Subgroups — C₇²×#107 #13096168

Order 1568.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇²×C₈	C₂	1	1	yes	maximal, characteristic
2	C₇²×C₄⋊C₄	C₂	1	2	yes	maximal
3	C₂×C₄×C₇²	—	2	2	no	—
4	C₂×C₄×C₇²	C₂²	1	1	yes	characteristic
5	C₇²×C₈	C₄	1	2	yes	complemented
6	C₇×#107	C₇	1	8	yes	maximal, direct product factor
7	C₂²×C₇²	D₄	1	1	yes	characteristic
8	C₄×C₇²	—	4	2	no	—
9	C₄×C₇²	C₂×C₄	1	1	yes	characteristic
10	C₄×C₇²	Q₈	1	1	yes	characteristic
11	C₂×C₇×C₈	C₂×C₇	1	8	yes	—
12	C₇×C₄⋊C₄	C₂×C₇	1	16	yes	—
13	C₂×C₇²	C₄⋊C₄	1	1	yes	characteristic
14	C₂×C₇²	D₈	1	1	yes	characteristic
15	C₂×C₇²	Q₁₆	1	1	yes	characteristic
16	C₂×C₄×C₇	—	2	16	no	—
17	C₂×C₄×C₇	C₂²×C₇	1	8	yes	—
18	C₇×C₈	C₄×C₇	1	16	yes	complemented
19	C₇²	#107	1	1	yes	characteristic, direct product factor, Sylow
20	#107	C₇²	1	1	yes	characteristic, direct product factor, Sylow
21	C₂²×C₇	C₇×D₄	1	8	yes	—
22	C₄×C₇	—	4	16	no	—
23	C₄×C₇	C₂×C₄×C₇	1	8	yes	—
24	C₄×C₇	C₇×Q₈	1	8	yes	—
25	C₂×C₈	C₂×C₇²	1	1	yes	characteristic
26	C₄⋊C₄	C₂×C₇²	1	2	yes	—
27	C₂×C₇	C₇×C₄⋊C₄	1	8	yes	—
28	C₂×C₇	C₇×D₈	1	8	yes	—
29	C₂×C₇	C₇×Q₁₆	1	8	yes	—
30	C₂×C₄	—	2	2	no	—
31	C₂×C₄	C₂²×C₇²	1	1	yes	characteristic
32	C₈	C₄×C₇²	1	2	yes	complemented
33	C₇	C₇×#107	1	8	yes	direct product factor
34	C₂²	C₇²×D₄	1	1	yes	characteristic
35	C₄	—	4	2	no	—
36	C₄	C₂×C₄×C₇²	1	1	yes	characteristic
37	C₄	C₇²×Q₈	1	1	yes	characteristic
38	C₂	C₇²×C₄⋊C₄	1	1	yes	characteristic
39	C₂	C₇²×D₈	1	1	yes	characteristic
40	C₂	C₇²×Q₁₆	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₇²	D₄
commutator	C₄	C₂×C₄×C₇²
Frattini, cosocle	C₂×C₄	C₂²×C₇²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇² ◁ C₂×C₄×C₇² ◁ G

Complements

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

N	H ≅ G/N	type
5. C₇²×C₈	C₄	semidirect
6. C₇×#107	C₇	direct (N × H)
18. C₇×C₈	C₄×C₇	semidirect
19. C₇²	#107	direct (N × H)
20. #107	C₇²	direct (N × H)
32. C₈	C₄×C₇²	semidirect
33. C₇	C₇×#107	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇²×C₈	2
C₇²×C₄⋊C₄	2
C₇×#107	7

Subgroups — C72×#107 #13096168

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇²×#107 #13096168