Subgroups — C₂×#22575527 #31771660

Order 17216.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22575527	C₂	1	4	yes	maximal, direct product factor
2	C₂×#16864405	C₂	1	2	yes	maximal, complemented
3	C₂²×#14524130	C₂	1	1	yes	maximal, characteristic
4	#16864405	C₂²	1	4	yes	complemented
5	C₂×#14524130	C₄	1	2	yes	—
6	C₂×#14524130	C₄	1	1	yes	characteristic
7	C₂×#14524130	C₂²	1	1	yes	characteristic
8	C₂×#14524130	C₂²	1	2	yes	—
9	C₂²×C₄×C₂₆₉	C₄	1	1	yes	characteristic
10	#14524130	C₂×C₄	1	1	yes	characteristic
11	#14524130	C₂×C₄	1	2	yes	—
12	#14524130	C₂³	1	1	yes	characteristic
13	C₂³×C₂₆₉	C₈	1	1	yes	characteristic
14	C₂×C₄×C₂₆₉	C₈	1	2	yes	—
15	C₂×C₄×C₂₆₉	C₈	1	1	yes	characteristic
16	C₂×C₄×C₂₆₉	C₂×C₄	1	2	yes	—
17	C₂×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₂₆₉	—	2	2	no	—
19	C₂²×C₂₆₉	C₂×C₈	1	2	yes	—
20	C₂²×C₂₆₉	C₂×C₈	1	1	yes	characteristic
21	C₄×C₂₆₉	C₂×C₈	1	2	yes	—
22	C₄×C₂₆₉	C₂×C₈	1	1	yes	characteristic
23	C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
24	C₂×C₂₆₉	—	2	2	no	—
25	C₂×C₂₆₉	M₃₂	1	2	yes	complemented
26	C₂×C₂₆₉	C₂²×C₈	1	1	yes	characteristic
27	C₂₆₉	C₂×M₃₂	1	1	yes	characteristic, complemented, Sylow
28	C₂×M₃₂	—	269	1	no	maximal, Sylow
29	C₂²×C₈	—	269	1	no	—
30	M₃₂	—	269	4	no	—
31	C₂×C₁₆	—	269	2	no	—
32	C₂²×C₄	#11761640	1	1	yes	characteristic
33	C₂×C₈	—	269	2	no	—
34	C₂×C₈	—	269	2	no	—
35	C₂×C₈	—	269	1	no	—
36	C₂×C₈	—	269	1	no	—
37	C₁₆	—	269	4	no	—
38	C₂³	#14524137	1	1	yes	characteristic
39	C₂×C₄	#14524137	1	1	yes	characteristic
40	C₂×C₄	#14524137	1	2	yes	—
41	C₂×C₄	C₂×#11761640	1	2	yes	—
42	C₂×C₄	C₂×#11761640	1	1	yes	characteristic
43	C₈	—	269	1	no	—
44	C₈	—	269	2	no	—
45	C₈	—	269	1	no	—
46	C₂²	—	2	2	no	—
47	C₂²	C₂×#14524137	1	2	yes	—
48	C₂²	C₂×#14524137	1	1	yes	characteristic
49	C₄	C₂×#14524137	1	2	yes	—
50	C₄	C₂×#14524137	1	1	yes	characteristic
51	C₄	C₂²×#11761640	1	1	yes	characteristic
52	C₂	—	2	2	no	—
53	C₂	#22575527	1	2	yes	direct product factor
54	C₂	C₂²×#14524137	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#11761640
commutator	C₂×C₂₆₉	C₂²×C₈
Frattini	C₄	C₂²×#11761640
Fitting	C₂²×C₄×C₂₆₉	C₄
socle	C₂²×C₂₆₉	C₂×C₈
prime radical	C₂³×C₂₆₉	C₈
cosocle	#14524130	C₂³
hypercenter	C₂²×C₄	#11761640
hypocenter	C₂₆₉	C₂×M₃₂
LCS	G ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
1. #22575527	C₂	direct (N × H)
2. C₂×#16864405	C₂	semidirect
4. #16864405	C₂²	semidirect
25. C₂×C₂₆₉	M₃₂	semidirect
27. C₂₆₉	C₂×M₃₂	semidirect
53. C₂	#22575527	direct (N × H)

Maximal subgroups

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

subgroup	index
#22575527	2
C₂×#16864405	2
C₂²×#14524130	2
C₂×M₃₂	269

Subgroups — C2×#22575527 #31771660

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22575527 #31771660