Subgroups — #31771551

Order 17216.

186 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#11761640	C₂	1	6	yes	maximal
2	C₄²×D₂₆₉	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11761640	C₂²	1	4	yes	—
4	C₄⋊#11761640	—	2	12	no	—
5	C₂×C₄×D₂₆₉	C₂²	1	3	yes	—
6	C₄×Dic₂₆₉	C₄	1	3	yes	complemented
7	C₄²×C₂₆₉	C₄	1	1	yes	characteristic, complemented
8	C₂×#11761640	—	2	12	no	—
9	C₂²×D₂₆₉	C₂³	1	1	yes	characteristic
10	C₂×Dic₂₆₉	C₂×C₄	1	3	yes	—
11	C₄×D₂₆₉	D₄	1	6	yes	—
12	C₄×D₂₆₉	Q₈	1	6	yes	—
13	C₂×C₄×C₂₆₉	C₂×C₄	1	3	yes	—
14	C₂²×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
15	C₂×D₂₆₉	C₂×D₄	1	3	yes	—
16	C₂×D₂₆₉	C₂×Q₈	1	3	yes	—
17	#11761640	—	4	4	no	—
18	C₄×C₂₆₉	C₄⋊C₄	1	6	yes	complemented
19	Dic₂₆₉	C₄⋊C₄	1	6	yes	complemented
20	C₂×C₂₆₉	C₂×C₄⋊C₄	1	3	yes	—
21	D₂₆₉	#127	1	1	yes	characteristic
22	D₂₆₉	C₄⋊Q₈	1	3	yes	—
23	C₂₆₉	#384	1	1	yes	characteristic, complemented, Sylow
24	#384	—	269	1	no	maximal, Sylow
25	C₂×C₄⋊C₄	—	269	6	no	—
26	C₂×C₄²	—	269	1	no	—
27	C₂²×C₄	—	269	3	no	—
28	C₂²×C₄	—	269	4	no	—
29	C₄⋊C₄	—	538	12	no	—
30	C₄²	—	269	3	no	—
31	C₄²	#11761640	1	1	yes	characteristic, complemented
32	C₂³	—	269	1	no	—
33	C₂×C₄	—	269	3	no	—
34	C₂×C₄	—	269	6	no	—
35	C₂×C₄	—	269	6	no	—
36	C₂×C₄	—	538	12	no	—
37	C₂×C₄	C₂×#11761640	1	3	yes	—
38	C₂²	—	269	3	no	—
39	C₂²	—	269	3	no	—
40	C₂²	C₂²×#11761640	1	1	yes	characteristic
41	C₄	—	269	6	no	—
42	C₄	—	1076	4	no	—
43	C₄	C₄⋊#11761640	1	6	yes	complemented
44	C₂	—	269	1	no	—
45	C₂	—	269	3	no	—
46	C₂	C₂×C₄⋊#11761640	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×#11761640
commutator, socle	C₂²×C₂₆₉	C₂²×C₄
Fitting	C₄²×C₂₆₉	C₄
prime radical, cosocle	C₂²×D₂₆₉	C₂³
hypercenter	C₄²	#11761640
hypocenter	C₂₆₉	#384
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
6. C₄×Dic₂₆₉	C₄	semidirect
7. C₄²×C₂₆₉	C₄	semidirect
18. C₄×C₂₆₉	C₄⋊C₄	semidirect
19. Dic₂₆₉	C₄⋊C₄	semidirect
23. C₂₆₉	#384	semidirect
31. C₄²	#11761640	semidirect
43. C₄	C₄⋊#11761640	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#11761640	2
C₄²×D₂₆₉	2
#384	269