Subgroups — #89253

Order 448.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇×C₈	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₇⋊C₈)	C₂	1	2	yes	maximal
3	C₂×C₇×C₈	C₄	1	1	yes	characteristic
4	C₂×C₇×C₈	C₄	1	1	yes	characteristic
5	C₄²×C₇	C₂²	1	1	yes	characteristic
6	C₂×C₇⋊C₈	—	2	2	no	—
7	#335	—	7	1	no	maximal, Sylow
8	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
9	C₂×C₄×C₇	D₄	1	1	yes	characteristic
10	C₂×C₄×C₇	Q₈	1	1	yes	characteristic
11	C₇×C₈	—	2	1	no	—
12	C₇×C₈	C₈	1	2	yes	complemented
13	C₇⋊C₈	—	4	2	no	—
14	C₄⋊C₈	—	7	2	no	—
15	C₄×C₈	D₇	1	1	yes	characteristic
16	C₂²×C₇	C₄⋊C₄	1	1	yes	characteristic
17	C₄×C₇	—	2	1	no	—
18	C₄×C₇	C₂×C₈	1	1	yes	characteristic
19	C₄×C₇	M₁₆	1	1	yes	characteristic
20	C₄×C₇	D₈	1	1	yes	characteristic
21	C₄×C₇	Q₁₆	1	1	yes	characteristic
22	C₂×C₈	—	14	2	no	—
23	C₂×C₈	Dic₇	1	1	yes	characteristic
24	C₂×C₈	Dic₇	1	1	yes	characteristic
25	C₄²	C₂×D₇	1	1	yes	characteristic
26	C₂×C₇	C₄⋊C₈	1	1	yes	characteristic
27	C₂×C₇	#107	1	1	yes	characteristic
28	C₂×C₇	#108	1	1	yes	characteristic
29	C₂×C₄	Dic₁₄	1	1	yes	characteristic
30	C₂×C₄	D₂₈	1	1	yes	characteristic
31	C₂×C₄	C₂×Dic₇	1	1	yes	characteristic
32	C₈	—	2	1	no	—
33	C₈	—	28	2	no	—
34	C₈	C₇⋊C₈	1	2	yes	complemented
35	C₇	#335	1	1	yes	characteristic, complemented, Sylow
36	C₂²	C₄⋊Dic₇	1	1	yes	characteristic
37	C₄	—	2	1	no	—
38	C₄	D₅₆	1	1	yes	characteristic
39	C₄	Dic₂₈	1	1	yes	characteristic
40	C₄	C₂×C₇⋊C₈	1	1	yes	characteristic
41	C₄	#1136	1	1	yes	characteristic
42	C₂	C₄⋊(C₇⋊C₈)	1	1	yes	characteristic
43	C₂	#6392	1	1	yes	characteristic
44	C₂	#6393	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₂₈
commutator	C₄×C₇	C₂×C₈
Frattini	C₄²	C₂×D₇
Fitting	C₄×C₇×C₈	C₂
socle, prime radical	C₂²×C₇	C₄⋊C₄
cosocle	C₄²×C₇	C₂²
hypercenter	C₄×C₈	D₇
hypocenter	C₇	#335
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
12. C₇×C₈	C₈	semidirect
34. C₈	C₇⋊C₈	semidirect
35. C₇	#335	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₇×C₈	2
C₄⋊(C₇⋊C₈)	2
#335	7