Subgroups — #31771684

Order 17216.

258 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22575541	C₂	1	2	yes	maximal, complemented
2	#22575540	C₂	1	6	yes	maximal, complemented
3	C₂×#16864412	C₂	1	3	yes	maximal, complemented
4	D₂₆₉⋊M₁₆	C₂	1	3	yes	maximal, complemented
5	#41×D₂₆₉	C₂	1	1	yes	maximal, characteristic
6	#16864412	C₂²	1	1	yes	characteristic
7	#16864412	C₂²	1	3	yes	—
8	#16864412	C₂²	1	6	yes	complemented
9	#16864411	C₂²	1	6	yes	complemented
10	D₂₆₉⋊C₈	C₂²	1	6	yes	complemented
11	C₂×#14524137	C₂²	1	6	yes	complemented
12	#16864403	C₂²	1	1	yes	characteristic
13	#16864402	C₄	1	3	yes	—
14	#16864401	C₂²	1	3	yes	—
15	Q₈×D₂₆₉	C₄	1	1	yes	characteristic
16	D₄×D₂₆₉	C₄	1	3	yes	—
17	C₂×C₄×D₂₆₉	C₂²	1	3	yes	—
18	#41×C₂₆₉	C₄	1	1	yes	characteristic
19	#14524137	C₂³	1	6	yes	—
20	#14524137	C₂³	1	2	yes	—
21	C₂²×D₂₆₉	C₂×C₄	1	3	yes	—
22	Dic₅₃₈	C₂×C₄	1	3	yes	—
23	D₁₀₇₆	C₂×C₄	1	3	yes	—
24	C₂²⋊D₂₆₉	C₂×C₄	1	6	yes	—
25	C₂×Dic₂₆₉	C₂³	1	3	yes	—
26	C₄×D₂₆₉	C₂×C₄	1	3	yes	—
27	C₄×D₂₆₉	C₂×C₄	1	3	yes	—
28	C₄×D₂₆₉	C₂³	1	1	yes	characteristic
29	C₄×D₂₆₉	C₂³	1	3	yes	—
30	Q₈×C₂₆₉	C₂×C₄	1	1	yes	characteristic
31	D₄×C₂₆₉	C₂×C₄	1	3	yes	—
32	C₂×C₄×C₂₆₉	C₂×C₄	1	3	yes	—
33	C₂²×C₂₆₉	C₂²×C₄	1	3	yes	—
34	C₂×D₂₆₉	—	4	3	no	—
35	C₂×D₂₆₉	C₂²×C₄	1	1	yes	characteristic
36	C₂×D₂₆₉	C₂²×C₄	1	3	yes	—
37	C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
38	C₄×C₂₆₉	C₂²×C₄	1	3	yes	—
39	Dic₂₆₉	C₂²×C₄	1	3	yes	—
40	Dic₂₆₉	C₂⁴	1	1	yes	characteristic
41	C₂×C₂₆₉	—	2	3	no	—
42	C₂×C₂₆₉	C₂³×C₄	1	1	yes	characteristic
43	D₂₆₉	—	2	1	no	—
44	D₂₆₉	—	2	3	no	—
45	C₂₆₉	#568	1	1	yes	characteristic, complemented, Sylow
46	#568	—	269	1	no	maximal, Sylow
47	C₂×#41	—	269	1	no	—
48	#131	—	269	6	no	—
49	#131	—	269	2	no	—
50	C₂×M₁₆	—	269	3	no	—
51	C₂×M₁₆	—	269	3	no	—
52	#41	—	269	1	no	—
53	#41	—	269	3	no	—
54	#41	—	269	3	no	—
55	#41	#11761640	1	1	yes	characteristic
56	C₂×Q₈	—	269	1	no	—
57	C₂×D₄	—	269	3	no	—
58	C₂²×C₄	—	269	3	no	—
59	M₁₆	—	269	1	no	—
60	M₁₆	—	269	3	no	—
61	M₁₆	—	269	6	no	—
62	M₁₆	—	269	6	no	—
63	C₂×C₈	—	269	6	no	—
64	C₂×C₈	—	269	6	no	—
65	C₂³	—	269	3	no	—
66	Q₈	—	269	3	no	—
67	Q₈	C₂×#11761640	1	1	yes	characteristic
68	D₄	—	269	6	no	—
69	D₄	—	269	3	no	—
70	D₄	C₂×#11761640	1	3	yes	—
71	C₂×C₄	—	269	3	no	—
72	C₂×C₄	—	269	3	no	—
73	C₂×C₄	—	269	1	no	—
74	C₂×C₄	—	269	3	no	—
75	C₂×C₄	—	269	3	no	—
76	C₂×C₄	C₂×#11761640	1	3	yes	—
77	C₈	—	269	6	no	—
78	C₈	—	269	2	no	—
79	C₂²	—	269	1	no	—
80	C₂²	—	269	3	no	—
81	C₂²	—	1076	3	no	—
82	C₂²	C₂²×#11761640	1	3	yes	—
83	C₄	—	269	1	no	—
84	C₄	—	269	3	no	—
85	C₄	C₂²×#11761640	1	1	yes	characteristic
86	C₄	C₂²×#11761640	1	3	yes	—
87	C₂	—	2	3	no	—
88	C₂	—	538	1	no	—
89	C₂	—	538	3	no	—
90	C₂	C₂³×#11761640	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×#11761640
commutator, socle	C₂×C₂₆₉	C₂³×C₄
Fitting	#41×C₂₆₉	C₄
prime radical	#41×D₂₆₉	C₂
cosocle	Dic₂₆₉	C₂⁴
hypercenter	#41	#11761640
hypocenter	C₂₆₉	#568
LCS	G ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂ ◁ #41

Complements

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

N	H ≅ G/N	type
1. #22575541	C₂	semidirect
2. #22575540	C₂	semidirect
3. C₂×#16864412	C₂	semidirect
4. D₂₆₉⋊M₁₆	C₂	semidirect
8. #16864412	C₂²	semidirect
9. #16864411	C₂²	semidirect
10. D₂₆₉⋊C₈	C₂²	semidirect
11. C₂×#14524137	C₂²	semidirect
45. C₂₆₉	#568	semidirect

Maximal subgroups

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

subgroup	index
#22575541	2
#22575540	2
C₂×#16864412	2
D₂₆₉⋊M₁₆	2
#41×D₂₆₉	2
#568	269