Subgroups — C₃×D₇×D₃₂₁ #41004130

Order 26964.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×D₇×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₇×D₃₂₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×D₂₂₄₇	C₂	1	1	yes	maximal, characteristic, complemented
4	D₇×D₃₂₁	C₃	1	1	yes	maximal, characteristic, direct product factor
5	C₃×D₇×D₁₀₇	—	3	1	no	maximal
6	C₃²×C₇×C₁₀₇	C₂²	1	1	yes	characteristic, complemented, Hall
7	D₂₂₄₇	C₂×C₃	1	1	yes	characteristic, complemented
8	C₇×D₃₂₁	C₂×C₃	1	1	yes	characteristic, complemented
9	C₃×D₇₄₉	—	3	1	no	—
10	C₃×D₇×C₁₀₇	—	2	1	no	—
11	C₃×D₇×C₁₀₇	D₃	1	1	yes	characteristic, complemented
12	C₃×D₇×C₁₀₇	C₂×C₃	1	1	yes	characteristic, complemented
13	C₃×C₇×D₁₀₇	—	3	1	no	—
14	C₂×C₃×D₃₂₁	—	7	1	no	maximal, Hall
15	D₇×D₁₀₇	—	3	1	no	Hall
16	C₃×C₇×C₁₀₇	—	2	1	no	—
17	C₃×C₇×C₁₀₇	C₂×D₃	1	1	yes	characteristic, complemented
18	C₃×C₇×C₁₀₇	C₂²×C₃	1	1	yes	characteristic, complemented
19	C₂×C₃²×C₁₀₇	—	7	1	no	—
20	C₃×D₃₂₁	—	7	1	no	—
21	C₃×D₃₂₁	D₇	1	1	yes	characteristic, direct product factor
22	D₇₄₉	—	3	1	no	—
23	C₇×D₁₀₇	—	3	1	no	—
24	D₇×C₁₀₇	C₃×D₃	1	1	yes	characteristic, complemented
25	C₂×D₃₂₁	—	7	1	no	—
26	C₂×C₃×D₁₀₇	—	21	1	no	—
27	C₃²×C₁₀₇	C₂×D₇	1	1	yes	characteristic, complemented, Hall
28	C₇×C₁₀₇	C₂×C₃×D₃	1	1	yes	characteristic, complemented, Hall
29	C₂×C₃×C₁₀₇	—	7	1	no	—
30	C₂×C₃×C₁₀₇	—	7	1	no	—
31	C₂×C₃×C₁₀₇	—	14	1	no	—
32	D₃₂₁	—	7	1	no	—
33	D₃₂₁	C₃×D₇	1	1	yes	characteristic, direct product factor
34	C₃×D₁₀₇	—	3	1	no	—
35	C₃×D₁₀₇	—	21	1	no	—
36	C₂×D₁₀₇	—	21	1	no	Hall
37	C₃×C₁₀₇	—	2	1	no	—
38	C₃×C₁₀₇	D₃×D₇	1	1	yes	characteristic, complemented
39	C₃×C₁₀₇	C₂×C₃×D₇	1	1	yes	characteristic, complemented
40	C₃×D₃×D₇	—	107	1	no	maximal, Hall
41	C₂×C₁₀₇	—	7	1	no	—
42	D₁₀₇	—	3	1	no	—
43	D₁₀₇	—	21	1	no	—
44	C₃×D₂₁	—	107	1	no	—
45	C₃×D₃×C₇	—	107	1	no	—
46	C₃²×D₇	D₁₀₇	1	1	yes	characteristic, complemented
47	C₁₀₇	C₃×D₃×D₇	1	1	yes	characteristic, complemented, Sylow
48	C₂×C₃×D₇	—	321	1	no	—
49	D₃×D₇	—	107	1	no	—
50	C₃²×C₇	C₂×D₁₀₇	1	1	yes	characteristic, complemented, Hall
51	C₂×C₃×C₇	—	321	1	no	—
52	D₂₁	—	107	1	no	—
53	C₃×D₇	—	2	1	no	—
54	C₃×D₇	—	321	1	no	—
55	C₃×D₇	C₃×D₁₀₇	1	1	yes	characteristic, complemented
56	C₃×D₇	D₃₂₁	1	1	yes	characteristic, direct product factor
57	D₃×C₇	—	107	1	no	—
58	C₂×C₃×D₃	—	749	1	no	Hall
59	C₂×D₇	—	321	1	no	Hall
60	C₃×C₇	—	2	1	no	—
61	C₃×C₇	C₂×C₃×D₁₀₇	1	1	yes	characteristic, complemented
62	C₃×C₇	C₂×D₃₂₁	1	1	yes	characteristic, complemented
63	C₂×C₃²	—	7	1	no	—
64	C₃×D₃	—	107	1	no	—
65	C₃×D₃	—	749	1	no	—
66	C₂×C₇	—	321	1	no	—
67	D₇	—	321	1	no	—
68	D₇	C₃×D₃₂₁	1	1	yes	characteristic, direct product factor
69	C₂²×C₃	—	2247	1	no	—
70	C₂×D₃	—	749	1	no	—
71	C₃²	D₇×D₁₀₇	1	1	yes	characteristic, complemented, Sylow
72	C₇	C₂×C₃×D₃₂₁	1	1	yes	characteristic, complemented, Sylow
73	C₂×C₃	—	7	1	no	—
74	C₂×C₃	—	7	1	no	—
75	C₂×C₃	—	14	1	no	—
76	C₂×C₃	—	321	1	no	—
77	C₂×C₃	—	2247	1	no	—
78	D₃	—	107	1	no	—
79	D₃	—	749	1	no	—
80	C₂²	—	2247	1	no	Sylow
81	C₃	—	2	1	no	—
82	C₃	C₃×D₇×D₁₀₇	1	1	yes	characteristic, complemented
83	C₃	D₇×D₃₂₁	1	1	yes	characteristic, direct product factor
84	C₂	—	7	1	no	—
85	C₂	—	321	1	no	—
86	C₂	—	2247	1	no	—

	subgroup	G/H
center, hypercenter	C₃	D₇×D₃₂₁
commutator, cosocle, hypocenter	C₃×C₇×C₁₀₇	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₃²×C₇×C₁₀₇	C₂²
LCS	G ▷ C₃×C₇×C₁₀₇
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
1. C₃²×D₇×C₁₀₇	C₂	semidirect
2. C₃×C₇×D₃₂₁	C₂	semidirect
3. C₃×D₂₂₄₇	C₂	semidirect
4. D₇×D₃₂₁	C₃	direct (N × H)
6. C₃²×C₇×C₁₀₇	C₂²	semidirect
7. D₂₂₄₇	C₂×C₃	semidirect
8. C₇×D₃₂₁	C₂×C₃	semidirect
11. C₃×D₇×C₁₀₇	D₃	semidirect
12. C₃×D₇×C₁₀₇	C₂×C₃	semidirect
17. C₃×C₇×C₁₀₇	C₂×D₃	semidirect
18. C₃×C₇×C₁₀₇	C₂²×C₃	semidirect
21. C₃×D₃₂₁	D₇	direct (N × H)
24. D₇×C₁₀₇	C₃×D₃	semidirect
27. C₃²×C₁₀₇	C₂×D₇	semidirect
28. C₇×C₁₀₇	C₂×C₃×D₃	semidirect
33. D₃₂₁	C₃×D₇	direct (N × H)
38. C₃×C₁₀₇	D₃×D₇	semidirect
39. C₃×C₁₀₇	C₂×C₃×D₇	semidirect
46. C₃²×D₇	D₁₀₇	semidirect
47. C₁₀₇	C₃×D₃×D₇	semidirect
50. C₃²×C₇	C₂×D₁₀₇	semidirect
55. C₃×D₇	C₃×D₁₀₇	semidirect
56. C₃×D₇	D₃₂₁	direct (N × H)
61. C₃×C₇	C₂×C₃×D₁₀₇	semidirect
62. C₃×C₇	C₂×D₃₂₁	semidirect
68. D₇	C₃×D₃₂₁	direct (N × H)
71. C₃²	D₇×D₁₀₇	semidirect
72. C₇	C₂×C₃×D₃₂₁	semidirect
82. C₃	C₃×D₇×D₁₀₇	semidirect
83. C₃	D₇×D₃₂₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×D₇×C₁₀₇	2
C₃×C₇×D₃₂₁	2
C₃×D₂₂₄₇	2
D₇×D₃₂₁	3
C₃×D₇×D₁₀₇	3
C₂×C₃×D₃₂₁	7
C₃×D₃×D₇	107

Subgroups — C3×D7×D321 #41004130

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₇×D₃₂₁ #41004130