Subgroups — #66677

Order 336.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₈₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₄×D₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#4246	C₂	1	2	yes	maximal, complemented
5	D₂₁⋊C₄	C₂	1	2	yes	maximal, complemented
6	#1161	—	3	1	no	maximal, Hall
7	C₂×D₂₁	C₂²	1	2	yes	—
8	C₂×C₃×D₇	C₂²	1	1	yes	characteristic
9	C₃×C₄×C₇	C₂²	1	1	yes	characteristic
10	C₃×Dic₇	C₂²	1	1	yes	characteristic
11	C₇×Dic₃	C₂²	1	2	yes	—
12	C₇×Q₈	—	3	1	no	—
13	D₂₈	—	3	2	no	—
14	D₂₈	—	3	1	no	—
15	C₄×D₇	—	3	2	no	—
16	C₄×D₇	D₃	1	1	yes	characteristic, complemented
17	#235	—	7	1	no	maximal, Hall
18	C₂×C₃×C₇	C₂³	1	1	yes	characteristic
19	D₂₁	—	2	2	no	—
20	C₃×D₇	—	2	1	no	—
21	C₂×D₇	—	3	2	no	—
22	C₂×D₇	C₂×D₃	1	1	yes	characteristic
23	C₄×C₇	—	3	2	no	—
24	C₄×C₇	C₂×D₃	1	1	yes	characteristic
25	Dic₇	C₂×D₃	1	1	yes	characteristic
26	C₂×C₃×C₄	—	7	1	no	—
27	#67	—	7	2	no	—
28	D₁₂	—	7	1	no	—
29	C₄×D₃	—	7	2	no	—
30	Dic₆	D₇	1	1	yes	characteristic, complemented
31	C₃×C₇	#41	1	1	yes	characteristic, complemented, Hall
32	#41	—	21	1	no	Sylow
33	C₂×C₇	C₂²×D₃	1	1	yes	characteristic
34	D₇	—	2	1	no	—
35	D₇	—	6	2	no	—
36	C₂²×C₃	—	7	1	no	—
37	C₂×D₃	—	7	2	no	—
38	C₃×C₄	—	7	1	no	—
39	C₃×C₄	C₂×D₇	1	1	yes	characteristic
40	Dic₃	C₂×D₇	1	2	yes	—
41	Q₈	—	3	1	no	—
42	D₄	—	21	2	no	—
43	D₄	—	21	1	no	—
44	C₂×C₄	—	7	1	no	—
45	C₂×C₄	—	21	2	no	—
46	C₇	#235	1	1	yes	characteristic, complemented, Sylow
47	C₂×C₃	—	14	1	no	—
48	C₂×C₃	C₂²×D₇	1	1	yes	characteristic
49	D₃	—	14	2	no	—
50	C₂²	—	7	1	no	—
51	C₂²	—	21	2	no	—
52	C₄	—	3	2	no	—
53	C₄	—	7	1	no	—
54	C₄	D₃×D₇	1	1	yes	characteristic
55	C₃	#1161	1	1	yes	characteristic, complemented, Sylow
56	C₂	—	14	1	no	—
57	C₂	—	42	2	no	—
58	C₂	C₂×D₃×D₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×D₃×D₇
commutator, socle, cosocle	C₂×C₃×C₇	C₂³
Fitting	C₃×C₄×C₇	C₂²
hypercenter	C₄	D₃×D₇
hypocenter	C₃×C₇	#41
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
1. D₈₄	C₂	semidirect
2. C₇×Dic₆	C₂	semidirect
3. C₃×C₄×D₇	C₂	semidirect
4. #4246	C₂	semidirect
5. D₂₁⋊C₄	C₂	semidirect
16. C₄×D₇	D₃	semidirect
30. Dic₆	D₇	semidirect
31. C₃×C₇	#41	semidirect
46. C₇	#235	semidirect
55. C₃	#1161	semidirect

Maximal subgroups

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

subgroup	index
D₈₄	2
C₇×Dic₆	2
C₃×C₄×D₇	2
#4246	2
D₂₁⋊C₄	2
#1161	3
#235	7