Subgroups — #66631

Order 336.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₂₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄×C₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×Dic₂₁	C₂	1	1	yes	maximal, characteristic, complemented
4	(C₂×D₇)⋊C₄	—	3	1	no	maximal, Hall
5	C₂²×C₃×C₇	C₂²	1	1	yes	characteristic
6	C₂×D₂₁	—	2	1	no	—
7	C₂×D₂₁	—	2	1	no	—
8	C₂×D₂₁	C₄	1	2	yes	complemented
9	C₃×C₄×C₇	—	2	1	no	—
10	Dic₂₁	—	2	1	no	—
11	C₂²×D₇	—	3	1	no	—
12	C₂×C₄×C₇	D₃	1	1	yes	characteristic, complemented
13	C₂×Dic₇	—	3	1	no	—
14	(C₂×D₃)⋊C₄	—	7	1	no	maximal, Hall
15	C₂×C₃×C₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₃×C₇	D₄	1	1	yes	characteristic
17	C₂×C₃×C₇	D₄	1	1	yes	characteristic
18	D₂₁	—	2	2	no	—
19	C₂²×C₇	C₂×D₃	1	1	yes	characteristic
20	C₂×D₇	—	3	2	no	—
21	C₂×D₇	—	6	1	no	—
22	C₂×D₇	—	6	1	no	—
23	C₄×C₇	—	2	1	no	—
24	Dic₇	—	6	1	no	—
25	C₂²×D₃	—	7	1	no	—
26	C₂×C₃×C₄	D₇	1	1	yes	characteristic, complemented
27	C₂×Dic₃	—	7	1	no	—
28	C₃×C₇	C₂²⋊C₄	1	1	yes	characteristic, complemented, Hall
29	C₂²⋊C₄	—	21	1	no	Sylow
30	C₂×C₇	C₄×D₃	1	1	yes	characteristic
31	C₂×C₇	D₁₂	1	1	yes	characteristic
32	C₂×C₇	#67	1	1	yes	characteristic
33	D₇	—	6	2	no	—
34	C₂²×C₃	C₂×D₇	1	1	yes	characteristic
35	C₂×D₃	—	7	2	no	—
36	C₂×D₃	—	14	1	no	—
37	C₂×D₃	—	14	1	no	—
38	C₃×C₄	—	2	1	no	—
39	Dic₃	—	14	1	no	—
40	C₂³	—	21	1	no	—
41	C₂×C₄	—	21	1	no	—
42	C₂×C₄	D₂₁	1	1	yes	characteristic, complemented
43	C₇	(C₂×D₃)⋊C₄	1	1	yes	characteristic, complemented, Sylow
44	C₂×C₃	C₄×D₇	1	1	yes	characteristic
45	C₂×C₃	D₂₈	1	1	yes	characteristic
46	C₂×C₃	C₂²⋊D₇	1	1	yes	characteristic
47	D₃	—	14	2	no	—
48	C₂²	—	21	2	no	—
49	C₂²	—	42	1	no	—
50	C₂²	—	42	1	no	—
51	C₂²	C₂×D₂₁	1	1	yes	characteristic
52	C₄	—	2	1	no	—
53	C₄	—	42	1	no	—
54	C₃	(C₂×D₇)⋊C₄	1	1	yes	characteristic, complemented, Sylow
55	C₂	—	42	2	no	—
56	C₂	C₄×D₂₁	1	1	yes	characteristic
57	C₂	D₈₄	1	1	yes	characteristic
58	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, cosocle	C₂²×C₃×C₇	C₂²
prime radical	C₂²×D₂₁	C₂
hypercenter	C₂×C₄	D₂₁
hypocenter	C₃×C₇	C₂²⋊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
2. C₂×C₃×C₄×C₇	C₂	semidirect
3. C₂×Dic₂₁	C₂	semidirect
8. C₂×D₂₁	C₄	semidirect
12. C₂×C₄×C₇	D₃	semidirect
26. C₂×C₃×C₄	D₇	semidirect
28. C₃×C₇	C₂²⋊C₄	semidirect
42. C₂×C₄	D₂₁	semidirect
43. C₇	(C₂×D₃)⋊C₄	semidirect
54. C₃	(C₂×D₇)⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₂₁	2
C₂×C₃×C₄×C₇	2
C₂×Dic₂₁	2
(C₂×D₇)⋊C₄	3
(C₂×D₃)⋊C₄	7