Subgroups — #31771610

Order 17216.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16864412	C₂	1	2	yes	maximal
2	#22575520	C₂	1	4	yes	maximal, complemented
3	C₄²×D₂₆₉	C₂	1	1	yes	maximal, characteristic
4	#16864412	—	2	4	no	—
5	C₂×#14524137	C₂²	1	4	yes	—
6	C₂×C₄×D₂₆₉	C₄	1	2	yes	—
7	C₂×C₄×D₂₆₉	C₂²	1	1	yes	characteristic
8	C₄×Dic₂₆₉	C₄	1	1	yes	characteristic
9	C₄×Dic₂₆₉	C₂²	1	2	yes	—
10	C₄²×C₂₆₉	C₄	1	1	yes	characteristic
11	#14524137	—	2	4	no	—
12	C₂²×D₂₆₉	C₂×C₄	1	1	yes	characteristic
13	C₂×Dic₂₆₉	C₂×C₄	1	2	yes	—
14	C₂×Dic₂₆₉	C₂³	1	1	yes	characteristic
15	C₄×D₂₆₉	—	2	4	no	—
16	C₄×D₂₆₉	D₄	1	2	yes	—
17	C₄×D₂₆₉	Q₈	1	2	yes	—
18	C₂×C₄×C₂₆₉	C₂×C₄	1	2	yes	—
19	C₂×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
21	C₂×D₂₆₉	—	2	2	no	—
22	C₂×D₂₆₉	C₄⋊C₄	1	2	yes	—
23	C₄×C₂₆₉	—	2	2	no	—
24	C₄×C₂₆₉	C₄⋊C₄	1	2	yes	—
25	Dic₂₆₉	M₁₆	1	4	yes	complemented
26	Dic₂₆₉	C₂×D₄	1	1	yes	characteristic
27	Dic₂₆₉	C₂×Q₈	1	1	yes	characteristic
28	C₂×C₂₆₉	C₂×C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₂₆₉	C₂×M₁₆	1	2	yes	—
30	D₂₆₉	—	2	2	no	—
31	C₂₆₉	#423	1	1	yes	characteristic, complemented, Sylow
32	#423	—	269	1	no	maximal, Sylow
33	C₂×M₁₆	—	269	2	no	—
34	C₂×C₄²	—	269	1	no	—
35	C₄⋊C₈	—	269	4	no	—
36	C₂²×C₄	—	269	1	no	—
37	C₂²×C₄	—	269	2	no	—
38	M₁₆	—	538	4	no	—
39	C₂×C₈	—	269	4	no	—
40	C₄²	—	269	2	no	—
41	C₄²	—	269	1	no	—
42	C₄²	#11761640	1	1	yes	characteristic
43	C₂³	—	269	1	no	—
44	C₂×C₄	—	269	1	no	—
45	C₂×C₄	—	269	2	no	—
46	C₂×C₄	—	269	2	no	—
47	C₂×C₄	—	269	2	no	—
48	C₂×C₄	—	538	4	no	—
49	C₂×C₄	C₂×#11761640	1	2	yes	—
50	C₂×C₄	C₂×#11761640	1	1	yes	characteristic
51	C₈	—	538	4	no	—
52	C₂²	—	269	2	no	—
53	C₂²	—	538	2	no	—
54	C₂²	C₂²×#11761640	1	1	yes	characteristic
55	C₄	—	2	2	no	—
56	C₄	—	269	1	no	—
57	C₄	—	269	1	no	—
58	C₄	—	269	4	no	—
59	C₄	C₄⋊#11761640	1	2	yes	—
60	C₂	—	538	2	no	—
61	C₂	C₂×C₄⋊#11761640	1	1	yes	characteristic
62	C₂	D₂₆₉⋊M₁₆	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×#11761640
commutator, socle	C₂²×C₂₆₉	C₂²×C₄
Fitting	C₄²×C₂₆₉	C₄
prime radical	C₂²×D₂₆₉	C₂×C₄
cosocle	C₂×Dic₂₆₉	C₂³
hypercenter	C₄²	#11761640
hypocenter	C₂₆₉	#423
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₄²

Complements

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

N	H ≅ G/N	type
2. #22575520	C₂	semidirect
25. Dic₂₆₉	M₁₆	semidirect
31. C₂₆₉	#423	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16864412	2
#22575520	2
C₄²×D₂₆₉	2
#423	269