Subgroups — C₂×C₃×C₈×D₇ #10621210

Order 672.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×D₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₈×D₇	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₈×D₇	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×D₇	C₄	1	1	yes	characteristic
7	C₂×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
8	C₂×C₃×Dic₇	C₄	1	1	yes	characteristic
9	C₃×C₄×D₇	C₄	1	2	yes	—
10	C₃×C₄×D₇	C₂²	1	2	yes	—
11	C₃×C₇×C₈	C₂²	1	2	yes	complemented
12	C₃×C₇⋊C₈	C₂²	1	2	yes	complemented
13	C₂×C₄×D₇	C₂×C₃	1	1	yes	characteristic
14	C₂×C₇×C₈	C₂×C₃	1	1	yes	characteristic, complemented
15	C₂×C₇⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
16	C₈×D₇	C₂×C₃	1	4	yes	direct product factor
17	C₂²×C₃×C₈	—	7	1	no	maximal, Hall
18	C₂²×C₃×C₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₃×D₇	C₈	1	4	yes	direct product factor
20	C₂×C₃×D₇	C₂×C₄	1	2	yes	—
21	C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
22	C₃×C₄×C₇	C₂³	1	1	yes	characteristic
23	C₃×Dic₇	C₂×C₄	1	2	yes	—
24	C₂²×D₇	C₃×C₄	1	1	yes	characteristic
25	C₂×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
26	C₂×Dic₇	C₃×C₄	1	1	yes	characteristic
27	C₄×D₇	C₃×C₄	1	2	yes	—
28	C₄×D₇	C₂²×C₃	1	2	yes	—
29	C₇×C₈	C₂²×C₃	1	2	yes	complemented
30	C₇⋊C₈	C₂²×C₃	1	2	yes	complemented
31	C₂²×C₃×C₄	—	7	1	no	—
32	C₂×C₃×C₈	—	7	1	no	—
33	C₂×C₃×C₈	—	7	4	no	—
34	C₂×C₃×C₈	D₇	1	1	yes	characteristic, direct product factor
35	C₂×C₃×C₇	C₂×C₈	1	2	yes	complemented
36	C₂×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
37	C₃×D₇	C₂×C₈	1	4	yes	direct product factor
38	C₂²×C₈	—	7	1	no	Sylow
39	C₂²×C₇	C₂×C₃×C₄	1	1	yes	characteristic
40	C₂×D₇	C₃×C₈	1	4	yes	direct product factor
41	C₂×D₇	C₂×C₃×C₄	1	2	yes	—
42	C₄×C₇	C₂×C₃×C₄	1	1	yes	characteristic
43	C₄×C₇	C₂³×C₃	1	1	yes	characteristic
44	Dic₇	C₂×C₃×C₄	1	2	yes	—
45	C₂³×C₃	—	7	1	no	—
46	C₂×C₃×C₄	—	7	1	no	—
47	C₂×C₃×C₄	—	7	2	no	—
48	C₂×C₃×C₄	—	7	2	no	—
49	C₂×C₃×C₄	C₂×D₇	1	1	yes	characteristic
50	C₃×C₈	—	7	2	no	—
51	C₃×C₈	C₂×D₇	1	2	yes	direct product factor
52	C₃×C₇	C₂²×C₈	1	1	yes	characteristic, complemented, Hall
53	C₂²×C₄	—	7	1	no	—
54	C₂×C₈	—	7	1	no	—
55	C₂×C₈	—	7	4	no	—
56	C₂×C₈	C₃×D₇	1	1	yes	characteristic, direct product factor
57	C₂×C₇	C₂×C₃×C₈	1	2	yes	complemented
58	C₂×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
59	D₇	C₂×C₃×C₈	1	4	yes	direct product factor
60	C₂²×C₃	—	7	4	no	—
61	C₂²×C₃	—	7	2	no	—
62	C₂²×C₃	C₄×D₇	1	1	yes	characteristic
63	C₃×C₄	—	7	2	no	—
64	C₃×C₄	C₄×D₇	1	1	yes	characteristic
65	C₃×C₄	C₂²×D₇	1	1	yes	characteristic
66	C₂³	—	7	1	no	—
67	C₂×C₄	—	7	1	no	—
68	C₂×C₄	—	7	2	no	—
69	C₂×C₄	—	7	2	no	—
70	C₂×C₄	C₂×C₃×D₇	1	1	yes	characteristic
71	C₈	—	7	2	no	—
72	C₈	C₂×C₃×D₇	1	2	yes	direct product factor
73	C₇	C₂²×C₃×C₈	1	1	yes	characteristic, complemented, Sylow
74	C₂×C₃	—	7	4	no	—
75	C₂×C₃	C₈×D₇	1	2	yes	direct product factor
76	C₂×C₃	C₂×C₄×D₇	1	1	yes	characteristic
77	C₂²	—	7	4	no	—
78	C₂²	—	7	2	no	—
79	C₂²	C₃×C₄×D₇	1	1	yes	characteristic
80	C₄	—	7	2	no	—
81	C₄	C₃×C₄×D₇	1	1	yes	characteristic
82	C₄	C₂²×C₃×D₇	1	1	yes	characteristic
83	C₃	C₂×C₈×D₇	1	1	yes	characteristic, direct product factor, Sylow
84	C₂	—	7	4	no	—
85	C₂	C₃×C₈×D₇	1	2	yes	direct product factor
86	C₂	C₂×C₃×C₄×D₇	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃×C₈	D₇
commutator, hypocenter	C₇	C₂²×C₃×C₈
Frattini	C₄	C₂²×C₃×D₇
Fitting	C₂×C₃×C₇×C₈	C₂
socle	C₂²×C₃×C₇	C₂×C₄
prime radical	C₂²×C₃×D₇	C₄
cosocle	C₄×C₇	C₂³×C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₂×C₃×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₇×C₈	C₂	semidirect
3. C₂×C₃×C₇⋊C₈	C₂	semidirect
4. C₃×C₈×D₇	C₂	direct (N × H)
5. C₂×C₈×D₇	C₃	direct (N × H)
11. C₃×C₇×C₈	C₂²	semidirect
12. C₃×C₇⋊C₈	C₂²	semidirect
14. C₂×C₇×C₈	C₂×C₃	semidirect
15. C₂×C₇⋊C₈	C₂×C₃	semidirect
16. C₈×D₇	C₂×C₃	direct (N × H)
19. C₂×C₃×D₇	C₈	direct (N × H)
29. C₇×C₈	C₂²×C₃	semidirect
30. C₇⋊C₈	C₂²×C₃	semidirect
34. C₂×C₃×C₈	D₇	direct (N × H)
35. C₂×C₃×C₇	C₂×C₈	semidirect
37. C₃×D₇	C₂×C₈	direct (N × H)
40. C₂×D₇	C₃×C₈	direct (N × H)
51. C₃×C₈	C₂×D₇	direct (N × H)
52. C₃×C₇	C₂²×C₈	semidirect
56. C₂×C₈	C₃×D₇	direct (N × H)
57. C₂×C₇	C₂×C₃×C₈	semidirect
59. D₇	C₂×C₃×C₈	direct (N × H)
72. C₈	C₂×C₃×D₇	direct (N × H)
73. C₇	C₂²×C₃×C₈	semidirect
75. C₂×C₃	C₈×D₇	direct (N × H)
83. C₃	C₂×C₈×D₇	direct (N × H)
85. C₂	C₃×C₈×D₇	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×C₃×C₄×D₇	2
C₂×C₃×C₇×C₈	2
C₂×C₃×C₇⋊C₈	2
C₃×C₈×D₇	2
C₂×C₈×D₇	3
C₂²×C₃×C₈	7

Subgroups — C2×C3×C8×D7 #10621210

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₈×D₇ #10621210