Subgroups — D₇×#674 #11764626

Order 1120.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₇×#168	C₂	1	1	yes	maximal, characteristic
2	C₇⋊#674	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#674	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₅×D₇	C₄	1	1	yes	characteristic
5	C₇⋊#168	C₄	1	1	yes	characteristic
6	C₇×#168	C₂²	1	1	yes	characteristic
7	D₇×C₁₆	—	5	1	no	maximal, Hall
8	C₂×#674	—	7	1	no	maximal, Hall
9	C₂×C₅×D₇	C₈	1	1	yes	characteristic
10	C₄×C₅×C₇	C₂×C₄	1	1	yes	characteristic
11	C₅×Dic₇	C₈	1	1	yes	characteristic
12	C₈×D₇	—	5	1	no	—
13	C₇×C₁₆	—	5	1	no	—
14	C₇⋊C₁₆	—	5	1	no	—
15	C₂×#168	—	7	1	no	—
16	#674	—	7	1	no	—
17	#674	D₇	1	1	yes	characteristic, direct product factor
18	C₂×C₅×C₇	C₂×C₈	1	1	yes	characteristic
19	C₅×D₇	C₁₆	1	2	yes	complemented
20	C₄×D₇	AGL(1,𝔽₅)	1	1	yes	characteristic
21	C₇×C₈	—	5	1	no	—
22	C₇⋊C₈	—	5	1	no	—
23	C₂×C₄×C₅	—	7	1	no	—
24	#168	—	7	1	no	—
25	#168	C₂×D₇	1	1	yes	characteristic
26	C₅×C₇	C₂×C₁₆	1	1	yes	characteristic, complemented, Hall
27	C₂×C₁₆	—	35	1	no	Sylow
28	C₂×D₇	#170	1	1	yes	characteristic
29	C₄×C₇	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
30	Dic₇	#170	1	1	yes	characteristic
31	C₂²×C₅	—	7	1	no	—
32	C₄×C₅	—	7	1	no	—
33	C₄×C₅	C₄×D₇	1	1	yes	characteristic
34	C₂×C₈	—	35	1	no	—
35	C₁₆	—	5	1	no	—
36	C₁₆	—	35	1	no	—
37	C₂×C₇	C₂×#170	1	1	yes	characteristic
38	D₇	#674	1	2	yes	direct product factor
39	C₂×C₅	—	7	2	no	—
40	C₂×C₅	C₈×D₇	1	1	yes	characteristic
41	C₂×C₄	—	7	1	no	—
42	C₈	—	5	1	no	—
43	C₈	—	35	1	no	—
44	C₇	C₂×#674	1	1	yes	characteristic, complemented, Sylow
45	C₅	D₇×C₁₆	1	1	yes	characteristic, complemented, Sylow
46	C₂²	—	7	1	no	—
47	C₄	—	7	1	no	—
48	C₄	D₇×AGL(1,𝔽₅)	1	1	yes	characteristic
49	C₂	—	7	2	no	—
50	C₂	D₇×#170	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₄	D₇×AGL(1,𝔽₅)
commutator, hypocenter	C₅×C₇	C₂×C₁₆
Fitting	C₄×C₅×C₇	C₂×C₄
socle	C₂×C₅×C₇	C₂×C₈
prime radical	C₂×C₅×D₇	C₈
cosocle	C₇×#168	C₂²
LCS	G ▷ C₅×C₇
UCS	C₁ ◁ C₄

Complements

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

N	H ≅ G/N	type
2. C₇⋊#674	C₂	semidirect
3. C₇×#674	C₂	semidirect
17. #674	D₇	direct (N × H)
19. C₅×D₇	C₁₆	semidirect
26. C₅×C₇	C₂×C₁₆	semidirect
38. D₇	#674	direct (N × H)
44. C₇	C₂×#674	semidirect
45. C₅	D₇×C₁₆	semidirect

Maximal subgroups

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

subgroup	index
D₇×#168	2
C₇⋊#674	2
C₇×#674	2
D₇×C₁₆	5
C₂×#674	7

Subgroups — D7×#674 #11764626

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₇×#674 #11764626