Subgroups — #31771611

Order 17216.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16864411	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14524137	C₂	1	1	yes	maximal, characteristic
3	#22575520	C₂	1	2	yes	maximal, complemented
4	C₄⋊#14524137	C₂	1	2	yes	maximal, complemented
5	#22575409	C₂	1	1	yes	maximal, characteristic
6	#16864411	—	2	2	no	—
7	C₂×#14524137	—	2	2	no	—
8	C₂×#14524137	C₂²	1	1	yes	characteristic
9	C₂×#14524137	C₂²	1	2	yes	—
10	C₂×#14524137	C₂²	1	1	yes	characteristic
11	C₂²×Dic₂₆₉	C₂²	1	1	yes	characteristic
12	#16864381	C₄	1	2	yes	—
13	C₂²⋊Dic₂₆₉	C₄	1	1	yes	characteristic
14	C₄×Dic₂₆₉	C₂²	1	2	yes	—
15	C₂²⋊C₄×C₂₆₉	C₄	1	1	yes	characteristic
16	#14524137	—	2	2	no	—
17	#14524137	—	2	1	no	—
18	#14524137	—	2	1	no	—
19	C₂×Dic₂₆₉	C₂×C₄	1	2	yes	—
20	C₂×Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
21	C₂×Dic₂₆₉	D₄	1	2	yes	—
22	C₂×Dic₂₆₉	Q₈	1	2	yes	—
23	C₂×Dic₂₆₉	C₂³	1	1	yes	characteristic
24	C₂³×C₂₆₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₂₆₉	C₂×C₄	1	2	yes	—
26	C₂²×C₂₆₉	—	2	2	no	—
27	C₂²×C₂₆₉	C₄⋊C₄	1	2	yes	—
28	C₂²×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
29	C₄×C₂₆₉	—	2	2	no	—
30	Dic₂₆₉	—	2	2	no	—
31	Dic₂₆₉	C₄⋊C₄	1	2	yes	—
32	Dic₂₆₉	C₂×D₄	1	1	yes	characteristic
33	Dic₂₆₉	C₂×Q₈	1	1	yes	characteristic
34	C₂×C₂₆₉	—	2	2	no	—
35	C₂×C₂₆₉	C₂×C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₂₆₉	#131	1	2	yes	—
37	C₂₆₉	#424	1	1	yes	characteristic, complemented, Sylow
38	#424	—	269	1	no	maximal, Sylow
39	C₂×M₁₆	—	269	1	no	—
40	C₂²×C₈	—	269	1	no	—
41	#117	—	269	1	no	—
42	C₄⋊C₈	—	269	2	no	—
43	C₄⋊C₈	—	269	2	no	—
44	C₂²×C₄	—	269	1	no	—
45	M₁₆	—	538	2	no	—
46	C₂×C₈	—	269	2	no	—
47	C₂×C₈	—	269	1	no	—
48	C₂×C₈	—	269	1	no	—
49	C₂×C₈	—	538	2	no	—
50	C₄⋊C₄	—	269	2	no	—
51	C₂²⋊C₄	—	269	1	no	—
52	C₂²⋊C₄	#11761640	1	1	yes	characteristic
53	C₄²	—	269	2	no	—
54	C₂³	C₂×#11761640	1	1	yes	characteristic
55	C₂×C₄	—	269	2	no	—
56	C₂×C₄	—	269	2	no	—
57	C₂×C₄	—	269	1	no	—
58	C₂×C₄	—	269	1	no	—
59	C₂×C₄	—	269	2	no	—
60	C₂×C₄	C₂×#11761640	1	2	yes	—
61	C₈	—	538	2	no	—
62	C₈	—	538	1	no	—
63	C₈	—	538	1	no	—
64	C₂²	—	2	2	no	—
65	C₂²	C₄⋊#11761640	1	2	yes	—
66	C₂²	C₂²×#11761640	1	1	yes	characteristic
67	C₄	—	2	2	no	—
68	C₄	—	269	1	no	—
69	C₄	—	269	2	no	—
70	C₄	—	269	1	no	—
71	C₄	—	538	2	no	—
72	C₂	—	2	2	no	—
73	C₂	C₂×C₄⋊#11761640	1	1	yes	characteristic
74	C₂	#22575540	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×#11761640
commutator, socle	C₂²×C₂₆₉	C₂²×C₄
Fitting	C₂²⋊C₄×C₂₆₉	C₄
prime radical	C₂³×C₂₆₉	C₂×C₄
cosocle	C₂×Dic₂₆₉	C₂³
hypercenter	C₂²⋊C₄	#11761640
hypocenter	C₂₆₉	#424
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
3. #22575520	C₂	semidirect
4. C₄⋊#14524137	C₂	semidirect
37. C₂₆₉	#424	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16864411	2
C₂²×#14524137	2
#22575520	2
C₄⋊#14524137	2
#22575409	2
#424	269