Subgroups — #31771664

Order 17216.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22575528	C₂	1	2	yes	maximal, complemented
2	#22575527	C₂	1	1	yes	maximal, characteristic, complemented
3	D₂₆₉⋊C₁₆	C₂	1	2	yes	maximal, complemented
4	C₂×#16864405	C₂	1	1	yes	maximal, characteristic, complemented
5	#22575461	C₂	1	1	yes	maximal, characteristic
6	#16864405	C₂²	1	1	yes	characteristic
7	#16864405	C₂²	1	1	yes	characteristic
8	#16864405	C₂²	1	2	yes	—
9	#16864402	C₄	1	1	yes	characteristic
10	#16864385	C₄	1	2	yes	—
11	C₂×#14524130	C₂²	1	1	yes	characteristic
12	C₈×D₂₆₉	C₂²	1	2	yes	—
13	M₁₆×C₂₆₉	C₄	1	1	yes	characteristic
14	Dic₅₃₈	C₈	1	1	yes	characteristic
15	D₁₀₇₆	C₈	1	1	yes	characteristic
16	C₂²⋊D₂₆₉	C₈	1	2	yes	—
17	C₄×D₂₆₉	C₂×C₄	1	2	yes	—
18	#14524130	C₂×C₄	1	1	yes	characteristic
19	#14524130	C₂³	1	1	yes	characteristic
20	C₂×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
21	C₈×C₂₆₉	C₂×C₄	1	2	yes	—
22	C₂²×C₂₆₉	C₂×C₈	1	1	yes	characteristic
23	C₂×D₂₆₉	C₂×C₈	1	2	yes	—
24	C₄×C₂₆₉	C₂×C₈	1	1	yes	characteristic
25	C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
26	Dic₂₆₉	C₂×C₈	1	2	yes	—
27	C₂×C₂₆₉	—	2	1	no	—
28	C₂×C₂₆₉	C₂²×C₈	1	1	yes	characteristic
29	D₂₆₉	—	2	2	no	—
30	C₂₆₉	#504	1	1	yes	characteristic, complemented, Sylow
31	#504	—	269	1	no	maximal, Sylow
32	#131	—	269	1	no	—
33	M₃₂	—	269	1	no	—
34	M₃₂	—	269	2	no	—
35	C₂×C₁₆	—	269	1	no	—
36	C₂×C₁₆	—	269	2	no	—
37	#41	—	269	1	no	—
38	M₁₆	—	269	2	no	—
39	M₁₆	#11761640	1	1	yes	characteristic
40	C₂×C₈	—	269	1	no	—
41	C₂×C₈	—	269	2	no	—
42	C₁₆	—	269	1	no	—
43	C₁₆	—	269	1	no	—
44	C₁₆	—	269	2	no	—
45	Q₈	—	269	1	no	—
46	D₄	—	269	2	no	—
47	D₄	—	269	1	no	—
48	C₂×C₄	—	269	2	no	—
49	C₂×C₄	C₂×#11761640	1	1	yes	characteristic
50	C₈	—	269	1	no	—
51	C₈	—	269	1	no	—
52	C₈	C₂×#11761640	1	2	yes	—
53	C₂²	—	269	2	no	—
54	C₂²	D₂₆₉⋊C₈	1	1	yes	characteristic
55	C₄	—	269	2	no	—
56	C₄	D₂₆₉⋊C₈	1	1	yes	characteristic
57	C₄	C₂²×#11761640	1	1	yes	characteristic
58	C₂	—	2	1	no	—
59	C₂	—	538	2	no	—
60	C₂	C₂×D₂₆₉⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂²×#11761640
commutator, socle	C₂×C₂₆₉	C₂²×C₈
Fitting	M₁₆×C₂₆₉	C₄
prime radical	#16864402	C₄
cosocle	#14524130	C₂³
hypercenter	M₁₆	#11761640
hypocenter	C₂₆₉	#504
LCS	G ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
1. #22575528	C₂	semidirect
2. #22575527	C₂	semidirect
3. D₂₆₉⋊C₁₆	C₂	semidirect
4. C₂×#16864405	C₂	semidirect
30. C₂₆₉	#504	semidirect

Maximal subgroups

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

subgroup	index
#22575528	2
#22575527	2
D₂₆₉⋊C₁₆	2
C₂×#16864405	2
#22575461	2
#504	269