Subgroups — #31771673

Order 17216.

266 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#11761640	C₂	1	8	yes	maximal, complemented
2	#22575532	C₂	1	6	yes	maximal
3	C₂×Q₈×D₂₆₉	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#11761640	C₂²	1	12	yes	—
5	C₂²⋊#11761640	C₂²	1	4	yes	—
6	C₄×#11761640	C₂²	1	12	yes	—
7	C₂×Dic₅₃₈	C₄	1	3	yes	complemented
8	Q₈×D₂₆₉	C₄	1	4	yes	complemented
9	Q₈×D₂₆₉	C₂²	1	4	yes	—
10	C₂×C₄×D₂₆₉	C₂²	1	3	yes	—
11	C₂×Q₈×C₂₆₉	C₄	1	1	yes	characteristic, complemented
12	C₂×#11761640	C₂³	1	8	yes	—
13	C₂²×D₂₆₉	C₂³	1	1	yes	characteristic
14	Dic₅₃₈	C₂×C₄	1	12	yes	—
15	C₂×Dic₂₆₉	C₂×C₄	1	3	yes	—
16	C₄×D₂₆₉	C₂×C₄	1	6	yes	—
17	C₄×D₂₆₉	C₂³	1	6	yes	—
18	Q₈×C₂₆₉	C₂×C₄	1	4	yes	—
19	C₂×C₄×C₂₆₉	C₂×C₄	1	3	yes	—
20	C₂²×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
21	C₂×D₂₆₉	—	2	2	no	—
22	C₂×D₂₆₉	C₂²×C₄	1	1	yes	characteristic
23	C₂×D₂₆₉	C₂⁴	1	1	yes	characteristic
24	#11761640	—	2	8	no	—
25	C₄×C₂₆₉	C₂²×C₄	1	6	yes	—
26	Dic₂₆₉	C₂²×C₄	1	6	yes	—
27	C₂×C₂₆₉	—	2	1	no	—
28	C₂×C₂₆₉	C₂³×C₄	1	1	yes	characteristic
29	D₂₆₉	—	2	1	no	—
30	D₂₆₉	2₋¹⁺⁴	1	2	yes	—
31	C₂₆₉	#519	1	1	yes	characteristic, complemented, Sylow
32	#519	—	269	1	no	maximal, Sylow
33	C₂²×Q₈	—	269	1	no	—
34	C₄×Q₈	—	269	8	no	—
35	#117	—	269	6	no	—
36	C₂×Q₈	—	269	4	no	—
37	C₂×Q₈	—	269	4	no	—
38	C₂×Q₈	—	269	3	no	—
39	C₂×Q₈	#11761640	1	1	yes	characteristic, complemented
40	C₂²×C₄	—	269	3	no	—
41	C₄⋊C₄	—	269	12	no	—
42	C₂²⋊C₄	—	269	4	no	—
43	C₄²	—	269	12	no	—
44	C₂³	—	269	1	no	—
45	Q₈	—	269	12	no	—
46	Q₈	C₂×#11761640	1	4	yes	—
47	C₂×C₄	—	269	3	no	—
48	C₂×C₄	—	269	6	no	—
49	C₂×C₄	—	269	6	no	—
50	C₂×C₄	—	269	8	no	—
51	C₂×C₄	C₂×#11761640	1	3	yes	—
52	C₂²	—	269	1	no	—
53	C₂²	—	269	1	no	—
54	C₂²	—	538	2	no	—
55	C₂²	C₂²×#11761640	1	1	yes	characteristic
56	C₄	—	269	6	no	—
57	C₄	—	538	8	no	—
58	C₄	C₂²×#11761640	1	6	yes	—
59	C₂	—	2	1	no	—
60	C₂	—	269	2	no	—
61	C₂	—	538	1	no	—
62	C₂	C₂³×#11761640	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×#11761640
commutator, socle	C₂×C₂₆₉	C₂³×C₄
Fitting	C₂×Q₈×C₂₆₉	C₄
prime radical	C₂²×D₂₆₉	C₂³
cosocle	C₂×D₂₆₉	C₂⁴
hypercenter	C₂×Q₈	#11761640
hypocenter	C₂₆₉	#519
LCS	G ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. Q₈×#11761640	C₂	semidirect
7. C₂×Dic₅₃₈	C₄	semidirect
8. Q₈×D₂₆₉	C₄	semidirect
11. C₂×Q₈×C₂₆₉	C₄	semidirect
31. C₂₆₉	#519	semidirect
39. C₂×Q₈	#11761640	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#11761640	2
#22575532	2
C₂×Q₈×D₂₆₉	2
#519	269