Subgroups — C₃×#6489 #10621237

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×C₂²⋊Dic₇	C₂	1	2	yes	maximal
3	C₃×C₄⋊Dic₇	C₂	1	2	yes	maximal, complemented
4	C₃×C₄×Dic₇	C₂	1	2	yes	maximal, complemented
5	#6489	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	1	yes	characteristic
9	C₂×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
10	C₂×C₃×Dic₇	C₂²	1	4	yes	—
11	C₂²×C₄×C₇	C₂×C₃	1	1	yes	characteristic
12	C₂²⋊Dic₇	C₂×C₃	1	2	yes	—
13	C₄⋊Dic₇	C₂×C₃	1	2	yes	complemented
14	C₄×Dic₇	C₂×C₃	1	2	yes	complemented
15	C₃×#117	—	7	1	no	maximal, 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	C₂×C₄	1	2	yes	—
20	C₃×C₄×C₇	C₂×C₄	1	2	yes	—
21	C₃×Dic₇	—	2	4	no	—
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	1	yes	characteristic
25	C₂×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
26	C₂×Dic₇	C₂²×C₃	1	4	yes	—
27	C₂²×C₃×C₄	D₇	1	1	yes	characteristic
28	C₃×C₄⋊C₄	—	7	2	no	—
29	C₃×C₂²⋊C₄	—	7	2	no	—
30	C₃×C₄²	—	7	2	no	—
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	—	7	1	no	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	C₂×C₃×C₄	1	2	yes	—
39	C₄×C₇	C₂×C₃×C₄	1	2	yes	—
40	Dic₇	—	2	4	no	—
41	C₂³×C₃	C₂×D₇	1	1	yes	characteristic
42	C₂×C₃×C₄	—	7	4	no	—
43	C₂×C₃×C₄	Dic₇	1	4	yes	complemented
44	C₂×C₃×C₄	C₂×D₇	1	1	yes	characteristic
45	C₂×C₃×C₄	C₂×D₇	1	1	yes	characteristic
46	C₃×C₇	#117	1	1	yes	characteristic, complemented, Hall
47	C₂²×C₄	C₃×D₇	1	1	yes	characteristic
48	C₄⋊C₄	—	7	2	no	—
49	C₂²⋊C₄	—	7	2	no	—
50	C₄²	—	7	2	no	—
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×Dic₇	1	2	yes	—
56	C₂²×C₃	C₂²×D₇	1	1	yes	characteristic
57	C₃×C₄	—	14	4	no	—
58	C₃×C₄	C₂×Dic₇	1	2	yes	—
59	C₃×C₄	C₂×Dic₇	1	2	yes	—
60	C₂³	C₂×C₃×D₇	1	1	yes	characteristic
61	C₂×C₄	—	7	4	no	—
62	C₂×C₄	C₃×Dic₇	1	4	yes	complemented
63	C₂×C₄	C₂×C₃×D₇	1	1	yes	characteristic
64	C₂×C₄	C₂×C₃×D₇	1	1	yes	characteristic
65	C₇	C₃×#117	1	1	yes	characteristic, complemented, Sylow
66	C₂×C₃	—	2	2	no	—
67	C₂×C₃	#1157	1	2	yes	—
68	C₂×C₃	C₂²×Dic₇	1	1	yes	characteristic
69	C₂²	—	2	2	no	—
70	C₂²	C₂×C₃×Dic₇	1	2	yes	—
71	C₂²	C₂²×C₃×D₇	1	1	yes	characteristic
72	C₄	—	14	4	no	—
73	C₄	C₂×C₃×Dic₇	1	2	yes	—
74	C₄	C₂×C₃×Dic₇	1	2	yes	—
75	C₃	#6489	1	1	yes	characteristic, direct product factor, Sylow
76	C₂	—	2	2	no	—
77	C₂	C₃×#1157	1	2	yes	—
78	C₂	C₂²×C₃×Dic₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂×D₇
commutator	C₂×C₇	C₂²×C₃×C₄
Frattini	C₂²	C₂²×C₃×D₇
Fitting	C₂²×C₃×C₄×C₇	C₂
socle	C₂²×C₃×C₇	C₂³
prime radical	C₂³×C₃×C₇	C₂²
cosocle	C₂²×C₇	C₂³×C₃
hypercenter	C₂²×C₃×C₄	D₇
hypocenter	C₇	C₃×#117
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂²×C₃×C₄

Complements

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

N	H ≅ G/N	type
3. C₃×C₄⋊Dic₇	C₂	semidirect
4. C₃×C₄×Dic₇	C₂	semidirect
5. #6489	C₃	direct (N × H)
7. C₂×C₃×C₄×C₇	C₄	semidirect
13. C₄⋊Dic₇	C₂×C₃	semidirect
14. C₄×Dic₇	C₂×C₃	semidirect
23. C₂×C₄×C₇	C₃×C₄	semidirect
43. C₂×C₃×C₄	Dic₇	semidirect
46. C₃×C₇	#117	semidirect
62. C₂×C₄	C₃×Dic₇	semidirect
65. C₇	C₃×#117	semidirect
75. C₃	#6489	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×C₂²⋊Dic₇	2
C₃×C₄⋊Dic₇	2
C₃×C₄×Dic₇	2
#6489	3
C₃×#117	7

Subgroups — C3×#6489 #10621237

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#6489 #10621237