Subgroups — #46809065

Order 34432.

74 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	C₂×#22575359	C₂	1	1	yes	maximal, characteristic
3	#366×C₂₆₉	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₁₀₇₆	—	2	1	no	—
5	Dic₂₁₅₂	—	2	1	no	—
6	Dic₂₁₅₂	C₄	1	2	yes	complemented
7	#22575359	—	2	1	no	—
8	C₂×C₁₆×C₂₆₉	C₂²	1	1	yes	characteristic
9	#107×C₂₆₉	—	2	1	no	—
10	C₂×Dic₅₃₈	—	4	1	no	—
11	Dic₁₀₇₆	—	2	2	no	—
12	Dic₁₀₇₆	—	4	1	no	—
13	C₂×C₈×C₂₆₉	D₄	1	1	yes	characteristic
14	C₄⋊C₄×C₂₆₉	—	4	1	no	—
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	C₂×C₄×C₂₆₉	—	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	D₁₆	1	1	yes	characteristic
25	C₄×C₂₆₉	—	16	1	no	—
26	C₄×C₂₆₉	#102	1	1	yes	characteristic
27	C₄×C₂₆₉	SD₃₂	1	1	yes	characteristic
28	Dic₂₆₉	—	8	2	no	—
29	C₂×C₂₆₉	#357	1	1	yes	characteristic
30	C₂×C₂₆₉	SD₆₄	1	1	yes	characteristic
31	C₂×C₂₆₉	Q₆₄	1	1	yes	characteristic
32	C₂₆₉	#1416	1	1	yes	characteristic, complemented, Sylow
33	#1416	—	269	1	no	maximal, Sylow
34	C₂×Q₃₂	—	269	1	no	—
35	C₂×C₃₂	—	269	1	no	—
36	#366	D₂₆₉	1	1	yes	characteristic
37	C₂×Q₁₆	—	538	1	no	—
38	Q₃₂	—	269	2	no	—
39	Q₃₂	—	538	1	no	—
40	C₂×C₁₆	C₂×D₂₆₉	1	1	yes	characteristic
41	#107	—	2	1	no	—
42	C₃₂	—	538	1	no	—
43	C₂×Q₈	—	1076	1	no	—
44	Q₁₆	—	538	2	no	—
45	Q₁₆	—	1076	1	no	—
46	C₂×C₈	C₂²⋊D₂₆₉	1	1	yes	characteristic
47	C₄⋊C₄	—	4	1	no	—
48	C₁₆	C₄×D₂₆₉	1	1	yes	characteristic
49	C₁₆	D₁₀₇₆	1	1	yes	characteristic
50	Q₈	—	1076	2	no	—
51	Q₈	—	2152	1	no	—
52	C₂×C₄	—	8	1	no	—
53	C₂×C₄	—	2152	1	no	—
54	C₂×C₄	#16864387	1	1	yes	characteristic
55	C₈	#16864379	1	1	yes	characteristic
56	C₈	#16864389	1	1	yes	characteristic
57	C₂²	#22575394	1	1	yes	characteristic
58	C₄	—	16	1	no	—
59	C₄	—	2152	2	no	—
60	C₄	#22575373	1	1	yes	characteristic
61	C₄	#22575396	1	1	yes	characteristic
62	C₂	#31770405	1	1	yes	characteristic
63	C₂	#31770443	1	1	yes	characteristic
64	C₂	#31770445	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#22575394
commutator	C₁₆×C₂₆₉	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₂₆₉
Fitting	#366×C₂₆₉	C₂
socle, prime radical	C₂²×C₂₆₉	D₁₆
cosocle	C₂×C₁₆×C₂₆₉	C₂²
hypercenter	#366	D₂₆₉
hypocenter	C₂₆₉	#1416
LCS	G ▷ C₁₆×C₂₆₉ ▷ C₈×C₂₆₉ ▷ C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #366

Complements

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

N	H ≅ G/N	type
6. Dic₂₁₅₂	C₄	semidirect
32. C₂₆₉	#1416	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
C₂×#22575359	2
#366×C₂₆₉	2
#1416	269