Subgroups — C₄×C₇×Dic₁₀ #11765112

Order 1120.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×Dic₁₀	C₂	1	1	yes	maximal, characteristic
2	C₄²×C₅×C₇	C₂	1	1	yes	maximal, characteristic
3	C₇×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic
4	C₇×#683	C₂	1	2	yes	maximal
5	C₄×C₇×Dic₅	C₂	1	2	yes	maximal
6	C₂×C₄×C₅×C₇	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₅×C₇	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₅×C₇	C₂²	1	1	yes	characteristic
9	C₂×C₇×Dic₅	C₂²	1	2	yes	—
10	C₂×C₇×Dic₅	C₂²	1	2	yes	—
11	C₇×Dic₁₀	C₄	1	4	yes	direct product factor
12	C₄×C₇×Q₈	—	5	1	no	maximal, Hall
13	C₄×Dic₁₀	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₂²×C₅×C₇	C₂³	1	1	yes	characteristic
15	C₄×C₅×C₇	—	2	1	no	—
16	C₄×C₅×C₇	C₂×C₄	1	2	yes	—
17	C₄×C₅×C₇	Q₈	1	2	yes	complemented
18	C₇×Dic₅	—	2	2	no	—
19	C₇×Dic₅	C₂×C₄	1	4	yes	—
20	C₂×C₇×Q₈	—	5	1	no	—
21	C₇×C₄⋊C₄	—	5	2	no	—
22	C₇×C₄⋊C₄	—	5	1	no	—
23	C₄²×C₇	—	5	2	no	—
24	C₄²×C₇	D₅	1	1	yes	characteristic
25	C₂×Dic₁₀	C₂×C₇	1	1	yes	characteristic
26	C₄²×C₅	C₂×C₇	1	1	yes	characteristic
27	C₄⋊Dic₅	C₂×C₇	1	1	yes	characteristic
28	#683	C₂×C₇	1	2	yes	—
29	C₄×Dic₅	C₂×C₇	1	2	yes	—
30	C₂×C₅×C₇	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₅×C₇	C₂×Q₈	1	1	yes	characteristic
32	C₂×C₅×C₇	#41	1	1	yes	characteristic
33	C₇×Q₈	—	5	4	no	—
34	C₂×C₄×C₇	—	5	2	no	—
35	C₂×C₄×C₇	—	5	2	no	—
36	C₂×C₄×C₇	C₂×D₅	1	1	yes	characteristic
37	C₂×C₄×C₇	C₂×D₅	1	1	yes	characteristic
38	C₂×C₄×C₇	C₂×D₅	1	1	yes	characteristic
39	C₂×C₄×C₅	C₂²×C₇	1	1	yes	characteristic
40	C₂×C₄×C₅	C₂²×C₇	1	1	yes	characteristic
41	C₂×C₄×C₅	C₂²×C₇	1	1	yes	characteristic
42	C₂×Dic₅	C₂²×C₇	1	2	yes	—
43	C₂×Dic₅	C₂²×C₇	1	2	yes	—
44	Dic₁₀	C₄×C₇	1	4	yes	direct product factor
45	C₅×C₇	C₄×Q₈	1	1	yes	characteristic, complemented, Hall
46	C₄×Q₈	—	5	1	no	Sylow
47	C₂²×C₇	C₂²×D₅	1	1	yes	characteristic
48	C₄×C₇	—	2	1	no	—
49	C₄×C₇	—	5	4	no	—
50	C₄×C₇	—	10	2	no	—
51	C₄×C₇	Dic₁₀	1	2	yes	direct product factor
52	C₄×C₇	C₄×D₅	1	2	yes	—
53	C₂²×C₅	C₂³×C₇	1	1	yes	characteristic
54	C₄×C₅	—	2	1	no	—
55	C₄×C₅	C₂×C₄×C₇	1	2	yes	—
56	C₄×C₅	C₇×Q₈	1	2	yes	complemented
57	Dic₅	—	2	2	no	—
58	Dic₅	C₂×C₄×C₇	1	4	yes	—
59	C₂×Q₈	—	5	1	no	—
60	C₄⋊C₄	—	5	2	no	—
61	C₄⋊C₄	—	5	1	no	—
62	C₄²	—	5	2	no	—
63	C₄²	C₇×D₅	1	1	yes	characteristic
64	C₂×C₇	C₂×Dic₁₀	1	1	yes	characteristic
65	C₂×C₇	C₂×C₄×D₅	1	1	yes	characteristic
66	C₂×C₇	#709	1	1	yes	characteristic
67	C₂×C₅	C₂²×C₄×C₇	1	1	yes	characteristic
68	C₂×C₅	C₂×C₇×Q₈	1	1	yes	characteristic
69	C₂×C₅	C₇×#41	1	1	yes	characteristic
70	Q₈	—	5	4	no	—
71	C₂×C₄	—	5	2	no	—
72	C₂×C₄	—	5	2	no	—
73	C₂×C₄	C₂×C₇×D₅	1	1	yes	characteristic
74	C₂×C₄	C₂×C₇×D₅	1	1	yes	characteristic
75	C₂×C₄	C₂×C₇×D₅	1	1	yes	characteristic
76	C₇	C₄×Dic₁₀	1	1	yes	characteristic, direct product factor, Sylow
77	C₅	C₄×C₇×Q₈	1	1	yes	characteristic, complemented, Sylow
78	C₂²	C₂²×C₇×D₅	1	1	yes	characteristic
79	C₄	—	2	1	no	—
80	C₄	—	5	4	no	—
81	C₄	—	10	2	no	—
82	C₄	C₇×Dic₁₀	1	2	yes	direct product factor
83	C₄	C₄×C₇×D₅	1	2	yes	—
84	C₂	C₂×C₇×Dic₁₀	1	1	yes	characteristic
85	C₂	C₂×C₄×C₇×D₅	1	1	yes	characteristic
86	C₂	C₇×#709	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	C₂×D₅
commutator	C₂×C₅	C₂²×C₄×C₇
Frattini	C₂²	C₂²×C₇×D₅
Fitting	C₄²×C₅×C₇	C₂
socle, prime radical	C₂²×C₅×C₇	C₂³
cosocle	C₂²×C₅	C₂³×C₇
hypercenter	C₄²×C₇	D₅
hypocenter	C₅	C₄×C₇×Q₈
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₄²×C₇

Complements

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

N	H ≅ G/N	type
11. C₇×Dic₁₀	C₄	direct (N × H)
13. C₄×Dic₁₀	C₇	direct (N × H)
17. C₄×C₅×C₇	Q₈	semidirect
44. Dic₁₀	C₄×C₇	direct (N × H)
45. C₅×C₇	C₄×Q₈	semidirect
51. C₄×C₇	Dic₁₀	direct (N × H)
56. C₄×C₅	C₇×Q₈	semidirect
76. C₇	C₄×Dic₁₀	direct (N × H)
77. C₅	C₄×C₇×Q₈	semidirect
82. C₄	C₇×Dic₁₀	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×C₇×Dic₁₀	2
C₄²×C₅×C₇	2
C₇×C₄⋊Dic₅	2
C₇×#683	2
C₄×C₇×Dic₅	2
C₄×C₇×Q₈	5
C₄×Dic₁₀	7

Subgroups — C4×C7×Dic10 #11765112

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₇×Dic₁₀ #11765112