Subgroups — #72245283

Order 3528.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×ΓL(1,𝔽₈)²	C₂	1	1	yes	maximal, characteristic
2	#11924149	C₃	1	1	yes	maximal, characteristic, complemented
3	#11924148	C₃	1	1	yes	maximal, characteristic, complemented
4	ΓL(1,𝔽₈)⋊(C₇⋊C₈)	C₃	1	2	yes	maximal, complemented
5	C₂×ΓL(1,𝔽₈)²	C₄	1	1	yes	characteristic
6	C₄×#3862	C₂×C₃	1	1	yes	characteristic
7	C₄×#3861	C₂×C₃	1	1	yes	characteristic
8	C₄×C₇×ΓL(1,𝔽₈)	C₂×C₃	1	2	yes	—
9	C₃×ΓL(1,𝔽₈)⋊C₈	—	7	2	no	maximal
10	ΓL(1,𝔽₈)²	C₈	1	1	yes	characteristic, complemented, Hall
11	C₇⋊(C₇⋊C₈)	C₃²	1	1	yes	characteristic, complemented, Hall
12	C₂×#3862	C₃×C₄	1	1	yes	characteristic
13	C₂×#3861	C₃×C₄	1	1	yes	characteristic
14	C₂×C₇×ΓL(1,𝔽₈)	C₃×C₄	1	2	yes	—
15	C₃×C₄×ΓL(1,𝔽₈)	—	7	2	no	—
16	C₄×C₇²	C₂×C₃²	1	1	yes	characteristic
17	C₃×C₇⋊C₈	—	7	2	no	—
18	ΓL(1,𝔽₈)⋊C₈	—	7	2	no	—
19	ΓL(1,𝔽₈)⋊C₈	—	7	2	no	—
20	ΓL(1,𝔽₈)⋊C₈	—	21	2	no	—
21	ΓL(1,𝔽₈)⋊C₈	—	21	2	no	—
22	#3862	C₃×C₈	1	1	yes	characteristic, complemented
23	#3861	C₃×C₈	1	1	yes	characteristic, complemented
24	C₇×ΓL(1,𝔽₈)	C₃×C₈	1	2	yes	complemented
25	C₂×C₃×ΓL(1,𝔽₈)	—	7	2	no	—
26	C₂×C₇²	C₃²×C₄	1	1	yes	characteristic
27	C₃×C₄×C₇	—	7	2	no	—
28	C₄×ΓL(1,𝔽₈)	—	7	2	no	—
29	C₄×ΓL(1,𝔽₈)	—	7	2	no	—
30	C₄×ΓL(1,𝔽₈)	—	21	2	no	—
31	C₄×ΓL(1,𝔽₈)	AGL(1,𝔽₇)	1	2	yes	—
32	C₃²×C₈	—	49	1	no	Hall
33	C₃×ΓL(1,𝔽₈)	—	7	2	no	—
34	C₇⋊C₈	—	21	2	no	—
35	C₇⋊C₈	—	63	2	no	—
36	C₇²	C₃²×C₈	1	1	yes	characteristic, complemented, Sylow
37	C₂×C₃×C₇	—	7	2	no	—
38	C₂×ΓL(1,𝔽₈)	—	7	2	no	—
39	C₂×ΓL(1,𝔽₈)	—	7	2	no	—
40	C₂×ΓL(1,𝔽₈)	—	21	2	no	—
41	C₂×ΓL(1,𝔽₈)	ΓL(1,𝔽₈)⋊C₄	1	2	yes	—
42	C₃²×C₄	—	49	1	no	—
43	C₄×C₇	—	3	2	no	—
44	C₄×C₇	C₃×AGL(1,𝔽₇)	1	2	yes	—
45	C₃×C₈	—	49	1	no	—
46	C₃×C₈	—	49	1	no	—
47	C₃×C₈	—	147	2	no	—
48	C₃×C₇	—	7	2	no	—
49	ΓL(1,𝔽₈)	—	7	2	no	—
50	ΓL(1,𝔽₈)	—	7	2	no	—
51	ΓL(1,𝔽₈)	—	21	2	no	—
52	ΓL(1,𝔽₈)	ΓL(1,𝔽₈)⋊C₈	1	2	yes	complemented
53	C₂×C₃²	—	49	1	no	—
54	C₂×C₇	—	3	2	no	—
55	C₂×C₇	C₃×ΓL(1,𝔽₈)⋊C₄	1	2	yes	—
56	C₃×C₄	—	7	2	no	—
57	C₃×C₄	—	49	1	no	—
58	C₃×C₄	—	49	1	no	—
59	C₃²	—	49	1	no	Sylow
60	C₈	—	441	1	no	Sylow
61	C₇	—	3	2	no	—
62	C₇	C₃×ΓL(1,𝔽₈)⋊C₈	1	2	yes	complemented
63	C₂×C₃	—	7	2	no	—
64	C₂×C₃	—	49	1	no	—
65	C₂×C₃	—	49	1	no	—
66	C₄	ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)	1	1	yes	characteristic
67	C₃	—	7	2	no	—
68	C₃	—	49	1	no	—
69	C₃	—	49	1	no	—
70	C₂	ΓL(1,𝔽₈)⋊(ΓL(1,𝔽₈)⋊C₄)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₄	ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)
commutator, hypocenter	C₇²	C₃²×C₈
Fitting, cosocle	C₄×C₇²	C₂×C₃²
socle	C₂×C₇²	C₃²×C₄
prime radical	C₂×ΓL(1,𝔽₈)²	C₄
LCS	G ▷ C₇²
UCS	C₁ ◁ C₄

Complements

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

N	H ≅ G/N	type
2. #11924149	C₃	semidirect
3. #11924148	C₃	semidirect
4. ΓL(1,𝔽₈)⋊(C₇⋊C₈)	C₃	semidirect
10. ΓL(1,𝔽₈)²	C₈	semidirect
11. C₇⋊(C₇⋊C₈)	C₃²	semidirect
22. #3862	C₃×C₈	semidirect
23. #3861	C₃×C₈	semidirect
24. C₇×ΓL(1,𝔽₈)	C₃×C₈	semidirect
36. C₇²	C₃²×C₈	semidirect
52. ΓL(1,𝔽₈)	ΓL(1,𝔽₈)⋊C₈	semidirect
62. C₇	C₃×ΓL(1,𝔽₈)⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×ΓL(1,𝔽₈)²	2
#11924149	3
#11924148	3
ΓL(1,𝔽₈)⋊(C₇⋊C₈)	3
C₃×ΓL(1,𝔽₈)⋊C₈	7