Subgroups — #31771585

Order 17216.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22575532	C₂	1	2	yes	maximal, complemented
2	#22575511	C₂	1	4	yes	maximal, complemented
3	C₂×#16864401	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#11761640	—	2	2	no	—
5	C₂²⋊#11761640	C₂²	1	4	yes	—
6	C₄×#11761640	—	2	2	no	—
7	#16864401	—	2	2	no	—
8	#16864401	—	2	2	no	—
9	C₂×Dic₅₃₈	C₄	1	1	yes	characteristic, complemented
10	C₂×C₂²⋊D₂₆₉	C₂²	1	2	yes	—
11	C₂²×Dic₂₆₉	C₄	1	2	yes	complemented
12	C₂×C₄×D₂₆₉	C₂²	1	1	yes	characteristic
13	C₂×D₄×C₂₆₉	C₄	1	1	yes	characteristic, complemented
14	C₂×#11761640	—	2	4	no	—
15	C₂²×D₂₆₉	C₂³	1	1	yes	characteristic
16	Dic₅₃₈	—	2	2	no	—
17	C₂²⋊D₂₆₉	—	2	4	no	—
18	C₂×Dic₂₆₉	—	2	2	no	—
19	C₂×Dic₂₆₉	—	2	2	no	—
20	C₂×Dic₂₆₉	C₂×C₄	1	2	yes	—
21	C₂×Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
22	C₄×D₂₆₉	D₄	1	2	yes	—
23	C₄×D₂₆₉	D₄	1	2	yes	—
24	C₂³×C₂₆₉	C₂×C₄	1	2	yes	—
25	D₄×C₂₆₉	—	2	2	no	—
26	C₂×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₂₆₉	—	2	2	no	—
28	C₂²×C₂₆₉	—	4	2	no	—
29	C₂²×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
30	C₂×D₂₆₉	—	4	1	no	—
31	C₂×D₂₆₉	C₂×D₄	1	2	yes	—
32	#11761640	—	4	4	no	—
33	C₄×C₂₆₉	C₂²⋊C₄	1	2	yes	—
34	Dic₂₆₉	—	2	2	no	—
35	Dic₂₆₉	C₂²⋊C₄	1	1	yes	characteristic
36	Dic₂₆₉	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₂₆₉	—	2	1	no	—
38	C₂×C₂₆₉	—	4	2	no	—
39	C₂×C₂₆₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
40	D₂₆₉	—	2	2	no	—
41	C₂₆₉	#410	1	1	yes	characteristic, complemented, Sylow
42	#410	—	269	1	no	maximal, Sylow
43	C₂×#41	—	269	1	no	—
44	#117	—	269	2	no	—
45	#99	—	269	4	no	—
46	#41	—	538	2	no	—
47	#41	—	538	2	no	—
48	C₂×Q₈	—	269	1	no	—
49	C₂×D₄	—	269	2	no	—
50	C₂×D₄	#11761640	1	1	yes	characteristic, complemented
51	C₂²×C₄	—	269	1	no	—
52	C₂²×C₄	—	269	2	no	—
53	C₄⋊C₄	—	538	2	no	—
54	C₂²⋊C₄	—	269	4	no	—
55	C₄²	—	538	2	no	—
56	C₂³	—	269	1	no	—
57	C₂³	C₂×#11761640	1	2	yes	—
58	Q₈	—	538	2	no	—
59	D₄	—	2	2	no	—
60	D₄	—	538	4	no	—
61	C₂×C₄	—	269	1	no	—
62	C₂×C₄	—	269	2	no	—
63	C₂×C₄	—	269	2	no	—
64	C₂×C₄	—	269	2	no	—
65	C₂×C₄	—	538	2	no	—
66	C₂×C₄	—	538	2	no	—
67	C₂×C₄	—	538	4	no	—
68	C₂×C₄	C₂×#11761640	1	1	yes	characteristic
69	C₂²	—	2	2	no	—
70	C₂²	—	4	2	no	—
71	C₂²	—	269	2	no	—
72	C₂²	—	1076	1	no	—
73	C₂²	C₂²×#11761640	1	1	yes	characteristic
74	C₄	—	269	1	no	—
75	C₄	—	269	1	no	—
76	C₄	—	538	2	no	—
77	C₄	—	1076	4	no	—
78	C₄	C₂²⋊#11761640	1	2	yes	—
79	C₂	—	2	1	no	—
80	C₂	—	4	2	no	—
81	C₂	—	538	2	no	—
82	C₂	C₂×C₂²⋊#11761640	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×C₂²⋊#11761640
commutator	C₂²×C₂₆₉	C₂²×C₄
Frattini	C₂²	C₂²×#11761640
Fitting	C₂×D₄×C₂₆₉	C₄
socle	C₂×C₂₆₉	C₂×C₂²⋊C₄
prime radical	C₂×#16864401	C₂
cosocle	C₂²×D₂₆₉	C₂³
hypercenter	C₂×D₄	#11761640
hypocenter	C₂₆₉	#410
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
1. #22575532	C₂	semidirect
2. #22575511	C₂	semidirect
9. C₂×Dic₅₃₈	C₄	semidirect
11. C₂²×Dic₂₆₉	C₄	semidirect
13. C₂×D₄×C₂₆₉	C₄	semidirect
41. C₂₆₉	#410	semidirect
50. C₂×D₄	#11761640	semidirect

Maximal subgroups

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

subgroup	index
#22575532	2
#22575511	2
C₂×#16864401	2
#410	269