Subgroups — C₇×#137 #6540

Order 224.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#41	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×Q₁₆	C₂	1	2	yes	maximal, complemented
4	C₇×SD₁₆	C₂	1	2	yes	maximal, complemented
5	C₇×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×Q₈	—	2	1	no	—
7	C₇×Q₈	C₂²	1	2	yes	—
8	C₇×Q₈	C₂²	1	1	yes	characteristic
9	C₇×D₄	—	2	1	no	—
10	C₇×D₄	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₇	—	2	1	no	—
12	C₂×C₄×C₇	—	2	1	no	—
13	C₂×C₄×C₇	C₂²	1	1	yes	characteristic
14	C₇×C₈	C₂²	1	2	yes	—
15	#137	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂²×C₇	—	2	1	no	—
17	C₂²×C₇	D₄	1	1	yes	characteristic
18	C₄×C₇	—	2	1	no	—
19	C₄×C₇	—	2	2	no	—
20	C₄×C₇	D₄	1	1	yes	characteristic
21	C₄×C₇	C₂³	1	1	yes	characteristic
22	#41	C₂×C₇	1	1	yes	characteristic
23	C₂×Q₈	C₂×C₇	1	1	yes	characteristic, complemented
24	Q₁₆	C₂×C₇	1	2	yes	complemented
25	SD₁₆	C₂×C₇	1	2	yes	complemented
26	M₁₆	C₂×C₇	1	1	yes	characteristic, complemented
27	C₂×C₇	—	2	1	no	—
28	C₂×C₇	—	4	1	no	—
29	C₂×C₇	C₂×D₄	1	1	yes	characteristic
30	Q₈	—	2	1	no	—
31	Q₈	C₂²×C₇	1	2	yes	—
32	Q₈	C₂²×C₇	1	1	yes	characteristic
33	D₄	—	2	1	no	—
34	D₄	C₂²×C₇	1	1	yes	characteristic
35	C₂×C₄	—	2	1	no	—
36	C₂×C₄	—	2	1	no	—
37	C₂×C₄	C₂²×C₇	1	1	yes	characteristic
38	C₈	C₂²×C₇	1	2	yes	—
39	C₇	#137	1	1	yes	characteristic, direct product factor, Sylow
40	C₂²	—	2	1	no	—
41	C₂²	C₇×D₄	1	1	yes	characteristic
42	C₄	—	2	1	no	—
43	C₄	—	2	2	no	—
44	C₄	C₇×D₄	1	1	yes	characteristic
45	C₄	C₂³×C₇	1	1	yes	characteristic
46	C₂	—	2	1	no	—
47	C₂	—	4	1	no	—
48	C₂	C₂×C₇×D₄	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₇	C₂×D₄
commutator, Frattini, cosocle	C₄	C₂³×C₇
prime radical	C₇×#41	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×Q₈	C₂	semidirect
3. C₇×Q₁₆	C₂	semidirect
4. C₇×SD₁₆	C₂	semidirect
5. C₇×M₁₆	C₂	semidirect
15. #137	C₇	direct (N × H)
23. C₂×Q₈	C₂×C₇	semidirect
24. Q₁₆	C₂×C₇	semidirect
25. SD₁₆	C₂×C₇	semidirect
26. M₁₆	C₂×C₇	semidirect
39. C₇	#137	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×#41	2
C₂×C₇×Q₈	2
C₇×Q₁₆	2
C₇×SD₁₆	2
C₇×M₁₆	2
#137	7

Subgroups — C7×#137 #6540

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#137 #6540