Subgroups — #31771481

Order 17216.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16864412	C₂	1	2	yes	maximal
2	C₂×C₈×D₂₆₉	C₂	1	1	yes	maximal, characteristic
3	#16864412	—	2	2	no	—
4	#16864412	C₄	1	4	yes	—
5	C₂×#14524137	—	2	2	no	—
6	C₂×C₄×D₂₆₉	C₂²	1	1	yes	characteristic
7	C₂×#14524130	C₄	1	1	yes	characteristic
8	C₈×D₂₆₉	—	2	2	no	—
9	C₂×C₈×C₂₆₉	C₄	1	1	yes	characteristic
10	#14524137	—	2	4	no	—
11	C₂²×D₂₆₉	Q₈	1	1	yes	characteristic
12	C₂×Dic₂₆₉	D₄	1	1	yes	characteristic
13	C₄×D₂₆₉	C₂×C₄	1	2	yes	—
14	C₄×D₂₆₉	D₄	1	2	yes	—
15	#14524130	—	2	1	no	—
16	C₂×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
17	C₈×C₂₆₉	—	2	1	no	—
18	C₂²×C₂₆₉	C₄⋊C₄	1	1	yes	characteristic
19	C₂×D₂₆₉	—	2	1	no	—
20	C₂×D₂₆₉	—	2	1	no	—
21	C₂×D₂₆₉	C₄⋊C₄	1	2	yes	—
22	C₄×C₂₆₉	C₄²	1	1	yes	characteristic
23	C₄×C₂₆₉	C₂²⋊C₄	1	1	yes	characteristic
24	Dic₂₆₉	C₂²⋊C₄	1	2	yes	—
25	C₂×C₂₆₉	#95	1	1	yes	characteristic
26	C₂×C₂₆₉	#108	1	1	yes	characteristic
27	C₂×C₂₆₉	#108	1	1	yes	characteristic
28	D₂₆₉	—	2	2	no	—
29	C₂₆₉	#341	1	1	yes	characteristic, complemented, Sylow
30	#341	—	269	1	no	maximal, Sylow
31	C₂×M₁₆	—	269	2	no	—
32	C₂²×C₈	—	269	1	no	—
33	C₂²×C₄	—	269	1	no	—
34	M₁₆	—	269	4	no	—
35	M₁₆	—	538	2	no	—
36	C₂×C₈	—	269	1	no	—
37	C₂×C₈	—	538	2	no	—
38	C₂×C₈	—	538	2	no	—
39	C₂×C₈	#11761640	1	1	yes	characteristic
40	C₂³	—	269	1	no	—
41	C₂×C₄	—	269	1	no	—
42	C₂×C₄	—	269	2	no	—
43	C₂×C₄	—	269	2	no	—
44	C₂×C₄	C₂×#11761640	1	1	yes	characteristic
45	C₈	—	2	1	no	—
46	C₈	—	538	4	no	—
47	C₈	—	538	1	no	—
48	C₂²	—	269	2	no	—
49	C₂²	—	538	1	no	—
50	C₂²	—	538	1	no	—
51	C₂²	C₄⋊#11761640	1	1	yes	characteristic
52	C₄	—	269	2	no	—
53	C₄	C₄×#11761640	1	1	yes	characteristic
54	C₄	C₂²⋊#11761640	1	1	yes	characteristic
55	C₂	—	538	2	no	—
56	C₂	(C₂×#11761640)⋊C₄	1	1	yes	characteristic
57	C₂	#22575523	1	1	yes	characteristic
58	C₂	#22575524	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₄⋊#11761640
commutator	C₄×C₂₆₉	C₄²
Frattini	C₂×C₄	C₂×#11761640
Fitting	C₂×C₈×C₂₆₉	C₄
socle	C₂²×C₂₆₉	C₄⋊C₄
prime radical	C₂²×D₂₆₉	Q₈
cosocle	C₂×C₄×D₂₆₉	C₂²
hypercenter	C₂×C₈	#11761640
hypocenter	C₂₆₉	#341
LCS	G ▷ C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
29. C₂₆₉	#341	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16864412	2
C₂×C₈×D₂₆₉	2
#341	269