Subgroups — C₇×D₇×D₁₂₁ #37557360

Order 23716.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×D₈₄₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×D₇×C₁₂₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇²×D₁₂₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇²×C₁₂₁	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×C₇×D₁₂₁	—	7	1	no	maximal
6	D₇×D₁₂₁	C₇	1	1	yes	maximal, characteristic, direct product factor
7	C₇×D₇×D₁₁	—	11	1	no	maximal
8	C₂×C₇×C₁₂₁	—	7	1	no	—
9	D₈₄₇	C₂×C₇	1	1	yes	characteristic, complemented
10	C₇×D₁₂₁	—	2	3	no	—
11	C₇×D₁₂₁	—	7	1	no	—
12	C₇×D₁₂₁	D₇	1	1	yes	characteristic, direct product factor
13	C₇×D₁₂₁	C₂×C₇	1	1	yes	characteristic, complemented
14	D₇×C₁₂₁	C₂×C₇	1	1	yes	characteristic, complemented
15	C₇×D₇₇	—	11	1	no	—
16	C₇×C₁₁×D₇	D₁₁	1	1	yes	characteristic
17	C₇²×D₁₁	—	11	1	no	—
18	C₇×C₁₂₁	—	2	3	no	—
19	C₇×C₁₂₁	C₂×D₇	1	1	yes	characteristic, complemented
20	C₇×C₁₂₁	C₂²×C₇	1	1	yes	characteristic, complemented
21	C₇²×C₁₁	C₂×D₁₁	1	1	yes	characteristic
22	C₂×D₁₂₁	—	7	1	no	Hall
23	C₂×C₇×D₁₁	—	77	1	no	—
24	D₇×D₁₁	—	11	1	no	—
25	C₂×C₁₂₁	—	7	1	no	—
26	D₁₂₁	—	7	1	no	—
27	D₁₂₁	C₇×D₇	1	1	yes	characteristic, direct product factor
28	C₂×C₇×D₇	—	121	1	no	Hall
29	C₂×C₇×C₁₁	—	7	1	no	—
30	D₇₇	—	11	1	no	—
31	C₇×D₁₁	—	11	1	no	—
32	C₇×D₁₁	—	11	1	no	—
33	C₇×D₁₁	—	22	3	no	—
34	C₇×D₁₁	—	77	1	no	—
35	C₁₁×D₇	C₇×D₁₁	1	1	yes	characteristic
36	C₁₂₁	C₂×C₇×D₇	1	1	yes	characteristic, complemented, Sylow
37	C₂×C₇²	—	121	1	no	—
38	C₇×D₇	—	121	1	no	—
39	C₇×D₇	D₁₂₁	1	1	yes	characteristic, direct product factor
40	C₇×C₁₁	—	2	3	no	—
41	C₇×C₁₁	D₇×D₁₁	1	1	yes	characteristic
42	C₇×C₁₁	C₂×C₇×D₁₁	1	1	yes	characteristic
43	C₇²	C₂×D₁₂₁	1	1	yes	characteristic, complemented, Sylow
44	C₂×D₁₁	—	77	1	no	—
45	C₂²×C₇	—	847	1	no	—
46	C₂×D₇	—	121	1	no	—
47	C₂×C₁₁	—	7	1	no	—
48	D₁₁	—	11	1	no	—
49	D₁₁	—	77	1	no	—
50	C₂×C₇	—	7	1	no	—
51	C₂×C₇	—	847	1	no	—
52	C₂×C₇	—	121	1	no	—
53	C₂×C₇	—	121	1	no	—
54	C₂×C₇	—	242	3	no	—
55	D₇	—	121	1	no	—
56	D₇	C₇×D₁₂₁	1	1	yes	characteristic, direct product factor
57	C₁₁	C₇×D₇×D₁₁	1	1	yes	characteristic
58	C₇	—	2	3	no	—
59	C₇	D₇×D₁₂₁	1	1	yes	characteristic, direct product factor
60	C₇	C₂×C₇×D₁₂₁	1	1	yes	characteristic, complemented
61	C₂²	—	847	1	no	Sylow
62	C₂	—	7	1	no	—
63	C₂	—	847	1	no	—
64	C₂	—	121	1	no	—

	subgroup	G/H
center, hypercenter	C₇	D₇×D₁₂₁
commutator, cosocle, hypocenter	C₇×C₁₂₁	C₂²×C₇
Frattini	C₁₁	C₇×D₇×D₁₁
Fitting	C₇²×C₁₂₁	C₂²
socle	C₇²×C₁₁	C₂×D₁₁
LCS	G ▷ C₇×C₁₂₁
UCS	C₁ ◁ C₇

Complements

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

N	H ≅ G/N	type
1. C₇×D₈₄₇	C₂	semidirect
2. C₇×D₇×C₁₂₁	C₂	semidirect
3. C₇²×D₁₂₁	C₂	semidirect
4. C₇²×C₁₂₁	C₂²	semidirect
6. D₇×D₁₂₁	C₇	direct (N × H)
9. D₈₄₇	C₂×C₇	semidirect
12. C₇×D₁₂₁	D₇	direct (N × H)
13. C₇×D₁₂₁	C₂×C₇	semidirect
14. D₇×C₁₂₁	C₂×C₇	semidirect
19. C₇×C₁₂₁	C₂×D₇	semidirect
20. C₇×C₁₂₁	C₂²×C₇	semidirect
27. D₁₂₁	C₇×D₇	direct (N × H)
36. C₁₂₁	C₂×C₇×D₇	semidirect
39. C₇×D₇	D₁₂₁	direct (N × H)
43. C₇²	C₂×D₁₂₁	semidirect
56. D₇	C₇×D₁₂₁	direct (N × H)
59. C₇	D₇×D₁₂₁	direct (N × H)
60. C₇	C₂×C₇×D₁₂₁	semidirect

Maximal subgroups

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

subgroup	index
C₇×D₈₄₇	2
C₇×D₇×C₁₂₁	2
C₇²×D₁₂₁	2
C₂×C₇×D₁₂₁	7
D₇×D₁₂₁	7
C₇×D₇×D₁₁	11

Subgroups — C7×D7×D121 #37557360

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×D₇×D₁₂₁ #37557360