Subgroups — C₇×#123 #6526

Order 224.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
4	C₇×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
6	C₂³×C₇	C₂²	1	1	yes	characteristic
7	C₂³×C₇	C₂²	1	1	yes	characteristic
8	C₇×D₄	—	2	2	no	—
9	C₂×C₄×C₇	—	2	2	no	—
10	C₂×C₄×C₇	C₂²	1	2	yes	—
11	C₂×C₄×C₇	C₂²	1	2	yes	—
12	C₂×C₄×C₇	C₂²	1	1	yes	characteristic
13	#123	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
14	C₂²×C₇	—	2	2	no	—
15	C₂²×C₇	—	2	1	no	—
16	C₂²×C₇	—	2	2	no	—
17	C₂²×C₇	D₄	1	2	yes	—
18	C₂²×C₇	C₂³	1	1	yes	characteristic
19	C₄×C₇	—	2	2	no	—
20	C₄×C₇	—	2	1	no	—
21	C₄×C₇	—	2	2	no	—
22	C₂×D₄	C₂×C₇	1	1	yes	characteristic
23	C₂²×C₄	C₂×C₇	1	1	yes	characteristic, complemented
24	C₄⋊C₄	C₂×C₇	1	2	yes	complemented
25	C₂²⋊C₄	C₂×C₇	1	1	yes	characteristic, complemented
26	C₂²⋊C₄	C₂×C₇	1	2	yes	complemented
27	C₂×C₇	—	2	2	no	—
28	C₂×C₇	—	4	1	no	—
29	C₂×C₇	C₂×D₄	1	1	yes	characteristic
30	C₂×C₇	#41	1	2	yes	—
31	C₂³	C₂²×C₇	1	1	yes	characteristic
32	C₂³	C₂²×C₇	1	1	yes	characteristic
33	D₄	—	2	2	no	—
34	C₂×C₄	—	2	2	no	—
35	C₂×C₄	C₂²×C₇	1	2	yes	—
36	C₂×C₄	C₂²×C₇	1	2	yes	—
37	C₂×C₄	C₂²×C₇	1	1	yes	characteristic
38	C₇	#123	1	1	yes	characteristic, direct product factor, Sylow
39	C₂²	—	2	2	no	—
40	C₂²	—	2	1	no	—
41	C₂²	—	2	2	no	—
42	C₂²	C₇×D₄	1	2	yes	—
43	C₂²	C₂³×C₇	1	1	yes	characteristic
44	C₄	—	2	2	no	—
45	C₄	—	2	1	no	—
46	C₄	—	2	2	no	—
47	C₂	—	2	2	no	—
48	C₂	—	4	1	no	—
49	C₂	C₂×C₇×D₄	1	1	yes	characteristic
50	C₂	C₇×#41	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂³
commutator, Frattini, cosocle	C₂²	C₂³×C₇
prime radical	C₂×C₇×D₄	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×C₇	C₂	semidirect
3. C₇×C₄⋊C₄	C₂	semidirect
4. C₇×C₂²⋊C₄	C₂	semidirect
5. C₇×C₂²⋊C₄	C₂	semidirect
13. #123	C₇	direct (N × H)
23. C₂²×C₄	C₂×C₇	semidirect
24. C₄⋊C₄	C₂×C₇	semidirect
25. C₂²⋊C₄	C₂×C₇	semidirect
26. C₂²⋊C₄	C₂×C₇	semidirect
38. C₇	#123	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×C₇×D₄	2
C₂²×C₄×C₇	2
C₇×C₄⋊C₄	2
C₇×C₂²⋊C₄	2
#123	7

Subgroups — C7×#123 #6526

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#123 #6526