Subgroups — #10620476

Order 672.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×D₂₄	C₂	1	1	yes	maximal, characteristic
2	C₃×Dic₂₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃⋊(C₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
4	#6401	—	3	1	no	maximal, Hall
5	C₇×D₁₂	—	2	1	no	—
6	C₃×Dic₁₄	—	2	1	no	—
7	C₃×C₇×C₈	C₂²	1	1	yes	characteristic
8	C₇×D₈	—	3	1	no	—
9	Dic₂₈	D₃	1	1	yes	characteristic, complemented
10	C₇⋊C₁₆	—	3	1	no	—
11	#828	—	7	1	no	maximal, Hall
12	C₂×D₃×C₇	—	4	1	no	—
13	C₃×C₄×C₇	D₄	1	1	yes	characteristic
14	C₃×Dic₇	—	4	1	no	—
15	C₇×D₄	—	6	1	no	—
16	Dic₁₄	—	2	1	no	—
17	C₇×C₈	C₂×D₃	1	1	yes	characteristic
18	C₃×Q₁₆	—	7	1	no	—
19	D₂₄	D₇	1	1	yes	characteristic
20	C₃⋊C₁₆	—	7	1	no	—
21	C₂×C₃×C₇	D₈	1	1	yes	characteristic
22	D₃×C₇	—	8	1	no	—
23	SD₃₂	—	21	1	no	Sylow
24	C₂²×C₇	—	12	1	no	—
25	C₄×C₇	#67	1	1	yes	characteristic
26	Dic₇	—	4	1	no	—
27	C₃×Q₈	—	14	1	no	—
28	D₁₂	—	2	1	no	—
29	C₃×C₈	C₂×D₇	1	1	yes	characteristic
30	C₃×C₇	SD₃₂	1	1	yes	characteristic, complemented, Hall
31	Q₁₆	—	7	1	no	—
32	D₈	—	3	1	no	—
33	C₁₆	—	21	1	no	—
34	C₂×C₇	—	24	1	no	—
35	C₂×C₇	#213	1	1	yes	characteristic
36	C₂×D₃	—	4	1	no	—
37	C₃×C₄	—	28	1	no	—
38	C₃×C₄	C₂²⋊D₇	1	1	yes	characteristic
39	Q₈	—	14	1	no	—
40	D₄	—	6	1	no	—
41	C₈	D₃×D₇	1	1	yes	characteristic
42	C₇	#828	1	1	yes	characteristic, complemented, Sylow
43	C₂×C₃	#1141	1	1	yes	characteristic
44	D₃	—	8	1	no	—
45	C₂²	—	12	1	no	—
46	C₄	—	28	1	no	—
47	C₄	#4245	1	1	yes	characteristic
48	C₃	#6401	1	1	yes	characteristic, complemented, Sylow
49	C₂	—	24	1	no	—
50	C₂	#66560	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#66560
commutator, Fitting, cosocle	C₃×C₇×C₈	C₂²
Frattini, hypercenter	C₈	D₃×D₇
socle	C₂×C₃×C₇	D₈
prime radical	C₇×D₂₄	C₂
hypocenter	C₃×C₇	SD₃₂
LCS	G ▷ C₃×C₇×C₈ ▷ C₃×C₄×C₇ ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

Complements

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

N	H ≅ G/N	type
2. C₃×Dic₂₈	C₂	semidirect
3. C₃⋊(C₇⋊C₁₆)	C₂	semidirect
9. Dic₂₈	D₃	semidirect
30. C₃×C₇	SD₃₂	semidirect
42. C₇	#828	semidirect
48. C₃	#6401	semidirect

Maximal subgroups

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

subgroup	index
C₇×D₂₄	2
C₃×Dic₂₈	2
C₃⋊(C₇⋊C₁₆)	2
#6401	3
#828	7