Subgroups — #46819283

Order 34432.

648 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22575453	C₂	1	6	yes	maximal
2	#31770475	C₂	1	8	yes	maximal
3	C₄³×C₂₆₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₅₃₈	C₂²	1	4	yes	—
5	#22575453	—	2	24	no	—
6	C₂×C₄⋊Dic₂₆₉	C₂²	1	24	yes	—
7	C₂×C₄²×C₂₆₉	C₂²	1	3	yes	—
8	C₂×C₄²×C₂₆₉	C₂²	1	4	yes	—
9	C₂×Dic₅₃₈	—	2	24	no	—
10	C₂²×Dic₂₆₉	C₂³	1	8	yes	—
11	C₄⋊Dic₂₆₉	—	2	48	no	—
12	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
13	C₂²×C₄×C₂₆₉	C₂³	1	6	yes	—
14	C₄²×C₂₆₉	D₄	1	12	yes	—
15	C₄²×C₂₆₉	Q₈	1	16	yes	complemented
16	Dic₅₃₈	—	4	16	no	—
17	C₂×Dic₂₆₉	—	2	24	no	—
18	C₂³×C₂₆₉	C₂⁴	1	1	yes	characteristic
19	C₂×C₄×C₂₆₉	C₂×D₄	1	6	yes	—
20	C₂×C₄×C₂₆₉	C₂×D₄	1	12	yes	—
21	C₂×C₄×C₂₆₉	C₂×Q₈	1	24	yes	—
22	C₂²×C₂₆₉	C₂²×D₄	1	3	yes	—
23	C₂²×C₂₆₉	C₂²×Q₈	1	4	yes	—
24	C₄×C₂₆₉	#127	1	4	yes	—
25	C₄×C₂₆₉	C₄⋊Q₈	1	24	yes	complemented
26	Dic₂₆₉	—	4	8	no	—
27	C₂×C₂₆₉	C₂×#127	1	1	yes	characteristic
28	C₂×C₂₆₉	C₂×C₄⋊Q₈	1	6	yes	—
29	C₂₆₉	#2868	1	1	yes	characteristic, complemented, Sylow
30	#2868	—	269	1	no	maximal, Sylow
31	C₂×C₄⋊Q₈	—	269	6	no	—
32	#384	—	269	8	no	—
33	C₄³	D₂₆₉	1	1	yes	characteristic
34	C₂²×Q₈	—	269	4	no	—
35	C₄⋊Q₈	—	538	24	no	—
36	C₂×C₄⋊C₄	—	269	24	no	—
37	C₂×C₄²	C₂×D₂₆₉	1	3	yes	—
38	C₂×C₄²	C₂×D₂₆₉	1	4	yes	—
39	C₂×Q₈	—	538	24	no	—
40	C₂²×C₄	—	269	8	no	—
41	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
42	C₂²×C₄	C₂²×D₂₆₉	1	6	yes	—
43	C₄⋊C₄	—	538	48	no	—
44	C₄²	D₁₀₇₆	1	12	yes	—
45	C₄²	Dic₅₃₈	1	16	yes	complemented
46	C₂³	C₂³×D₂₆₉	1	1	yes	characteristic
47	Q₈	—	1076	16	no	—
48	C₂×C₄	—	538	24	no	—
49	C₂×C₄	C₂×D₁₀₇₆	1	6	yes	—
50	C₂×C₄	C₂×D₁₀₇₆	1	12	yes	—
51	C₂×C₄	C₂×Dic₅₃₈	1	24	yes	—
52	C₂²	C₂²×D₁₀₇₆	1	3	yes	—
53	C₂²	C₂²×Dic₅₃₈	1	4	yes	—
54	C₄	—	1076	8	no	—
55	C₄	#22575450	1	4	yes	—
56	C₄	#22575453	1	24	yes	complemented
57	C₂	C₂×#22575450	1	1	yes	characteristic
58	C₂	C₂×#22575453	1	6	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₆₉
commutator, socle, prime radical, cosocle	C₂³×C₂₆₉	C₂⁴
Fitting	C₄³×C₂₆₉	C₂
hypercenter	C₄³	D₂₆₉
hypocenter	C₂₆₉	#2868
LCS	G ▷ C₂³×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂³ ◁ C₄³

Complements

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

N	H ≅ G/N	type
15. C₄²×C₂₆₉	Q₈	semidirect
25. C₄×C₂₆₉	C₄⋊Q₈	semidirect
29. C₂₆₉	#2868	semidirect
45. C₄²	Dic₅₃₈	semidirect
56. C₄	#22575453	semidirect

Maximal subgroups

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

subgroup	index
C₂×#22575453	2
#31770475	2
C₄³×C₂₆₉	2
#2868	269