Subgroups — #31770958

Order 17216.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22575444	C₂	1	1	yes	maximal, characteristic, complemented
2	#22575440	C₂	1	1	yes	maximal, characteristic
3	#22575378	C₂	1	1	yes	maximal, characteristic, complemented
4	#22575376	C₂	1	1	yes	maximal, characteristic, complemented
5	#22575374	C₂	1	1	yes	maximal, characteristic
6	C₈×Dic₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
7	#102×C₂₆₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₅₃₈	C₂²	1	1	yes	characteristic
9	C₂×#14524130	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₂₆₉	—	2	1	no	—
11	C₄⋊Dic₂₆₉	C₂²	1	1	yes	characteristic
12	#16864381	—	2	1	no	—
13	C₂²⋊Dic₂₆₉	—	2	1	no	—
14	C₂²⋊Dic₂₆₉	—	2	1	no	—
15	C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
16	C₂×D₄×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	Dic₅₃₈	—	2	1	no	—
20	Dic₅₃₈	—	2	1	no	—
21	C₂×Dic₂₆₉	—	2	1	no	—
22	C₂×Dic₂₆₉	—	2	1	no	—
23	C₂×Dic₂₆₉	D₄	1	1	yes	characteristic
24	C₂×Dic₂₆₉	D₄	1	1	yes	characteristic
25	#14524130	—	2	1	no	—
26	C₂³×C₂₆₉	—	2	1	no	—
27	D₄×C₂₆₉	—	2	1	no	—
28	D₄×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	—	4	1	no	—
33	C₂²×C₂₆₉	—	4	1	no	—
34	C₂²×C₂₆₉	—	4	1	no	—
35	C₂²×C₂₆₉	C₂×D₄	1	1	yes	characteristic
36	C₄×C₂₆₉	—	4	1	no	—
37	C₄×C₂₆₉	#41	1	1	yes	characteristic
38	C₄×C₂₆₉	#41	1	1	yes	characteristic
39	Dic₂₆₉	—	2	1	no	—
40	Dic₂₆₉	—	2	1	no	—
41	Dic₂₆₉	—	4	1	no	—
42	Dic₂₆₉	—	4	1	no	—
43	C₂×C₂₆₉	—	8	1	no	—
44	C₂×C₂₆₉	#124	1	1	yes	characteristic
45	C₂×C₂₆₉	#135	1	1	yes	characteristic
46	C₂×C₂₆₉	#135	1	1	yes	characteristic
47	C₂₆₉	#488	1	1	yes	characteristic, complemented, Sylow
48	#488	—	269	1	no	maximal, Sylow
49	#125	—	269	1	no	—
50	#124	—	269	1	no	—
51	#103	—	269	1	no	—
52	#103	—	269	1	no	—
53	#102	—	269	1	no	—
54	#102	D₂₆₉	1	1	yes	characteristic
55	C₄×C₈	—	269	1	no	—
56	C₂×Q₈	—	269	1	no	—
57	C₂×D₄	C₂×D₂₆₉	1	1	yes	characteristic
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	—	538	1	no	—
63	C₄⋊C₄	C₂×D₂₆₉	1	1	yes	characteristic
64	C₂²⋊C₄	—	538	1	no	—
65	C₂²⋊C₄	—	538	1	no	—
66	C₄²	—	269	1	no	—
67	C₂³	—	2	1	no	—
68	Q₈	—	538	1	no	—
69	Q₈	—	538	1	no	—
70	D₄	—	2	1	no	—
71	D₄	—	2	1	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	269	1	no	—
74	C₂×C₄	—	269	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	—	4	1	no	—
81	C₂²	—	4	1	no	—
82	C₂²	—	4	1	no	—
83	C₂²	D₄×D₂₆₉	1	1	yes	characteristic
84	C₄	—	4	1	no	—
85	C₄	—	538	1	no	—
86	C₄	—	538	1	no	—
87	C₄	—	1076	1	no	—
88	C₄	—	1076	1	no	—
89	C₄	#16864401	1	1	yes	characteristic
90	C₄	#16864402	1	1	yes	characteristic
91	C₂	—	8	1	no	—
92	C₂	#22575439	1	1	yes	characteristic
93	C₂	#22575472	1	1	yes	characteristic
94	C₂	#22575474	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₆₉
commutator, cosocle	C₂×C₄×C₂₆₉	C₂³
Frattini	C₂×C₄	C₂²×D₂₆₉
Fitting	#102×C₂₆₉	C₂
socle	C₂²×C₂₆₉	C₂×D₄
prime radical	C₂×D₄×C₂₆₉	C₂²
hypercenter	#102	D₂₆₉
hypocenter	C₂₆₉	#488
LCS	G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #102

Complements

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

N	H ≅ G/N	type
1. #22575444	C₂	semidirect
3. #22575378	C₂	semidirect
4. #22575376	C₂	semidirect
6. C₈×Dic₂₆₉	C₂	semidirect
47. C₂₆₉	#488	semidirect

Maximal subgroups

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

subgroup	index
#22575444	2
#22575440	2
#22575378	2
#22575376	2
#22575374	2
C₈×Dic₂₆₉	2
#102×C₂₆₉	2
#488	269