Subgroups — D₇×#6891 #72249103

Order 3402.

504 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#6891	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₃⁴×D₇	C₃	1	1	yes	maximal, characteristic
3	C₃×D₇×3₋¹⁺²	C₃	1	3	yes	maximal, complemented
4	D₇×C₃²⋊C₉	C₃	1	9	yes	maximal, complemented
5	C₃⁴×C₇	C₂×C₃	1	1	yes	characteristic
6	C₃×C₇×3₋¹⁺²	C₂×C₃	1	3	yes	complemented
7	C₇×C₃²⋊C₉	C₂×C₃	1	9	yes	complemented
8	C₂×#6891	—	7	1	no	maximal, Hall
9	C₃³×D₇	—	3	3	no	—
10	C₃³×D₇	—	3	9	no	—
11	C₃³×D₇	C₃²	1	1	yes	characteristic
12	C₃³×D₇	C₃²	1	3	yes	—
13	D₇×3₋¹⁺²	—	3	9	no	—
14	C₃×C₉×D₇	C₃²	1	9	yes	complemented
15	#6891	D₇	1	1	yes	characteristic, direct product factor, Sylow
16	C₃³×C₇	—	3	3	no	—
17	C₃³×C₇	—	3	9	no	—
18	C₃³×C₇	C₂×C₃²	1	1	yes	characteristic
19	C₃³×C₇	C₂×C₃²	1	3	yes	—
20	C₇×3₋¹⁺²	—	3	9	no	—
21	C₃×C₇×C₉	C₂×C₃²	1	9	yes	complemented
22	C₂×C₃⁴	—	7	1	no	—
23	C₂×C₃×3₋¹⁺²	—	7	3	no	—
24	C₂×C₃²⋊C₉	—	7	9	no	—
25	C₃²×D₇	—	3	27	no	—
26	C₃²×D₇	—	3	3	no	—
27	C₃²×D₇	—	3	3	no	—
28	C₃²×D₇	—	3	6	no	—
29	C₃²×D₇	H(3,𝔽₃)	1	3	yes	—
30	C₃²×D₇	3₋¹⁺²	1	9	yes	complemented
31	C₃²×D₇	C₃³	1	1	yes	characteristic
32	C₉×D₇	—	3	9	no	—
33	C₃⁴	C₃×D₇	1	1	yes	characteristic
34	C₃×3₋¹⁺²	C₃×D₇	1	3	yes	complemented
35	C₃²⋊C₉	C₃×D₇	1	9	yes	complemented
36	C₃²×C₇	—	3	27	no	—
37	C₃²×C₇	—	3	3	no	—
38	C₃²×C₇	—	3	3	no	—
39	C₃²×C₇	—	3	6	no	—
40	C₃²×C₇	C₂×H(3,𝔽₃)	1	3	yes	—
41	C₃²×C₇	C₂×3₋¹⁺²	1	9	yes	complemented
42	C₃²×C₇	C₂×C₃³	1	1	yes	characteristic
43	C₇×C₉	—	3	9	no	—
44	C₂×C₃³	—	7	3	no	—
45	C₂×C₃³	—	7	1	no	—
46	C₂×C₃³	—	21	3	no	—
47	C₂×C₃³	—	21	9	no	—
48	C₂×3₋¹⁺²	—	21	9	no	—
49	C₂×C₃×C₉	—	7	9	no	—
50	C₃×D₇	—	3	3	no	—
51	C₃×D₇	—	3	9	no	—
52	C₃×D₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
53	C₃×D₇	C₃×3₋¹⁺²	1	3	yes	—
54	C₃³	—	3	3	no	—
55	C₃³	—	3	9	no	—
56	C₃³	C₃²×D₇	1	1	yes	characteristic
57	C₃³	C₃²×D₇	1	3	yes	—
58	3₋¹⁺²	—	3	9	no	—
59	C₃×C₉	C₃²×D₇	1	9	yes	complemented
60	C₃×C₇	—	3	3	no	—
61	C₃×C₇	—	3	9	no	—
62	C₃×C₇	C₂×C₃×H(3,𝔽₃)	1	1	yes	characteristic
63	C₃×C₇	C₂×C₃×3₋¹⁺²	1	3	yes	—
64	C₂×C₃²	—	7	1	no	—
65	C₂×C₃²	—	7	3	no	—
66	C₂×C₃²	—	7	9	no	—
67	C₂×C₃²	—	21	3	no	—
68	C₂×C₃²	—	21	6	no	—
69	C₂×C₃²	—	21	3	no	—
70	C₂×C₃²	—	21	27	no	—
71	C₂×C₉	—	21	9	no	—
72	D₇	#6891	1	1	yes	characteristic, direct product factor, Hall
73	C₃²	—	3	27	no	—
74	C₃²	—	3	3	no	—
75	C₃²	—	3	3	no	—
76	C₃²	—	3	6	no	—
77	C₃²	D₇×H(3,𝔽₃)	1	3	yes	—
78	C₃²	D₇×3₋¹⁺²	1	9	yes	complemented
79	C₃²	C₃³×D₇	1	1	yes	characteristic
80	C₉	—	3	9	no	—
81	C₇	C₂×#6891	1	1	yes	characteristic, complemented, Sylow
82	C₂×C₃	—	7	1	no	—
83	C₂×C₃	—	7	3	no	—
84	C₂×C₃	—	21	3	no	—
85	C₂×C₃	—	21	9	no	—
86	C₃	—	3	3	no	—
87	C₃	—	3	9	no	—
88	C₃	C₃×D₇×H(3,𝔽₃)	1	1	yes	characteristic
89	C₃	C₃×D₇×3₋¹⁺²	1	3	yes	—
90	C₂	—	7	1	no	Sylow

	subgroup	G/H
center, Frattini	C₃²	C₃³×D₇
commutator, socle, cosocle	C₃²×C₇	C₂×C₃³
Fitting	C₇×#6891	C₂
prime radical	C₃⁴×D₇	C₃
hypercenter	#6891	D₇
hypocenter	C₇	C₂×#6891
LCS	G ▷ C₃²×C₇ ▷ C₇
UCS	C₁ ◁ C₃² ◁ #6891

Complements

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

N	H ≅ G/N	type
1. C₇×#6891	C₂	semidirect
3. C₃×D₇×3₋¹⁺²	C₃	semidirect
4. D₇×C₃²⋊C₉	C₃	semidirect
6. C₃×C₇×3₋¹⁺²	C₂×C₃	semidirect
7. C₇×C₃²⋊C₉	C₂×C₃	semidirect
14. C₃×C₉×D₇	C₃²	semidirect
15. #6891	D₇	direct (N × H)
21. C₃×C₇×C₉	C₂×C₃²	semidirect
30. C₃²×D₇	3₋¹⁺²	semidirect
34. C₃×3₋¹⁺²	C₃×D₇	semidirect
35. C₃²⋊C₉	C₃×D₇	semidirect
41. C₃²×C₇	C₂×3₋¹⁺²	semidirect
59. C₃×C₉	C₃²×D₇	semidirect
72. D₇	#6891	direct (N × H)
78. C₃²	D₇×3₋¹⁺²	semidirect
81. C₇	C₂×#6891	semidirect

Maximal subgroups

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

subgroup	index
C₇×#6891	2
C₃⁴×D₇	3
C₃×D₇×3₋¹⁺²	3
D₇×C₃²⋊C₉	3
C₂×#6891	7

Subgroups — D7×#6891 #72249103

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₇×#6891 #72249103