Subgroups — C₇×#543 #90564

Order 448.

242 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₂²⋊Q₈	C₂	1	12	yes	maximal, complemented
2	C₂×C₇×C₂²⋊C₄	C₂	1	3	yes	maximal
3	C₂⁴×C₇	C₂²	1	1	yes	characteristic
4	C₂×C₇×Q₈	C₂²	1	4	yes	complemented
5	C₂²×C₄×C₇	C₂²	1	6	yes	—
6	C₇×C₄⋊C₄	C₂²	1	12	yes	complemented
7	C₇×C₂²⋊C₄	C₂²	1	12	yes	—
8	#543	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₂³×C₇	—	4	2	no	—
10	C₂³×C₇	Q₈	1	4	yes	complemented
11	C₂³×C₇	C₂³	1	3	yes	—
12	C₇×Q₈	—	4	4	no	—
13	C₂×C₄×C₇	—	4	6	no	—
14	C₂×C₄×C₇	C₂³	1	12	yes	—
15	C₂²⋊Q₈	C₂×C₇	1	12	yes	complemented
16	C₂×C₂²⋊C₄	C₂×C₇	1	3	yes	—
17	C₂²×C₇	—	2	6	no	—
18	C₂²×C₇	—	4	4	no	—
19	C₂²×C₇	C₂×Q₈	1	6	yes	—
20	C₂²×C₇	C₂⁴	1	1	yes	characteristic
21	C₄×C₇	—	2	12	no	—
22	C₂⁴	C₂²×C₇	1	1	yes	characteristic
23	C₂×Q₈	C₂²×C₇	1	4	yes	complemented
24	C₂²×C₄	C₂²×C₇	1	6	yes	—
25	C₄⋊C₄	C₂²×C₇	1	12	yes	complemented
26	C₂²⋊C₄	C₂²×C₇	1	12	yes	—
27	C₂×C₇	—	2	6	no	—
28	C₂×C₇	C₂²×Q₈	1	1	yes	characteristic
29	C₂×C₇	H(5,𝔽₂)	1	2	yes	—
30	C₂³	—	4	2	no	—
31	C₂³	C₇×Q₈	1	4	yes	complemented
32	C₂³	C₂³×C₇	1	3	yes	—
33	Q₈	—	4	4	no	—
34	C₂×C₄	—	4	6	no	—
35	C₂×C₄	C₂³×C₇	1	12	yes	—
36	C₇	#543	1	1	yes	characteristic, direct product factor, Sylow
37	C₂²	—	2	6	no	—
38	C₂²	—	4	4	no	—
39	C₂²	C₂×C₇×Q₈	1	6	yes	—
40	C₂²	C₂⁴×C₇	1	1	yes	characteristic
41	C₄	—	2	12	no	—
42	C₂	—	2	6	no	—
43	C₂	C₂²×C₇×Q₈	1	1	yes	characteristic
44	C₂	C₇×H(5,𝔽₂)	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₇
prime radical	C₂⁴×C₇	C₂²
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
1. C₇×C₂²⋊Q₈	C₂	semidirect
4. C₂×C₇×Q₈	C₂²	semidirect
6. C₇×C₄⋊C₄	C₂²	semidirect
8. #543	C₇	direct (N × H)
10. C₂³×C₇	Q₈	semidirect
15. C₂²⋊Q₈	C₂×C₇	semidirect
23. C₂×Q₈	C₂²×C₇	semidirect
25. C₄⋊C₄	C₂²×C₇	semidirect
31. C₂³	C₇×Q₈	semidirect
36. C₇	#543	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×C₂²⋊Q₈	2
C₂×C₇×C₂²⋊C₄	2
#543	7

Subgroups — C7×#543 #90564

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#543 #90564