Subgroups — #31771348

Order 17216.

260 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22575440	C₂	1	3	yes	maximal, complemented
2	#22575439	C₂	1	6	yes	maximal, complemented
3	#22575422	C₂	1	2	yes	maximal, complemented
4	#22575421	C₂	1	3	yes	maximal, complemented
5	#120×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₂₆₉	C₂²	1	1	yes	characteristic, complemented
7	C₂×Dic₅₃₈	C₂²	1	3	yes	complemented
8	C₂×C₂²⋊D₂₆₉	C₂²	1	6	yes	—
9	C₂²⋊Dic₂₆₉	C₂²	1	3	yes	complemented
10	C₂²⋊Dic₂₆₉	C₂²	1	6	yes	complemented
11	#16864379	C₂²	1	6	yes	complemented
12	C₄×Dic₂₆₉	C₂²	1	3	yes	complemented
13	C₂⁴×C₂₆₉	C₂²	1	1	yes	characteristic, complemented
14	C₂×D₄×C₂₆₉	C₂²	1	3	yes	—
15	C₂²⋊C₄×C₂₆₉	C₂²	1	3	yes	complemented
16	C₂²×D₂₆₉	—	4	3	no	—
17	C₂²×D₂₆₉	C₂³	1	2	yes	—
18	Dic₅₃₈	—	4	3	no	—
19	C₂²⋊D₂₆₉	—	4	6	no	—
20	C₂×Dic₂₆₉	C₂³	1	6	yes	—
21	C₂³×C₂₆₉	—	4	3	no	—
22	C₂³×C₂₆₉	C₂³	1	3	yes	—
23	C₂³×C₂₆₉	C₂³	1	1	yes	characteristic
24	D₄×C₂₆₉	—	4	3	no	—
25	C₂×C₄×C₂₆₉	C₂³	1	3	yes	—
26	C₂²×C₂₆₉	—	2	3	no	—
27	C₂²×C₂₆₉	—	2	3	no	—
28	C₂²×C₂₆₉	—	2	6	no	—
29	C₂²×C₂₆₉	—	4	4	no	—
30	C₂²×C₂₆₉	C₂⁴	1	1	yes	characteristic
31	C₂×D₂₆₉	—	2	6	no	—
32	C₂×D₂₆₉	—	4	4	no	—
33	C₄×C₂₆₉	—	2	3	no	—
34	Dic₂₆₉	—	2	6	no	—
35	C₂×C₂₆₉	—	4	1	no	—
36	C₂×C₂₆₉	—	4	3	no	—
37	C₂×C₂₆₉	H(5,𝔽₂)	1	3	yes	—
38	D₂₆₉	—	4	2	no	—
39	C₂₆₉	H(3,𝔽₄)	1	1	yes	characteristic, complemented, Sylow
40	H(3,𝔽₄)	—	269	1	no	maximal, Sylow
41	#124	—	269	3	no	—
42	#124	—	269	6	no	—
43	#120	—	269	2	no	—
44	#120	—	269	3	no	—
45	#120	D₂₆₉	1	1	yes	characteristic, complemented
46	C₂⁴	—	269	1	no	—
47	C₂⁴	C₂×D₂₆₉	1	1	yes	characteristic, complemented
48	C₂×Q₈	—	269	3	no	—
49	C₂×D₄	—	269	6	no	—
50	C₂×D₄	C₂×D₂₆₉	1	3	yes	—
51	C₂²⋊C₄	—	269	3	no	—
52	C₂²⋊C₄	—	269	6	no	—
53	C₂²⋊C₄	—	269	6	no	—
54	C₂²⋊C₄	C₂×D₂₆₉	1	3	yes	complemented
55	C₄²	—	269	3	no	—
56	C₂³	—	4	3	no	—
57	C₂³	—	269	2	no	—
58	C₂³	—	1076	3	no	—
59	C₂³	C₂²×D₂₆₉	1	1	yes	characteristic
60	C₂³	C₂²×D₂₆₉	1	3	yes	—
61	Q₈	—	1076	3	no	—
62	D₄	—	4	3	no	—
63	D₄	—	1076	6	no	—
64	C₂×C₄	—	269	6	no	—
65	C₂×C₄	C₂²×D₂₆₉	1	3	yes	—
66	C₂²	—	2	3	no	—
67	C₂²	—	2	3	no	—
68	C₂²	—	2	6	no	—
69	C₂²	—	4	4	no	—
70	C₂²	—	538	6	no	—
71	C₂²	—	1076	4	no	—
72	C₂²	C₂³×D₂₆₉	1	1	yes	characteristic
73	C₄	—	2	3	no	—
74	C₄	—	538	6	no	—
75	C₂	—	4	3	no	—
76	C₂	—	4	1	no	—
77	C₂	—	1076	2	no	—
78	C₂	#22575498	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₆₉
commutator, socle, cosocle	C₂²×C₂₆₉	C₂⁴
Fitting	#120×C₂₆₉	C₂
hypercenter	#120	D₂₆₉
hypocenter	C₂₆₉	H(3,𝔽₄)
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ #120

Complements

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

N	H ≅ G/N	type
1. #22575440	C₂	semidirect
2. #22575439	C₂	semidirect
3. #22575422	C₂	semidirect
4. #22575421	C₂	semidirect
5. #120×C₂₆₉	C₂	semidirect
6. C₂³×D₂₆₉	C₂²	semidirect
7. C₂×Dic₅₃₈	C₂²	semidirect
9. C₂²⋊Dic₂₆₉	C₂²	semidirect
10. C₂²⋊Dic₂₆₉	C₂²	semidirect
11. #16864379	C₂²	semidirect
12. C₄×Dic₂₆₉	C₂²	semidirect
13. C₂⁴×C₂₆₉	C₂²	semidirect
15. C₂²⋊C₄×C₂₆₉	C₂²	semidirect
39. C₂₆₉	H(3,𝔽₄)	semidirect
45. #120	D₂₆₉	semidirect
47. C₂⁴	C₂×D₂₆₉	semidirect
54. C₂²⋊C₄	C₂×D₂₆₉	semidirect

Maximal subgroups

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

subgroup	index
#22575440	2
#22575439	2
#22575422	2
#22575421	2
#120×C₂₆₉	2
H(3,𝔽₄)	269