Subgroups — #31771355

Order 17216.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22575451	C₂	1	2	yes	maximal, complemented
2	#22575446	C₂	1	4	yes	maximal, complemented
3	#22575445	C₂	1	2	yes	maximal, complemented
4	#22575434	C₂	1	2	yes	maximal
5	#22575430	C₂	1	4	yes	maximal, complemented
6	#124×C₂₆₉	C₂	1	1	yes	maximal, characteristic
7	C₂×Dic₅₃₈	C₂²	1	2	yes	—
8	C₂²×Dic₂₆₉	C₂²	1	2	yes	—
9	C₄⋊Dic₂₆₉	C₂²	1	2	yes	—
10	C₄⋊Dic₂₆₉	C₂²	1	2	yes	—
11	#16864381	C₂²	1	4	yes	—
12	#16864381	C₂²	1	4	yes	—
13	#16864381	C₂²	1	4	yes	—
14	C₂²⋊Dic₂₆₉	C₂²	1	4	yes	—
15	C₂²⋊Dic₂₆₉	C₂²	1	2	yes	—
16	C₄×Dic₂₆₉	C₂²	1	2	yes	—
17	C₂×Q₈×C₂₆₉	C₂²	1	1	yes	characteristic
18	C₂×D₄×C₂₆₉	C₂²	1	1	yes	characteristic
19	C₂²⋊C₄×C₂₆₉	C₂²	1	4	yes	—
20	C₄²×C₂₆₉	C₂²	1	1	yes	characteristic
21	Dic₅₃₈	—	4	2	no	—
22	C₂×Dic₂₆₉	—	4	2	no	—
23	C₂×Dic₂₆₉	C₂³	1	4	yes	—
24	C₂×Dic₂₆₉	C₂³	1	4	yes	—
25	C₂³×C₂₆₉	C₂³	1	2	yes	—
26	Q₈×C₂₆₉	—	4	1	no	—
27	D₄×C₂₆₉	—	4	1	no	—
28	C₂×C₄×C₂₆₉	C₂³	1	2	yes	—
29	C₂×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
30	C₂×C₄×C₂₆₉	C₂³	1	2	yes	—
31	C₂²×C₂₆₉	—	2	2	no	—
32	C₂²×C₂₆₉	—	2	4	no	—
33	C₂²×C₂₆₉	C₂⁴	1	1	yes	characteristic
34	C₄×C₂₆₉	—	2	1	no	—
35	C₄×C₂₆₉	—	2	2	no	—
36	C₄×C₂₆₉	—	2	2	no	—
37	Dic₂₆₉	—	2	4	no	—
38	Dic₂₆₉	—	2	4	no	—
39	C₂×C₂₆₉	—	4	2	no	—
40	C₂×C₂₆₉	H(5,𝔽₂)	1	1	yes	characteristic
41	C₂×C₂₆₉	2₋¹⁺⁴	1	2	yes	—
42	C₂₆₉	#563	1	1	yes	characteristic, complemented, Sylow
43	#563	—	269	1	no	maximal, Sylow
44	C₄⋊Q₈	—	269	2	no	—
45	#126	—	269	4	no	—
46	#125	—	269	2	no	—
47	#124	D₂₆₉	1	1	yes	characteristic
48	#123	—	269	2	no	—
49	C₂²⋊Q₈	—	269	4	no	—
50	C₂×Q₈	—	269	2	no	—
51	C₂×Q₈	C₂×D₂₆₉	1	1	yes	characteristic
52	C₂×D₄	C₂×D₂₆₉	1	1	yes	characteristic
53	C₂²×C₄	—	269	2	no	—
54	C₄⋊C₄	—	269	2	no	—
55	C₄⋊C₄	—	269	4	no	—
56	C₄⋊C₄	—	269	4	no	—
57	C₄⋊C₄	—	269	4	no	—
58	C₄⋊C₄	—	269	2	no	—
59	C₂²⋊C₄	—	269	4	no	—
60	C₂²⋊C₄	—	269	2	no	—
61	C₂²⋊C₄	C₂×D₂₆₉	1	4	yes	—
62	C₄²	—	269	2	no	—
63	C₄²	C₂×D₂₆₉	1	1	yes	characteristic
64	C₂³	C₂²×D₂₆₉	1	2	yes	—
65	Q₈	—	4	1	no	—
66	Q₈	—	1076	2	no	—
67	D₄	—	4	1	no	—
68	C₂×C₄	—	269	4	no	—
69	C₂×C₄	—	269	4	no	—
70	C₂×C₄	—	1076	2	no	—
71	C₂×C₄	C₂²×D₂₆₉	1	2	yes	—
72	C₂×C₄	C₂²×D₂₆₉	1	2	yes	—
73	C₂×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
74	C₂²	—	2	2	no	—
75	C₂²	—	2	4	no	—
76	C₂²	C₂³×D₂₆₉	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	—	2	2	no	—
79	C₄	—	2	2	no	—
80	C₄	—	538	4	no	—
81	C₄	—	538	4	no	—
82	C₂	—	4	2	no	—
83	C₂	#22575498	1	1	yes	characteristic
84	C₂	#22575500	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₆₉
commutator, socle, cosocle	C₂²×C₂₆₉	C₂⁴
Fitting	#124×C₂₆₉	C₂
prime radical	C₂×D₄×C₂₆₉	C₂²
hypercenter	#124	D₂₆₉
hypocenter	C₂₆₉	#563
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ #124

Complements

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

N	H ≅ G/N	type
1. #22575451	C₂	semidirect
2. #22575446	C₂	semidirect
3. #22575445	C₂	semidirect
5. #22575430	C₂	semidirect
42. C₂₆₉	#563	semidirect

Maximal subgroups

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

subgroup	index
#22575451	2
#22575446	2
#22575445	2
#22575434	2
#22575430	2
#124×C₂₆₉	2
#563	269