Subgroups — #66728

Order 336.

80 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, complemented
2	#4268	C₂	1	2	yes	maximal, complemented
3	D₈₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×D₂₁	C₂	1	2	yes	maximal, complemented
5	Dic₄₂	C₂	1	1	yes	maximal, characteristic, complemented
6	#1157	—	3	1	no	maximal, Hall
7	C₂²×C₃×C₇	C₂²	1	1	yes	characteristic
8	C₂×D₂₁	C₂²	1	2	yes	—
9	C₃×C₄×C₇	C₂²	1	1	yes	characteristic
10	C₃×C₄×C₇	C₂²	1	1	yes	characteristic
11	Dic₂₁	C₂²	1	2	yes	—
12	C₂×C₄×C₇	D₃	1	1	yes	characteristic, complemented
13	C₂²⋊D₇	—	3	2	no	—
14	D₂₈	—	3	1	no	—
15	C₄×D₇	—	3	2	no	—
16	Dic₁₄	—	3	1	no	—
17	#235	—	7	1	no	maximal, Hall
18	C₂×C₃×C₇	—	2	1	no	—
19	C₂×C₃×C₇	C₂³	1	1	yes	characteristic
20	D₂₁	—	2	2	no	—
21	C₂²×C₇	C₂×D₃	1	1	yes	characteristic
22	C₂×D₇	—	3	2	no	—
23	C₄×C₇	C₂×D₃	1	1	yes	characteristic
24	C₄×C₇	C₂×D₃	1	1	yes	characteristic
25	Dic₇	—	3	2	no	—
26	C₂×C₃×C₄	D₇	1	1	yes	characteristic, complemented
27	#67	—	7	2	no	—
28	D₁₂	—	7	1	no	—
29	C₄×D₃	—	7	2	no	—
30	Dic₆	—	7	1	no	—
31	C₃×C₇	#41	1	1	yes	characteristic, complemented, Hall
32	#41	—	21	1	no	Sylow
33	C₂×C₇	—	2	1	no	—
34	C₂×C₇	C₂²×D₃	1	1	yes	characteristic
35	D₇	—	6	2	no	—
36	C₂²×C₃	C₂×D₇	1	1	yes	characteristic
37	C₂×D₃	—	7	2	no	—
38	C₃×C₄	C₂×D₇	1	1	yes	characteristic
39	C₃×C₄	C₂×D₇	1	1	yes	characteristic
40	Dic₃	—	7	2	no	—
41	Q₈	—	21	1	no	—
42	D₄	—	21	2	no	—
43	D₄	—	21	1	no	—
44	C₂×C₄	—	21	2	no	—
45	C₂×C₄	D₂₁	1	1	yes	characteristic, complemented
46	C₇	#235	1	1	yes	characteristic, complemented, Sylow
47	C₂×C₃	—	2	1	no	—
48	C₂×C₃	C₂²×D₇	1	1	yes	characteristic
49	D₃	—	14	2	no	—
50	C₂²	—	21	2	no	—
51	C₂²	C₂×D₂₁	1	1	yes	characteristic
52	C₄	—	21	2	no	—
53	C₄	C₂×D₂₁	1	1	yes	characteristic
54	C₄	C₂×D₂₁	1	1	yes	characteristic
55	C₃	#1157	1	1	yes	characteristic, complemented, Sylow
56	C₂	—	2	1	no	—
57	C₂	—	42	2	no	—
58	C₂	C₂²×D₂₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₂₁
commutator, socle, cosocle	C₂×C₃×C₇	C₂³
Frattini	C₂	C₂²×D₂₁
Fitting	C₂×C₃×C₄×C₇	C₂
hypercenter	C₂×C₄	D₂₁
hypocenter	C₃×C₇	#41
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₄×C₇	C₂	semidirect
2. #4268	C₂	semidirect
3. D₈₄	C₂	semidirect
4. C₄×D₂₁	C₂	semidirect
5. Dic₄₂	C₂	semidirect
12. C₂×C₄×C₇	D₃	semidirect
26. C₂×C₃×C₄	D₇	semidirect
31. C₃×C₇	#41	semidirect
45. C₂×C₄	D₂₁	semidirect
46. C₇	#235	semidirect
55. C₃	#1157	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
#4268	2
D₈₄	2
C₄×D₂₁	2
Dic₄₂	2
#1157	3
#235	7