Subgroups — #31770419

Order 17216.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₂₁₅₂	C₂	1	1	yes	maximal, characteristic
2	#22575389	C₂	1	1	yes	maximal, characteristic, complemented
3	M₃₂×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₀₇₆	—	2	1	no	—
5	D₂₁₅₂	—	2	1	no	—
6	D₂₁₅₂	C₄	1	2	yes	—
7	#16864386	—	2	1	no	—
8	C₂×C₈×C₂₆₉	C₂²	1	1	yes	characteristic
9	C₁₆×C₂₆₉	—	2	1	no	—
10	C₂²×D₂₆₉	—	4	1	no	—
11	D₁₀₇₆	—	2	2	no	—
12	D₁₀₇₆	—	4	1	no	—
13	#14524130	—	4	1	no	—
14	C₂×C₄×C₂₆₉	D₄	1	1	yes	characteristic
15	C₈×C₂₆₉	C₂×C₄	1	1	yes	characteristic
16	C₈×C₂₆₉	D₄	1	1	yes	characteristic
17	C₂²×C₂₆₉	SD₁₆	1	1	yes	characteristic
18	C₂×D₂₆₉	—	4	2	no	—
19	C₂×D₂₆₉	—	8	2	no	—
20	C₄×C₂₆₉	C₂²⋊C₄	1	1	yes	characteristic
21	C₄×C₂₆₉	D₈	1	1	yes	characteristic
22	C₂×C₂₆₉	—	2	1	no	—
23	C₂×C₂₆₉	#102	1	1	yes	characteristic
24	D₂₆₉	—	8	2	no	—
25	C₂₆₉	#361	1	1	yes	characteristic, complemented, Sylow
26	#361	—	269	1	no	maximal, Sylow
27	C₂×D₈	—	269	1	no	—
28	M₃₂	D₂₆₉	1	1	yes	characteristic, complemented
29	#108	—	269	1	no	—
30	C₂×D₄	—	538	1	no	—
31	D₈	—	269	2	no	—
32	D₈	—	538	1	no	—
33	M₁₆	—	538	1	no	—
34	C₂×C₈	C₂×D₂₆₉	1	1	yes	characteristic
35	C₁₆	—	2	1	no	—
36	C₂³	—	1076	1	no	—
37	D₄	—	538	2	no	—
38	D₄	—	1076	1	no	—
39	C₂×C₄	D₁₀₇₆	1	1	yes	characteristic
40	C₈	—	1076	1	no	—
41	C₈	C₄×D₂₆₉	1	1	yes	characteristic
42	C₈	C₂²⋊D₂₆₉	1	1	yes	characteristic
43	C₂²	—	1076	2	no	—
44	C₂²	—	2152	2	no	—
45	C₂²	#16864391	1	1	yes	characteristic
46	C₄	#16864379	1	1	yes	characteristic
47	C₄	D₂₁₅₂	1	1	yes	characteristic
48	C₂	—	2	1	no	—
49	C₂	—	2152	2	no	—
50	C₂	#22575375	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22575375
commutator	C₈×C₂₆₉	C₂×C₄
Frattini	C₂×C₈	C₂×D₂₆₉
Fitting	M₃₂×C₂₆₉	C₂
socle	C₂×C₂₆₉	#102
prime radical	C₂×D₂₁₅₂	C₂
cosocle	C₂×C₈×C₂₆₉	C₂²
hypercenter	M₃₂	D₂₆₉
hypocenter	C₂₆₉	#361
LCS	G ▷ C₈×C₂₆₉ ▷ C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
2. #22575389	C₂	semidirect
3. M₃₂×C₂₆₉	C₂	semidirect
25. C₂₆₉	#361	semidirect
28. M₃₂	D₂₆₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₂₁₅₂	2
#22575389	2
M₃₂×C₂₆₉	2
#361	269