Subgroups — #31770988

Order 17216.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₀₇₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16864390	C₂	1	2	yes	maximal, complemented
3	#22575440	C₂	1	2	yes	maximal, complemented
4	C₈×Dic₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₈×C₂₆₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₃₈	C₂²	1	2	yes	—
7	Dic₁₀₇₆	—	2	2	no	—
8	#16864390	—	2	4	no	—
9	C₂×#14524130	C₂²	1	1	yes	characteristic
10	C₂²⋊Dic₂₆₉	—	2	4	no	—
11	C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
12	C₂×D₄×C₂₆₉	C₂²	1	2	yes	—
13	D₈×C₂₆₉	—	2	2	no	—
14	C₂×C₈×C₂₆₉	C₂²	1	1	yes	characteristic
15	Dic₅₃₈	—	2	2	no	—
16	Dic₅₃₈	—	2	2	no	—
17	C₂×Dic₂₆₉	—	2	2	no	—
18	C₂×Dic₂₆₉	D₄	1	2	yes	—
19	#14524130	D₄	1	2	yes	—
20	C₂³×C₂₆₉	—	2	2	no	—
21	D₄×C₂₆₉	—	2	2	no	—
22	D₄×C₂₆₉	—	2	2	no	—
23	C₂×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
24	C₈×C₂₆₉	D₄	1	2	yes	—
25	C₂²×C₂₆₉	—	4	2	no	—
26	C₂²×C₂₆₉	—	4	4	no	—
27	C₂²×C₂₆₉	C₂×D₄	1	1	yes	characteristic
28	C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
29	C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
30	Dic₂₆₉	—	2	2	no	—
31	Dic₂₆₉	—	4	2	no	—
32	C₂×C₂₆₉	—	8	2	no	—
33	C₂×C₂₆₉	#127	1	1	yes	characteristic
34	C₂×C₂₆₉	#135	1	2	yes	—
35	C₂₆₉	#495	1	1	yes	characteristic, complemented, Sylow
36	#495	—	269	1	no	maximal, Sylow
37	C₂×Q₁₆	—	269	1	no	—
38	C₂×SD₁₆	—	269	2	no	—
39	C₂×D₈	D₂₆₉	1	1	yes	characteristic
40	#124	—	269	2	no	—
41	C₄×C₈	—	269	1	no	—
42	C₂×Q₈	—	269	2	no	—
43	C₂×D₄	C₂×D₂₆₉	1	2	yes	—
44	Q₁₆	—	538	2	no	—
45	SD₁₆	—	538	4	no	—
46	D₈	—	2	2	no	—
47	C₂×C₈	—	269	1	no	—
48	C₂×C₈	C₂×D₂₆₉	1	1	yes	characteristic
49	C₂²⋊C₄	—	538	4	no	—
50	C₄²	—	269	1	no	—
51	C₂³	—	2	2	no	—
52	Q₈	—	538	2	no	—
53	Q₈	—	538	2	no	—
54	D₄	—	2	2	no	—
55	D₄	—	2	2	no	—
56	C₂×C₄	—	269	2	no	—
57	C₂×C₄	—	538	2	no	—
58	C₂×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
59	C₈	—	269	2	no	—
60	C₈	C₂²⋊D₂₆₉	1	2	yes	—
61	C₂²	—	4	2	no	—
62	C₂²	—	4	4	no	—
63	C₂²	D₄×D₂₆₉	1	1	yes	characteristic
64	C₄	—	538	2	no	—
65	C₄	—	1076	2	no	—
66	C₄	D₄×D₂₆₉	1	1	yes	characteristic
67	C₄	C₂×C₂²⋊D₂₆₉	1	1	yes	characteristic
68	C₂	—	8	2	no	—
69	C₂	#22575449	1	1	yes	characteristic
70	C₂	#22575472	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₂₆₉
commutator, cosocle	C₂×C₄×C₂₆₉	C₂³
Frattini	C₂×C₄	C₂²×D₂₆₉
Fitting, prime radical	C₂×D₈×C₂₆₉	C₂
socle	C₂²×C₂₆₉	C₂×D₄
hypercenter	C₂×D₈	D₂₆₉
hypocenter	C₂₆₉	#495
LCS	G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

Complements

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

N	H ≅ G/N	type
1. C₂×Dic₁₀₇₆	C₂	semidirect
2. C₂×#16864390	C₂	semidirect
3. #22575440	C₂	semidirect
4. C₈×Dic₂₆₉	C₂	semidirect
35. C₂₆₉	#495	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₁₀₇₆	2
C₂×#16864390	2
#22575440	2
C₈×Dic₂₆₉	2
C₂×D₈×C₂₆₉	2
#495	269