Subgroups — #13060786

Order 1344.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#120⋊ΓL(1,𝔽₈)	C₂	1	3	yes	maximal, complemented
2	C₇×H(3,𝔽₄)	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₂²⋊(C₂²⋊ΓL(1,𝔽₈))	C₂²	1	1	yes	characteristic, complemented
4	C₂²×C₂²⋊ΓL(1,𝔽₈)	—	4	1	no	maximal
5	C₇×#124	—	3	3	no	—
6	C₇×#120	—	3	1	no	—
7	C₇×#120	C₂×C₃	1	3	yes	complemented
8	#5450	—	7	1	no	maximal, Hall
9	C₂×C₂²⋊ΓL(1,𝔽₈)	—	4	3	no	—
10	C₂⁴×C₇	A₄	1	1	yes	characteristic, complemented
11	C₂⁴×C₇	C₂²×C₃	1	1	yes	characteristic, complemented
12	C₂×C₇×Q₈	—	3	1	no	—
13	C₂×C₇×D₄	—	3	3	no	—
14	C₇×C₂²⋊C₄	—	3	3	no	—
15	C₇×C₂²⋊C₄	—	3	3	no	—
16	C₄²×C₇	—	3	1	no	—
17	#120⋊C₃	—	7	3	no	—
18	C₂²⋊ΓL(1,𝔽₈)	—	4	1	no	—
19	C₂²⋊ΓL(1,𝔽₈)	—	16	1	no	—
20	C₂²×ΓL(1,𝔽₈)	—	16	1	no	—
21	H(3,𝔽₄)	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
22	C₂³×C₇	—	3	1	no	—
23	C₂³×C₇	—	12	1	no	—
24	C₂³×C₇	—	12	1	no	—
25	C₂³×C₇	C₂×A₄	1	3	yes	—
26	C₇×Q₈	—	12	1	no	—
27	C₇×D₄	—	12	3	no	—
28	C₂×C₄×C₇	—	3	3	no	—
29	C₂²⋊A₄	—	7	1	no	—
30	C₂²×A₄	—	28	1	no	—
31	C₂×ΓL(1,𝔽₈)	—	16	3	no	—
32	#124	—	3	3	no	—
33	#120	—	3	1	no	—
34	#120	C₂×ΓL(1,𝔽₈)	1	3	yes	complemented
35	C₂²×C₇	—	4	1	no	—
36	C₂²×C₇	—	4	1	no	—
37	C₂²×C₇	—	6	3	no	—
38	C₂²×C₇	—	6	3	no	—
39	C₂²×C₇	—	12	1	no	—
40	C₂²×C₇	—	12	1	no	—
41	C₂²×C₇	C₂²×A₄	1	1	yes	characteristic
42	C₄×C₇	—	6	3	no	—
43	C₂×A₄	—	28	3	no	—
44	ΓL(1,𝔽₈)	—	16	1	no	Hall
45	C₂⁴	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
46	C₂⁴	C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
47	C₂×Q₈	—	3	1	no	—
48	C₂×D₄	—	3	3	no	—
49	C₂²⋊C₄	—	3	3	no	—
50	C₂²⋊C₄	—	3	3	no	—
51	C₄²	—	3	1	no	—
52	C₂×C₇	—	3	1	no	—
53	C₂×C₇	—	4	3	no	—
54	C₂×C₇	—	12	1	no	—
55	C₂²×C₃	—	112	1	no	—
56	A₄	—	28	1	no	—
57	A₄	—	112	1	no	—
58	C₂³	—	3	1	no	—
59	C₂³	—	12	1	no	—
60	C₂³	—	12	1	no	—
61	C₂³	C₂×C₂²⋊ΓL(1,𝔽₈)	1	3	yes	—
62	Q₈	—	12	1	no	—
63	D₄	—	12	3	no	—
64	C₂×C₄	—	3	3	no	—
65	C₇	#5450	1	1	yes	characteristic, complemented, Sylow
66	C₂×C₃	—	112	3	no	—
67	C₂²	—	4	1	no	—
68	C₂²	—	4	1	no	—
69	C₂²	—	6	3	no	—
70	C₂²	—	6	3	no	—
71	C₂²	—	12	1	no	—
72	C₂²	—	12	1	no	—
73	C₂²	C₂²×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
74	C₄	—	6	3	no	—
75	C₃	—	112	1	no	Sylow
76	C₂	—	3	1	no	—
77	C₂	—	4	3	no	—
78	C₂	—	12	1	no	—

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₂⁴×C₇	C₂²×C₃
Frattini	C₂²	C₂²×C₂²⋊ΓL(1,𝔽₈)
Fitting	C₇×H(3,𝔽₄)	C₃
socle	C₂²×C₇	C₂²×A₄
LCS	G ▷ C₂⁴×C₇
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. #120⋊ΓL(1,𝔽₈)	C₂	semidirect
2. C₇×H(3,𝔽₄)	C₃	semidirect
3. C₂²⋊(C₂²⋊ΓL(1,𝔽₈))	C₂²	semidirect
7. C₇×#120	C₂×C₃	semidirect
10. C₂⁴×C₇	A₄	semidirect
11. C₂⁴×C₇	C₂²×C₃	semidirect
21. H(3,𝔽₄)	ΓL(1,𝔽₈)	semidirect
34. #120	C₂×ΓL(1,𝔽₈)	semidirect
45. C₂⁴	C₂²×ΓL(1,𝔽₈)	semidirect
46. C₂⁴	C₂²⋊ΓL(1,𝔽₈)	semidirect
65. C₇	#5450	semidirect

Maximal subgroups

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

subgroup	index
#120⋊ΓL(1,𝔽₈)	2
C₇×H(3,𝔽₄)	3
C₂²×C₂²⋊ΓL(1,𝔽₈)	4
#5450	7