Subgroups — C₇×#519 #90540

Order 448.

266 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₄×C₇×Q₈	C₂	1	8	yes	maximal, complemented
3	C₇×#117	C₂	1	6	yes	maximal
4	C₂×C₇×Q₈	C₄	1	8	yes	complemented
5	C₂×C₇×Q₈	C₂²	1	4	yes	—
6	C₂²×C₄×C₇	C₂²	1	3	yes	—
7	C₇×C₄⋊C₄	C₂²	1	12	yes	—
8	C₇×C₂²⋊C₄	C₂²	1	4	yes	—
9	C₄²×C₇	C₂²	1	12	yes	—
10	#519	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂³×C₇	C₂³	1	1	yes	characteristic
12	C₇×Q₈	C₂×C₄	1	16	yes	—
13	C₂×C₄×C₇	C₂×C₄	1	12	yes	—
14	C₂×C₄×C₇	C₂³	1	8	yes	—
15	C₂×C₄×C₇	C₂³	1	6	yes	—
16	C₂²×Q₈	C₂×C₇	1	1	yes	characteristic
17	C₄×Q₈	C₂×C₇	1	8	yes	complemented
18	#117	C₂×C₇	1	6	yes	—
19	C₂²×C₇	—	2	2	no	—
20	C₂²×C₇	C₂²×C₄	1	2	yes	—
21	C₂²×C₇	C₂⁴	1	1	yes	characteristic
22	C₄×C₇	—	2	8	no	—
23	C₄×C₇	C₂²×C₄	1	12	yes	—
24	C₂×Q₈	C₄×C₇	1	8	yes	complemented
25	C₂×Q₈	C₂²×C₇	1	4	yes	—
26	C₂²×C₄	C₂²×C₇	1	3	yes	—
27	C₄⋊C₄	C₂²×C₇	1	12	yes	—
28	C₂²⋊C₄	C₂²×C₇	1	4	yes	—
29	C₄²	C₂²×C₇	1	12	yes	—
30	C₂×C₇	—	2	2	no	—
31	C₂×C₇	C₂³×C₄	1	1	yes	characteristic
32	C₂×C₇	2₋¹⁺⁴	1	2	yes	—
33	C₂³	C₂³×C₇	1	1	yes	characteristic
34	Q₈	C₂×C₄×C₇	1	16	yes	—
35	C₂×C₄	C₂×C₄×C₇	1	12	yes	—
36	C₂×C₄	C₂³×C₇	1	8	yes	—
37	C₂×C₄	C₂³×C₇	1	6	yes	—
38	C₇	#519	1	1	yes	characteristic, direct product factor, Sylow
39	C₂²	—	2	2	no	—
40	C₂²	C₂²×C₄×C₇	1	2	yes	—
41	C₂²	C₂⁴×C₇	1	1	yes	characteristic
42	C₄	—	2	8	no	—
43	C₄	C₂²×C₄×C₇	1	12	yes	—
44	C₂	—	2	2	no	—
45	C₂	C₂³×C₄×C₇	1	1	yes	characteristic
46	C₂	C₇×2₋¹⁺⁴	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂⁴
commutator	C₂	C₂³×C₄×C₇
Frattini, cosocle	C₂²	C₂⁴×C₇
prime radical	C₂³×C₇	C₂³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₇×Q₈	C₂	semidirect
4. C₂×C₇×Q₈	C₄	semidirect
10. #519	C₇	direct (N × H)
17. C₄×Q₈	C₂×C₇	semidirect
24. C₂×Q₈	C₄×C₇	semidirect
38. C₇	#519	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₇×Q₈	2
C₄×C₇×Q₈	2
C₇×#117	2
#519	7

Subgroups — C7×#519 #90540

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#519 #90540