Subgroups — #31770713

Order 17216.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₀₇₆	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₅₃₈	C₂	1	2	yes	maximal
3	#22575385	C₂	1	1	yes	maximal, characteristic
4	#22575378	C₂	1	2	yes	maximal
5	#97×C₂₆₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₃₈	C₂²	1	2	yes	—
7	Dic₁₀₇₆	C₄	1	4	yes	complemented
8	C₄⋊Dic₂₆₉	C₂²	1	2	yes	—
9	#16864381	—	2	2	no	—
10	C₄×Dic₂₆₉	—	2	2	no	—
11	C₂×C₈×C₂₆₉	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₂₆₉	C₂²	1	1	yes	characteristic
13	C₄²×C₂₆₉	C₂²	1	1	yes	characteristic
14	Dic₅₃₈	—	2	2	no	—
15	Dic₅₃₈	C₂×C₄	1	4	yes	—
16	C₂×Dic₂₆₉	—	2	2	no	—
17	C₂×Dic₂₆₉	—	2	2	no	—
18	C₂×C₄×C₂₆₉	D₄	1	2	yes	—
19	C₂×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
20	C₈×C₂₆₉	—	2	1	no	—
21	C₈×C₂₆₉	C₂×C₄	1	2	yes	—
22	C₂²×C₂₆₉	C₂×D₄	1	1	yes	characteristic
23	C₄×C₂₆₉	—	2	2	no	—
24	C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
25	C₄×C₂₆₉	#41	1	1	yes	characteristic
26	Dic₂₆₉	—	2	4	no	—
27	Dic₂₆₉	—	4	2	no	—
28	C₂×C₂₆₉	C₄×D₄	1	1	yes	characteristic
29	C₂×C₂₆₉	#137	1	2	yes	—
30	C₂₆₉	#441	1	1	yes	characteristic, complemented, Sylow
31	#441	—	269	1	no	maximal, Sylow
32	C₂×Q₁₆	—	269	1	no	—
33	C₄×Q₈	—	269	2	no	—
34	#106	—	269	1	no	—
35	#103	—	269	2	no	—
36	#97	D₂₆₉	1	1	yes	characteristic
37	C₂×Q₈	—	269	2	no	—
38	Q₁₆	—	269	4	no	—
39	C₂×C₈	C₂×D₂₆₉	1	1	yes	characteristic
40	C₂×C₈	C₂×D₂₆₉	1	1	yes	characteristic
41	C₄⋊C₄	—	269	2	no	—
42	C₄⋊C₄	—	538	2	no	—
43	C₄²	—	538	2	no	—
44	C₄²	C₂×D₂₆₉	1	1	yes	characteristic
45	Q₈	—	269	4	no	—
46	Q₈	—	538	2	no	—
47	C₂×C₄	—	538	2	no	—
48	C₂×C₄	—	538	2	no	—
49	C₂×C₄	D₁₀₇₆	1	2	yes	—
50	C₂×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
51	C₈	—	2	1	no	—
52	C₈	C₄×D₂₆₉	1	2	yes	—
53	C₂²	C₂×D₁₀₇₆	1	1	yes	characteristic
54	C₄	—	2	2	no	—
55	C₄	—	538	4	no	—
56	C₄	—	1076	2	no	—
57	C₄	C₂×C₄×D₂₆₉	1	1	yes	characteristic
58	C₄	#16864402	1	1	yes	characteristic
59	C₂	C₄×D₁₀₇₆	1	1	yes	characteristic
60	C₂	#22575486	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₀₇₆
commutator	C₄×C₂₆₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₆₉
Fitting	#97×C₂₆₉	C₂
socle, prime radical	C₂²×C₂₆₉	C₂×D₄
cosocle	C₂×C₄×C₂₆₉	C₂³
hypercenter	#97	D₂₆₉
hypocenter	C₂₆₉	#441
LCS	G ▷ C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

Complements

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

N	H ≅ G/N	type
7. Dic₁₀₇₆	C₄	semidirect
30. C₂₆₉	#441	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₁₀₇₆	2
C₄×Dic₅₃₈	2
#22575385	2
#22575378	2
#97×C₂₆₉	2
#441	269