Subgroups — #89352

Order 448.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×M₁₆	C₂	1	1	yes	maximal, characteristic
2	#6489	C₂	1	2	yes	maximal
3	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
4	C₇×M₁₆	—	2	2	no	—
5	C₂×C₇×C₈	C₄	1	2	yes	complemented
6	C₂²⋊Dic₇	—	2	2	no	—
7	C₄⋊Dic₇	—	2	2	no	—
8	C₄×Dic₇	C₄	1	4	yes	complemented
9	H(3,ℤ/4)	—	7	1	no	maximal, Sylow
10	C₂³×C₇	D₄	1	1	yes	characteristic
11	C₂×C₄×C₇	C₂×C₄	1	2	yes	—
12	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄×C₇	D₄	1	2	yes	—
14	C₂×C₄×C₇	Q₈	1	1	yes	characteristic
15	C₂×Dic₇	—	2	4	no	—
16	C₇×C₈	—	2	2	no	—
17	C₂×M₁₆	D₇	1	1	yes	characteristic
18	#117	—	7	2	no	—
19	C₂²×C₇	—	4	1	no	—
20	C₂²×C₇	C₂²⋊C₄	1	2	yes	—
21	C₂²×C₇	C₄⋊C₄	1	1	yes	characteristic
22	C₄×C₇	C₄²	1	1	yes	characteristic
23	C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
24	C₄×C₇	C₄⋊C₄	1	2	yes	—
25	Dic₇	—	4	4	no	—
26	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
27	M₁₆	—	2	2	no	—
28	C₂×C₈	Dic₇	1	2	yes	complemented
29	C₄⋊C₄	—	14	2	no	—
30	C₂²⋊C₄	—	14	2	no	—
31	C₄²	—	7	4	no	—
32	C₂×C₇	—	2	2	no	—
33	C₂×C₇	—	2	1	no	—
34	C₂×C₇	#95	1	1	yes	characteristic
35	C₂³	D₂₈	1	1	yes	characteristic
36	C₂×C₄	—	14	4	no	—
37	C₂×C₄	Dic₁₄	1	1	yes	characteristic
38	C₂×C₄	C₄×D₇	1	2	yes	—
39	C₂×C₄	C₂×Dic₇	1	1	yes	characteristic
40	C₂×C₄	C₂²⋊D₇	1	2	yes	—
41	C₈	—	2	2	no	—
42	C₇	H(3,ℤ/4)	1	1	yes	characteristic, complemented, Sylow
43	C₂²	—	4	1	no	—
44	C₂²	C₄⋊Dic₇	1	1	yes	characteristic
45	C₂²	(C₂×D₇)⋊C₄	1	2	yes	—
46	C₄	—	28	4	no	—
47	C₄	C₄×Dic₇	1	1	yes	characteristic
48	C₄	Dic₇⋊C₄	1	2	yes	—
49	C₄	C₂²⋊Dic₇	1	1	yes	characteristic
50	C₂	—	2	2	no	—
51	C₂	—	2	1	no	—
52	C₂	C₇⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄×Dic₇
commutator	C₄×C₇	C₄²
Frattini	C₂²×C₄	C₂×D₇
Fitting	C₂×C₇×M₁₆	C₂
socle	C₂×C₇	#95
prime radical	C₂³×C₇	D₄
cosocle	C₂²×C₄×C₇	C₂²
hypercenter	C₂×M₁₆	D₇
hypocenter	C₇	H(3,ℤ/4)
LCS	G ▷ C₄×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
5. C₂×C₇×C₈	C₄	semidirect
8. C₄×Dic₇	C₄	semidirect
28. C₂×C₈	Dic₇	semidirect
42. C₇	H(3,ℤ/4)	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×M₁₆	2
#6489	2
H(3,ℤ/4)	7