Subgroups — #89469

Order 448.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6438	C₂	1	1	yes	maximal, characteristic
2	#6433	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#6395	C₂	1	2	yes	maximal
5	#6390	C₂	1	2	yes	maximal, complemented
6	C₂×D₂₈	C₂²	1	1	yes	characteristic
7	C₂×Dic₁₄	C₂²	1	1	yes	characteristic
8	C₂×C₇×C₈	C₂²	1	2	yes	—
9	C₄²×C₇	C₂²	1	1	yes	characteristic
10	(C₂×D₇)⋊C₄	—	2	2	no	—
11	C₄⋊Dic₇	C₂²	1	2	yes	—
12	Dic₇⋊C₄	—	2	2	no	—
13	#488	—	7	1	no	maximal, Sylow
14	C₂²×D₇	—	2	1	no	—
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	1	no	—
19	D₂₈	—	2	2	no	—
20	Dic₁₄	—	2	2	no	—
21	C₇×C₈	—	2	2	no	—
22	#125	—	7	1	no	—
23	#124	—	7	1	no	—
24	#103	—	7	2	no	—
25	#102	—	7	2	no	—
26	C₄×C₈	D₇	1	1	yes	characteristic, complemented
27	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
28	C₂×D₇	—	4	1	no	—
29	C₂×D₇	—	4	2	no	—
30	C₄×C₇	—	2	2	no	—
31	C₄×C₇	#41	1	2	yes	—
32	Dic₇	—	4	1	no	—
33	Dic₇	—	4	2	no	—
34	C₂×Q₈	—	7	1	no	—
35	C₂×D₄	—	7	1	no	—
36	C₂×C₈	C₂×D₇	1	2	yes	—
37	C₄⋊C₄	—	7	2	no	—
38	C₄⋊C₄	—	14	2	no	—
39	C₂²⋊C₄	—	14	2	no	—
40	C₄²	C₂×D₇	1	1	yes	characteristic
41	C₂×C₇	#124	1	1	yes	characteristic
42	C₂×C₇	#135	1	2	yes	—
43	D₇	—	8	1	no	—
44	C₂³	—	14	1	no	—
45	Q₈	—	14	2	no	—
46	D₄	—	14	2	no	—
47	C₂×C₄	—	14	1	no	—
48	C₂×C₄	—	14	2	no	—
49	C₂×C₄	D₂₈	1	2	yes	—
50	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
51	C₈	—	2	2	no	—
52	C₇	#488	1	1	yes	characteristic, complemented, Sylow
53	C₂²	—	28	1	no	—
54	C₂²	—	28	2	no	—
55	C₂²	C₂×D₂₈	1	1	yes	characteristic
56	C₄	—	2	2	no	—
57	C₄	—	28	1	no	—
58	C₄	—	28	2	no	—
59	C₄	#1157	1	2	yes	—
60	C₂	—	56	1	no	—
61	C₂	#6438	1	1	yes	characteristic
62	C₂	#6467	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₂₈
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₄×C₇×C₈	C₂
socle	C₂²×C₇	C₂×D₄
prime radical	C₂×D₂₈	C₂²
hypercenter	C₄×C₈	D₇
hypocenter	C₇	#488
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #6433	C₂	semidirect
3. C₄×C₇×C₈	C₂	semidirect
5. #6390	C₂	semidirect
26. C₄×C₈	D₇	semidirect
52. C₇	#488	semidirect

Maximal subgroups

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

subgroup	index
#6438	2
#6433	2
C₄×C₇×C₈	2
#6395	2
#6390	2
#488	7