Subgroups — #31770949

Order 17216.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22575433	C₂	1	1	yes	maximal, characteristic
2	#22575411	C₂	1	1	yes	maximal, characteristic
3	#22575386	C₂	1	1	yes	maximal, characteristic, complemented
4	#22575378	C₂	1	1	yes	maximal, characteristic, complemented
5	#22575376	C₂	1	1	yes	maximal, characteristic, complemented
6	#22575365	C₂	1	1	yes	maximal, characteristic
7	#106×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₅₃₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₂₆₉	C₂²	1	1	yes	characteristic
10	C₂×#14524130	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₂₆₉	—	2	1	no	—
12	C₄⋊Dic₂₆₉	C₂²	1	1	yes	characteristic
13	#16864379	—	2	1	no	—
14	#16864379	—	2	1	no	—
15	C₄×Dic₂₆₉	—	2	1	no	—
16	C₂×C₈×C₂₆₉	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₂₆₉	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₂₆₉	C₂²	1	1	yes	characteristic
19	C₂²×D₂₆₉	D₄	1	1	yes	characteristic
20	Dic₅₃₈	—	2	1	no	—
21	Dic₅₃₈	—	2	1	no	—
22	C₂×Dic₂₆₉	—	2	1	no	—
23	C₂×Dic₂₆₉	—	2	1	no	—
24	C₂×Dic₂₆₉	D₄	1	1	yes	characteristic
25	C₄×D₂₆₉	—	2	1	no	—
26	C₄×D₂₆₉	—	2	1	no	—
27	#14524130	—	2	1	no	—
28	C₂×C₄×C₂₆₉	—	2	1	no	—
29	C₂×C₄×C₂₆₉	—	2	1	no	—
30	C₂×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
31	C₈×C₂₆₉	—	2	1	no	—
32	C₂²×C₂₆₉	C₂×D₄	1	1	yes	characteristic
33	C₂×D₂₆₉	—	2	1	no	—
34	C₂×D₂₆₉	—	2	1	no	—
35	C₂×D₂₆₉	—	2	1	no	—
36	C₄×C₂₆₉	—	4	1	no	—
37	C₄×C₂₆₉	—	4	1	no	—
38	C₄×C₂₆₉	#41	1	1	yes	characteristic
39	C₄×C₂₆₉	#41	1	1	yes	characteristic
40	Dic₂₆₉	—	2	1	no	—
41	Dic₂₆₉	—	4	1	no	—
42	Dic₂₆₉	—	4	1	no	—
43	C₂×C₂₆₉	#123	1	1	yes	characteristic
44	C₂×C₂₆₉	#135	1	1	yes	characteristic
45	C₂×C₂₆₉	#137	1	1	yes	characteristic
46	D₂₆₉	—	4	1	no	—
47	C₂₆₉	#485	1	1	yes	characteristic, complemented, Sylow
48	#485	—	269	1	no	maximal, Sylow
49	C₂²⋊Q₈	—	269	1	no	—
50	#117	—	269	1	no	—
51	#107	—	269	1	no	—
52	#106	D₂₆₉	1	1	yes	characteristic, complemented
53	#103	—	269	1	no	—
54	#103	—	269	1	no	—
55	C₂²⋊C₈	—	269	1	no	—
56	C₂×Q₈	—	269	1	no	—
57	C₂²×C₄	—	269	1	no	—
58	C₂×C₈	—	269	1	no	—
59	C₂×C₈	C₂×D₂₆₉	1	1	yes	characteristic
60	C₄⋊C₄	—	269	1	no	—
61	C₄⋊C₄	—	538	1	no	—
62	C₄⋊C₄	C₂×D₂₆₉	1	1	yes	characteristic
63	C₄⋊C₄	C₂×D₂₆₉	1	1	yes	characteristic
64	C₂²⋊C₄	—	538	1	no	—
65	C₂²⋊C₄	—	538	1	no	—
66	C₄²	—	538	1	no	—
67	C₂³	—	269	1	no	—
68	Q₈	—	538	1	no	—
69	Q₈	—	538	1	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	269	1	no	—
73	C₂×C₄	—	538	1	no	—
74	C₂×C₄	—	538	1	no	—
75	C₂×C₄	—	538	1	no	—
76	C₂×C₄	—	538	1	no	—
77	C₂×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
78	C₈	—	2	1	no	—
79	C₈	—	538	1	no	—
80	C₂²	—	538	1	no	—
81	C₂²	—	538	1	no	—
82	C₂²	—	538	1	no	—
83	C₂²	D₄×D₂₆₉	1	1	yes	characteristic
84	C₄	—	4	1	no	—
85	C₄	—	4	1	no	—
86	C₄	—	538	1	no	—
87	C₄	—	1076	1	no	—
88	C₄	—	1076	1	no	—
89	C₄	#16864402	1	1	yes	characteristic
90	C₄	#16864403	1	1	yes	characteristic
91	C₂	—	1076	1	no	—
92	C₂	#22575438	1	1	yes	characteristic
93	C₂	#22575474	1	1	yes	characteristic
94	C₂	#22575482	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₆₉
commutator, cosocle	C₂×C₄×C₂₆₉	C₂³
Frattini	C₂×C₄	C₂²×D₂₆₉
Fitting	#106×C₂₆₉	C₂
socle	C₂²×C₂₆₉	C₂×D₄
prime radical	C₂²×D₂₆₉	D₄
hypercenter	#106	D₂₆₉
hypocenter	C₂₆₉	#485
LCS	G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

Complements

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

N	H ≅ G/N	type
3. #22575386	C₂	semidirect
4. #22575378	C₂	semidirect
5. #22575376	C₂	semidirect
7. #106×C₂₆₉	C₂	semidirect
47. C₂₆₉	#485	semidirect
52. #106	D₂₆₉	semidirect

Maximal subgroups

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

subgroup	index
#22575433	2
#22575411	2
#22575386	2
#22575378	2
#22575376	2
#22575365	2
#106×C₂₆₉	2
#485	269