Subgroups — C₇×#1747 #11730216

Order 896.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#405	C₂	1	3	yes	maximal
2	C₇×#338	C₂	1	1	yes	maximal, characteristic
3	C₇×H(3,ℤ/4)	C₂	1	3	yes	maximal
4	C₂×C₇×M₁₆	C₂²	1	3	yes	—
5	C₂²×C₇×C₈	C₂²	1	1	yes	characteristic
6	C₇×#117	C₂²	1	3	yes	—
7	C₇×#97	C₄	1	6	yes	complemented
8	C₄×C₇×C₈	C₄	1	6	yes	complemented
9	#1747	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
11	C₇×M₁₆	—	2	6	no	—
12	C₂×C₇×C₈	C₂×C₄	1	6	yes	—
13	C₂×C₇×C₈	C₂×C₄	1	3	yes	—
14	C₂×C₇×C₈	C₂×C₄	1	3	yes	—
15	C₇×C₄⋊C₄	—	2	3	no	—
16	C₇×C₂²⋊C₄	—	2	3	no	—
17	C₄²×C₇	C₂×C₄	1	6	yes	—
18	#405	C₂×C₇	1	3	yes	—
19	#338	C₂×C₇	1	1	yes	characteristic
20	H(3,ℤ/4)	C₂×C₇	1	3	yes	—
21	C₂³×C₇	#41	1	1	yes	characteristic
22	C₂×C₄×C₇	—	2	6	no	—
23	C₂×C₄×C₇	C₂²×C₄	1	3	yes	—
24	C₂×C₄×C₇	#41	1	3	yes	—
25	C₇×C₈	—	2	6	no	—
26	C₇×C₈	C₄²	1	3	yes	—
27	C₇×C₈	C₄²	1	1	yes	characteristic
28	C₂×M₁₆	C₂²×C₇	1	3	yes	—
29	C₂²×C₈	C₂²×C₇	1	1	yes	characteristic
30	#117	C₂²×C₇	1	3	yes	—
31	#97	C₄×C₇	1	6	yes	complemented
32	C₄×C₈	C₄×C₇	1	6	yes	complemented
33	C₂²×C₇	—	4	1	no	—
34	C₂²×C₇	#117	1	3	yes	—
35	C₄×C₇	—	4	6	no	—
36	C₄×C₇	C₂×C₄²	1	1	yes	characteristic
37	C₄×C₇	#117	1	3	yes	—
38	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
39	M₁₆	—	2	6	no	—
40	C₂×C₈	C₂×C₄×C₇	1	6	yes	—
41	C₂×C₈	C₂×C₄×C₇	1	3	yes	—
42	C₂×C₈	C₂×C₄×C₇	1	3	yes	—
43	C₄⋊C₄	—	2	3	no	—
44	C₂²⋊C₄	—	2	3	no	—
45	C₄²	C₂×C₄×C₇	1	6	yes	—
46	C₂×C₇	—	2	3	no	—
47	C₂×C₇	#376	1	1	yes	characteristic
48	C₂³	C₇×#41	1	1	yes	characteristic
49	C₂×C₄	—	2	6	no	—
50	C₂×C₄	C₂²×C₄×C₇	1	3	yes	—
51	C₂×C₄	C₇×#41	1	3	yes	—
52	C₈	—	2	6	no	—
53	C₈	C₄²×C₇	1	3	yes	—
54	C₈	C₄²×C₇	1	1	yes	characteristic
55	C₇	#1747	1	1	yes	characteristic, direct product factor, Sylow
56	C₂²	—	4	1	no	—
57	C₂²	C₇×#117	1	3	yes	—
58	C₄	—	4	6	no	—
59	C₄	C₂×C₄²×C₇	1	1	yes	characteristic
60	C₄	C₇×#117	1	3	yes	—
61	C₂	—	2	3	no	—
62	C₂	C₇×#376	1	1	yes	characteristic

	subgroup	G/H
center	C₇×C₈	C₄²
commutator	C₄	C₂×C₄²×C₇
Frattini, cosocle	C₂²×C₄	C₂³×C₇
socle	C₂×C₇	#376
prime radical	C₂³×C₇	#41
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
7. C₇×#97	C₄	semidirect
8. C₄×C₇×C₈	C₄	semidirect
9. #1747	C₇	direct (N × H)
31. #97	C₄×C₇	semidirect
32. C₄×C₈	C₄×C₇	semidirect
55. C₇	#1747	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×#405	2
C₇×#338	2
C₇×H(3,ℤ/4)	2
#1747	7

Subgroups — C7×#1747 #11730216

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#1747 #11730216