Subgroups — #41×Dic₇ #90517

Order 448.

310 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#41	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₇	C₂	1	2	yes	maximal, complemented
3	D₄×Dic₇	C₂	1	6	yes	maximal, complemented
4	#6489	C₂	1	3	yes	maximal, complemented
5	C₂×C₄×Dic₇	C₂	1	3	yes	maximal, complemented
6	C₇×#41	C₄	1	8	yes	complemented
7	C₂×C₇×Q₈	C₂²	1	1	yes	characteristic
8	C₂×C₇×D₄	C₂²	1	3	yes	—
9	C₂²×C₄×C₇	C₂²	1	3	yes	—
10	C₂²×Dic₇	C₂²	1	6	yes	—
11	C₂²⋊Dic₇	C₂²	1	6	yes	—
12	C₄⋊Dic₇	C₂²	1	6	yes	—
13	C₄×Dic₇	C₂²	1	1	yes	characteristic
14	C₄×Dic₇	C₂²	1	3	yes	—
15	C₄×Dic₇	C₂²	1	6	yes	—
16	C₄×#41	—	7	1	no	maximal, Sylow
17	C₂³×C₇	C₂³	1	3	yes	—
18	C₇×Q₈	C₂×C₄	1	4	yes	complemented
19	C₇×D₄	C₂×C₄	1	12	yes	complemented
20	C₂×C₄×C₇	C₂×C₄	1	12	yes	complemented
21	C₂×C₄×C₇	C₂³	1	3	yes	—
22	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
23	C₂×Dic₇	—	2	12	no	—
24	C₂×Dic₇	C₂³	1	6	yes	—
25	C₂×Dic₇	C₂³	1	2	yes	—
26	C₂×#41	D₇	1	1	yes	characteristic
27	C₄×Q₈	—	7	2	no	—
28	C₄×D₄	—	7	6	no	—
29	#117	—	7	3	no	—
30	C₂×C₄²	—	7	3	no	—
31	C₂²×C₇	—	2	6	no	—
32	C₂²×C₇	C₂²×C₄	1	6	yes	—
33	C₂²×C₇	C₂⁴	1	1	yes	characteristic
34	C₄×C₇	C₂²×C₄	1	2	yes	—
35	C₄×C₇	C₂²×C₄	1	6	yes	—
36	Dic₇	—	2	6	no	—
37	Dic₇	#41	1	4	yes	direct product factor
38	#41	Dic₇	1	8	yes	direct product factor
39	C₂×Q₈	C₂×D₇	1	1	yes	characteristic
40	C₂×D₄	C₂×D₇	1	3	yes	—
41	C₂²×C₄	—	7	6	no	—
42	C₂²×C₄	C₂×D₇	1	3	yes	—
43	C₄⋊C₄	—	7	6	no	—
44	C₂²⋊C₄	—	7	6	no	—
45	C₄²	—	7	6	no	—
46	C₄²	—	7	1	no	—
47	C₄²	—	7	3	no	—
48	C₂×C₇	—	2	6	no	—
49	C₂×C₇	C₂³×C₄	1	1	yes	characteristic
50	C₂×C₇	C₂×#41	1	2	yes	—
51	C₂³	C₂²×D₇	1	3	yes	—
52	Q₈	C₂×Dic₇	1	4	yes	complemented
53	D₄	C₂×Dic₇	1	12	yes	complemented
54	C₂×C₄	—	7	2	no	—
55	C₂×C₄	—	7	6	no	—
56	C₂×C₄	—	14	12	no	—
57	C₂×C₄	C₂×Dic₇	1	12	yes	complemented
58	C₂×C₄	C₂²×D₇	1	3	yes	—
59	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
60	C₇	C₄×#41	1	1	yes	characteristic, complemented, Sylow
61	C₂²	—	2	6	no	—
62	C₂²	C₂²×Dic₇	1	6	yes	—
63	C₂²	C₂³×D₇	1	1	yes	characteristic
64	C₄	—	7	4	no	—
65	C₄	—	14	6	no	—
66	C₄	C₂²×Dic₇	1	2	yes	—
67	C₄	C₂²×Dic₇	1	6	yes	—
68	C₂	—	2	6	no	—
69	C₂	D₇×#41	1	2	yes	—
70	C₂	C₂³×Dic₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₇
commutator	C₂×C₇	C₂³×C₄
Frattini	C₂²	C₂³×D₇
Fitting, prime radical	C₂×C₇×#41	C₂
socle, cosocle	C₂²×C₇	C₂⁴
hypercenter	C₂×#41	D₇
hypocenter	C₇	C₄×#41
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
2. Q₈×Dic₇	C₂	semidirect
3. D₄×Dic₇	C₂	semidirect
4. #6489	C₂	semidirect
5. C₂×C₄×Dic₇	C₂	semidirect
6. C₇×#41	C₄	semidirect
18. C₇×Q₈	C₂×C₄	semidirect
19. C₇×D₄	C₂×C₄	semidirect
20. C₂×C₄×C₇	C₂×C₄	semidirect
37. Dic₇	#41	direct (N × H)
38. #41	Dic₇	direct (N × H)
52. Q₈	C₂×Dic₇	semidirect
53. D₄	C₂×Dic₇	semidirect
57. C₂×C₄	C₂×Dic₇	semidirect
60. C₇	C₄×#41	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#41	2
Q₈×Dic₇	2
D₄×Dic₇	2
#6489	2
C₂×C₄×Dic₇	2
C₄×#41	7

Subgroups — #41×Dic7 #90517

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #41×Dic₇ #90517