Subgroups — #31770913

Order 17216.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22575445	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₅₃₈	C₂	1	1	yes	maximal, characteristic
3	#22575387	C₂	1	1	yes	maximal, characteristic
4	#22575385	C₂	1	1	yes	maximal, characteristic
5	#22575378	C₂	1	1	yes	maximal, characteristic
6	#22575378	C₂	1	1	yes	maximal, characteristic
7	C₄⋊C₈×C₂₆₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₅₃₈	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₂₆₉	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₂₆₉	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₂₆₉	C₂²	1	1	yes	characteristic
12	#16864381	—	2	1	no	—
13	#16864381	—	2	1	no	—
14	#16864381	—	2	1	no	—
15	C₄×Dic₂₆₉	—	2	1	no	—
16	C₂×C₈×C₂₆₉	C₂²	1	1	yes	characteristic
17	C₂×C₈×C₂₆₉	C₂²	1	1	yes	characteristic
18	C₄²×C₂₆₉	C₂²	1	1	yes	characteristic
19	Dic₅₃₈	—	2	1	no	—
20	Dic₅₃₈	Q₈	1	2	yes	—
21	C₂×Dic₂₆₉	—	2	1	no	—
22	C₂×Dic₂₆₉	—	2	1	no	—
23	C₂×Dic₂₆₉	—	2	1	no	—
24	C₂×Dic₂₆₉	—	2	1	no	—
25	C₂×C₄×C₂₆₉	D₄	1	1	yes	characteristic
26	C₂×C₄×C₂₆₉	D₄	1	1	yes	characteristic
27	C₂×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
28	C₈×C₂₆₉	—	2	1	no	—
29	C₈×C₂₆₉	—	2	1	no	—
30	C₂²×C₂₆₉	C₂×D₄	1	1	yes	characteristic
31	C₄×C₂₆₉	—	2	1	no	—
32	C₄×C₂₆₉	—	2	1	no	—
33	C₄×C₂₆₉	C₂×Q₈	1	1	yes	characteristic
34	C₄×C₂₆₉	#41	1	1	yes	characteristic
35	Dic₂₆₉	—	2	2	no	—
36	Dic₂₆₉	—	4	1	no	—
37	Dic₂₆₉	—	4	1	no	—
38	Dic₂₆₉	—	4	1	no	—
39	C₂×C₂₆₉	C₂²⋊Q₈	1	1	yes	characteristic
40	C₂×C₂₆₉	#135	1	1	yes	characteristic
41	C₂×C₂₆₉	#137	1	1	yes	characteristic
42	C₂₆₉	#479	1	1	yes	characteristic, complemented, Sylow
43	#479	—	269	1	no	maximal, Sylow
44	#125	—	269	1	no	—
45	C₄×Q₈	—	269	1	no	—
46	#107	—	269	1	no	—
47	#106	—	269	1	no	—
48	C₄⋊C₈	D₂₆₉	1	1	yes	characteristic
49	#103	—	269	1	no	—
50	#103	—	269	1	no	—
51	C₂×Q₈	—	269	1	no	—
52	C₂×C₈	C₂×D₂₆₉	1	1	yes	characteristic
53	C₂×C₈	C₂×D₂₆₉	1	1	yes	characteristic
54	C₄⋊C₄	—	269	1	no	—
55	C₄⋊C₄	—	269	1	no	—
56	C₄⋊C₄	—	269	1	no	—
57	C₄⋊C₄	—	538	1	no	—
58	C₄⋊C₄	—	538	1	no	—
59	C₄⋊C₄	—	538	1	no	—
60	C₄²	—	538	1	no	—
61	C₄²	C₂×D₂₆₉	1	1	yes	characteristic
62	Q₈	—	269	2	no	—
63	Q₈	—	538	1	no	—
64	C₂×C₄	—	538	1	no	—
65	C₂×C₄	—	538	1	no	—
66	C₂×C₄	—	538	1	no	—
67	C₂×C₄	—	538	1	no	—
68	C₂×C₄	D₁₀₇₆	1	1	yes	characteristic
69	C₂×C₄	D₁₀₇₆	1	1	yes	characteristic
70	C₂×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
71	C₈	—	2	1	no	—
72	C₈	—	2	1	no	—
73	C₂²	C₂×D₁₀₇₆	1	1	yes	characteristic
74	C₄	—	2	1	no	—
75	C₄	—	2	1	no	—
76	C₄	—	538	2	no	—
77	C₄	—	1076	1	no	—
78	C₄	—	1076	1	no	—
79	C₄	—	1076	1	no	—
80	C₄	Q₈×D₂₆₉	1	1	yes	characteristic
81	C₄	#16864401	1	1	yes	characteristic
82	C₂	#22575433	1	1	yes	characteristic
83	C₂	#22575475	1	1	yes	characteristic
84	C₂	#22575486	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₀₇₆
commutator, cosocle	C₂×C₄×C₂₆₉	C₂³
Frattini	C₂×C₄	C₂²×D₂₆₉
Fitting	C₄⋊C₈×C₂₆₉	C₂
socle, prime radical	C₂²×C₂₆₉	C₂×D₄
hypercenter	C₄⋊C₈	D₂₆₉
hypocenter	C₂₆₉	#479
LCS	G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
42. C₂₆₉	#479	semidirect

Maximal subgroups

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

subgroup	index
#22575445	2
C₄×Dic₅₃₈	2
#22575387	2
#22575385	2
#22575378	2
C₄⋊C₈×C₂₆₉	2
#479	269