Subgroups — C₃×C₇×#117 #10621440

Order 672.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₄×C₇	C₂	1	1	yes	maximal, characteristic
2	C₃×C₇×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
3	C₃×C₇×C₂²⋊C₄	C₂	1	2	yes	maximal
4	C₃×C₄²×C₇	C₂	1	2	yes	maximal, complemented
5	C₇×#117	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂³×C₃×C₇	C₂²	1	1	yes	characteristic
7	C₂×C₃×C₄×C₇	C₄	1	4	yes	complemented
8	C₂×C₃×C₄×C₇	C₂²	1	4	yes	—
9	C₂×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₇	C₂×C₃	1	1	yes	characteristic
12	C₇×C₄⋊C₄	C₂×C₃	1	2	yes	complemented
13	C₇×C₂²⋊C₄	C₂×C₃	1	2	yes	—
14	C₄²×C₇	C₂×C₃	1	2	yes	complemented
15	C₃×#117	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂²×C₃×C₇	—	2	2	no	—
17	C₂²×C₃×C₇	C₂×C₄	1	2	yes	—
18	C₂²×C₃×C₇	C₂³	1	1	yes	characteristic
19	C₃×C₄×C₇	—	2	4	no	—
20	C₃×C₄×C₇	C₂×C₄	1	2	yes	—
21	C₃×C₄×C₇	C₂×C₄	1	2	yes	—
22	C₂³×C₇	C₂²×C₃	1	1	yes	characteristic
23	C₂×C₄×C₇	C₃×C₄	1	4	yes	complemented
24	C₂×C₄×C₇	C₂²×C₃	1	4	yes	—
25	C₂×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
26	C₂×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
27	C₂²×C₃×C₄	C₂×C₇	1	1	yes	characteristic
28	C₃×C₄⋊C₄	C₂×C₇	1	2	yes	complemented
29	C₃×C₂²⋊C₄	C₂×C₇	1	2	yes	—
30	C₃×C₄²	C₂×C₇	1	2	yes	complemented
31	C₂×C₃×C₇	—	2	2	no	—
32	C₂×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₃×C₇	#41	1	2	yes	—
34	#117	C₃×C₇	1	1	yes	characteristic, direct product factor, Sylow
35	C₂²×C₇	—	2	2	no	—
36	C₂²×C₇	C₂×C₃×C₄	1	2	yes	—
37	C₂²×C₇	C₂³×C₃	1	1	yes	characteristic
38	C₄×C₇	—	2	4	no	—
39	C₄×C₇	C₂×C₃×C₄	1	2	yes	—
40	C₄×C₇	C₂×C₃×C₄	1	2	yes	—
41	C₂³×C₃	C₂²×C₇	1	1	yes	characteristic
42	C₂×C₃×C₄	C₄×C₇	1	4	yes	complemented
43	C₂×C₃×C₄	C₂²×C₇	1	4	yes	—
44	C₂×C₃×C₄	C₂²×C₇	1	1	yes	characteristic
45	C₂×C₃×C₄	C₂²×C₇	1	1	yes	characteristic
46	C₃×C₇	#117	1	1	yes	characteristic, direct product factor, Hall
47	C₂²×C₄	C₂×C₃×C₇	1	1	yes	characteristic
48	C₄⋊C₄	C₂×C₃×C₇	1	2	yes	complemented
49	C₂²⋊C₄	C₂×C₃×C₇	1	2	yes	—
50	C₄²	C₂×C₃×C₇	1	2	yes	complemented
51	C₂×C₇	—	2	2	no	—
52	C₂×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
53	C₂×C₇	C₃×#41	1	2	yes	—
54	C₂²×C₃	—	2	2	no	—
55	C₂²×C₃	C₂×C₄×C₇	1	2	yes	—
56	C₂²×C₃	C₂³×C₇	1	1	yes	characteristic
57	C₃×C₄	—	2	4	no	—
58	C₃×C₄	C₂×C₄×C₇	1	2	yes	—
59	C₃×C₄	C₂×C₄×C₇	1	2	yes	—
60	C₂³	C₂²×C₃×C₇	1	1	yes	characteristic
61	C₂×C₄	C₃×C₄×C₇	1	4	yes	complemented
62	C₂×C₄	C₂²×C₃×C₇	1	4	yes	—
63	C₂×C₄	C₂²×C₃×C₇	1	1	yes	characteristic
64	C₂×C₄	C₂²×C₃×C₇	1	1	yes	characteristic
65	C₇	C₃×#117	1	1	yes	characteristic, direct product factor, Sylow
66	C₂×C₃	—	2	2	no	—
67	C₂×C₃	C₂²×C₄×C₇	1	1	yes	characteristic
68	C₂×C₃	C₇×#41	1	2	yes	—
69	C₂²	—	2	2	no	—
70	C₂²	C₂×C₃×C₄×C₇	1	2	yes	—
71	C₂²	C₂³×C₃×C₇	1	1	yes	characteristic
72	C₄	—	2	4	no	—
73	C₄	C₂×C₃×C₄×C₇	1	2	yes	—
74	C₄	C₂×C₃×C₄×C₇	1	2	yes	—
75	C₃	C₇×#117	1	1	yes	characteristic, direct product factor, Sylow
76	C₂	—	2	2	no	—
77	C₂	C₂²×C₃×C₄×C₇	1	1	yes	characteristic
78	C₂	C₃×C₇×#41	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₃×C₇×C₄⋊C₄	C₂	semidirect
4. C₃×C₄²×C₇	C₂	semidirect
5. C₇×#117	C₃	direct (N × H)
7. C₂×C₃×C₄×C₇	C₄	semidirect
12. C₇×C₄⋊C₄	C₂×C₃	semidirect
14. C₄²×C₇	C₂×C₃	semidirect
15. C₃×#117	C₇	direct (N × H)
23. C₂×C₄×C₇	C₃×C₄	semidirect
28. C₃×C₄⋊C₄	C₂×C₇	semidirect
30. C₃×C₄²	C₂×C₇	semidirect
34. #117	C₃×C₇	direct (N × H)
42. C₂×C₃×C₄	C₄×C₇	semidirect
46. C₃×C₇	#117	direct (N × H)
48. C₄⋊C₄	C₂×C₃×C₇	semidirect
50. C₄²	C₂×C₃×C₇	semidirect
61. C₂×C₄	C₃×C₄×C₇	semidirect
65. C₇	C₃×#117	direct (N × H)
75. C₃	C₇×#117	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×C₃×C₄×C₇	2
C₃×C₇×C₄⋊C₄	2
C₃×C₇×C₂²⋊C₄	2
C₃×C₄²×C₇	2
C₇×#117	3
C₃×#117	7

Subgroups — C3×C7×#117 #10621440

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₇×#117 #10621440