Subgroups — #34067316

Order 19328.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23752042	C₂	1	1	yes	maximal, characteristic
2	#23752040	C₂	1	1	yes	maximal, characteristic
3	M₆₄×C₁₅₁	C₂	1	1	yes	maximal, characteristic
4	#18012675	—	2	1	no	—
5	#18012673	—	2	1	no	—
6	C₂×C₁₆×C₁₅₁	C₂²	1	1	yes	characteristic
7	C₃₂×C₁₅₁	C₄	1	2	yes	complemented
8	#14558988	—	4	1	no	—
9	C₄⋊Dic₁₅₁	—	4	1	no	—
10	C₂×C₈×C₁₅₁	D₄	1	1	yes	characteristic
11	C₁₆×C₁₅₁	C₂×C₄	1	1	yes	characteristic
12	C₁₆×C₁₅₁	Q₈	1	1	yes	characteristic
13	C₂×Dic₁₅₁	—	8	1	no	—
14	C₁₅₁⋊C₈	—	8	1	no	—
15	C₂×C₄×C₁₅₁	D₈	1	1	yes	characteristic
16	C₈×C₁₅₁	C₄⋊C₄	1	1	yes	characteristic
17	C₈×C₁₅₁	Q₁₆	1	1	yes	characteristic
18	C₂²×C₁₅₁	SD₃₂	1	1	yes	characteristic
19	C₄×C₁₅₁	#107	1	1	yes	characteristic
20	C₄×C₁₅₁	SD₃₂	1	1	yes	characteristic
21	Dic₁₅₁	—	16	1	no	—
22	C₂×C₁₅₁	—	2	1	no	—
23	C₂×C₁₅₁	#367	1	1	yes	characteristic
24	C₁₅₁	#1426	1	1	yes	characteristic, complemented, Sylow
25	#1426	—	151	1	no	maximal, Sylow
26	M₆₄	D₁₅₁	1	1	yes	characteristic
27	#368	—	151	1	no	—
28	#367	—	151	1	no	—
29	C₂×C₁₆	C₂×D₁₅₁	1	1	yes	characteristic
30	#108	—	302	1	no	—
31	#107	—	302	1	no	—
32	C₃₂	Dic₁₅₁	1	2	yes	complemented
33	M₁₆	—	604	1	no	—
34	C₂×C₈	D₆₀₄	1	1	yes	characteristic
35	C₄⋊C₄	—	604	1	no	—
36	C₁₆	C₂×Dic₁₅₁	1	1	yes	characteristic
37	C₁₆	Dic₃₀₂	1	1	yes	characteristic
38	C₂×C₄	—	1208	1	no	—
39	C₂×C₄	D₁₂₀₈	1	1	yes	characteristic
40	C₈	—	1208	1	no	—
41	C₈	C₄⋊Dic₁₅₁	1	1	yes	characteristic
42	C₈	Dic₆₀₄	1	1	yes	characteristic
43	C₂²	#18012684	1	1	yes	characteristic
44	C₄	—	2416	1	no	—
45	C₄	#18012673	1	1	yes	characteristic
46	C₄	#18012684	1	1	yes	characteristic
47	C₂	—	2	1	no	—
48	C₂	#23752040	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#23752040
commutator	C₁₆×C₁₅₁	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₅₁
Fitting	M₆₄×C₁₅₁	C₂
socle	C₂×C₁₅₁	#367
prime radical	C₂²×C₁₅₁	SD₃₂
cosocle	C₂×C₁₆×C₁₅₁	C₂²
hypercenter	M₆₄	D₁₅₁
hypocenter	C₁₅₁	#1426
LCS	G ▷ C₁₆×C₁₅₁ ▷ C₈×C₁₅₁ ▷ C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
7. C₃₂×C₁₅₁	C₄	semidirect
24. C₁₅₁	#1426	semidirect
32. C₃₂	Dic₁₅₁	semidirect

Maximal subgroups

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

subgroup	index
#23752042	2
#23752040	2
M₆₄×C₁₅₁	2
#1426	151