Subgroups — C₃⋊#20301294 #36348016

Order 22050.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅²⋊#3862	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₇×C₃⋊(C₅⋊D₃₅)	C₃	1	1	yes	maximal, characteristic, complemented
3	#20301294	—	3	1	no	maximal
4	C₅²⋊#3862	—	2	1	no	—
5	C₅²⋊#3862	D₃	1	1	yes	characteristic, complemented
6	C₃×C₅²×C₇²	C₂×C₃	1	1	yes	characteristic, complemented
7	C₃⋊((C₅⋊D₅)⋊ΓL(1,𝔽₈))	—	7	1	no	maximal
8	C₃⋊#11760386	—	7	1	no	maximal
9	C₇×C₅⋊D₃₅	—	3	1	no	Hall
10	C₃×C₇⋊(C₅²⋊C₃)	—	7	1	no	—
11	C₃×C₇⋊(C₅²⋊C₃)	—	7	1	no	—
12	C₇×D₁₀₅	—	15	2	no	—
13	C₅²×C₇²	C₃×D₃	1	1	yes	characteristic, complemented, Hall
14	C₃⋊(C₅⋊D₃₅)	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
15	C₇×C₃⋊(C₅⋊D₅)	—	7	1	no	—
16	#11760386	—	21	1	no	—
17	(C₅⋊D₅)⋊ΓL(1,𝔽₈)	—	21	1	no	—
18	C₃⋊#64422	—	25	1	no	maximal, Hall
19	C₃×C₅×C₇²	—	3	2	no	—
20	C₃×C₅²×C₇	—	6	1	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	1	no	—
24	C₇⋊(C₅²⋊C₃)	—	7	1	no	—
25	C₇⋊(C₅²⋊C₃)	—	14	1	no	—
26	C₇⋊(C₅²⋊C₃)	—	14	1	no	—
27	C₇×D₃₅	—	45	2	no	—
28	C₃⋊((C₅⋊D₅)⋊C₃)	—	49	1	no	Hall
29	C₃×#3862	—	25	1	no	Hall
30	C₅⋊D₃₅	—	3	1	no	—
31	C₇×C₅⋊D₅	—	21	1	no	—
32	C₇×D₂₁	—	25	1	no	—
33	#64422	—	75	1	no	—
34	C₅×C₇²	—	3	2	no	—
35	C₃×C₅²⋊C₃	—	49	1	no	Hall
36	D₁₀₅	—	15	2	no	—
37	C₇×D₁₅	—	105	2	no	—
38	C₅²×C₇	—	6	1	no	—
39	C₅²×C₇	D₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
40	C₅²×C₇	C₃⋊AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
41	C₃⋊(C₅⋊D₅)	—	7	1	no	—
42	(C₅⋊D₅)⋊C₃	—	147	1	no	—
43	C₃×C₇²	(C₅⋊D₅)⋊C₃	1	1	yes	characteristic, complemented
44	#3862	—	25	1	no	—
45	#3862	—	50	1	no	—
46	C₃⋊AGL(1,𝔽₇)	—	175	1	no	—
47	D₃×ΓL(1,𝔽₈)	—	175	1	no	—
48	C₃×C₅×C₇	—	3	2	no	—
49	C₃×C₅×C₇	—	3	2	no	—
50	C₃×C₅×C₇	—	6	6	no	—
51	C₇×D₇	—	75	1	no	Hall
52	C₃×C₅²	#64422	1	1	yes	characteristic, complemented
53	C₅²⋊C₃	—	49	1	no	—
54	C₅²⋊C₃	—	98	1	no	—
55	D₃₅	—	45	2	no	—
56	C₇×D₅	—	315	2	no	—
57	C₃×ΓL(1,𝔽₈)	—	175	1	no	—
58	C₃×ΓL(1,𝔽₈)	—	175	1	no	—
59	C₅⋊D₅	—	21	1	no	Hall
60	C₇²	C₃⋊((C₅⋊D₅)⋊C₃)	1	1	yes	characteristic, complemented, Sylow
61	D₂₁	—	25	1	no	—
62	D₃×C₇	—	175	1	no	—
63	C₂×ΓL(1,𝔽₈)	—	525	1	no	—
64	AGL(1,𝔽₇)	—	525	1	no	—
65	C₅×C₇	—	3	2	no	—
66	C₅×C₇	—	3	2	no	—
67	C₅×C₇	—	6	6	no	—
68	D₁₅	—	105	2	no	—
69	C₅²	C₃⋊#64422	1	1	yes	characteristic, complemented, Sylow
70	C₃×C₇	—	6	1	no	—
71	C₃×C₇	(C₅⋊D₅)⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
72	C₃×C₇	#11760386	1	1	yes	characteristic, complemented
73	ΓL(1,𝔽₈)	—	350	1	no	—
74	ΓL(1,𝔽₈)	—	350	1	no	—
75	ΓL(1,𝔽₈)	—	175	1	no	—
76	ΓL(1,𝔽₈)	—	175	1	no	—
77	C₃×D₃	—	1225	1	no	Hall
78	C₃×C₅	—	3	2	no	—
79	C₂×C₇	—	525	1	no	—
80	D₇	—	75	1	no	—
81	D₅	—	315	2	no	—
82	C₃²	—	1225	1	no	Sylow
83	C₇	—	6	1	no	—
84	C₇	C₃⋊#11760386	1	1	yes	characteristic, complemented
85	C₇	C₃⋊((C₅⋊D₅)⋊ΓL(1,𝔽₈))	1	1	yes	characteristic, complemented
86	C₂×C₃	—	3675	1	no	—
87	D₃	—	175	1	no	—
88	C₅	—	3	2	no	—
89	C₃	—	2450	1	no	—
90	C₃	—	1225	1	no	—
91	C₃	#20301294	1	1	yes	characteristic, complemented
92	C₂	—	525	1	no	Sylow

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

Complements

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

N	H ≅ G/N	type
1. C₃×C₅²⋊#3862	C₂	semidirect
2. C₇×C₃⋊(C₅⋊D₃₅)	C₃	semidirect
5. C₅²⋊#3862	D₃	semidirect
6. C₃×C₅²×C₇²	C₂×C₃	semidirect
13. C₅²×C₇²	C₃×D₃	semidirect
14. C₃⋊(C₅⋊D₃₅)	ΓL(1,𝔽₈)	semidirect
21. C₃×C₅²×C₇	AGL(1,𝔽₇)	semidirect
22. C₃×C₅²×C₇	C₂×ΓL(1,𝔽₈)	semidirect
39. C₅²×C₇	D₃×ΓL(1,𝔽₈)	semidirect
40. C₅²×C₇	C₃⋊AGL(1,𝔽₇)	semidirect
43. C₃×C₇²	(C₅⋊D₅)⋊C₃	semidirect
52. C₃×C₅²	#64422	semidirect
60. C₇²	C₃⋊((C₅⋊D₅)⋊C₃)	semidirect
69. C₅²	C₃⋊#64422	semidirect
71. C₃×C₇	(C₅⋊D₅)⋊ΓL(1,𝔽₈)	semidirect
72. C₃×C₇	#11760386	semidirect
84. C₇	C₃⋊#11760386	semidirect
85. C₇	C₃⋊((C₅⋊D₅)⋊ΓL(1,𝔽₈))	semidirect
91. C₃	#20301294	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅²⋊#3862	2
C₇×C₃⋊(C₅⋊D₃₅)	3
#20301294	3
C₃⋊((C₅⋊D₅)⋊ΓL(1,𝔽₈))	7
C₃⋊#11760386	7
C₃⋊#64422	25

Subgroups — C3⋊#20301294 #36348016

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃⋊#20301294 #36348016