Subgroups — #31770939

Order 17216.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22575429	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₄×D₂₆₉	C₂	1	1	yes	maximal, characteristic
3	#22575387	C₂	1	1	yes	maximal, characteristic, complemented
4	#22575386	C₂	1	1	yes	maximal, characteristic, complemented
5	#22575377	C₂	1	1	yes	maximal, characteristic, complemented
6	#22575365	C₂	1	1	yes	maximal, characteristic
7	#103×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₂₆₉	—	2	1	no	—
9	C₂×C₄×D₂₆₉	C₂²	1	1	yes	characteristic
10	C₂×#14524130	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₂₆₉	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₂₆₉	C₂²	1	1	yes	characteristic
13	#16864381	—	2	1	no	—
14	#16864381	—	2	1	no	—
15	#16864379	—	2	1	no	—
16	C₂×Q₈×C₂₆₉	C₂²	1	1	yes	characteristic
17	C₂×C₈×C₂₆₉	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₂₆₉	C₂²	1	1	yes	characteristic
19	C₂²×D₂₆₉	D₄	1	1	yes	characteristic
20	C₂×Dic₂₆₉	—	2	1	no	—
21	C₂×Dic₂₆₉	—	2	1	no	—
22	C₂×Dic₂₆₉	D₄	1	1	yes	characteristic
23	C₄×D₂₆₉	—	2	1	no	—
24	C₄×D₂₆₉	—	2	1	no	—
25	C₄×D₂₆₉	—	4	2	no	—
26	#14524130	—	2	1	no	—
27	Q₈×C₂₆₉	—	2	1	no	—
28	Q₈×C₂₆₉	—	2	1	no	—
29	C₂×C₄×C₂₆₉	—	2	1	no	—
30	C₂×C₄×C₂₆₉	—	2	1	no	—
31	C₂×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
32	C₈×C₂₆₉	—	2	1	no	—
33	C₂²×C₂₆₉	C₂×D₄	1	1	yes	characteristic
34	C₂×D₂₆₉	—	2	1	no	—
35	C₂×D₂₆₉	—	2	1	no	—
36	C₂×D₂₆₉	Q₁₆	1	2	yes	—
37	C₄×C₂₆₉	—	4	1	no	—
38	C₄×C₂₆₉	—	4	1	no	—
39	C₄×C₂₆₉	#41	1	1	yes	characteristic
40	C₄×C₂₆₉	#41	1	1	yes	characteristic
41	Dic₂₆₉	—	2	1	no	—
42	Dic₂₆₉	—	4	1	no	—
43	Dic₂₆₉	—	4	1	no	—
44	C₂×C₂₆₉	#123	1	1	yes	characteristic
45	C₂×C₂₆₉	C₂×Q₁₆	1	1	yes	characteristic
46	C₂×C₂₆₉	AGL(1,ℤ/8)	1	1	yes	characteristic
47	D₂₆₉	—	2	2	no	—
48	C₂₆₉	#484	1	1	yes	characteristic, complemented, Sylow
49	#484	—	269	1	no	maximal, Sylow
50	C₂²⋊Q₈	—	269	1	no	—
51	C₂×C₄⋊C₄	—	269	1	no	—
52	#107	—	269	1	no	—
53	#107	—	269	1	no	—
54	#103	—	269	1	no	—
55	#103	D₂₆₉	1	1	yes	characteristic, complemented
56	C₂²⋊C₈	—	269	1	no	—
57	C₂×Q₈	C₂×D₂₆₉	1	1	yes	characteristic
58	C₂²×C₄	—	269	1	no	—
59	C₂²×C₄	—	538	1	no	—
60	C₂×C₈	—	269	1	no	—
61	C₂×C₈	C₂×D₂₆₉	1	1	yes	characteristic
62	C₄⋊C₄	—	269	1	no	—
63	C₄⋊C₄	—	269	1	no	—
64	C₄⋊C₄	—	538	1	no	—
65	C₄⋊C₄	—	538	1	no	—
66	C₄⋊C₄	C₂×D₂₆₉	1	1	yes	characteristic
67	C₂²⋊C₄	—	538	1	no	—
68	C₂³	—	269	1	no	—
69	Q₈	—	2	1	no	—
70	Q₈	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	269	1	no	—
74	C₂×C₄	—	538	1	no	—
75	C₂×C₄	—	538	1	no	—
76	C₂×C₄	—	538	1	no	—
77	C₂×C₄	—	538	1	no	—
78	C₂×C₄	—	1076	2	no	—
79	C₂×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
80	C₈	—	2	1	no	—
81	C₈	—	538	1	no	—
82	C₂²	—	269	2	no	—
83	C₂²	—	538	1	no	—
84	C₂²	—	538	1	no	—
85	C₂²	D₄×D₂₆₉	1	1	yes	characteristic
86	C₄	—	4	1	no	—
87	C₄	—	4	1	no	—
88	C₄	—	538	1	no	—
89	C₄	—	1076	1	no	—
90	C₄	—	1076	1	no	—
91	C₄	#16864401	1	1	yes	characteristic
92	C₄	#16864402	1	1	yes	characteristic
93	C₂	—	538	2	no	—
94	C₂	#22575435	1	1	yes	characteristic
95	C₂	Q₁₆×D₂₆₉	1	1	yes	characteristic
96	C₂	#22575480	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₆₉
commutator, cosocle	C₂×C₄×C₂₆₉	C₂³
Frattini	C₂×C₄	C₂²×D₂₆₉
Fitting	#103×C₂₆₉	C₂
socle	C₂²×C₂₆₉	C₂×D₄
prime radical	C₂²×D₂₆₉	D₄
hypercenter	#103	D₂₆₉
hypocenter	C₂₆₉	#484
LCS	G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

Complements

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

N	H ≅ G/N	type
3. #22575387	C₂	semidirect
4. #22575386	C₂	semidirect
5. #22575377	C₂	semidirect
7. #103×C₂₆₉	C₂	semidirect
48. C₂₆₉	#484	semidirect
55. #103	D₂₆₉	semidirect

Maximal subgroups

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

subgroup	index
#22575429	2
C₄⋊C₄×D₂₆₉	2
#22575387	2
#22575386	2
#22575377	2
#22575365	2
#103×C₂₆₉	2
#484	269