Subgroups — #90443

Order 448.

268 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6553	C₂	1	2	yes	maximal, complemented
2	C₇×#131	C₂	1	1	yes	maximal, characteristic, complemented
3	#6471	C₂	1	6	yes	maximal, complemented
4	#6467	C₂	1	3	yes	maximal, complemented
5	C₂×D₅₆	C₂	1	3	yes	maximal, complemented
6	C₇×#41	C₂²	1	1	yes	characteristic
7	#1161	—	2	2	no	—
8	D₄×D₇	—	2	6	no	—
9	#1157	C₂²	1	6	yes	complemented
10	C₂×D₂₈	C₂²	1	6	yes	complemented
11	C₇×M₁₆	C₂²	1	3	yes	complemented
12	C₂×C₇×C₈	C₂²	1	3	yes	complemented
13	Dic₂₈	C₂²	1	1	yes	characteristic, complemented
14	D₅₆	C₂²	1	3	yes	complemented
15	D₅₆	C₂²	1	6	yes	complemented
16	#1132	C₂²	1	6	yes	complemented
17	#576	—	7	1	no	maximal, Sylow
18	C₂²×D₇	—	2	6	no	—
19	C₇×Q₈	D₄	1	1	yes	characteristic
20	C₇×D₄	D₄	1	3	yes	—
21	C₂×C₄×C₇	C₂³	1	3	yes	—
22	C₂²⋊D₇	—	2	6	no	—
23	D₂₈	—	2	6	no	—
24	D₂₈	C₂³	1	6	yes	—
25	C₄×D₇	—	2	6	no	—
26	Dic₁₄	C₂³	1	2	yes	—
27	C₇×C₈	C₂³	1	3	yes	—
28	C₇×C₈	C₂³	1	1	yes	characteristic
29	H(5,𝔽₂)	—	7	2	no	—
30	AGL(1,ℤ/8)	—	7	6	no	—
31	#135	—	7	3	no	—
32	C₂×D₈	—	7	3	no	—
33	#131	D₇	1	1	yes	characteristic, complemented
34	C₂²×C₇	C₂×D₄	1	3	yes	—
35	C₂×D₇	—	2	6	no	—
36	C₂×D₇	—	8	6	no	—
37	C₄×C₇	C₂×D₄	1	3	yes	—
38	C₄×C₇	C₂⁴	1	1	yes	characteristic
39	Dic₇	—	2	2	no	—
40	#41	—	7	6	no	—
41	#41	—	14	2	no	—
42	#41	C₂×D₇	1	1	yes	characteristic
43	C₂×D₄	—	7	6	no	—
44	C₂×D₄	—	14	6	no	—
45	Q₁₆	—	7	1	no	—
46	SD₁₆	—	7	6	no	—
47	D₈	—	7	3	no	—
48	D₈	—	7	6	no	—
49	M₁₆	C₂×D₇	1	3	yes	complemented
50	C₂×C₈	C₂×D₇	1	3	yes	complemented
51	C₂×C₇	—	2	3	no	—
52	C₂×C₇	C₂²×D₄	1	1	yes	characteristic
53	D₇	—	4	6	no	—
54	C₂³	—	14	6	no	—
55	Q₈	—	7	2	no	—
56	Q₈	D₂₈	1	1	yes	characteristic
57	D₄	—	7	6	no	—
58	D₄	—	14	6	no	—
59	D₄	—	14	6	no	—
60	D₄	D₂₈	1	3	yes	—
61	C₂×C₄	—	14	6	no	—
62	C₂×C₄	C₂²×D₇	1	3	yes	—
63	C₈	C₂²×D₇	1	1	yes	characteristic
64	C₈	C₂²×D₇	1	3	yes	—
65	C₇	#576	1	1	yes	characteristic, complemented, Sylow
66	C₂²	—	14	6	no	—
67	C₂²	—	56	6	no	—
68	C₂²	C₂×D₂₈	1	3	yes	—
69	C₄	—	14	2	no	—
70	C₄	C₂×D₂₈	1	3	yes	—
71	C₄	C₂³×D₇	1	1	yes	characteristic
72	C₂	—	2	3	no	—
73	C₂	—	28	6	no	—
74	C₂	C₂²×D₂₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₂₈
commutator, cosocle	C₄×C₇	C₂⁴
Frattini	C₄	C₂³×D₇
Fitting	C₇×#131	C₂
socle	C₂×C₇	C₂²×D₄
hypercenter	#131	D₇
hypocenter	C₇	#576
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

Complements

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

N	H ≅ G/N	type
1. #6553	C₂	semidirect
2. C₇×#131	C₂	semidirect
3. #6471	C₂	semidirect
4. #6467	C₂	semidirect
5. C₂×D₅₆	C₂	semidirect
9. #1157	C₂²	semidirect
10. C₂×D₂₈	C₂²	semidirect
11. C₇×M₁₆	C₂²	semidirect
12. C₂×C₇×C₈	C₂²	semidirect
13. Dic₂₈	C₂²	semidirect
14. D₅₆	C₂²	semidirect
15. D₅₆	C₂²	semidirect
16. #1132	C₂²	semidirect
33. #131	D₇	semidirect
49. M₁₆	C₂×D₇	semidirect
50. C₂×C₈	C₂×D₇	semidirect
65. C₇	#576	semidirect

Maximal subgroups

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

subgroup	index
#6553	2
C₇×#131	2
#6471	2
#6467	2
C₂×D₅₆	2
#576	7