Subgroups — C₃×#20301304 #36347943

Order 22050.

176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅²⋊#3861	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₃×C₅²×C₇×D₇	C₃	1	1	yes	maximal, characteristic, complemented
3	#20301304	C₃	1	3	yes	maximal, direct product factor
4	C₅²⋊#3861	C₂×C₃	1	3	yes	complemented
5	C₃×C₅²×C₇²	C₂×C₃	1	1	yes	characteristic, complemented
6	C₂×C₃×C₇⋊(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)	—	7	1	no	—
10	C₃×C₇⋊(C₅²⋊C₃)	—	7	1	no	—
11	C₃×C₇⋊(C₅²⋊C₃)	—	14	3	no	—
12	C₃×C₅×C₇×D₇	—	3	2	no	—
13	C₅²×C₇²	C₂×C₃²	1	1	yes	characteristic, complemented, Hall
14	C₂×C₃×C₅²×C₇	—	7	1	no	—
15	C₃×C₅²×D₇	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
16	C₂×C₇⋊(C₅²⋊C₃)	—	7	3	no	—
17	D₇⋊(C₅²⋊C₃)	—	7	3	no	—
18	C₃×#64423	—	25	1	no	maximal, Hall
19	C₃×C₅×C₇²	—	3	2	no	—
20	C₃×C₅²×C₇	—	2	3	no	—
21	C₃×C₅²×C₇	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
22	C₃×C₅²×C₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
23	C₇⋊(C₅²⋊C₃)	—	7	3	no	—
24	C₇⋊(C₅²⋊C₃)	—	7	3	no	—
25	C₇⋊(C₅²⋊C₃)	—	14	9	no	—
26	C₅×C₇×D₇	—	3	2	no	—
27	C₂×C₃×C₅²⋊C₃	—	49	1	no	Hall
28	C₃×#3861	—	25	1	no	Hall
29	C₂×C₅²×C₇	—	7	1	no	—
30	C₅²×D₇	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
31	C₃×C₇×D₇	C₅²⋊C₃	1	1	yes	characteristic, complemented
32	#64423	—	25	3	no	—
33	C₅×C₇²	—	3	2	no	—
34	C₃×C₅²⋊C₃	—	49	1	no	Hall
35	C₂×C₃×C₅×C₇	—	21	2	no	—
36	C₃×C₅×D₇	—	3	2	no	—
37	C₅²×C₇	—	2	3	no	—
38	C₅²×C₇	C₃×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
39	C₅²×C₇	C₂×C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
40	C₂×C₃×C₅²	—	21	1	no	—
41	C₂×C₅²⋊C₃	—	49	3	no	—
42	C₃×C₇²	C₂×C₅²⋊C₃	1	1	yes	characteristic, complemented
43	#3861	—	25	3	no	—
44	C₂×C₃×ΓL(1,𝔽₈)	—	175	1	no	—
45	C₃×AGL(1,𝔽₇)	—	175	1	no	—
46	C₃×C₅×C₇	—	3	2	no	—
47	C₃×C₅×C₇	—	3	2	no	—
48	C₃×C₅×C₇	—	6	6	no	—
49	C₇×D₇	C₃×C₅²⋊C₃	1	1	yes	characteristic, complemented, Hall
50	C₃×C₅²	#64423	1	1	yes	characteristic, complemented
51	C₅²⋊C₃	—	49	3	no	—
52	C₂×C₅×C₇	—	21	2	no	—
53	C₅×D₇	—	3	2	no	—
54	C₃×ΓL(1,𝔽₈)	—	350	3	no	—
55	C₃×ΓL(1,𝔽₈)	—	175	1	no	—
56	C₃×ΓL(1,𝔽₈)	—	175	1	no	—
57	C₂×C₅²	—	21	1	no	Hall
58	C₇²	C₂×C₃×C₅²⋊C₃	1	1	yes	characteristic, complemented, Sylow
59	C₂×C₃×C₇	—	21	1	no	—
60	C₃×D₇	C₇⋊(C₅²⋊C₃)	1	1	yes	characteristic, complemented
61	C₂×ΓL(1,𝔽₈)	—	175	3	no	—
62	AGL(1,𝔽₇)	—	175	3	no	—
63	C₅×C₇	—	3	2	no	—
64	C₅×C₇	—	3	2	no	—
65	C₅×C₇	—	6	6	no	—
66	C₂×C₃×C₅	—	21	2	no	—
67	C₅²	C₃×#64423	1	1	yes	characteristic, complemented, Sylow
68	C₃×C₇	—	2	3	no	—
69	C₃×C₇	D₇⋊(C₅²⋊C₃)	1	1	yes	characteristic, complemented
70	C₃×C₇	C₂×C₇⋊(C₅²⋊C₃)	1	1	yes	characteristic, complemented
71	ΓL(1,𝔽₈)	—	350	9	no	—
72	ΓL(1,𝔽₈)	—	175	3	no	—
73	ΓL(1,𝔽₈)	—	175	3	no	—
74	C₂×C₃²	—	1225	1	no	Hall
75	C₃×C₅	—	3	2	no	—
76	C₂×C₇	—	21	1	no	—
77	D₇	C₃×C₇⋊(C₅²⋊C₃)	1	1	yes	characteristic, complemented
78	C₂×C₅	—	21	2	no	—
79	C₃²	—	1225	1	no	Sylow
80	C₇	—	2	3	no	—
81	C₇	C₃×D₇⋊(C₅²⋊C₃)	1	1	yes	characteristic, complemented
82	C₇	C₂×C₃×C₇⋊(C₅²⋊C₃)	1	1	yes	characteristic, complemented
83	C₂×C₃	—	21	1	no	—
84	C₂×C₃	—	1225	3	no	—
85	C₅	—	3	2	no	—
86	C₃	—	1225	3	no	—
87	C₃	#20301304	1	1	yes	characteristic, direct product factor
88	C₂	—	21	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃	#20301304
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×C₅²⋊#3861	C₂	semidirect
2. C₃×C₅²×C₇×D₇	C₃	semidirect
3. #20301304	C₃	direct (N × H)
4. C₅²⋊#3861	C₂×C₃	semidirect
5. C₃×C₅²×C₇²	C₂×C₃	semidirect
8. C₅²×C₇×D₇	C₃²	semidirect
13. C₅²×C₇²	C₂×C₃²	semidirect
15. C₃×C₅²×D₇	ΓL(1,𝔽₈)	semidirect
21. C₃×C₅²×C₇	AGL(1,𝔽₇)	semidirect
22. C₃×C₅²×C₇	C₂×ΓL(1,𝔽₈)	semidirect
30. C₅²×D₇	C₃×ΓL(1,𝔽₈)	semidirect
31. C₃×C₇×D₇	C₅²⋊C₃	semidirect
38. C₅²×C₇	C₃×AGL(1,𝔽₇)	semidirect
39. C₅²×C₇	C₂×C₃×ΓL(1,𝔽₈)	semidirect
42. C₃×C₇²	C₂×C₅²⋊C₃	semidirect
49. C₇×D₇	C₃×C₅²⋊C₃	semidirect
50. C₃×C₅²	#64423	semidirect
58. C₇²	C₂×C₃×C₅²⋊C₃	semidirect
60. C₃×D₇	C₇⋊(C₅²⋊C₃)	semidirect
67. C₅²	C₃×#64423	semidirect
69. C₃×C₇	D₇⋊(C₅²⋊C₃)	semidirect
70. C₃×C₇	C₂×C₇⋊(C₅²⋊C₃)	semidirect
77. D₇	C₃×C₇⋊(C₅²⋊C₃)	semidirect
81. C₇	C₃×D₇⋊(C₅²⋊C₃)	semidirect
82. C₇	C₂×C₃×C₇⋊(C₅²⋊C₃)	semidirect
87. C₃	#20301304	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×C₅²⋊#3861	2
C₃×C₅²×C₇×D₇	3
#20301304	3
C₂×C₃×C₇⋊(C₅²⋊C₃)	7
C₃×D₇⋊(C₅²⋊C₃)	7
C₃×#64423	25

Subgroups — C3×#20301304 #36347943

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#20301304 #36347943