Subgroups — C₂×C₅×#4248 #13132681

Order 1680.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×Dic₂₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₇×Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₅×Dic₇	C₂	1	1	yes	maximal, characteristic
4	C₅×#4248	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₅×Dic₁₄	—	3	1	no	maximal, Hall
6	C₂²×C₃×C₅×C₇	C₂²	1	1	yes	characteristic
7	C₅×Dic₂₁	C₂²	1	2	yes	—
8	C₅×C₇×Dic₃	C₂²	1	2	yes	—
9	C₃×C₅×Dic₇	C₂²	1	2	yes	—
10	C₂×#4248	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂×C₄×C₅×C₇	—	3	1	no	—
12	C₂×C₅×Dic₇	—	3	1	no	—
13	C₂×C₅×Dic₇	D₃	1	1	yes	characteristic
14	C₅×Dic₁₄	—	3	4	no	—
15	C₂×C₅×Dic₆	—	7	1	no	maximal, Hall
16	C₂×C₃×C₅×C₇	Q₈	1	2	yes	complemented
17	C₂×C₃×C₅×C₇	C₂³	1	1	yes	characteristic
18	C₂×Dic₂₁	C₂×C₅	1	1	yes	characteristic
19	C₂×C₇×Dic₃	C₂×C₅	1	1	yes	characteristic
20	C₂×C₃×Dic₇	C₂×C₅	1	1	yes	characteristic
21	#4248	C₂×C₅	1	4	yes	direct product factor
22	C₂²×C₅×C₇	C₂×D₃	1	1	yes	characteristic
23	C₄×C₅×C₇	—	3	2	no	—
24	C₅×Dic₇	—	3	2	no	—
25	C₅×Dic₇	C₂×D₃	1	2	yes	—
26	C₂×C₃×C₄×C₅	—	7	1	no	—
27	C₂×C₅×Dic₃	—	7	1	no	—
28	C₂×C₅×Dic₃	D₇	1	1	yes	characteristic
29	C₅×Dic₆	—	7	4	no	—
30	C₂×Dic₁₄	—	3	1	no	Hall
31	C₃×C₅×C₇	C₂×Q₈	1	1	yes	characteristic, complemented, Hall
32	C₂²×C₃×C₇	C₂²×C₅	1	1	yes	characteristic
33	Dic₂₁	C₂²×C₅	1	2	yes	—
34	C₃×Dic₇	C₂²×C₅	1	2	yes	—
35	C₇×Dic₃	C₂²×C₅	1	2	yes	—
36	C₂×C₅×Q₈	—	21	1	no	Hall
37	C₂×C₅×C₇	Dic₆	1	2	yes	complemented
38	C₂×C₅×C₇	C₂²×D₃	1	1	yes	characteristic
39	C₂²×C₃×C₅	C₂×D₇	1	1	yes	characteristic
40	C₃×C₄×C₅	—	7	2	no	—
41	C₅×Dic₃	—	7	2	no	—
42	C₅×Dic₃	C₂×D₇	1	2	yes	—
43	C₂×C₄×C₇	—	3	1	no	—
44	C₂×Dic₇	—	3	1	no	—
45	C₂×Dic₇	C₅×D₃	1	1	yes	characteristic
46	Dic₁₄	—	3	4	no	—
47	C₂×Dic₆	—	7	1	no	Hall
48	C₂×C₃×C₇	C₅×Q₈	1	2	yes	complemented
49	C₂×C₃×C₇	C₂³×C₅	1	1	yes	characteristic
50	C₅×Q₈	—	21	4	no	—
51	C₂×C₄×C₅	—	3	1	no	—
52	C₂×C₄×C₅	—	7	1	no	—
53	C₂×C₄×C₅	—	21	1	no	—
54	C₅×C₇	C₂×Dic₆	1	1	yes	characteristic, complemented, Hall
55	C₂×C₃×C₅	Dic₁₄	1	2	yes	complemented
56	C₂×C₃×C₅	C₂²×D₇	1	1	yes	characteristic
57	C₂²×C₇	C₂×C₅×D₃	1	1	yes	characteristic
58	C₄×C₇	—	3	2	no	—
59	Dic₇	—	3	2	no	—
60	Dic₇	C₂×C₅×D₃	1	2	yes	—
61	C₂×C₃×C₄	—	7	1	no	—
62	C₂×Dic₃	—	7	1	no	—
63	C₂×Dic₃	C₅×D₇	1	1	yes	characteristic
64	Dic₆	—	7	4	no	—
65	C₃×C₇	C₂×C₅×Q₈	1	1	yes	characteristic, complemented, Hall
66	C₂²×C₅	D₃×D₇	1	1	yes	characteristic
67	C₄×C₅	—	3	2	no	—
68	C₄×C₅	—	7	2	no	—
69	C₄×C₅	—	21	2	no	—
70	C₂×Q₈	—	21	1	no	Sylow
71	C₃×C₅	C₂×Dic₁₄	1	1	yes	characteristic, complemented, Hall
72	C₂×C₇	C₅×Dic₆	1	2	yes	complemented
73	C₂×C₇	C₂²×C₅×D₃	1	1	yes	characteristic
74	C₂²×C₃	C₂×C₅×D₇	1	1	yes	characteristic
75	C₃×C₄	—	7	2	no	—
76	Dic₃	—	7	2	no	—
77	Dic₃	C₂×C₅×D₇	1	2	yes	—
78	C₂×C₅	#4248	1	2	yes	direct product factor
79	C₂×C₅	C₂×D₃×D₇	1	1	yes	characteristic
80	Q₈	—	21	4	no	—
81	C₂×C₄	—	3	1	no	—
82	C₂×C₄	—	7	1	no	—
83	C₂×C₄	—	21	1	no	—
84	C₇	C₂×C₅×Dic₆	1	1	yes	characteristic, complemented, Sylow
85	C₂×C₃	C₅×Dic₁₄	1	2	yes	complemented
86	C₂×C₃	C₂²×C₅×D₇	1	1	yes	characteristic
87	C₅	C₂×#4248	1	1	yes	characteristic, direct product factor, Sylow
88	C₂²	C₅×D₃×D₇	1	1	yes	characteristic
89	C₄	—	3	2	no	—
90	C₄	—	7	2	no	—
91	C₄	—	21	2	no	—
92	C₃	C₂×C₅×Dic₁₄	1	1	yes	characteristic, complemented, Sylow
93	C₂	C₅×#4248	1	2	yes	direct product factor
94	C₂	C₂×C₅×D₃×D₇	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₅	D₃×D₇
commutator, cosocle	C₂×C₃×C₇	C₂³×C₅
Frattini	C₂	C₂×C₅×D₃×D₇
Fitting, socle, prime radical	C₂²×C₃×C₅×C₇	C₂²
hypocenter	C₃×C₇	C₂×C₅×Q₈
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂²×C₅

Complements

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

N	H ≅ G/N	type
4. C₅×#4248	C₂	direct (N × H)
10. C₂×#4248	C₅	direct (N × H)
16. C₂×C₃×C₅×C₇	Q₈	semidirect
21. #4248	C₂×C₅	direct (N × H)
31. C₃×C₅×C₇	C₂×Q₈	semidirect
37. C₂×C₅×C₇	Dic₆	semidirect
48. C₂×C₃×C₇	C₅×Q₈	semidirect
54. C₅×C₇	C₂×Dic₆	semidirect
55. C₂×C₃×C₅	Dic₁₄	semidirect
65. C₃×C₇	C₂×C₅×Q₈	semidirect
71. C₃×C₅	C₂×Dic₁₄	semidirect
72. C₂×C₇	C₅×Dic₆	semidirect
78. C₂×C₅	#4248	direct (N × H)
84. C₇	C₂×C₅×Dic₆	semidirect
85. C₂×C₃	C₅×Dic₁₄	semidirect
87. C₅	C₂×#4248	direct (N × H)
92. C₃	C₂×C₅×Dic₁₄	semidirect
93. C₂	C₅×#4248	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×Dic₂₁	2
C₂×C₅×C₇×Dic₃	2
C₂×C₃×C₅×Dic₇	2
C₅×#4248	2
C₂×C₅×Dic₁₄	3
C₂×#4248	5
C₂×C₅×Dic₆	7

Subgroups — C2×C5×#4248 #13132681

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#4248 #13132681