Subgroups — #89488

Order 448.

108 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	C₄×Dic₁₄	C₂	1	2	yes	maximal
3	C₇×#97	C₂	1	1	yes	maximal, characteristic
4	#6391	C₂	1	1	yes	maximal, characteristic
5	#6390	C₂	1	2	yes	maximal
6	C₂×Dic₁₄	C₂²	1	2	yes	—
7	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
9	C₄²×C₇	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₇	C₂²	1	2	yes	—
11	Dic₇⋊C₄	—	2	2	no	—
12	C₄×Dic₇	—	2	2	no	—
13	Dic₂₈	C₄	1	4	yes	complemented
14	#441	—	7	1	no	maximal, Sylow
15	C₂×C₄×C₇	D₄	1	2	yes	—
16	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
17	C₂×Dic₇	—	2	2	no	—
18	C₂×Dic₇	—	2	2	no	—
19	Dic₁₄	—	2	2	no	—
20	Dic₁₄	C₂×C₄	1	4	yes	—
21	C₇×C₈	—	2	1	no	—
22	C₇×C₈	C₂×C₄	1	2	yes	—
23	C₂×Q₁₆	—	7	1	no	—
24	C₄×Q₈	—	7	2	no	—
25	#106	—	7	1	no	—
26	#103	—	7	2	no	—
27	#97	D₇	1	1	yes	characteristic
28	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
29	C₄×C₇	—	2	2	no	—
30	C₄×C₇	C₂²×C₄	1	1	yes	characteristic
31	C₄×C₇	#41	1	1	yes	characteristic
32	Dic₇	—	2	4	no	—
33	Dic₇	—	4	2	no	—
34	C₂×Q₈	—	7	2	no	—
35	Q₁₆	—	7	4	no	—
36	C₂×C₈	C₂×D₇	1	1	yes	characteristic
37	C₂×C₈	C₂×D₇	1	1	yes	characteristic
38	C₄⋊C₄	—	7	2	no	—
39	C₄⋊C₄	—	14	2	no	—
40	C₄²	—	14	2	no	—
41	C₄²	C₂×D₇	1	1	yes	characteristic
42	C₂×C₇	C₄×D₄	1	1	yes	characteristic
43	C₂×C₇	#137	1	2	yes	—
44	Q₈	—	7	4	no	—
45	Q₈	—	14	2	no	—
46	C₂×C₄	—	14	2	no	—
47	C₂×C₄	—	14	2	no	—
48	C₂×C₄	D₂₈	1	2	yes	—
49	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
50	C₈	—	2	1	no	—
51	C₈	C₄×D₇	1	2	yes	—
52	C₇	#441	1	1	yes	characteristic, complemented, Sylow
53	C₂²	C₂×D₂₈	1	1	yes	characteristic
54	C₄	—	2	2	no	—
55	C₄	—	14	4	no	—
56	C₄	—	28	2	no	—
57	C₄	C₂×C₄×D₇	1	1	yes	characteristic
58	C₄	#1157	1	1	yes	characteristic
59	C₂	C₄×D₂₈	1	1	yes	characteristic
60	C₂	#6472	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₂₈
commutator	C₄×C₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×#97	C₂
socle, prime radical	C₂²×C₇	C₂×D₄
cosocle	C₂×C₄×C₇	C₂³
hypercenter	#97	D₇
hypocenter	C₇	#441
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

Complements

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

N	H ≅ G/N	type
13. Dic₂₈	C₄	semidirect
52. C₇	#441	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
C₄×Dic₁₄	2
C₇×#97	2
#6391	2
#6390	2
#441	7