Subgroups — C₃×Q₈×Dic₇ #10621256

Order 672.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₇×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄⋊Dic₇	C₂	1	3	yes	maximal
3	C₃×C₄×Dic₇	C₂	1	3	yes	maximal
4	Q₈×Dic₇	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₃×C₇×Q₈	C₄	1	4	yes	complemented
6	C₂×C₃×C₄×C₇	C₂²	1	3	yes	—
7	C₂×C₃×Dic₇	C₂²	1	3	yes	—
8	C₂×C₃×Dic₇	C₂²	1	1	yes	characteristic
9	C₂×C₇×Q₈	C₂×C₃	1	1	yes	characteristic
10	C₄⋊Dic₇	C₂×C₃	1	3	yes	—
11	C₄×Dic₇	C₂×C₃	1	3	yes	—
12	C₃×C₄×Q₈	—	7	1	no	maximal, Hall
13	C₂²×C₃×C₇	C₂³	1	1	yes	characteristic
14	C₃×C₄×C₇	C₂×C₄	1	6	yes	—
15	C₃×Dic₇	—	2	3	no	—
16	C₃×Dic₇	Q₈	1	2	yes	direct product factor
17	C₇×Q₈	C₃×C₄	1	4	yes	complemented
18	C₂×C₄×C₇	C₂²×C₃	1	3	yes	—
19	C₂×Dic₇	C₂²×C₃	1	3	yes	—
20	C₂×Dic₇	C₂²×C₃	1	1	yes	characteristic
21	C₂×C₃×Q₈	D₇	1	1	yes	characteristic
22	C₃×C₄⋊C₄	—	7	3	no	—
23	C₃×C₄²	—	7	3	no	—
24	C₂×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
25	C₂×C₃×C₇	C₂×Q₈	1	1	yes	characteristic
26	C₂×C₃×C₇	#41	1	1	yes	characteristic
27	C₄×Q₈	—	7	1	no	Sylow
28	C₂²×C₇	C₂³×C₃	1	1	yes	characteristic
29	C₄×C₇	C₂×C₃×C₄	1	6	yes	—
30	Dic₇	—	2	3	no	—
31	Dic₇	C₃×Q₈	1	2	yes	direct product factor
32	C₃×Q₈	Dic₇	1	4	yes	direct product factor
33	C₂×C₃×C₄	—	7	1	no	—
34	C₂×C₃×C₄	—	7	3	no	—
35	C₂×C₃×C₄	C₂×D₇	1	3	yes	—
36	C₃×C₇	C₄×Q₈	1	1	yes	characteristic, complemented, Hall
37	C₂×Q₈	C₃×D₇	1	1	yes	characteristic
38	C₄⋊C₄	—	7	3	no	—
39	C₄²	—	7	3	no	—
40	C₂×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
41	C₂×C₇	C₂×C₃×Q₈	1	1	yes	characteristic
42	C₂×C₇	C₃×#41	1	1	yes	characteristic
43	C₂²×C₃	C₂²×D₇	1	1	yes	characteristic
44	C₃×C₄	—	7	2	no	—
45	C₃×C₄	—	14	3	no	—
46	C₃×C₄	C₂×Dic₇	1	6	yes	—
47	Q₈	C₃×Dic₇	1	4	yes	direct product factor
48	C₂×C₄	—	7	1	no	—
49	C₂×C₄	—	7	3	no	—
50	C₂×C₄	C₂×C₃×D₇	1	3	yes	—
51	C₇	C₃×C₄×Q₈	1	1	yes	characteristic, complemented, Sylow
52	C₂×C₃	Q₈×D₇	1	1	yes	characteristic
53	C₂×C₃	#1161	1	1	yes	characteristic
54	C₂×C₃	C₂²×Dic₇	1	1	yes	characteristic
55	C₂²	C₂²×C₃×D₇	1	1	yes	characteristic
56	C₄	—	7	2	no	—
57	C₄	—	14	3	no	—
58	C₄	C₂×C₃×Dic₇	1	6	yes	—
59	C₃	Q₈×Dic₇	1	1	yes	characteristic, direct product factor, Sylow
60	C₂	C₃×Q₈×D₇	1	1	yes	characteristic
61	C₂	C₃×#1161	1	1	yes	characteristic
62	C₂	C₂²×C₃×Dic₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂²×D₇
commutator	C₂×C₇	C₂²×C₃×C₄
Frattini	C₂²	C₂²×C₃×D₇
Fitting	C₂×C₃×C₇×Q₈	C₂
socle, prime radical	C₂²×C₃×C₇	C₂³
cosocle	C₂²×C₇	C₂³×C₃
hypercenter	C₂×C₃×Q₈	D₇
hypocenter	C₇	C₃×C₄×Q₈
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×Q₈

Complements

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

N	H ≅ G/N	type
4. Q₈×Dic₇	C₃	direct (N × H)
5. C₃×C₇×Q₈	C₄	semidirect
16. C₃×Dic₇	Q₈	direct (N × H)
17. C₇×Q₈	C₃×C₄	semidirect
31. Dic₇	C₃×Q₈	direct (N × H)
32. C₃×Q₈	Dic₇	direct (N × H)
36. C₃×C₇	C₄×Q₈	semidirect
47. Q₈	C₃×Dic₇	direct (N × H)
51. C₇	C₃×C₄×Q₈	semidirect
59. C₃	Q₈×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
C₂×C₃×C₇×Q₈	2
C₃×C₄⋊Dic₇	2
C₃×C₄×Dic₇	2
Q₈×Dic₇	3
C₃×C₄×Q₈	7

Subgroups — C3×Q8×Dic7 #10621256

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×Q₈×Dic₇ #10621256