Subgroups — #66629

Order 336.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₇	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₂₁	C₂	1	1	yes	maximal, characteristic
3	C₂×Dic₂₁	C₂	1	1	yes	maximal, characteristic
4	Dic₇⋊C₄	—	3	1	no	maximal, Hall
5	C₂²×C₃×C₇	C₂²	1	1	yes	characteristic
6	C₃×C₄×C₇	—	2	1	no	—
7	Dic₂₁	—	2	1	no	—
8	Dic₂₁	C₄	1	2	yes	complemented
9	C₂×C₄×C₇	D₃	1	1	yes	characteristic
10	C₂×Dic₇	—	3	1	no	—
11	C₂×Dic₇	—	3	1	no	—
12	Dic₃⋊C₄	—	7	1	no	maximal, Hall
13	C₂×C₃×C₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₃×C₇	D₄	1	1	yes	characteristic
15	C₂×C₃×C₇	Q₈	1	1	yes	characteristic
16	C₂²×C₇	C₂×D₃	1	1	yes	characteristic
17	C₄×C₇	—	2	1	no	—
18	Dic₇	—	3	2	no	—
19	Dic₇	—	6	1	no	—
20	C₂×C₃×C₄	D₇	1	1	yes	characteristic
21	C₂×Dic₃	—	7	1	no	—
22	C₂×Dic₃	—	7	1	no	—
23	C₃×C₇	C₄⋊C₄	1	1	yes	characteristic, complemented, Hall
24	C₄⋊C₄	—	21	1	no	Sylow
25	C₂×C₇	Dic₆	1	1	yes	characteristic
26	C₂×C₇	C₄×D₃	1	1	yes	characteristic
27	C₂×C₇	#67	1	1	yes	characteristic
28	C₂²×C₃	C₂×D₇	1	1	yes	characteristic
29	C₃×C₄	—	2	1	no	—
30	Dic₃	—	7	2	no	—
31	Dic₃	—	14	1	no	—
32	C₂×C₄	—	21	1	no	—
33	C₂×C₄	—	21	1	no	—
34	C₂×C₄	D₂₁	1	1	yes	characteristic
35	C₇	Dic₃⋊C₄	1	1	yes	characteristic, complemented, Sylow
36	C₂×C₃	Dic₁₄	1	1	yes	characteristic
37	C₂×C₃	C₄×D₇	1	1	yes	characteristic
38	C₂×C₃	C₂²⋊D₇	1	1	yes	characteristic
39	C₂²	C₂×D₂₁	1	1	yes	characteristic
40	C₄	—	2	1	no	—
41	C₄	—	21	2	no	—
42	C₄	—	42	1	no	—
43	C₃	Dic₇⋊C₄	1	1	yes	characteristic, complemented, Sylow
44	C₂	Dic₄₂	1	1	yes	characteristic
45	C₂	C₄×D₂₁	1	1	yes	characteristic
46	C₂	#4268	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂×D₂₁
commutator	C₂×C₃×C₇	C₂×C₄
Fitting	C₂×C₃×C₄×C₇	C₂
socle, prime radical, cosocle	C₂²×C₃×C₇	C₂²
hypercenter	C₂×C₄	D₂₁
hypocenter	C₃×C₇	C₄⋊C₄
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
8. Dic₂₁	C₄	semidirect
23. C₃×C₇	C₄⋊C₄	semidirect
35. C₇	Dic₃⋊C₄	semidirect
43. C₃	Dic₇⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×C₇	2
C₂×Dic₂₁	2
Dic₇⋊C₄	3
Dic₃⋊C₄	7