Subgroups — C₂×#20301298 #28413445

Order 14700.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×C₇⋊(C₅²⋊C₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	#20301298	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₇×C₅⋊D₃₅	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₇×C₇⋊(C₅²⋊C₃)	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₇×C₅⋊D₃₅	C₂×C₃	1	2	yes	complemented
6	C₂×C₅²×C₇²	C₂×C₃	1	1	yes	characteristic, complemented
7	C₂×#11760384	—	7	1	no	maximal
8	C₂×(C₅⋊D₅)⋊ΓL(1,𝔽₈)	—	7	1	no	maximal
9	C₅²×C₇²	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
10	C₂×C₇⋊(C₅²⋊C₃)	D₇	1	1	yes	characteristic, complemented
11	C₂×C₇×C₅²⋊C₃	—	7	1	no	—
12	(C₅⋊D₅)⋊ΓL(1,𝔽₈)	—	7	2	no	—
13	#11760384	—	7	2	no	—
14	C₂×C₇×D₃₅	—	15	2	no	—
15	C₂×C₅⋊D₃₅	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
16	C₂×C₇×C₅⋊D₅	—	7	1	no	—
17	C₂×D₇×ΓL(1,𝔽₈)	—	25	1	no	maximal, Hall
18	C₇⋊(C₅²⋊C₃)	C₂×D₇	1	1	yes	characteristic, complemented
19	C₇×C₅²⋊C₃	—	7	1	no	—
20	C₂×C₅×C₇²	—	3	2	no	—
21	C₇×D₃₅	—	15	4	no	—
22	C₂×C₅²×C₇	—	6	1	no	—
23	C₂×C₅²×C₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
24	C₂×C₅²×C₇	C₃×D₇	1	1	yes	characteristic, complemented
25	C₅⋊D₃₅	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
26	C₇×C₅⋊D₅	—	7	2	no	—
27	C₂×(C₅⋊D₅)⋊C₃	—	49	1	no	Hall
28	C₂×C₇×ΓL(1,𝔽₈)	—	25	1	no	—
29	D₇×ΓL(1,𝔽₈)	—	25	2	no	—
30	C₅×C₇²	—	3	2	no	—
31	C₂×C₇×D₇	—	25	1	no	Hall
32	C₅²×C₇	—	6	1	no	—
33	C₅²×C₇	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
34	C₅²×C₇	C₂×C₃×D₇	1	1	yes	characteristic, complemented
35	C₂×C₅²⋊C₃	—	7	1	no	—
36	(C₅⋊D₅)⋊C₃	—	49	2	no	—
37	C₇×ΓL(1,𝔽₈)	—	25	1	no	Hall
38	C₂×D₃₅	—	15	2	no	—
39	C₂×C₇×D₅	—	105	2	no	—
40	C₂×C₅⋊D₅	—	7	1	no	Hall
41	C₂×C₇²	(C₅⋊D₅)⋊C₃	1	1	yes	characteristic, complemented
42	C₇×D₇	—	25	2	no	—
43	C₂×C₃×D₇	—	175	1	no	—
44	C₂²×ΓL(1,𝔽₈)	—	175	1	no	—
45	C₅²⋊C₃	—	7	1	no	Hall
46	C₂×C₅×C₇	—	3	2	no	—
47	C₂×C₅×C₇	—	3	2	no	—
48	C₂×C₅×C₇	—	6	6	no	—
49	D₃₅	—	15	4	no	—
50	C₇×D₅	—	105	4	no	—
51	C₂×C₅²	D₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
52	C₅⋊D₅	—	7	2	no	—
53	C₇²	C₂×(C₅⋊D₅)⋊C₃	1	1	yes	characteristic, complemented, Sylow
54	C₂×C₃×C₇	—	175	1	no	—
55	C₃×D₇	—	175	2	no	—
56	C₂×ΓL(1,𝔽₈)	—	25	1	no	—
57	C₂×ΓL(1,𝔽₈)	—	175	2	no	—
58	C₅×C₇	—	3	2	no	—
59	C₅×C₇	—	3	2	no	—
60	C₅×C₇	—	6	6	no	—
61	C₂²×C₇	—	175	1	no	—
62	C₂×D₇	—	25	1	no	—
63	C₅²	C₂×D₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
64	C₃×C₇	—	175	1	no	—
65	ΓL(1,𝔽₈)	—	25	1	no	—
66	C₂×D₅	—	105	2	no	—
67	C₂×C₇	—	6	1	no	—
68	C₂×C₇	—	175	2	no	—
69	C₂×C₇	#11760384	1	1	yes	characteristic, complemented
70	C₂×C₇	(C₅⋊D₅)⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
71	D₇	—	25	2	no	—
72	C₂²×C₃	—	1225	1	no	Hall
73	C₂×C₅	—	3	2	no	—
74	D₅	—	105	4	no	—
75	C₇	—	6	1	no	—
76	C₇	C₂×(C₅⋊D₅)⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
77	C₇	C₂×#11760384	1	1	yes	characteristic, complemented
78	C₂×C₃	—	175	1	no	—
79	C₂×C₃	—	1225	2	no	—
80	C₅	—	3	2	no	—
81	C₂²	—	175	1	no	Sylow
82	C₃	—	175	1	no	Sylow
83	C₂	—	175	2	no	—
84	C₂	#20301298	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂	#20301298
commutator, cosocle, hypocenter	C₅²×C₇²	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₅²×C₇²	C₂×C₃
LCS	G ▷ C₅²×C₇²
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×C₇⋊(C₅²⋊C₃)	C₂	semidirect
2. #20301298	C₂	direct (N × H)
3. C₂×C₇×C₅⋊D₃₅	C₃	semidirect
4. C₇×C₇⋊(C₅²⋊C₃)	C₂²	semidirect
5. C₇×C₅⋊D₃₅	C₂×C₃	semidirect
6. C₂×C₅²×C₇²	C₂×C₃	semidirect
9. C₅²×C₇²	C₂²×C₃	semidirect
10. C₂×C₇⋊(C₅²⋊C₃)	D₇	semidirect
15. C₂×C₅⋊D₃₅	ΓL(1,𝔽₈)	semidirect
18. C₇⋊(C₅²⋊C₃)	C₂×D₇	semidirect
23. C₂×C₅²×C₇	C₂×ΓL(1,𝔽₈)	semidirect
24. C₂×C₅²×C₇	C₃×D₇	semidirect
25. C₅⋊D₃₅	C₂×ΓL(1,𝔽₈)	semidirect
33. C₅²×C₇	C₂²×ΓL(1,𝔽₈)	semidirect
34. C₅²×C₇	C₂×C₃×D₇	semidirect
41. C₂×C₇²	(C₅⋊D₅)⋊C₃	semidirect
51. C₂×C₅²	D₇×ΓL(1,𝔽₈)	semidirect
53. C₇²	C₂×(C₅⋊D₅)⋊C₃	semidirect
63. C₅²	C₂×D₇×ΓL(1,𝔽₈)	semidirect
69. C₂×C₇	#11760384	semidirect
70. C₂×C₇	(C₅⋊D₅)⋊ΓL(1,𝔽₈)	semidirect
76. C₇	C₂×(C₅⋊D₅)⋊ΓL(1,𝔽₈)	semidirect
77. C₇	C₂×#11760384	semidirect
84. C₂	#20301298	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇×C₇⋊(C₅²⋊C₃)	2
#20301298	2
C₂×C₇×C₅⋊D₃₅	3
C₂×#11760384	7
C₂×(C₅⋊D₅)⋊ΓL(1,𝔽₈)	7
C₂×D₇×ΓL(1,𝔽₈)	25

Subgroups — C2×#20301298 #28413445

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20301298 #28413445