Subgroups — #13055064

Order 1344.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇⋊(C₄²⋊C₃)	C₂	1	1	yes	maximal, characteristic
2	C₇×#383	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₄×C₂²⋊ΓL(1,𝔽₈)	—	4	1	no	maximal
4	C₇⋊(C₄²⋊C₃)	C₄	1	1	yes	characteristic, complemented
5	C₂×C₄²×C₇	C₂×C₃	1	1	yes	characteristic
6	C₇×#95	—	3	1	no	—
7	C₇×#95	—	3	1	no	—
8	#383⋊C₃	—	7	1	no	maximal, Hall
9	C₂×C₂²⋊ΓL(1,𝔽₈)	—	4	1	no	—
10	C₂²×C₄×C₇	—	3	1	no	—
11	C₂²×C₄×C₇	—	3	1	no	—
12	C₂²×C₄×C₇	A₄	1	1	yes	characteristic
13	C₄²×C₇	—	3	1	no	—
14	C₄²×C₇	C₃×C₄	1	1	yes	characteristic, complemented
15	C₂×C₄²⋊C₃	—	7	1	no	—
16	C₂²⋊ΓL(1,𝔽₈)	—	4	1	no	—
17	C₄×ΓL(1,𝔽₈)	—	16	1	no	—
18	#383	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
19	C₂³×C₇	C₂×A₄	1	1	yes	characteristic
20	C₂×C₄×C₇	—	3	1	no	—
21	C₂×C₄×C₇	—	3	1	no	—
22	C₂×C₄×C₇	—	6	1	no	—
23	C₂×C₄×C₇	—	6	1	no	—
24	C₂×C₄×C₇	—	6	1	no	—
25	C₂×C₄×C₇	—	6	2	no	—
26	C₂×C₄×C₇	—	6	1	no	—
27	C₄×A₄	—	28	1	no	—
28	C₄²⋊C₃	—	7	1	no	—
29	C₂×ΓL(1,𝔽₈)	—	16	1	no	—
30	C₂×C₄²	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
31	#95	—	3	1	no	—
32	#95	—	3	1	no	—
33	C₂²×C₇	—	3	1	no	—
34	C₂²×C₇	—	3	1	no	—
35	C₂²×C₇	C₄×A₄	1	1	yes	characteristic
36	C₄×C₇	—	4	1	no	—
37	C₄×C₇	—	6	1	no	—
38	C₄×C₇	—	6	1	no	—
39	C₄×C₇	—	12	1	no	—
40	C₂×A₄	—	28	1	no	—
41	ΓL(1,𝔽₈)	—	16	1	no	Hall
42	C₂²×C₄	—	3	1	no	—
43	C₂²×C₄	—	3	1	no	—
44	C₂²×C₄	C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
45	C₄²	—	3	1	no	—
46	C₄²	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
47	C₂×C₇	—	3	1	no	—
48	C₂×C₇	—	3	1	no	—
49	C₂×C₇	#126⋊C₃	1	1	yes	characteristic
50	A₄	—	28	1	no	—
51	C₃×C₄	—	112	1	no	—
52	C₂³	C₂×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
53	C₂×C₄	—	3	1	no	—
54	C₂×C₄	—	3	1	no	—
55	C₂×C₄	—	6	1	no	—
56	C₂×C₄	—	6	1	no	—
57	C₂×C₄	—	6	1	no	—
58	C₂×C₄	—	6	2	no	—
59	C₂×C₄	—	6	1	no	—
60	C₇	#383⋊C₃	1	1	yes	characteristic, complemented, Sylow
61	C₂×C₃	—	112	1	no	—
62	C₂²	—	3	1	no	—
63	C₂²	—	3	1	no	—
64	C₂²	C₄×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
65	C₄	—	4	1	no	—
66	C₄	—	6	1	no	—
67	C₄	—	6	1	no	—
68	C₄	—	12	1	no	—
69	C₃	—	112	1	no	Sylow
70	C₂	—	3	1	no	—
71	C₂	—	3	1	no	—
72	C₂	#10621175	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂	#10621175
commutator, hypocenter	C₄²×C₇	C₃×C₄
Frattini	C₂³	C₂×C₂²⋊ΓL(1,𝔽₈)
Fitting	C₇×#383	C₃
socle	C₂³×C₇	C₂×A₄
prime radical	C₂×C₇⋊(C₄²⋊C₃)	C₂
cosocle	C₂×C₄²×C₇	C₂×C₃
LCS	G ▷ C₄²×C₇
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
2. C₇×#383	C₃	semidirect
4. C₇⋊(C₄²⋊C₃)	C₄	semidirect
14. C₄²×C₇	C₃×C₄	semidirect
18. #383	ΓL(1,𝔽₈)	semidirect
46. C₄²	C₄×ΓL(1,𝔽₈)	semidirect
60. C₇	#383⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇⋊(C₄²⋊C₃)	2
C₇×#383	3
C₄×C₂²⋊ΓL(1,𝔽₈)	4
#383⋊C₃	7