Subgroups — D₇×#127 #90405

Order 448.

498 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×D₇	C₂	1	6	yes	maximal, complemented
2	C₇×#127	C₂	1	1	yes	maximal, characteristic, complemented
3	#6503	C₂	1	6	yes	maximal, complemented
4	#6437	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄²×D₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₇	C₂²	1	4	yes	—
7	C₂×C₇×D₄	C₂²	1	6	yes	complemented
8	C₂×C₂²⋊D₇	C₂²	1	12	yes	complemented
9	D₄×D₇	—	2	24	no	—
10	C₂×D₂₈	C₂²	1	6	yes	complemented
11	C₂×C₄×D₇	C₂²	1	3	yes	—
12	C₄²×C₇	C₂²	1	1	yes	characteristic, complemented
13	C₄×Dic₇	C₂²	1	3	yes	complemented
14	C₂×#127	—	7	1	no	maximal, Sylow
15	C₂³×C₇	C₂³	1	4	yes	—
16	C₂²×D₇	—	2	24	no	—
17	C₂²×D₇	C₂³	1	4	yes	—
18	C₂²×D₇	C₂³	1	1	yes	characteristic
19	C₇×D₄	—	2	12	no	—
20	C₂×C₄×C₇	C₂³	1	3	yes	—
21	C₂²⋊D₇	—	2	24	no	—
22	C₂×Dic₇	C₂³	1	3	yes	—
23	D₂₈	—	2	12	no	—
24	C₄×D₇	D₄	1	12	yes	complemented
25	C₂²×D₄	—	7	6	no	—
26	#127	—	7	6	no	—
27	#127	—	7	1	no	—
28	#127	D₇	1	1	yes	characteristic, direct product factor
29	C₂×C₄²	—	7	1	no	—
30	C₂²×C₇	—	2	12	no	—
31	C₂²×C₇	C₂⁴	1	1	yes	characteristic
32	C₂×D₇	—	2	12	no	—
33	C₂×D₇	—	4	16	no	—
34	C₂×D₇	C₂×D₄	1	6	yes	—
35	C₄×C₇	C₂×D₄	1	6	yes	complemented
36	Dic₇	C₂×D₄	1	6	yes	complemented
37	C₂⁴	—	7	4	no	—
38	C₂×D₄	—	7	12	no	—
39	C₂×D₄	—	7	6	no	—
40	C₂×D₄	—	14	24	no	—
41	C₂×D₄	C₂×D₇	1	6	yes	complemented
42	C₂²×C₄	—	7	3	no	—
43	C₄²	—	7	3	no	—
44	C₄²	C₂×D₇	1	1	yes	characteristic, complemented
45	C₂×C₇	—	4	4	no	—
46	C₂×C₇	C₂²×D₄	1	3	yes	—
47	D₇	—	4	4	no	—
48	D₇	#127	1	4	yes	direct product factor
49	C₂³	—	7	1	no	—
50	C₂³	—	7	4	no	—
51	C₂³	—	14	24	no	—
52	C₂³	C₂²×D₇	1	4	yes	—
53	D₄	—	2	12	no	—
54	D₄	—	14	24	no	—
55	D₄	—	14	12	no	—
56	C₂×C₄	—	7	3	no	—
57	C₂×C₄	—	7	12	no	—
58	C₂×C₄	C₂²×D₇	1	3	yes	—
59	C₇	C₂×#127	1	1	yes	characteristic, complemented, Sylow
60	C₂²	—	2	12	no	—
61	C₂²	—	7	6	no	—
62	C₂²	—	14	12	no	—
63	C₂²	—	28	16	no	—
64	C₂²	C₂³×D₇	1	1	yes	characteristic
65	C₄	—	7	6	no	—
66	C₄	D₄×D₇	1	6	yes	complemented
67	C₂	—	4	4	no	—
68	C₂	—	7	4	no	—
69	C₂	—	28	4	no	—
70	C₂	C₂×D₄×D₇	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇
commutator, socle, cosocle	C₂²×C₇	C₂⁴
Fitting	C₇×#127	C₂
hypercenter	#127	D₇
hypocenter	C₇	C₂×#127
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #127

Complements

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

N	H ≅ G/N	type
1. C₂×D₄×D₇	C₂	semidirect
2. C₇×#127	C₂	semidirect
3. #6503	C₂	semidirect
4. #6437	C₂	semidirect
5. C₄²×D₇	C₂	semidirect
7. C₂×C₇×D₄	C₂²	semidirect
8. C₂×C₂²⋊D₇	C₂²	semidirect
10. C₂×D₂₈	C₂²	semidirect
12. C₄²×C₇	C₂²	semidirect
13. C₄×Dic₇	C₂²	semidirect
24. C₄×D₇	D₄	semidirect
28. #127	D₇	direct (N × H)
35. C₄×C₇	C₂×D₄	semidirect
36. Dic₇	C₂×D₄	semidirect
41. C₂×D₄	C₂×D₇	semidirect
44. C₄²	C₂×D₇	semidirect
48. D₇	#127	direct (N × H)
59. C₇	C₂×#127	semidirect
66. C₄	D₄×D₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×D₇	2
C₇×#127	2
#6503	2
#6437	2
C₄²×D₇	2
C₂×#127	7

Subgroups — D7×#127 #90405

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₇×#127 #90405