Subgroups — ΓL(1,𝔽₈)²×D₂₃ #35202607

Order 20286.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	ΓL(1,𝔽₈)²×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	D₂₃×#3862	C₃	1	1	yes	maximal, characteristic, complemented
3	D₂₃×#3861	C₃	1	1	yes	maximal, characteristic, complemented
4	C₇×ΓL(1,𝔽₈)×D₂₃	C₃	1	2	yes	maximal, complemented
5	C₂₃×#3862	C₂×C₃	1	1	yes	characteristic, complemented
6	C₂₃×#3861	C₂×C₃	1	1	yes	characteristic, complemented
7	C₇×ΓL(1,𝔽₈)×C₂₃	C₂×C₃	1	2	yes	complemented
8	C₃×ΓL(1,𝔽₈)×D₂₃	—	7	2	no	maximal
9	C₇²×D₂₃	C₃²	1	1	yes	characteristic, complemented, Hall
10	C₃×ΓL(1,𝔽₈)×C₂₃	—	7	2	no	—
11	C₇²×C₂₃	C₂×C₃²	1	1	yes	characteristic, complemented, Hall
12	C₃×C₇×D₂₃	—	7	2	no	—
13	ΓL(1,𝔽₈)×D₂₃	—	7	2	no	—
14	ΓL(1,𝔽₈)×D₂₃	—	7	2	no	—
15	ΓL(1,𝔽₈)×D₂₃	—	21	2	no	—
16	ΓL(1,𝔽₈)×D₂₃	ΓL(1,𝔽₈)	1	2	yes	direct product factor
17	C₂×ΓL(1,𝔽₈)²	—	23	1	no	maximal, Hall
18	C₃×C₇×C₂₃	—	7	2	no	—
19	ΓL(1,𝔽₈)×C₂₃	—	7	2	no	—
20	ΓL(1,𝔽₈)×C₂₃	—	7	2	no	—
21	ΓL(1,𝔽₈)×C₂₃	—	21	2	no	—
22	ΓL(1,𝔽₈)×C₂₃	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
23	ΓL(1,𝔽₈)²	D₂₃	1	1	yes	characteristic, direct product factor, Hall
24	C₃²×D₂₃	—	49	1	no	Hall
25	C₇×D₂₃	—	3	2	no	—
26	C₇×D₂₃	C₃×ΓL(1,𝔽₈)	1	2	yes	complemented
27	C₂×#3862	—	23	1	no	—
28	C₂×#3861	—	23	1	no	—
29	C₂×C₇×ΓL(1,𝔽₈)	—	23	2	no	—
30	C₃²×C₂₃	—	49	1	no	Hall
31	C₇×C₂₃	—	3	2	no	—
32	C₇×C₂₃	C₂×C₃×ΓL(1,𝔽₈)	1	2	yes	complemented
33	#3862	C₃×D₂₃	1	1	yes	characteristic, complemented
34	#3861	C₃×D₂₃	1	1	yes	characteristic, complemented
35	C₇×ΓL(1,𝔽₈)	C₃×D₂₃	1	2	yes	complemented
36	C₃×D₂₃	—	7	2	no	—
37	C₃×D₂₃	—	49	1	no	—
38	C₃×D₂₃	—	49	1	no	—
39	C₂×C₃×ΓL(1,𝔽₈)	—	161	2	no	—
40	C₂×C₇²	—	23	1	no	Hall
41	C₃×C₂₃	—	7	2	no	—
42	C₃×C₂₃	—	49	1	no	—
43	C₃×C₂₃	—	49	1	no	—
44	C₃×ΓL(1,𝔽₈)	—	7	2	no	—
45	C₇²	C₃²×D₂₃	1	1	yes	characteristic, complemented, Sylow
46	D₂₃	ΓL(1,𝔽₈)²	1	1	yes	characteristic, direct product factor, Hall
47	C₂×C₃×C₇	—	161	2	no	—
48	C₂×ΓL(1,𝔽₈)	—	23	2	no	—
49	C₂×ΓL(1,𝔽₈)	—	161	2	no	—
50	C₂×ΓL(1,𝔽₈)	—	161	2	no	—
51	C₂×ΓL(1,𝔽₈)	—	483	2	no	—
52	C₂₃	C₂×ΓL(1,𝔽₈)²	1	1	yes	characteristic, complemented, Sylow
53	C₃×C₇	—	7	2	no	—
54	ΓL(1,𝔽₈)	—	7	2	no	—
55	ΓL(1,𝔽₈)	—	7	2	no	—
56	ΓL(1,𝔽₈)	—	21	2	no	—
57	ΓL(1,𝔽₈)	ΓL(1,𝔽₈)×D₂₃	1	2	yes	direct product factor
58	C₂×C₃²	—	1127	1	no	Hall
59	C₂×C₇	—	23	2	no	—
60	C₂×C₇	—	69	2	no	—
61	C₃²	—	49	1	no	Sylow
62	C₇	—	3	2	no	—
63	C₇	C₃×ΓL(1,𝔽₈)×D₂₃	1	2	yes	complemented
64	C₂×C₃	—	1127	1	no	—
65	C₂×C₃	—	1127	1	no	—
66	C₂×C₃	—	161	2	no	—
67	C₃	—	7	2	no	—
68	C₃	—	49	1	no	—
69	C₃	—	49	1	no	—
70	C₂	—	23	1	no	Sylow

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇²×C₂₃	C₂×C₃²
LCS	G ▷ C₇²×C₂₃
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. ΓL(1,𝔽₈)²×C₂₃	C₂	semidirect
2. D₂₃×#3862	C₃	semidirect
3. D₂₃×#3861	C₃	semidirect
4. C₇×ΓL(1,𝔽₈)×D₂₃	C₃	semidirect
5. C₂₃×#3862	C₂×C₃	semidirect
6. C₂₃×#3861	C₂×C₃	semidirect
7. C₇×ΓL(1,𝔽₈)×C₂₃	C₂×C₃	semidirect
9. C₇²×D₂₃	C₃²	semidirect
11. C₇²×C₂₃	C₂×C₃²	semidirect
16. ΓL(1,𝔽₈)×D₂₃	ΓL(1,𝔽₈)	direct (N × H)
22. ΓL(1,𝔽₈)×C₂₃	C₂×ΓL(1,𝔽₈)	semidirect
23. ΓL(1,𝔽₈)²	D₂₃	direct (N × H)
26. C₇×D₂₃	C₃×ΓL(1,𝔽₈)	semidirect
32. C₇×C₂₃	C₂×C₃×ΓL(1,𝔽₈)	semidirect
33. #3862	C₃×D₂₃	semidirect
34. #3861	C₃×D₂₃	semidirect
35. C₇×ΓL(1,𝔽₈)	C₃×D₂₃	semidirect
45. C₇²	C₃²×D₂₃	semidirect
46. D₂₃	ΓL(1,𝔽₈)²	direct (N × H)
52. C₂₃	C₂×ΓL(1,𝔽₈)²	semidirect
57. ΓL(1,𝔽₈)	ΓL(1,𝔽₈)×D₂₃	direct (N × H)
63. C₇	C₃×ΓL(1,𝔽₈)×D₂₃	semidirect

Maximal subgroups

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

subgroup	index
ΓL(1,𝔽₈)²×C₂₃	2
D₂₃×#3862	3
D₂₃×#3861	3
C₇×ΓL(1,𝔽₈)×D₂₃	3
C₃×ΓL(1,𝔽₈)×D₂₃	7
C₂×ΓL(1,𝔽₈)²	23

Subgroups — ΓL(1,𝔽8)2×D23 #35202607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)²×D₂₃ #35202607