Subgroups — #89346

Order 448.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₇	C₂	1	2	yes	maximal
3	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
4	C₂²×Dic₇	—	2	2	no	—
5	C₂×C₇×C₈	—	2	2	no	—
6	C₂×C₇×C₈	C₄	1	2	yes	complemented
7	C₄⋊Dic₇	—	2	2	no	—
8	C₄⋊Dic₇	C₄	1	4	yes	complemented
9	#340	—	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	C₂×Dic₇	—	4	2	no	—
17	C₂×Dic₇	—	4	2	no	—
18	C₇×C₈	—	2	2	no	—
19	C₂²×C₈	D₇	1	1	yes	characteristic
20	C₂×C₄⋊C₄	—	7	2	no	—
21	C₂²×C₇	C₂²⋊C₄	1	2	yes	—
22	C₂²×C₇	C₄⋊C₄	1	1	yes	characteristic
23	C₂²×C₇	D₈	1	1	yes	characteristic
24	C₂²×C₇	SD₁₆	1	2	yes	—
25	C₂²×C₇	Q₁₆	1	1	yes	characteristic
26	C₄×C₇	C₄²	1	1	yes	characteristic
27	C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
28	C₄×C₇	C₄⋊C₄	1	2	yes	—
29	Dic₇	—	4	4	no	—
30	C₂²×C₄	—	14	2	no	—
31	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
32	C₂×C₈	—	2	2	no	—
33	C₂×C₈	Dic₇	1	2	yes	complemented
34	C₄⋊C₄	—	7	4	no	—
35	C₄⋊C₄	—	14	2	no	—
36	C₂×C₇	#95	1	1	yes	characteristic
37	C₂×C₇	#102	1	2	yes	—
38	C₂×C₇	#103	1	2	yes	—
39	C₂×C₇	#106	1	1	yes	characteristic
40	C₂×C₇	#107	1	1	yes	characteristic
41	C₂³	D₂₈	1	1	yes	characteristic
42	C₂×C₄	—	14	4	no	—
43	C₂×C₄	—	28	2	no	—
44	C₂×C₄	—	28	2	no	—
45	C₂×C₄	Dic₁₄	1	1	yes	characteristic
46	C₂×C₄	C₄×D₇	1	2	yes	—
47	C₂×C₄	C₂×Dic₇	1	1	yes	characteristic
48	C₂×C₄	C₂²⋊D₇	1	2	yes	—
49	C₈	—	2	2	no	—
50	C₇	#340	1	1	yes	characteristic, complemented, Sylow
51	C₂²	#1132	1	2	yes	—
52	C₂²	D₅₆	1	1	yes	characteristic
53	C₂²	Dic₂₈	1	1	yes	characteristic
54	C₂²	C₄⋊Dic₇	1	1	yes	characteristic
55	C₂²	(C₂×D₇)⋊C₄	1	2	yes	—
56	C₄	—	28	4	no	—
57	C₄	C₄×Dic₇	1	1	yes	characteristic
58	C₄	Dic₇⋊C₄	1	2	yes	—
59	C₄	C₂²⋊Dic₇	1	1	yes	characteristic
60	C₂	#6390	1	2	yes	—
61	C₂	#6391	1	1	yes	characteristic
62	C₂	#6392	1	1	yes	characteristic
63	C₂	#6395	1	2	yes	—
64	C₂	C₇⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₂₈
commutator	C₄×C₇	C₄²
Frattini	C₂²×C₄	C₂×D₇
Fitting	C₂²×C₇×C₈	C₂
socle, prime radical	C₂³×C₇	D₄
cosocle	C₂²×C₄×C₇	C₂²
hypercenter	C₂²×C₈	D₇
hypocenter	C₇	#340
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
6. C₂×C₇×C₈	C₄	semidirect
8. C₄⋊Dic₇	C₄	semidirect
33. C₂×C₈	Dic₇	semidirect
50. C₇	#340	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₇×C₈	2
C₂×C₄⋊Dic₇	2
#340	7