Subgroups — #31771524

Order 17216.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22575528	C₂	1	1	yes	maximal, characteristic
2	#22575523	C₂	1	1	yes	maximal, characteristic
3	Q₁₆×D₂₆₉	C₂	1	1	yes	maximal, characteristic
4	#16864412	—	2	1	no	—
5	#16864405	—	2	1	no	—
6	Q₈×D₂₆₉	—	2	1	no	—
7	Dic₁₀₇₆	C₄	1	1	yes	characteristic
8	#16864392	—	2	1	no	—
9	C₈×D₂₆₉	C₂²	1	1	yes	characteristic
10	Q₁₆×C₂₆₉	C₄	1	1	yes	characteristic
11	#14524137	—	4	1	no	—
12	Dic₅₃₈	—	2	1	no	—
13	Dic₅₃₈	—	4	1	no	—
14	C₄×D₂₆₉	—	4	1	no	—
15	C₄×D₂₆₉	D₄	1	1	yes	characteristic
16	#14524130	D₄	1	1	yes	characteristic
17	Q₈×C₂₆₉	—	2	1	no	—
18	C₈×C₂₆₉	C₂×C₄	1	1	yes	characteristic
19	C₂×D₂₆₉	SD₁₆	1	1	yes	characteristic
20	C₄×C₂₆₉	—	4	1	no	—
21	C₄×C₂₆₉	C₂²⋊C₄	1	1	yes	characteristic
22	Dic₂₆₉	—	4	1	no	—
23	Dic₂₆₉	D₈	1	1	yes	characteristic
24	C₂×C₂₆₉	#102	1	1	yes	characteristic
25	D₂₆₉	—	2	1	no	—
26	C₂₆₉	#362	1	1	yes	characteristic, complemented, Sylow
27	#362	—	269	1	no	maximal, Sylow
28	C₂×Q₁₆	—	269	1	no	—
29	M₃₂	—	269	1	no	—
30	#108	—	269	1	no	—
31	C₂×Q₈	—	538	1	no	—
32	Q₁₆	—	269	1	no	—
33	Q₁₆	—	538	1	no	—
34	Q₁₆	#11761640	1	1	yes	characteristic
35	M₁₆	—	538	1	no	—
36	C₂×C₈	—	269	1	no	—
37	C₁₆	—	538	1	no	—
38	Q₈	—	2	1	no	—
39	Q₈	—	538	1	no	—
40	Q₈	—	1076	1	no	—
41	C₂×C₄	—	269	1	no	—
42	C₂×C₄	—	1076	1	no	—
43	C₈	—	269	1	no	—
44	C₈	—	1076	1	no	—
45	C₈	C₂×#11761640	1	1	yes	characteristic
46	C₂²	—	269	1	no	—
47	C₄	—	4	1	no	—
48	C₄	—	269	1	no	—
49	C₄	—	1076	1	no	—
50	C₄	C₂²⋊#11761640	1	1	yes	characteristic
51	C₂	—	538	1	no	—
52	C₂	#22575515	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22575515
commutator	C₈×C₂₆₉	C₂×C₄
Frattini	C₈	C₂×#11761640
Fitting	Q₁₆×C₂₆₉	C₄
socle	C₂×C₂₆₉	#102
prime radical	C₂×D₂₆₉	SD₁₆
cosocle	C₈×D₂₆₉	C₂²
hypercenter	Q₁₆	#11761640
hypocenter	C₂₆₉	#362
LCS	G ▷ C₈×C₂₆₉ ▷ C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
26. C₂₆₉	#362	semidirect

Maximal subgroups

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

subgroup	index
#22575528	2
#22575523	2
Q₁₆×D₂₆₉	2
#362	269