Subgroups — C₃×#6470 #10621218

Order 672.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#1157	C₂	1	1	yes	maximal, characteristic
2	C₃×C₇×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#1131	C₂	1	2	yes	maximal, complemented
5	C₃×C₈×D₇	C₂	1	2	yes	maximal, complemented
6	#6470	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
8	C₃×C₂²⋊D₇	C₄	1	2	yes	—
9	C₃×D₂₈	C₄	1	1	yes	characteristic
10	C₃×C₄×D₇	C₂²	1	2	yes	—
11	C₃×Dic₁₄	C₄	1	1	yes	characteristic
12	C₃×C₇×C₈	C₂²	1	2	yes	—
13	C₃×C₇⋊C₈	C₂²	1	1	yes	characteristic
14	C₃×C₇⋊C₈	C₂²	1	1	yes	characteristic
15	#1157	C₂×C₃	1	1	yes	characteristic
16	C₇×M₁₆	C₂×C₃	1	1	yes	characteristic, complemented
17	C₂×C₇⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
18	#1131	C₂×C₃	1	2	yes	complemented
19	C₈×D₇	C₂×C₃	1	2	yes	complemented
20	C₃×#131	—	7	1	no	maximal, Hall
21	C₂²×C₃×C₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₃×D₇	C₂×C₄	1	2	yes	—
23	C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
24	C₃×C₄×C₇	C₂³	1	1	yes	characteristic
25	C₃×Dic₇	C₂×C₄	1	2	yes	—
26	C₂×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
27	C₂²⋊D₇	C₃×C₄	1	2	yes	—
28	D₂₈	C₃×C₄	1	1	yes	characteristic
29	C₄×D₇	C₂²×C₃	1	2	yes	—
30	Dic₁₄	C₃×C₄	1	1	yes	characteristic
31	C₇×C₈	C₂²×C₃	1	2	yes	—
32	C₇⋊C₈	C₂²×C₃	1	1	yes	characteristic
33	C₇⋊C₈	C₂²×C₃	1	1	yes	characteristic
34	C₃×#41	—	7	1	no	—
35	C₃×M₁₆	—	7	2	no	—
36	C₃×M₁₆	D₇	1	1	yes	characteristic, complemented
37	C₂×C₃×C₈	—	7	1	no	—
38	C₂×C₃×C₈	—	7	2	no	—
39	C₂×C₃×C₇	—	2	1	no	—
40	C₂×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
41	C₃×D₇	—	2	2	no	—
42	#131	—	7	1	no	Sylow
43	C₂²×C₇	C₂×C₃×C₄	1	1	yes	characteristic
44	C₂×D₇	C₂×C₃×C₄	1	2	yes	—
45	C₄×C₇	C₂×C₃×C₄	1	1	yes	characteristic
46	C₄×C₇	C₂³×C₃	1	1	yes	characteristic
47	Dic₇	C₂×C₃×C₄	1	2	yes	—
48	C₃×Q₈	—	7	1	no	—
49	C₃×D₄	—	7	2	no	—
50	C₃×D₄	—	7	1	no	—
51	C₂×C₃×C₄	—	7	2	no	—
52	C₂×C₃×C₄	C₂×D₇	1	1	yes	characteristic
53	C₃×C₈	—	7	1	no	—
54	C₃×C₈	—	7	1	no	—
55	C₃×C₈	C₂×D₇	1	2	yes	—
56	C₃×C₇	#131	1	1	yes	characteristic, complemented, Hall
57	#41	—	7	1	no	—
58	M₁₆	—	7	2	no	—
59	M₁₆	C₃×D₇	1	1	yes	characteristic, complemented
60	C₂×C₈	—	7	1	no	—
61	C₂×C₈	—	7	2	no	—
62	C₂×C₇	—	2	1	no	—
63	C₂×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
64	D₇	—	2	2	no	—
65	C₂²×C₃	—	7	2	no	—
66	C₂²×C₃	C₄×D₇	1	1	yes	characteristic
67	C₃×C₄	—	7	2	no	—
68	C₃×C₄	C₄×D₇	1	1	yes	characteristic
69	C₃×C₄	C₂²×D₇	1	1	yes	characteristic
70	Q₈	—	7	1	no	—
71	D₄	—	7	2	no	—
72	D₄	—	7	1	no	—
73	C₂×C₄	—	7	2	no	—
74	C₂×C₄	C₂×C₃×D₇	1	1	yes	characteristic
75	C₈	—	7	1	no	—
76	C₈	—	7	1	no	—
77	C₈	C₂×C₃×D₇	1	2	yes	—
78	C₇	C₃×#131	1	1	yes	characteristic, complemented, Sylow
79	C₂×C₃	—	2	1	no	—
80	C₂×C₃	—	14	2	no	—
81	C₂×C₃	C₂×C₄×D₇	1	1	yes	characteristic
82	C₂²	—	7	2	no	—
83	C₂²	C₃×C₄×D₇	1	1	yes	characteristic
84	C₄	—	7	2	no	—
85	C₄	C₃×C₄×D₇	1	1	yes	characteristic
86	C₄	C₂²×C₃×D₇	1	1	yes	characteristic
87	C₃	#6470	1	1	yes	characteristic, direct product factor, Sylow
88	C₂	—	2	1	no	—
89	C₂	—	14	2	no	—
90	C₂	C₂×C₃×C₄×D₇	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	C₂²×D₇
commutator	C₂×C₇	C₂²×C₃×C₄
Frattini	C₄	C₂²×C₃×D₇
Fitting	C₃×C₇×M₁₆	C₂
socle	C₂×C₃×C₇	C₂²×C₄
prime radical	C₃×#1157	C₂
cosocle	C₄×C₇	C₂³×C₃
hypercenter	C₃×M₁₆	D₇
hypocenter	C₇	C₃×#131
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₃×C₄ ◁ C₃×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₃×C₇×M₁₆	C₂	semidirect
3. C₂×C₃×C₇⋊C₈	C₂	semidirect
4. C₃×#1131	C₂	semidirect
5. C₃×C₈×D₇	C₂	semidirect
6. #6470	C₃	direct (N × H)
16. C₇×M₁₆	C₂×C₃	semidirect
17. C₂×C₇⋊C₈	C₂×C₃	semidirect
18. #1131	C₂×C₃	semidirect
19. C₈×D₇	C₂×C₃	semidirect
36. C₃×M₁₆	D₇	semidirect
56. C₃×C₇	#131	semidirect
59. M₁₆	C₃×D₇	semidirect
78. C₇	C₃×#131	semidirect
87. C₃	#6470	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#1157	2
C₃×C₇×M₁₆	2
C₂×C₃×C₇⋊C₈	2
C₃×#1131	2
C₃×C₈×D₇	2
#6470	3
C₃×#131	7

Subgroups — C3×#6470 #10621218

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#6470 #10621218