	C₂	1	1	yes	maximal, characteristic
3	C₅×C₇×SD₁₆	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₅×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×C₅×C₇	—	2	1	no	—
6	C₅×C₇×Q₈	—	2	1	no	—
7	C₅×C₇×Q₈	C₂²	1	2	yes	complemented
8	C₅×C₇×D₄	—	2	1	no	—
9	C₅×C₇×D₄	C₂²	1	2	yes	—
10	C₂×C₄×C₅×C₇	—	2	1	no	—
11	C₂×C₄×C₅×C₇	C₂²	1	1	yes	characteristic
12	C₅×C₇×C₈	C₂²	1	2	yes	complemented
13	C₂×C₇×SD₁₆	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₂×C₅×SD₁₆	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₂²×C₅×C₇	—	2	2	no	—
16	C₂²×C₅×C₇	—	4	2	no	—
17	C₂²×C₅×C₇	D₄	1	1	yes	characteristic
18	C₄×C₅×C₇	—	2	2	no	—
19	C₄×C₅×C₇	D₄	1	1	yes	characteristic
20	C₄×C₅×C₇	C₂³	1	1	yes	characteristic
21	C₂×C₇×Q₈	C₂×C₅	1	1	yes	characteristic, complemented
22	C₂×C₇×D₄	C₂×C₅	1	1	yes	characteristic
23	C₇×SD₁₆	C₂×C₅	1	4	yes	direct product factor
24	C₂×C₇×C₈	C₂×C₅	1	1	yes	characteristic, complemented
25	C₂×C₅×Q₈	C₂×C₇	1	1	yes	characteristic, complemented
26	C₂×C₅×D₄	C₂×C₇	1	1	yes	characteristic
27	C₅×SD₁₆	C₂×C₇	1	4	yes	direct product factor
28	C₂×C₅×C₈	C₂×C₇	1	1	yes	characteristic, complemented
29	C₂×C₅×C₇	—	4	2	no	—
30	C₂×C₅×C₇	SD₁₆	1	2	yes	direct product factor
31	C₂×C₅×C₇	C₂×D₄	1	1	yes	characteristic
32	C₂³×C₇	—	2	1	no	—
33	C₇×Q₈	—	2	1	no	—
34	C₇×Q₈	C₂²×C₅	1	2	yes	complemented
35	C₇×D₄	—	2	1	no	—
36	C₇×D₄	C₂²×C₅	1	2	yes	—
37	C₂×C₄×C₇	—	2	1	no	—
38	C₂×C₄×C₇	C₂²×C₅	1	1	yes	characteristic
39	C₇×C₈	C₂²×C₅	1	2	yes	complemented
40	C₂³×C₅	—	2	1	no	—
41	C₅×Q₈	—	2	1	no	—
42	C₅×Q₈	C₂²×C₇	1	2	yes	complemented
43	C₅×D₄	—	2	1	no	—
44	C₅×D₄	C₂²×C₇	1	2	yes	—
45	C₂×C₄×C₅	—	2	1	no	—
46	C₂×C₄×C₅	C₂²×C₇	1	1	yes	characteristic
47	C₅×C₈	C₂²×C₇	1	2	yes	complemented
48	C₅×C₇	C₂×SD₁₆	1	1	yes	characteristic, direct product factor, Hall
49	C₂×SD₁₆	C₅×C₇	1	1	yes	characteristic, direct product factor, Sylow
50	C₂²×C₇	—	2	2	no	—
51	C₂²×C₇	—	4	2	no	—
52	C₂²×C₇	C₅×D₄	1	1	yes	characteristic
53	C₄×C₇	—	2	2	no	—
54	C₄×C₇	C₅×D₄	1	1	yes	characteristic
55	C₄×C₇	C₂³×C₅	1	1	yes	characteristic
56	C₂²×C₅	—	2	2	no	—
57	C₂²×C₅	—	4	2	no	—
58	C₂²×C₅	C₇×D₄	1	1	yes	characteristic
59	C₄×C₅	—	2	2	no	—
60	C₄×C₅	C₇×D₄	1	1	yes	characteristic
61	C₄×C₅	C₂³×C₇	1	1	yes	characteristic
62	C₂×Q₈	C₂×C₅×C₇	1	1	yes	characteristic, complemented
63	C₂×D₄	C₂×C₅×C₇	1	1	yes	characteristic
64	SD₁₆	C₂×C₅×C₇	1	4	yes	direct product factor
65	C₂×C₈	C₂×C₅×C₇	1	1	yes	characteristic, complemented
66	C₂×C₇	—	4	2	no	—
67	C₂×C₇	C₅×SD₁₆	1	2	yes	direct product factor
68	C₂×C₇	C₂×C₅×D₄	1	1	yes	characteristic
69	C₂×C₅	—	4	2	no	—
70	C₂×C₅	C₇×SD₁₆	1	2	yes	direct product factor
71	C₂×C₅	C₂×C₇×D₄	1	1	yes	characteristic
72	C₂³	—	2	1	no	—
73	Q₈	—	2	1	no	—
74	Q₈	C₂²×C₅×C₇	1	2	yes	complemented
75	D₄	—	2	1	no	—
76	D₄	C₂²×C₅×C₇	1	2	yes	—
77	C₂×C₄	—	2	1	no	—
78	C₂×C₄	C₂²×C₅×C₇	1	1	yes	characteristic
79	C₈	C₂²×C₅×C₇	1	2	yes	complemented
80	C₇	C₂×C₅×SD₁₆	1	1	yes	characteristic, direct product factor, Sylow
81	C₅	C₂×C₇×SD₁₆	1	1	yes	characteristic, direct product factor, Sylow
82	C₂²	—	2	2	no	—
83	C₂²	—	4	2	no	—
84	C₂²	C₅×C₇×D₄	1	1	yes	characteristic
85	C₄	—	2	2	no	—
86	C₄	C₅×C₇×D₄	1	1	yes	characteristic
87	C₄	C₂³×C₅×C₇	1	1	yes	characteristic
88	C₂	—	4	2	no	—
89	C₂	C₅×C₇×SD₁₆	1	2	yes	direct product factor
90	C₂	C₂×C₅×C₇×D₄	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅×C₇	D₄
commutator, Frattini, cosocle	C₄	C₂³×C₅×C₇
prime radical	C₂×C₅×C₇×D₄	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅×C₇ ◁ C₂×C₄×C₅×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×C₇×Q₈	C₂	semidirect
3. C₅×C₇×SD₁₆	C₂	direct (N × H)
4. C₂×C₅×C₇×C₈	C₂	semidirect
7. C₅×C₇×Q₈	C₂²	semidirect
12. C₅×C₇×C₈	C₂²	semidirect
13. C₂×C₇×SD₁₆	C₅	direct (N × H)
14. C₂×C₅×SD₁₆	C₇	direct (N × H)
21. C₂×C₇×Q₈	C₂×C₅	semidirect
23. C₇×SD₁₆	C₂×C₅	direct (N × H)
24. C₂×C₇×C₈	C₂×C₅	semidirect
25. C₂×C₅×Q₈	C₂×C₇	semidirect
27. C₅×SD₁₆	C₂×C₇	direct (N × H)
28. C₂×C₅×C₈	C₂×C₇	semidirect
30. C₂×C₅×C₇	SD₁₆	direct (N × H)
34. C₇×Q₈	C₂²×C₅	semidirect
39. C₇×C₈	C₂²×C₅	semidirect
42. C₅×Q₈	C₂²×C₇	semidirect
47. C₅×C₈	C₂²×C₇	semidirect
48. C₅×C₇	C₂×SD₁₆	direct (N × H)
49. C₂×SD₁₆	C₅×C₇	direct (N × H)
62. C₂×Q₈	C₂×C₅×C₇	semidirect
64. SD₁₆	C₂×C₅×C₇	direct (N × H)
65. C₂×C₈	C₂×C₅×C₇	semidirect
67. C₂×C₇	C₅×SD₁₆	direct (N × H)
70. C₂×C₅	C₇×SD₁₆	direct (N × H)
74. Q₈	C₂²×C₅×C₇	semidirect
79. C₈	C₂²×C₅×C₇	semidirect
80. C₇	C₂×C₅×SD₁₆	direct (N × H)
81. C₅	C₂×C₇×SD₁₆	direct (N × H)
89. C₂	C₅×C₇×SD₁₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×C₇×Q₈	2
C₂×C₅×C₇×D₄	2
C₅×C₇×SD₁₆	2
C₂×C₅×C₇×C₈	2
C₂×C₇×SD₁₆	5
C₂×C₅×SD₁₆	7

Subgroups — C2×C5×C7×SD16 #11765303

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×C₇×SD₁₆ #11765303