Subgroups — #31771608

Order 17216.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16864411	C₂	1	2	yes	maximal
2	#22575520	C₂	1	2	yes	maximal, complemented
3	C₄⋊#14524137	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×Dic₂₆₉	C₂	1	1	yes	maximal, characteristic
5	#16864411	—	2	4	no	—
6	C₂×#14524137	C₂²	1	4	yes	—
7	C₂²×Dic₂₆₉	C₄	1	1	yes	characteristic
8	C₂²×Dic₂₆₉	C₂²	1	1	yes	characteristic
9	C₄×Dic₂₆₉	C₄	1	1	yes	characteristic
10	C₄×Dic₂₆₉	C₄	1	1	yes	characteristic
11	C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
12	C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₂₆₉	C₄	1	1	yes	characteristic
14	#14524137	—	2	4	no	—
15	C₂×Dic₂₆₉	—	2	2	no	—
16	C₂×Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
17	C₂×Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
18	C₂×Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
19	C₂×Dic₂₆₉	D₄	1	2	yes	—
20	C₂×Dic₂₆₉	Q₈	1	2	yes	—
21	C₂×Dic₂₆₉	C₂³	1	1	yes	characteristic
22	C₂³×C₂₆₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₂₆₉	—	2	2	no	—
24	C₂×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₂₆₉	—	2	1	no	—
27	C₂²×C₂₆₉	—	2	1	no	—
28	C₂²×C₂₆₉	C₄⋊C₄	1	2	yes	—
29	C₂²×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
30	C₄×C₂₆₉	—	2	1	no	—
31	C₄×C₂₆₉	M₁₆	1	2	yes	complemented
32	Dic₂₆₉	—	2	1	no	—
33	Dic₂₆₉	C₄⋊C₄	1	2	yes	—
34	Dic₂₆₉	M₁₆	1	2	yes	complemented
35	Dic₂₆₉	C₂×D₄	1	1	yes	characteristic
36	Dic₂₆₉	C₂×Q₈	1	1	yes	characteristic
37	C₂×C₂₆₉	—	2	2	no	—
38	C₂×C₂₆₉	C₂×C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₂₆₉	C₂×M₁₆	1	1	yes	characteristic
40	C₂×C₂₆₉	C₂×M₁₆	1	1	yes	characteristic
41	C₂₆₉	#423	1	1	yes	characteristic, complemented, Sylow
42	#423	—	269	1	no	maximal, Sylow
43	C₂×M₁₆	—	269	2	no	—
44	C₂×C₄²	—	269	1	no	—
45	C₄⋊C₈	—	269	2	no	—
46	C₄⋊C₈	—	269	2	no	—
47	C₂²×C₄	—	269	1	no	—
48	C₂²×C₄	—	269	1	no	—
49	C₂²×C₄	#11761640	1	1	yes	characteristic
50	M₁₆	—	538	4	no	—
51	C₂×C₈	—	269	4	no	—
52	C₄²	—	269	1	no	—
53	C₄²	—	269	1	no	—
54	C₄²	—	269	1	no	—
55	C₄²	—	269	1	no	—
56	C₂³	C₂×#11761640	1	1	yes	characteristic
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	—	269	2	no	—
59	C₂×C₄	—	269	2	no	—
60	C₂×C₄	—	269	1	no	—
61	C₂×C₄	—	269	1	no	—
62	C₂×C₄	—	269	1	no	—
63	C₂×C₄	—	269	1	no	—
64	C₂×C₄	—	538	2	no	—
65	C₂×C₄	C₂×#11761640	1	1	yes	characteristic
66	C₂×C₄	C₂×#11761640	1	1	yes	characteristic
67	C₈	—	538	4	no	—
68	C₂²	—	2	1	no	—
69	C₂²	—	2	1	no	—
70	C₂²	C₄⋊#11761640	1	2	yes	—
71	C₂²	C₂²×#11761640	1	1	yes	characteristic
72	C₄	—	2	1	no	—
73	C₄	—	269	1	no	—
74	C₄	—	269	2	no	—
75	C₄	—	269	1	no	—
76	C₄	—	269	2	no	—
77	C₄	—	538	1	no	—
78	C₄	#16864411	1	2	yes	complemented
79	C₂	—	2	2	no	—
80	C₂	C₂×C₄⋊#11761640	1	1	yes	characteristic
81	C₂	C₂×#16864411	1	1	yes	characteristic
82	C₂	D₂₆₉⋊M₁₆	1	1	yes	characteristic

	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	#423
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
2. #22575520	C₂	semidirect
3. C₄⋊#14524137	C₂	semidirect
31. C₄×C₂₆₉	M₁₆	semidirect
34. Dic₂₆₉	M₁₆	semidirect
41. C₂₆₉	#423	semidirect
78. C₄	#16864411	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
#22575520	2
C₄⋊#14524137	2
C₂×C₄×Dic₂₆₉	2
#423	269