Subgroups — C₂×D₃×C₇⋊Dic₇ #72245481

Order 2352.

540 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₃×C₇²	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃⋊(C₇⋊Dic₇)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₇⋊Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
4	D₃×C₇⋊Dic₇	C₂	1	4	yes	maximal, direct product factor
5	C₂²×C₇⋊Dic₇	—	3	1	no	maximal, Hall
6	C₂²×C₃×C₇²	C₂²	1	1	yes	characteristic
7	C₂×D₃×C₇²	C₄	1	4	yes	complemented
8	C₂×D₃×C₇²	C₂²	1	2	yes	—
9	C₃⋊(C₇⋊Dic₇)	C₂²	1	2	yes	complemented
10	C₃×C₇⋊Dic₇	C₂²	1	2	yes	complemented
11	C₂³×C₇²	—	3	1	no	—
12	C₂×C₇⋊Dic₇	—	3	1	no	—
13	C₂×C₇⋊Dic₇	—	3	4	no	—
14	C₂×C₇⋊Dic₇	D₃	1	1	yes	characteristic, direct product factor
15	C₂×D₃×Dic₇	—	7	8	no	maximal
16	C₂×C₃×C₇²	C₂×C₄	1	2	yes	complemented
17	C₂×C₃×C₇²	C₂³	1	1	yes	characteristic
18	D₃×C₇²	C₂×C₄	1	4	yes	complemented
19	C₂²×C₇²	—	3	4	no	—
20	C₂²×C₇²	—	3	2	no	—
21	C₂²×C₇²	C₂×D₃	1	1	yes	characteristic
22	C₇⋊Dic₇	—	3	2	no	—
23	C₇⋊Dic₇	C₂×D₃	1	2	yes	direct product factor
24	C₂²×D₃×C₇	D₇	1	8	yes	—
25	C₂×Dic₂₁	—	7	8	no	—
26	C₂×C₃×Dic₇	—	7	8	no	—
27	D₃×Dic₇	—	7	32	no	—
28	C₃×C₇²	C₂²×C₄	1	1	yes	characteristic, complemented, Hall
29	C₂²×Dic₇	—	21	8	no	—
30	C₂×C₇²	—	3	4	no	—
31	C₂×C₇²	C₄×D₃	1	2	yes	complemented
32	C₂×C₇²	C₂²×D₃	1	1	yes	characteristic
33	C₂²×C₃×C₇	C₂×D₇	1	8	yes	—
34	C₂×D₃×C₇	Dic₇	1	32	yes	complemented
35	C₂×D₃×C₇	C₂×D₇	1	16	yes	—
36	Dic₂₁	—	7	16	no	—
37	C₃×Dic₇	—	7	16	no	—
38	C₂³×C₇	—	3	8	no	—
39	C₂×Dic₇	—	7	8	no	—
40	C₂×Dic₇	—	21	32	no	—
41	C₂×Dic₇	—	21	8	no	—
42	C₇²	C₂×C₄×D₃	1	1	yes	characteristic, complemented, Sylow
43	C₂×C₄×D₃	—	49	1	no	Hall
44	C₂×C₃×C₇	C₂×Dic₇	1	16	yes	complemented
45	C₂×C₃×C₇	C₂²×D₇	1	8	yes	—
46	D₃×C₇	C₂×Dic₇	1	32	yes	complemented
47	C₂²×C₇	—	3	32	no	—
48	C₂²×C₇	—	3	16	no	—
49	C₂²×C₇	D₃×D₇	1	8	yes	—
50	Dic₇	—	7	16	no	—
51	Dic₇	—	21	16	no	—
52	C₂²×D₃	C₇⋊D₇	1	1	yes	characteristic
53	C₂×C₃×C₄	—	49	1	no	—
54	C₂×Dic₃	—	49	1	no	—
55	C₄×D₃	—	49	4	no	—
56	C₃×C₇	C₂²×Dic₇	1	8	yes	complemented
57	C₂²×C₄	—	147	1	no	Sylow
58	C₂×C₇	—	3	32	no	—
59	C₂×C₇	D₃×Dic₇	1	16	yes	complemented
60	C₂×C₇	C₂×D₃×D₇	1	8	yes	—
61	C₂²×C₃	C₂×C₇⋊D₇	1	1	yes	characteristic
62	C₂×D₃	C₇⋊Dic₇	1	4	yes	direct product factor
63	C₂×D₃	C₂×C₇⋊D₇	1	2	yes	—
64	C₃×C₄	—	49	2	no	—
65	Dic₃	—	49	2	no	—
66	C₂³	—	3	1	no	—
67	C₂×C₄	—	49	1	no	—
68	C₂×C₄	—	147	4	no	—
69	C₂×C₄	—	147	1	no	—
70	C₇	C₂×D₃×Dic₇	1	8	yes	complemented
71	C₂×C₃	C₂×C₇⋊Dic₇	1	2	yes	complemented
72	C₂×C₃	C₂²×C₇⋊D₇	1	1	yes	characteristic
73	D₃	C₂×C₇⋊Dic₇	1	4	yes	direct product factor
74	C₂²	—	3	4	no	—
75	C₂²	—	3	2	no	—
76	C₂²	D₃×C₇⋊D₇	1	1	yes	characteristic
77	C₄	—	49	2	no	—
78	C₄	—	147	2	no	—
79	C₃	C₂²×C₇⋊Dic₇	1	1	yes	characteristic, complemented, Sylow
80	C₂	—	3	4	no	—
81	C₂	D₃×C₇⋊Dic₇	1	2	yes	direct product factor
82	C₂	C₂×D₃×C₇⋊D₇	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²	D₃×C₇⋊D₇
commutator, hypocenter	C₃×C₇²	C₂²×C₄
Frattini	C₂	C₂×D₃×C₇⋊D₇
Fitting, socle	C₂²×C₃×C₇²	C₂²
prime radical	C₂²×D₃×C₇²	C₂
cosocle	C₂×C₃×C₇²	C₂³
LCS	G ▷ C₃×C₇²
UCS	C₁ ◁ C₂²

Complements

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

N	H ≅ G/N	type
2. C₂×C₃⋊(C₇⋊Dic₇)	C₂	semidirect
3. C₂×C₃×C₇⋊Dic₇	C₂	semidirect
4. D₃×C₇⋊Dic₇	C₂	direct (N × H)
7. C₂×D₃×C₇²	C₄	semidirect
9. C₃⋊(C₇⋊Dic₇)	C₂²	semidirect
10. C₃×C₇⋊Dic₇	C₂²	semidirect
14. C₂×C₇⋊Dic₇	D₃	direct (N × H)
16. C₂×C₃×C₇²	C₂×C₄	semidirect
18. D₃×C₇²	C₂×C₄	semidirect
23. C₇⋊Dic₇	C₂×D₃	direct (N × H)
28. C₃×C₇²	C₂²×C₄	semidirect
31. C₂×C₇²	C₄×D₃	semidirect
34. C₂×D₃×C₇	Dic₇	semidirect
42. C₇²	C₂×C₄×D₃	semidirect
44. C₂×C₃×C₇	C₂×Dic₇	semidirect
46. D₃×C₇	C₂×Dic₇	semidirect
56. C₃×C₇	C₂²×Dic₇	semidirect
59. C₂×C₇	D₃×Dic₇	semidirect
62. C₂×D₃	C₇⋊Dic₇	direct (N × H)
70. C₇	C₂×D₃×Dic₇	semidirect
71. C₂×C₃	C₂×C₇⋊Dic₇	semidirect
73. D₃	C₂×C₇⋊Dic₇	direct (N × H)
79. C₃	C₂²×C₇⋊Dic₇	semidirect
81. C₂	D₃×C₇⋊Dic₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×D₃×C₇²	2
C₂×C₃⋊(C₇⋊Dic₇)	2
C₂×C₃×C₇⋊Dic₇	2
D₃×C₇⋊Dic₇	2
C₂²×C₇⋊Dic₇	3
C₂×D₃×Dic₇	7

Subgroups — C2×D3×C7⋊Dic7 #72245481

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₃×C₇⋊Dic₇ #72245481