Subgroups — #6393

Order 224.

30 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×C₈	C₂	1	1	yes	maximal, characteristic
2	#1136	C₂	1	2	yes	maximal
3	C₂×C₄×C₇	C₂²	1	1	yes	characteristic
4	C₇×C₈	C₄	1	2	yes	—
5	C₇⋊C₈	—	2	2	no	—
6	#108	—	7	1	no	maximal, Sylow
7	C₂²×C₇	Q₈	1	1	yes	characteristic
8	C₄×C₇	C₂×C₄	1	1	yes	characteristic
9	C₄×C₇	D₄	1	1	yes	characteristic
10	M₁₆	—	7	2	no	—
11	C₂×C₈	D₇	1	1	yes	characteristic
12	C₂×C₇	—	2	1	no	—
13	C₂×C₇	C₄⋊C₄	1	1	yes	characteristic
14	C₂×C₄	C₂×D₇	1	1	yes	characteristic
15	C₈	—	14	2	no	—
16	C₈	Dic₇	1	2	yes	—
17	C₇	#108	1	1	yes	characteristic, complemented, Sylow
18	C₂²	Dic₁₄	1	1	yes	characteristic
19	C₄	D₂₈	1	1	yes	characteristic
20	C₄	C₂×Dic₇	1	1	yes	characteristic
21	C₂	—	2	1	no	—
22	C₂	C₄⋊Dic₇	1	1	yes	characteristic

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

N	H ≅ G/N	type
17. C₇	#108	semidirect

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