Subgroups — C₇×#443 #90089

Order 448.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#135	C₂	1	1	yes	maximal, characteristic
2	C₇×#131	C₂	1	2	yes	maximal, complemented
3	C₇×#108	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#104	C₂	1	2	yes	maximal, complemented
5	C₄×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#41	C₂²	1	2	yes	—
7	C₇×Q₁₆	C₄	1	1	yes	characteristic, complemented
8	C₇×SD₁₆	C₄	1	2	yes	complemented
9	C₇×D₈	C₄	1	1	yes	characteristic, complemented
10	C₇×M₁₆	—	2	2	no	—
11	C₇×M₁₆	C₂²	1	2	yes	—
12	C₂×C₇×C₈	—	2	2	no	—
13	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
15	C₄²×C₇	C₂²	1	1	yes	characteristic
16	#443	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₇×Q₈	C₂×C₄	1	2	yes	—
18	C₇×D₄	—	2	2	no	—
19	C₇×D₄	C₂×C₄	1	2	yes	—
20	C₂×C₄×C₇	—	2	1	no	—
21	C₂×C₄×C₇	—	2	2	no	—
22	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
23	C₇×C₈	—	2	2	no	—
24	C₇×C₈	C₂×C₄	1	1	yes	characteristic
25	C₇×C₈	C₂×C₄	1	1	yes	characteristic
26	C₇×C₈	D₄	1	1	yes	characteristic
27	C₇×C₈	D₄	1	1	yes	characteristic
28	#135	C₂×C₇	1	1	yes	characteristic
29	#131	C₂×C₇	1	2	yes	complemented
30	#108	C₂×C₇	1	1	yes	characteristic, complemented
31	#104	C₂×C₇	1	2	yes	complemented
32	C₄×C₈	C₂×C₇	1	1	yes	characteristic, complemented
33	C₂²×C₇	—	2	2	no	—
34	C₂²×C₇	#41	1	1	yes	characteristic
35	C₄×C₇	—	2	2	no	—
36	C₄×C₇	—	2	2	no	—
37	C₄×C₇	C₂²×C₄	1	1	yes	characteristic
38	C₄×C₇	C₂×D₄	1	1	yes	characteristic
39	#41	C₂²×C₇	1	2	yes	—
40	Q₁₆	C₄×C₇	1	1	yes	characteristic, complemented
41	SD₁₆	C₄×C₇	1	2	yes	complemented
42	D₈	C₄×C₇	1	1	yes	characteristic, complemented
43	M₁₆	—	2	2	no	—
44	M₁₆	C₂²×C₇	1	2	yes	—
45	C₂×C₈	—	2	2	no	—
46	C₂×C₈	C₂²×C₇	1	1	yes	characteristic
47	C₂×C₈	C₂²×C₇	1	1	yes	characteristic
48	C₄²	C₂²×C₇	1	1	yes	characteristic
49	C₂×C₇	—	2	1	no	—
50	C₂×C₇	—	4	2	no	—
51	C₂×C₇	C₄×D₄	1	1	yes	characteristic
52	Q₈	C₂×C₄×C₇	1	2	yes	—
53	D₄	—	2	2	no	—
54	D₄	C₂×C₄×C₇	1	2	yes	—
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
58	C₈	—	2	2	no	—
59	C₈	C₂×C₄×C₇	1	1	yes	characteristic
60	C₈	C₂×C₄×C₇	1	1	yes	characteristic
61	C₈	C₇×D₄	1	1	yes	characteristic
62	C₈	C₇×D₄	1	1	yes	characteristic
63	C₇	#443	1	1	yes	characteristic, direct product factor, Sylow
64	C₂²	—	2	2	no	—
65	C₂²	C₇×#41	1	1	yes	characteristic
66	C₄	—	2	2	no	—
67	C₄	—	2	2	no	—
68	C₄	C₂²×C₄×C₇	1	1	yes	characteristic
69	C₄	C₂×C₇×D₄	1	1	yes	characteristic
70	C₂	—	2	1	no	—
71	C₂	—	4	2	no	—
72	C₂	C₄×C₇×D₄	1	1	yes	characteristic

	subgroup	G/H
center	C₇×C₈	D₄
commutator	C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₄	C₂³×C₇
socle	C₂×C₇	C₄×D₄
prime radical	C₇×#135	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₇×C₈ ◁ C₂×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#131	C₂	semidirect
3. C₇×#108	C₂	semidirect
4. C₇×#104	C₂	semidirect
5. C₄×C₇×C₈	C₂	semidirect
7. C₇×Q₁₆	C₄	semidirect
8. C₇×SD₁₆	C₄	semidirect
9. C₇×D₈	C₄	semidirect
16. #443	C₇	direct (N × H)
29. #131	C₂×C₇	semidirect
30. #108	C₂×C₇	semidirect
31. #104	C₂×C₇	semidirect
32. C₄×C₈	C₂×C₇	semidirect
40. Q₁₆	C₄×C₇	semidirect
41. SD₁₆	C₄×C₇	semidirect
42. D₈	C₄×C₇	semidirect
63. C₇	#443	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×#135	2
C₇×#131	2
C₇×#108	2
C₇×#104	2
C₄×C₇×C₈	2
#443	7

Subgroups — C7×#443 #90089

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#443 #90089