Subgroups — #13051409

Order 1344.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₇×C₁₆	C₂	1	1	yes	maximal, characteristic
2	#10620658	C₂	1	2	yes	maximal
3	#89301	—	3	1	no	maximal, Hall
4	C₂×C₃×C₇×C₈	C₂²	1	1	yes	characteristic
5	#66627	—	2	2	no	—
6	C₃×C₇×C₁₆	C₄	1	2	yes	—
7	C₂×C₇×C₁₆	D₃	1	1	yes	characteristic
8	#6393	—	3	2	no	—
9	#4506	—	7	1	no	maximal, Hall
10	C₂×C₃×C₄×C₇	D₄	1	1	yes	characteristic
11	C₃×C₇×C₈	C₂×C₄	1	1	yes	characteristic
12	C₃×C₇×C₈	Q₈	1	1	yes	characteristic
13	C₃⋊(C₇⋊C₈)	—	4	2	no	—
14	C₂×C₇×C₈	C₂×D₃	1	1	yes	characteristic
15	#1136	—	6	2	no	—
16	C₇×C₁₆	Dic₃	1	2	yes	—
17	C₂×C₃×C₁₆	D₇	1	1	yes	characteristic
18	#819	—	7	2	no	—
19	C₂²×C₃×C₇	Q₁₆	1	1	yes	characteristic
20	C₃×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
21	C₃×C₄×C₇	D₈	1	1	yes	characteristic
22	#368	—	21	1	no	Sylow
23	C₂×C₄×C₇	D₁₂	1	1	yes	characteristic
24	C₇×C₈	Dic₆	1	1	yes	characteristic
25	C₇×C₈	C₂×Dic₃	1	1	yes	characteristic
26	C₇⋊C₈	—	12	2	no	—
27	C₂×C₃×C₈	C₂×D₇	1	1	yes	characteristic
28	#208	—	14	2	no	—
29	C₃×C₁₆	Dic₇	1	2	yes	—
30	C₂×C₃×C₇	—	2	1	no	—
31	C₂×C₃×C₇	#107	1	1	yes	characteristic
32	C₂×C₁₆	D₂₁	1	1	yes	characteristic
33	#108	—	21	2	no	—
34	C₂²×C₇	Dic₁₂	1	1	yes	characteristic
35	C₄×C₇	D₂₄	1	1	yes	characteristic
36	C₄×C₇	C₄⋊Dic₃	1	1	yes	characteristic
37	C₂×C₃×C₄	D₂₈	1	1	yes	characteristic
38	C₃×C₈	Dic₁₄	1	1	yes	characteristic
39	C₃×C₈	C₂×Dic₇	1	1	yes	characteristic
40	C₃⋊C₈	—	28	2	no	—
41	C₃×C₇	#368	1	1	yes	characteristic, complemented, Hall
42	M₁₆	—	42	2	no	—
43	C₂×C₈	C₂×D₂₁	1	1	yes	characteristic
44	C₁₆	Dic₂₁	1	2	yes	—
45	C₂×C₇	—	2	1	no	—
46	C₂×C₇	#818	1	1	yes	characteristic
47	C₂²×C₃	Dic₂₈	1	1	yes	characteristic
48	C₃×C₄	D₅₆	1	1	yes	characteristic
49	C₃×C₄	C₄⋊Dic₇	1	1	yes	characteristic
50	C₂×C₄	D₈₄	1	1	yes	characteristic
51	C₈	—	84	2	no	—
52	C₈	Dic₄₂	1	1	yes	characteristic
53	C₈	C₂×Dic₂₁	1	1	yes	characteristic
54	C₇	#4506	1	1	yes	characteristic, complemented, Sylow
55	C₂×C₃	—	2	1	no	—
56	C₂×C₃	#6392	1	1	yes	characteristic
57	C₂²	Dic₈₄	1	1	yes	characteristic
58	C₄	D₁₆₈	1	1	yes	characteristic
59	C₄	C₄⋊Dic₂₁	1	1	yes	characteristic
60	C₃	#89301	1	1	yes	characteristic, complemented, Sylow
61	C₂	—	2	1	no	—
62	C₂	#10620657	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₁₆₈
commutator	C₃×C₇×C₈	C₂×C₄
Frattini	C₂×C₈	C₂×D₂₁
Fitting	C₂×C₃×C₇×C₁₆	C₂
socle	C₂×C₃×C₇	#107
prime radical	C₂²×C₃×C₇	Q₁₆
cosocle	C₂×C₃×C₇×C₈	C₂²
hypercenter	C₂×C₁₆	D₂₁
hypocenter	C₃×C₇	#368
LCS	G ▷ C₃×C₇×C₈ ▷ C₃×C₄×C₇ ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
41. C₃×C₇	#368	semidirect
54. C₇	#4506	semidirect
60. C₃	#89301	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₇×C₁₆	2
#10620658	2
#89301	3
#4506	7