Subgroups — #46809028

Order 34432.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31771021	C₂	1	1	yes	maximal, characteristic
2	C₂×#22575359	C₂	1	1	yes	maximal, characteristic, complemented
3	M₆₄×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#22575459	C₄	1	1	yes	characteristic
5	#22575393	C₄	1	1	yes	characteristic
6	#22575392	—	2	1	no	—
7	C₁₆×D₂₆₉	—	2	1	no	—
8	#22575359	—	2	1	no	—
9	C₂×C₁₆×C₂₆₉	C₂²	1	1	yes	characteristic
10	C₃₂×C₂₆₉	—	2	1	no	—
11	#16864402	C₈	1	1	yes	characteristic
12	#16864386	C₈	1	1	yes	characteristic
13	#16864385	—	2	1	no	—
14	C₈×D₂₆₉	—	2	1	no	—
15	#16864377	—	2	1	no	—
16	C₂×C₈×C₂₆₉	C₂×C₄	1	1	yes	characteristic
17	C₁₆×C₂₆₉	D₄	1	1	yes	characteristic
18	C₁₆×C₂₆₉	D₄	1	1	yes	characteristic
19	Dic₅₃₈	C₁₆	1	1	yes	characteristic
20	D₁₀₇₆	C₁₆	1	1	yes	characteristic
21	C₂²⋊D₂₆₉	—	2	1	no	—
22	C₄×D₂₆₉	—	2	1	no	—
23	#14524130	—	2	1	no	—
24	C₂×C₄×C₂₆₉	C₂×C₈	1	1	yes	characteristic
25	C₈×C₂₆₉	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₂₆₉	M₁₆	1	1	yes	characteristic
27	C₂²×C₂₆₉	M₃₂	1	1	yes	characteristic
28	C₂×D₂₆₉	—	2	1	no	—
29	C₄×C₂₆₉	C₂²⋊C₈	1	1	yes	characteristic
30	C₄×C₂₆₉	C₂×C₁₆	1	1	yes	characteristic
31	Dic₂₆₉	—	2	1	no	—
32	C₂×C₂₆₉	—	2	1	no	—
33	C₂×C₂₆₉	C₂²⋊C₁₆	1	1	yes	characteristic
34	D₂₆₉	—	4	1	no	—
35	C₂₆₉	#1401	1	1	yes	characteristic, complemented, Sylow
36	#1401	—	269	1	no	maximal, Sylow
37	#504	—	269	1	no	—
38	M₆₄	D₂₆₉	1	1	yes	characteristic, complemented
39	C₂×C₃₂	—	269	1	no	—
40	#131	—	269	1	no	—
41	M₃₂	—	269	1	no	—
42	M₃₂	—	538	1	no	—
43	C₂×C₁₆	—	538	1	no	—
44	C₂×C₁₆	C₂×D₂₆₉	1	1	yes	characteristic
45	C₃₂	—	2	1	no	—
46	C₃₂	—	538	1	no	—
47	#41	—	269	1	no	—
48	M₁₆	—	269	1	no	—
49	M₁₆	—	538	1	no	—
50	C₂×C₈	—	538	1	no	—
51	C₂×C₈	C₄×D₂₆₉	1	1	yes	characteristic
52	C₁₆	—	538	1	no	—
53	C₁₆	C₂²⋊D₂₆₉	1	1	yes	characteristic
54	C₁₆	D₁₀₇₆	1	1	yes	characteristic
55	Q₈	—	269	1	no	—
56	D₄	—	269	1	no	—
57	D₄	—	538	1	no	—
58	C₂×C₄	—	538	1	no	—
59	C₂×C₄	C₈×D₂₆₉	1	1	yes	characteristic
60	C₈	—	538	1	no	—
61	C₈	#16864379	1	1	yes	characteristic
62	C₈	#16864385	1	1	yes	characteristic
63	C₂²	—	538	1	no	—
64	C₂²	#22575392	1	1	yes	characteristic
65	C₄	—	538	1	no	—
66	C₄	#22575365	1	1	yes	characteristic
67	C₄	C₁₆×D₂₆₉	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	1076	1	no	—
70	C₂	#31770384	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₂²⋊D₂₆₉
commutator	C₄×C₂₆₉	C₂×C₁₆
Frattini	C₂×C₁₆	C₂×D₂₆₉
Fitting	M₆₄×C₂₆₉	C₂
socle	C₂×C₂₆₉	C₂²⋊C₁₆
prime radical	#16864402	C₈
cosocle	C₂×C₁₆×C₂₆₉	C₂²
hypercenter	M₆₄	D₂₆₉
hypocenter	C₂₆₉	#1401
LCS	G ▷ C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
2. C₂×#22575359	C₂	semidirect
3. M₆₄×C₂₆₉	C₂	semidirect
35. C₂₆₉	#1401	semidirect
38. M₆₄	D₂₆₉	semidirect

Maximal subgroups

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

subgroup	index
#31771021	2
C₂×#22575359	2
M₆₄×C₂₆₉	2
#1401	269