Subgroups — C₇×#441 #90087

Order 448.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₄×C₇×Q₈	C₂	1	2	yes	maximal
3	C₇×#106	C₂	1	1	yes	maximal, characteristic
4	C₇×#103	C₂	1	2	yes	maximal
5	C₇×#97	C₂	1	1	yes	maximal, characteristic
6	C₂×C₇×Q₈	C₂²	1	2	yes	—
7	C₇×Q₁₆	C₄	1	4	yes	complemented
8	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
10	C₇×C₄⋊C₄	—	2	2	no	—
11	C₇×C₄⋊C₄	C₂²	1	2	yes	—
12	C₄²×C₇	—	2	2	no	—
13	C₄²×C₇	C₂²	1	1	yes	characteristic
14	#441	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
15	C₇×Q₈	—	2	2	no	—
16	C₇×Q₈	C₂×C₄	1	4	yes	—
17	C₂×C₄×C₇	—	2	2	no	—
18	C₂×C₄×C₇	—	2	2	no	—
19	C₂×C₄×C₇	D₄	1	2	yes	—
20	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
21	C₇×C₈	—	2	1	no	—
22	C₇×C₈	C₂×C₄	1	2	yes	—
23	C₂×Q₁₆	C₂×C₇	1	1	yes	characteristic
24	C₄×Q₈	C₂×C₇	1	2	yes	—
25	#106	C₂×C₇	1	1	yes	characteristic
26	#103	C₂×C₇	1	2	yes	—
27	#97	C₂×C₇	1	1	yes	characteristic
28	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
29	C₄×C₇	—	2	4	no	—
30	C₄×C₇	—	2	2	no	—
31	C₄×C₇	—	4	2	no	—
32	C₄×C₇	C₂²×C₄	1	1	yes	characteristic
33	C₄×C₇	#41	1	1	yes	characteristic
34	C₂×Q₈	C₂²×C₇	1	2	yes	—
35	Q₁₆	C₄×C₇	1	4	yes	complemented
36	C₂×C₈	C₂²×C₇	1	1	yes	characteristic
37	C₂×C₈	C₂²×C₇	1	1	yes	characteristic
38	C₄⋊C₄	—	2	2	no	—
39	C₄⋊C₄	C₂²×C₇	1	2	yes	—
40	C₄²	—	2	2	no	—
41	C₄²	C₂²×C₇	1	1	yes	characteristic
42	C₂×C₇	C₄×D₄	1	1	yes	characteristic
43	C₂×C₇	#137	1	2	yes	—
44	Q₈	—	2	2	no	—
45	Q₈	C₂×C₄×C₇	1	4	yes	—
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	—	2	2	no	—
48	C₂×C₄	C₇×D₄	1	2	yes	—
49	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
50	C₈	—	2	1	no	—
51	C₈	C₂×C₄×C₇	1	2	yes	—
52	C₇	#441	1	1	yes	characteristic, direct product factor, Sylow
53	C₂²	C₂×C₇×D₄	1	1	yes	characteristic
54	C₄	—	2	4	no	—
55	C₄	—	2	2	no	—
56	C₄	—	4	2	no	—
57	C₄	C₂²×C₄×C₇	1	1	yes	characteristic
58	C₄	C₇×#41	1	1	yes	characteristic
59	C₂	C₄×C₇×D₄	1	1	yes	characteristic
60	C₂	C₇×#137	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂²×C₇	C₂×D₄
commutator	C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₄	C₂³×C₇
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
7. C₇×Q₁₆	C₄	semidirect
14. #441	C₇	direct (N × H)
35. Q₁₆	C₄×C₇	semidirect
52. C₇	#441	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×Q₁₆	2
C₄×C₇×Q₈	2
C₇×#106	2
C₇×#103	2
C₇×#97	2
#441	7

Subgroups — C7×#441 #90087

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#441 #90087