Subgroups — #46825796

Order 34432.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31771491	C₂	1	1	yes	maximal, characteristic
2	C₈×#14524137	C₂	1	1	yes	maximal, characteristic
3	C₁₆×Dic₂₆₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#16864405	C₄	1	2	yes	—
5	C₄×#14524137	C₄	1	2	yes	—
6	C₂×#16864377	C₄	1	1	yes	characteristic
7	C₈×Dic₂₆₉	C₂²	1	1	yes	characteristic
8	C₂×C₁₆×C₂₆₉	C₄	1	1	yes	characteristic
9	C₂×#14524137	C₈	1	4	yes	—
10	#16864405	C₈	1	4	yes	complemented
11	C₂×#14524130	C₂×C₄	1	1	yes	characteristic
12	C₄×Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
13	#16864377	C₈	1	2	yes	complemented
14	C₂×C₈×C₂₆₉	C₂×C₄	1	1	yes	characteristic
15	C₁₆×C₂₆₉	C₈	1	2	yes	complemented
16	#14524137	—	2	4	no	—
17	C₂×Dic₂₆₉	C₂×C₈	1	2	yes	—
18	#14524130	C₂×C₈	1	2	yes	—
19	C₂×C₄×C₂₆₉	C₄²	1	1	yes	characteristic
20	C₈×C₂₆₉	C₂×C₈	1	1	yes	characteristic
21	C₈×C₂₆₉	C₂×C₈	1	1	yes	characteristic
22	C₂²×C₂₆₉	C₄×C₈	1	1	yes	characteristic
23	C₄×C₂₆₉	C₄×C₈	1	1	yes	characteristic
24	C₄×C₂₆₉	C₄×C₈	1	1	yes	characteristic
25	Dic₂₆₉	M₃₂	1	4	yes	—
26	C₂×C₂₆₉	C₈²	1	1	yes	characteristic
27	C₂×C₂₆₉	#346	1	2	yes	—
28	C₂₆₉	#1311	1	1	yes	characteristic, complemented, Sylow
29	#1311	—	269	1	no	maximal, Sylow
30	C₄×C₁₆	—	269	1	no	—
31	C₄×C₁₆	—	269	1	no	—
32	C₈²	—	269	1	no	—
33	C₂×C₁₆	—	269	2	no	—
34	C₂×C₁₆	—	269	1	no	—
35	C₂×C₁₆	#11761640	1	1	yes	characteristic
36	C₄×C₈	—	269	2	no	—
37	C₄×C₈	—	269	1	no	—
38	C₂×C₈	—	269	1	no	—
39	C₂×C₈	—	269	4	no	—
40	C₂×C₈	C₂×#11761640	1	1	yes	characteristic
41	C₄²	—	269	1	no	—
42	C₁₆	—	269	4	no	—
43	C₁₆	—	269	2	no	—
44	C₁₆	#14524137	1	2	yes	complemented
45	C₂×C₄	—	269	2	no	—
46	C₂×C₄	C₄×#11761640	1	1	yes	characteristic
47	C₈	—	269	2	no	—
48	C₈	—	538	4	no	—
49	C₈	C₂×#14524137	1	1	yes	characteristic
50	C₈	D₂₆₉⋊C₈	1	1	yes	characteristic
51	C₂²	C₈×#11761640	1	1	yes	characteristic
52	C₄	—	269	4	no	—
53	C₄	C₈×#11761640	1	1	yes	characteristic
54	C₄	C₄×#14524137	1	1	yes	characteristic
55	C₂	C₈×#14524137	1	1	yes	characteristic
56	C₂	#31771492	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂×#11761640
commutator	C₂×C₂₆₉	C₈²
Fitting	C₂×C₁₆×C₂₆₉	C₄
socle, prime radical	C₂²×C₂₆₉	C₄×C₈
cosocle	C₈×Dic₂₆₉	C₂²
hypercenter	C₂×C₁₆	#11761640
hypocenter	C₂₆₉	#1311
LCS	G ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
10. #16864405	C₈	semidirect
13. #16864377	C₈	semidirect
15. C₁₆×C₂₆₉	C₈	semidirect
28. C₂₆₉	#1311	semidirect
44. C₁₆	#14524137	semidirect

Maximal subgroups

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

subgroup	index
#31771491	2
C₈×#14524137	2
C₁₆×Dic₂₆₉	2
#1311	269