Subgroups — C₃×#20301306 #36347939

Order 22050.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇×C₇⋊(C₅²⋊C₃)	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₃×C₅²×C₇⋊D₇	C₃	1	1	yes	maximal, characteristic, complemented
3	#20301306	C₃	1	3	yes	maximal, direct product factor
4	C₇×C₇⋊(C₅²⋊C₃)	C₂×C₃	1	3	yes	complemented
5	C₃×C₅²×C₇²	C₂×C₃	1	1	yes	characteristic, complemented
6	C₃×D₇×C₅²⋊C₃	—	7	1	no	maximal
7	C₃×D₇⋊(C₅²⋊C₃)	—	7	1	no	maximal
8	C₅²×C₇⋊D₇	C₃²	1	1	yes	characteristic, complemented, Hall
9	C₃×C₇⋊(C₅²⋊C₃)	D₇	1	1	yes	characteristic, complemented
10	C₃×C₇×C₅²⋊C₃	—	7	1	no	—
11	C₃×C₅×C₇⋊D₇	—	3	2	no	—
12	C₅²×C₇²	C₂×C₃²	1	1	yes	characteristic, complemented, Hall
13	C₃×C₅²×D₇	—	7	1	no	—
14	C₃×C₅²×D₇	—	21	2	no	—
15	C₃×C₅²×D₇	—	21	1	no	—
16	D₇⋊(C₅²⋊C₃)	—	7	3	no	—
17	D₇×C₅²⋊C₃	—	7	3	no	—
18	C₃×ΓL(1,𝔽₈)⋊D₇	—	25	1	no	maximal, Hall
19	C₃×C₅×C₇²	—	3	2	no	—
20	C₃×C₅²×C₇	—	3	2	no	—
21	C₃×C₅²×C₇	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
22	C₃×C₅²×C₇	C₃×D₇	1	1	yes	characteristic, complemented
23	C₇⋊(C₅²⋊C₃)	C₃×D₇	1	3	yes	complemented
24	C₇×C₅²⋊C₃	—	7	3	no	—
25	C₅×C₇⋊D₇	—	3	2	no	—
26	C₂×C₃×C₅²⋊C₃	—	49	1	no	Hall
27	C₃×C₇×ΓL(1,𝔽₈)	—	25	1	no	Hall
28	C₅²×D₇	—	7	1	no	—
29	C₅²×D₇	—	21	2	no	—
30	C₅²×D₇	—	21	1	no	—
31	C₃×C₇⋊D₇	C₅²⋊C₃	1	1	yes	characteristic, complemented
32	ΓL(1,𝔽₈)⋊D₇	—	25	3	no	—
33	C₅×C₇²	—	3	2	no	—
34	C₃×C₅²⋊C₃	—	7	1	no	Hall
35	C₃×C₅×D₇	—	21	2	no	—
36	C₃×C₅×D₇	—	21	12	no	—
37	C₃×C₅×D₇	—	21	2	no	—
38	C₅²×C₇	—	3	2	no	—
39	C₅²×C₇	C₃×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
40	C₅²×C₇	C₃²×D₇	1	1	yes	characteristic, complemented
41	C₂×C₃×C₅²	—	147	1	no	—
42	C₂×C₅²⋊C₃	—	49	3	no	—
43	C₃×C₇²	C₂×C₅²⋊C₃	1	1	yes	characteristic, complemented
44	C₇×ΓL(1,𝔽₈)	—	25	3	no	—
45	C₃²×D₇	—	175	1	no	—
46	C₃×AGL(1,𝔽₇)	—	175	1	no	—
47	C₃×C₅×C₇	—	3	12	no	—
48	C₃×C₅×C₇	—	3	2	no	—
49	C₃×C₅×C₇	—	3	2	no	—
50	C₇⋊D₇	C₃×C₅²⋊C₃	1	1	yes	characteristic, complemented, Hall
51	C₃×C₅²	ΓL(1,𝔽₈)⋊D₇	1	1	yes	characteristic, complemented
52	C₅²⋊C₃	—	7	3	no	—
53	C₅×D₇	—	21	2	no	—
54	C₅×D₇	—	21	12	no	—
55	C₅×D₇	—	21	2	no	—
56	C₃²×C₇	—	175	1	no	—
57	C₃×ΓL(1,𝔽₈)	—	25	1	no	—
58	C₂×C₅²	—	147	1	no	Hall
59	C₇²	C₂×C₃×C₅²⋊C₃	1	1	yes	characteristic, complemented, Sylow
60	C₃×D₇	—	7	1	no	—
61	C₃×D₇	—	21	2	no	—
62	C₃×D₇	—	21	1	no	—
63	C₃×D₇	—	175	3	no	—
64	AGL(1,𝔽₇)	—	175	3	no	—
65	C₅×C₇	—	3	12	no	—
66	C₅×C₇	—	3	2	no	—
67	C₅×C₇	—	3	2	no	—
68	C₂×C₃×C₅	—	147	2	no	—
69	C₅²	C₃×ΓL(1,𝔽₈)⋊D₇	1	1	yes	characteristic, complemented, Sylow
70	C₃×C₇	—	3	2	no	—
71	C₃×C₇	—	175	3	no	—
72	C₃×C₇	D₇×C₅²⋊C₃	1	1	yes	characteristic, complemented
73	C₃×C₇	D₇⋊(C₅²⋊C₃)	1	1	yes	characteristic, complemented
74	ΓL(1,𝔽₈)	—	25	3	no	—
75	C₂×C₃²	—	1225	1	no	Hall
76	C₃×C₅	—	3	2	no	—
77	D₇	—	7	1	no	—
78	D₇	—	21	2	no	—
79	D₇	—	21	1	no	—
80	C₂×C₅	—	147	2	no	—
81	C₃²	—	175	1	no	Sylow
82	C₇	—	3	2	no	—
83	C₇	C₃×D₇⋊(C₅²⋊C₃)	1	1	yes	characteristic, complemented
84	C₇	C₃×D₇×C₅²⋊C₃	1	1	yes	characteristic, complemented
85	C₂×C₃	—	147	1	no	—
86	C₂×C₃	—	1225	3	no	—
87	C₅	—	3	2	no	—
88	C₃	—	175	3	no	—
89	C₃	#20301306	1	1	yes	characteristic, direct product factor
90	C₂	—	147	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃	#20301306
commutator, cosocle, hypocenter	C₅²×C₇²	C₂×C₃²
Frattini	C₁	G
Fitting, socle	C₃×C₅²×C₇²	C₂×C₃
LCS	G ▷ C₅²×C₇²
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
1. C₃×C₇×C₇⋊(C₅²⋊C₃)	C₂	semidirect
2. C₃×C₅²×C₇⋊D₇	C₃	semidirect
3. #20301306	C₃	direct (N × H)
4. C₇×C₇⋊(C₅²⋊C₃)	C₂×C₃	semidirect
5. C₃×C₅²×C₇²	C₂×C₃	semidirect
8. C₅²×C₇⋊D₇	C₃²	semidirect
9. C₃×C₇⋊(C₅²⋊C₃)	D₇	semidirect
12. C₅²×C₇²	C₂×C₃²	semidirect
21. C₃×C₅²×C₇	AGL(1,𝔽₇)	semidirect
22. C₃×C₅²×C₇	C₃×D₇	semidirect
23. C₇⋊(C₅²⋊C₃)	C₃×D₇	semidirect
31. C₃×C₇⋊D₇	C₅²⋊C₃	semidirect
39. C₅²×C₇	C₃×AGL(1,𝔽₇)	semidirect
40. C₅²×C₇	C₃²×D₇	semidirect
43. C₃×C₇²	C₂×C₅²⋊C₃	semidirect
50. C₇⋊D₇	C₃×C₅²⋊C₃	semidirect
51. C₃×C₅²	ΓL(1,𝔽₈)⋊D₇	semidirect
59. C₇²	C₂×C₃×C₅²⋊C₃	semidirect
69. C₅²	C₃×ΓL(1,𝔽₈)⋊D₇	semidirect
72. C₃×C₇	D₇×C₅²⋊C₃	semidirect
73. C₃×C₇	D₇⋊(C₅²⋊C₃)	semidirect
83. C₇	C₃×D₇⋊(C₅²⋊C₃)	semidirect
84. C₇	C₃×D₇×C₅²⋊C₃	semidirect
89. C₃	#20301306	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₇×C₇⋊(C₅²⋊C₃)	2
C₃×C₅²×C₇⋊D₇	3
#20301306	3
C₃×D₇×C₅²⋊C₃	7
C₃×D₇⋊(C₅²⋊C₃)	7
C₃×ΓL(1,𝔽₈)⋊D₇	25

Subgroups — C3×#20301306 #36347939

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#20301306 #36347939