Subgroups — #89298

Order 448.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₂₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×C₁₆	C₂	1	1	yes	maximal, characteristic
3	#6392	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₁₄	—	2	1	no	—
5	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
6	C₄⋊Dic₇	—	2	1	no	—
7	Dic₂₈	—	2	1	no	—
8	Dic₂₈	C₄	1	2	yes	complemented
9	C₇×C₁₆	—	2	1	no	—
10	#358	—	7	1	no	maximal, Sylow
11	C₂×C₄×C₇	D₄	1	1	yes	characteristic
12	C₂×Dic₇	—	4	1	no	—
13	C₂×Dic₇	—	4	1	no	—
14	Dic₁₄	—	2	2	no	—
15	Dic₁₄	—	4	1	no	—
16	C₇×C₈	C₂×C₄	1	1	yes	characteristic
17	C₇×C₈	D₄	1	1	yes	characteristic
18	C₂×Q₁₆	—	7	1	no	—
19	C₂×C₁₆	D₇	1	1	yes	characteristic
20	#107	—	7	1	no	—
21	C₂²×C₇	D₈	1	1	yes	characteristic
22	C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₄×C₇	SD₁₆	1	1	yes	characteristic
24	Dic₇	—	4	2	no	—
25	Dic₇	—	8	1	no	—
26	C₂×Q₈	—	14	1	no	—
27	Q₁₆	—	7	2	no	—
28	Q₁₆	—	14	1	no	—
29	C₂×C₈	C₂×D₇	1	1	yes	characteristic
30	C₄⋊C₄	—	14	1	no	—
31	C₁₆	—	2	1	no	—
32	C₂×C₇	#102	1	1	yes	characteristic
33	C₂×C₇	SD₃₂	1	1	yes	characteristic
34	C₂×C₇	Q₃₂	1	1	yes	characteristic
35	Q₈	—	14	2	no	—
36	Q₈	—	28	1	no	—
37	C₂×C₄	—	28	1	no	—
38	C₂×C₄	—	28	1	no	—
39	C₂×C₄	D₂₈	1	1	yes	characteristic
40	C₈	C₄×D₇	1	1	yes	characteristic
41	C₈	C₂²⋊D₇	1	1	yes	characteristic
42	C₇	#358	1	1	yes	characteristic, complemented, Sylow
43	C₂²	D₅₆	1	1	yes	characteristic
44	C₄	—	28	2	no	—
45	C₄	—	56	1	no	—
46	C₄	#1132	1	1	yes	characteristic
47	C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
48	C₂	#6374	1	1	yes	characteristic
49	C₂	Dic₅₆	1	1	yes	characteristic
50	C₂	#6395	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₅₆
commutator	C₇×C₈	C₂×C₄
Frattini	C₂×C₈	C₂×D₇
Fitting	C₂×C₇×C₁₆	C₂
socle, prime radical	C₂²×C₇	D₈
cosocle	C₂×C₇×C₈	C₂²
hypercenter	C₂×C₁₆	D₇
hypocenter	C₇	#358
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
8. Dic₂₈	C₄	semidirect
42. C₇	#358	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₂₈	2
C₂×C₇×C₁₆	2
#6392	2
#358	7