Subgroups — #46809079

Order 34432.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₂₁₅₂	C₂	1	1	yes	maximal, characteristic
2	#31770435	C₂	1	1	yes	maximal, characteristic
3	M₆₄×C₂₆₉	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₁₀₇₆	—	2	1	no	—
5	Dic₂₁₅₂	—	2	1	no	—
6	Dic₂₁₅₂	C₄	1	2	yes	—
7	#22575389	—	2	1	no	—
8	C₂×C₁₆×C₂₆₉	C₂²	1	1	yes	characteristic
9	C₃₂×C₂₆₉	—	2	1	no	—
10	C₂×Dic₅₃₈	—	4	1	no	—
11	Dic₁₀₇₆	—	2	2	no	—
12	Dic₁₀₇₆	—	4	1	no	—
13	#16864386	—	4	1	no	—
14	C₂×C₈×C₂₆₉	D₄	1	1	yes	characteristic
15	C₁₆×C₂₆₉	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₂₆₉	D₄	1	1	yes	characteristic
17	Dic₅₃₈	—	4	2	no	—
18	Dic₅₃₈	—	8	1	no	—
19	C₂×Dic₂₆₉	—	8	1	no	—
20	#14524130	—	8	1	no	—
21	C₂×C₄×C₂₆₉	D₈	1	1	yes	characteristic
22	C₈×C₂₆₉	C₂²⋊C₄	1	1	yes	characteristic
23	C₈×C₂₆₉	SD₁₆	1	1	yes	characteristic
24	C₂²×C₂₆₉	SD₃₂	1	1	yes	characteristic
25	C₄×C₂₆₉	#102	1	1	yes	characteristic
26	C₄×C₂₆₉	D₁₆	1	1	yes	characteristic
27	Dic₂₆₉	—	8	2	no	—
28	C₂×C₂₆₉	—	2	1	no	—
29	C₂×C₂₆₉	#357	1	1	yes	characteristic
30	C₂₆₉	#1420	1	1	yes	characteristic, complemented, Sylow
31	#1420	—	269	1	no	maximal, Sylow
32	C₂×Q₃₂	—	269	1	no	—
33	M₆₄	D₂₆₉	1	1	yes	characteristic
34	#368	—	269	1	no	—
35	C₂×Q₁₆	—	538	1	no	—
36	Q₃₂	—	269	2	no	—
37	Q₃₂	—	538	1	no	—
38	C₂×C₁₆	C₂×D₂₆₉	1	1	yes	characteristic
39	#108	—	538	1	no	—
40	C₃₂	—	2	1	no	—
41	C₂×Q₈	—	1076	1	no	—
42	Q₁₆	—	538	2	no	—
43	Q₁₆	—	1076	1	no	—
44	M₁₆	—	1076	1	no	—
45	C₂×C₈	D₁₀₇₆	1	1	yes	characteristic
46	C₁₆	C₄×D₂₆₉	1	1	yes	characteristic
47	C₁₆	C₂²⋊D₂₆₉	1	1	yes	characteristic
48	Q₈	—	1076	2	no	—
49	Q₈	—	2152	1	no	—
50	C₂×C₄	—	2152	1	no	—
51	C₂×C₄	D₂₁₅₂	1	1	yes	characteristic
52	C₈	—	2152	1	no	—
53	C₈	#16864379	1	1	yes	characteristic
54	C₈	#16864391	1	1	yes	characteristic
55	C₂²	#22575398	1	1	yes	characteristic
56	C₄	—	2152	2	no	—
57	C₄	#22575375	1	1	yes	characteristic
58	C₄	D₄₃₀₄	1	1	yes	characteristic
59	C₂	—	2	1	no	—
60	C₂	#31770407	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#31770407
commutator	C₁₆×C₂₆₉	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₂₆₉
Fitting	M₆₄×C₂₆₉	C₂
socle	C₂×C₂₆₉	#357
prime radical	C₂²×C₂₆₉	SD₃₂
cosocle	C₂×C₁₆×C₂₆₉	C₂²
hypercenter	M₆₄	D₂₆₉
hypocenter	C₂₆₉	#1420
LCS	G ▷ C₁₆×C₂₆₉ ▷ C₈×C₂₆₉ ▷ C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
30. C₂₆₉	#1420	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₂₁₅₂	2
#31770435	2
M₆₄×C₂₆₉	2
#1420	269