Subgroups — #31771615

Order 17216.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22575532	C₂	1	2	yes	maximal
2	#22575522	C₂	1	2	yes	maximal, complemented
3	#22575521	C₂	1	2	yes	maximal, complemented
4	C₂×C₈×D₂₆₉	C₂	1	1	yes	maximal, characteristic
5	C₄⋊#11761640	C₂²	1	4	yes	—
6	C₂²⋊#11761640	—	2	2	no	—
7	C₄×#11761640	—	2	2	no	—
8	C₂×C₄×D₂₆₉	C₂²	1	1	yes	characteristic
9	C₂×#14524130	C₄	1	1	yes	characteristic, complemented
10	C₈×D₂₆₉	C₄	1	1	yes	characteristic, complemented
11	C₈×D₂₆₉	C₄	1	1	yes	characteristic, complemented
12	C₈×D₂₆₉	C₂²	1	1	yes	characteristic
13	C₈×D₂₆₉	C₂²	1	1	yes	characteristic
14	C₂×C₈×C₂₆₉	C₄	1	1	yes	characteristic, complemented
15	C₂×#11761640	—	2	4	no	—
16	C₂²×D₂₆₉	D₄	1	1	yes	characteristic
17	C₂×Dic₂₆₉	Q₈	1	1	yes	characteristic
18	C₄×D₂₆₉	C₂×C₄	1	1	yes	characteristic
19	C₄×D₂₆₉	D₄	1	1	yes	characteristic
20	C₄×D₂₆₉	Q₈	1	1	yes	characteristic
21	C₄×D₂₆₉	C₂³	1	1	yes	characteristic
22	#14524130	C₂×C₄	1	1	yes	characteristic
23	#14524130	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
25	C₈×C₂₆₉	C₂×C₄	1	1	yes	characteristic
26	C₈×C₂₆₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₂₆₉	C₄⋊C₄	1	1	yes	characteristic
28	C₂×D₂₆₉	—	2	2	no	—
29	C₂×D₂₆₉	C₄⋊C₄	1	1	yes	characteristic
30	C₂×D₂₆₉	C₂×D₄	1	1	yes	characteristic
31	#11761640	—	4	4	no	—
32	C₄×C₂₆₉	C₄⋊C₄	1	1	yes	characteristic
33	C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
34	Dic₂₆₉	C₄⋊C₄	1	1	yes	characteristic
35	Dic₂₆₉	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₂₆₉	—	2	1	no	—
37	C₂×C₂₆₉	C₂×C₄⋊C₄	1	1	yes	characteristic
38	D₂₆₉	—	2	1	no	—
39	D₂₆₉	#135	1	2	yes	—
40	C₂₆₉	#427	1	1	yes	characteristic, complemented, Sylow
41	#427	—	269	1	no	maximal, Sylow
42	C₂²×C₈	—	269	1	no	—
43	#117	—	269	2	no	—
44	#107	—	269	2	no	—
45	#106	—	269	2	no	—
46	C₂²×C₄	—	269	1	no	—
47	C₂×C₈	—	269	1	no	—
48	C₂×C₈	—	269	1	no	—
49	C₂×C₈	—	269	1	no	—
50	C₂×C₈	—	269	1	no	—
51	C₂×C₈	—	269	1	no	—
52	C₂×C₈	#11761640	1	1	yes	characteristic, complemented
53	C₄⋊C₄	—	269	4	no	—
54	C₂²⋊C₄	—	538	2	no	—
55	C₄²	—	538	2	no	—
56	C₂³	—	269	1	no	—
57	C₂×C₄	—	269	1	no	—
58	C₂×C₄	—	269	1	no	—
59	C₂×C₄	—	269	1	no	—
60	C₂×C₄	—	269	1	no	—
61	C₂×C₄	—	269	1	no	—
62	C₂×C₄	—	538	4	no	—
63	C₂×C₄	C₂×#11761640	1	1	yes	characteristic
64	C₈	—	269	1	no	—
65	C₈	—	269	1	no	—
66	C₈	C₂×#11761640	1	1	yes	characteristic
67	C₈	C₂×#11761640	1	1	yes	characteristic
68	C₂²	—	269	1	no	—
69	C₂²	—	269	1	no	—
70	C₂²	—	538	2	no	—
71	C₂²	C₄⋊#11761640	1	1	yes	characteristic
72	C₄	—	269	1	no	—
73	C₄	—	269	1	no	—
74	C₄	—	1076	4	no	—
75	C₄	C₄⋊#11761640	1	1	yes	characteristic
76	C₄	C₂²×#11761640	1	1	yes	characteristic
77	C₂	—	2	1	no	—
78	C₂	—	269	2	no	—
79	C₂	—	538	1	no	—
80	C₂	C₂×C₄⋊#11761640	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄⋊#11761640
commutator	C₄×C₂₆₉	C₂²×C₄
Frattini	C₄	C₂²×#11761640
Fitting	C₂×C₈×C₂₆₉	C₄
socle	C₂×C₂₆₉	C₂×C₄⋊C₄
prime radical	C₂²×D₂₆₉	D₄
cosocle	C₄×D₂₆₉	C₂³
hypercenter	C₂×C₈	#11761640
hypocenter	C₂₆₉	#427
LCS	G ▷ C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
2. #22575522	C₂	semidirect
3. #22575521	C₂	semidirect
9. C₂×#14524130	C₄	semidirect
10. C₈×D₂₆₉	C₄	semidirect
11. C₈×D₂₆₉	C₄	semidirect
14. C₂×C₈×C₂₆₉	C₄	semidirect
40. C₂₆₉	#427	semidirect
52. C₂×C₈	#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
#22575522	2
#22575521	2
C₂×C₈×D₂₆₉	2
#427	269