Subgroups — #31771042

Order 17216.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22575476	C₂	1	1	yes	maximal, characteristic, complemented
2	D₈×D₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#22575396	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄₃₀₄	C₂	1	1	yes	maximal, characteristic, complemented
5	#22575394	C₂	1	1	yes	maximal, characteristic, complemented
6	#22575392	C₂	1	1	yes	maximal, characteristic, complemented
7	SD₃₂×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#16864403	—	2	1	no	—
9	D₄×D₂₆₉	—	2	1	no	—
10	#16864389	—	2	1	no	—
11	D₂₁₅₂	C₂²	1	1	yes	characteristic
12	D₂₁₅₂	C₂²	1	1	yes	characteristic, complemented
13	#16864387	—	2	1	no	—
14	C₈×D₂₆₉	C₂²	1	1	yes	characteristic
15	#16864377	C₂²	1	1	yes	characteristic, complemented
16	Q₁₆×C₂₆₉	C₂²	1	1	yes	characteristic, complemented
17	D₈×C₂₆₉	C₂²	1	1	yes	characteristic
18	C₁₆×C₂₆₉	C₂²	1	1	yes	characteristic, complemented
19	C₂²×D₂₆₉	—	4	1	no	—
20	D₁₀₇₆	—	2	1	no	—
21	D₁₀₇₆	—	2	1	no	—
22	D₁₀₇₆	—	4	1	no	—
23	C₂²⋊D₂₆₉	—	4	1	no	—
24	C₄×D₂₆₉	—	4	1	no	—
25	C₄×D₂₆₉	D₄	1	1	yes	characteristic
26	#14524130	D₄	1	1	yes	characteristic
27	Q₈×C₂₆₉	—	2	1	no	—
28	D₄×C₂₆₉	—	2	1	no	—
29	C₈×C₂₆₉	C₂³	1	1	yes	characteristic
30	C₂²×C₂₆₉	—	4	1	no	—
31	C₂×D₂₆₉	—	4	1	no	—
32	C₂×D₂₆₉	—	4	1	no	—
33	C₂×D₂₆₉	—	8	2	no	—
34	C₂×D₂₆₉	D₈	1	1	yes	characteristic
35	C₄×C₂₆₉	—	4	1	no	—
36	C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
37	Dic₂₆₉	D₈	1	1	yes	characteristic
38	C₂×C₂₆₉	—	8	1	no	—
39	C₂×C₂₆₉	C₂×D₈	1	1	yes	characteristic
40	D₂₆₉	—	2	1	no	—
41	D₂₆₉	—	8	1	no	—
42	D₂₆₉	—	8	1	no	—
43	C₂₆₉	#509	1	1	yes	characteristic, complemented, Sylow
44	#509	—	269	1	no	maximal, Sylow
45	#135	—	269	1	no	—
46	C₂×D₈	—	269	1	no	—
47	SD₃₂	—	269	1	no	—
48	SD₃₂	D₂₆₉	1	1	yes	characteristic, complemented
49	D₁₆	—	269	1	no	—
50	D₁₆	—	269	1	no	—
51	M₃₂	—	269	1	no	—
52	#41	—	538	1	no	—
53	C₂×D₄	—	538	1	no	—
54	Q₁₆	C₂×D₂₆₉	1	1	yes	characteristic, complemented
55	SD₁₆	—	538	1	no	—
56	D₈	—	269	1	no	—
57	D₈	—	269	1	no	—
58	D₈	—	538	1	no	—
59	D₈	C₂×D₂₆₉	1	1	yes	characteristic
60	C₂×C₈	—	269	1	no	—
61	C₁₆	—	269	1	no	—
62	C₁₆	C₂×D₂₆₉	1	1	yes	characteristic, complemented
63	C₂³	—	1076	1	no	—
64	Q₈	—	2	1	no	—
65	D₄	—	2	1	no	—
66	D₄	—	538	1	no	—
67	D₄	—	538	1	no	—
68	D₄	—	1076	1	no	—
69	D₄	—	1076	1	no	—
70	C₂×C₄	—	269	1	no	—
71	C₂×C₄	—	1076	1	no	—
72	C₈	—	269	1	no	—
73	C₈	C₂²×D₂₆₉	1	1	yes	characteristic
74	C₂²	—	4	1	no	—
75	C₂²	—	269	1	no	—
76	C₂²	—	1076	1	no	—
77	C₂²	—	1076	1	no	—
78	C₂²	—	2152	2	no	—
79	C₄	—	4	1	no	—
80	C₄	—	269	1	no	—
81	C₄	D₄×D₂₆₉	1	1	yes	characteristic
82	C₂	—	8	1	no	—
83	C₂	—	538	1	no	—
84	C₂	—	2152	1	no	—
85	C₂	—	2152	1	no	—
86	C₂	D₈×D₂₆₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×D₂₆₉
commutator, cosocle	C₈×C₂₆₉	C₂³
Frattini	C₈	C₂²×D₂₆₉
Fitting	SD₃₂×C₂₆₉	C₂
socle	C₂×C₂₆₉	C₂×D₈
hypercenter	SD₃₂	D₂₆₉
hypocenter	C₂₆₉	#509
LCS	G ▷ C₈×C₂₆₉ ▷ C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ SD₃₂

Complements

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

N	H ≅ G/N	type
1. #22575476	C₂	semidirect
2. D₈×D₂₆₉	C₂	semidirect
3. #22575396	C₂	semidirect
4. D₄₃₀₄	C₂	semidirect
5. #22575394	C₂	semidirect
6. #22575392	C₂	semidirect
7. SD₃₂×C₂₆₉	C₂	semidirect
12. D₂₁₅₂	C₂²	semidirect
15. #16864377	C₂²	semidirect
16. Q₁₆×C₂₆₉	C₂²	semidirect
18. C₁₆×C₂₆₉	C₂²	semidirect
43. C₂₆₉	#509	semidirect
48. SD₃₂	D₂₆₉	semidirect
54. Q₁₆	C₂×D₂₆₉	semidirect
62. C₁₆	C₂×D₂₆₉	semidirect

Maximal subgroups

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

subgroup	index
#22575476	2
D₈×D₂₆₉	2
#22575396	2
D₄₃₀₄	2
#22575394	2
#22575392	2
SD₃₂×C₂₆₉	2
#509	269