Subgroups — #38760471

Order 25284.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27162422	C₂	1	1	yes	maximal, characteristic, complemented
2	#27162421	C₂	1	1	yes	maximal, characteristic, complemented
3	#27162410	C₂	1	1	yes	maximal, characteristic, complemented
4	D₇×D₃₀₁	C₃	1	1	yes	maximal, characteristic, complemented, Hall
5	C₄₃⋊#3862	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₇⋊D₃₀₁	C₂×C₃	1	1	yes	characteristic, complemented
7	C₇×D₃₀₁	C₂×C₃	1	1	yes	characteristic, complemented
8	C₇×D₇×C₄₃	C₂×C₃	1	1	yes	characteristic, complemented
9	C₂×#14268593	—	7	1	no	maximal
10	#14268588⋊C₂	—	7	1	no	maximal
11	C₇²×C₄₃	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
12	C₂×#11744748	—	7	1	no	—
13	#14268594	—	7	1	no	—
14	#14268593	—	7	1	no	—
15	#14268593	—	14	1	no	—
16	#14268592	—	14	1	no	—
17	#14268588	—	7	1	no	—
18	C₂×D₃₀₁	—	7	1	no	—
19	D₇×D₄₃	—	7	1	no	—
20	#11744749	—	14	1	no	—
21	#11744748	—	14	1	no	—
22	C₂×C₇×C₄₃	—	7	1	no	—
23	D₃₀₁	—	7	1	no	—
24	D₃₀₁	—	7	1	no	—
25	D₃₀₁	—	42	1	no	—
26	D₃₀₁	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
27	C₇×D₄₃	—	7	1	no	—
28	D₇×C₄₃	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
29	#64422⋊C₂	—	43	1	no	maximal, Hall
30	C₂×D₄₃⋊C₃	—	49	1	no	Hall
31	C₇×C₄₃	—	6	1	no	—
32	C₇×C₄₃	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
33	C₇×C₄₃	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
34	#64425	—	43	1	no	—
35	#64422	—	43	1	no	—
36	#64422	—	43	1	no	—
37	C₂×C₄₃⋊C₃	—	49	1	no	—
38	D₄₃⋊C₃	—	49	1	no	—
39	D₄₃⋊C₃	—	98	1	no	—
40	D₇²	—	43	1	no	Hall
41	C₂×D₄₃	—	49	1	no	Hall
42	#3862	—	43	1	no	Hall
43	C₄₃⋊C₃	—	98	1	no	Hall
44	C₇⋊D₇	—	43	1	no	—
45	C₇×D₇	—	43	1	no	—
46	C₇×D₇	D₄₃⋊C₃	1	1	yes	characteristic, complemented
47	C₂×C₄₃	—	7	1	no	—
48	D₄₃	—	7	1	no	—
49	D₄₃	—	49	1	no	—
50	C₂×AGL(1,𝔽₇)	—	301	1	no	—
51	C₂×AGL(1,𝔽₇)	—	301	1	no	—
52	C₇²	C₂×D₄₃⋊C₃	1	1	yes	characteristic, complemented, Sylow
53	C₄₃	#64422⋊C₂	1	1	yes	characteristic, complemented, Sylow
54	C₂×ΓL(1,𝔽₈)	—	301	1	no	—
55	C₂×ΓL(1,𝔽₈)	—	602	1	no	—
56	AGL(1,𝔽₇)	—	301	1	no	—
57	AGL(1,𝔽₇)	—	301	1	no	—
58	AGL(1,𝔽₇)	—	301	1	no	—
59	AGL(1,𝔽₇)	—	602	1	no	—
60	C₂×D₇	—	301	1	no	—
61	C₂×D₇	—	301	1	no	—
62	ΓL(1,𝔽₈)	—	602	1	no	—
63	ΓL(1,𝔽₈)	—	1204	1	no	—
64	C₂×C₇	—	7	1	no	—
65	C₂×C₇	—	301	1	no	—
66	D₇	—	1806	1	no	—
67	D₇	—	43	1	no	—
68	D₇	—	301	1	no	—
69	D₇	—	301	1	no	—
70	D₇	#14268593	1	1	yes	characteristic, complemented
71	C₂²×C₃	—	2107	1	no	Hall
72	C₇	—	6	1	no	—
73	C₇	#14268588⋊C₂	1	1	yes	characteristic, complemented
74	C₇	C₂×#14268593	1	1	yes	characteristic, complemented
75	C₂×C₃	—	2107	1	no	—
76	C₂×C₃	—	4214	1	no	—
77	C₂×C₃	—	4214	1	no	—
78	C₂²	—	2107	1	no	Sylow
79	C₃	—	8428	1	no	Sylow
80	C₂	—	7	1	no	—
81	C₂	—	301	1	no	—
82	C₂	—	2107	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇²×C₄₃	C₂²×C₃
LCS	G ▷ C₇²×C₄₃
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. #27162422	C₂	semidirect
2. #27162421	C₂	semidirect
3. #27162410	C₂	semidirect
4. D₇×D₃₀₁	C₃	semidirect
5. C₄₃⋊#3862	C₂²	semidirect
6. C₇⋊D₃₀₁	C₂×C₃	semidirect
7. C₇×D₃₀₁	C₂×C₃	semidirect
8. C₇×D₇×C₄₃	C₂×C₃	semidirect
11. C₇²×C₄₃	C₂²×C₃	semidirect
26. D₃₀₁	AGL(1,𝔽₇)	semidirect
28. D₇×C₄₃	AGL(1,𝔽₇)	semidirect
32. C₇×C₄₃	C₂×AGL(1,𝔽₇)	semidirect
33. C₇×C₄₃	C₂×AGL(1,𝔽₇)	semidirect
46. C₇×D₇	D₄₃⋊C₃	semidirect
52. C₇²	C₂×D₄₃⋊C₃	semidirect
53. C₄₃	#64422⋊C₂	semidirect
70. D₇	#14268593	semidirect
73. C₇	#14268588⋊C₂	semidirect
74. C₇	C₂×#14268593	semidirect

Maximal subgroups

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

subgroup	index
#27162422	2
#27162421	2
#27162410	2
D₇×D₃₀₁	3
C₂×#14268593	7
#14268588⋊C₂	7
#64422⋊C₂	43