Subgroups — #41×#11761640 #31771671

Order 17216.

310 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#11761640	C₂	1	2	yes	maximal, complemented
2	D₄×#11761640	C₂	1	6	yes	maximal, complemented
3	#22575532	C₂	1	3	yes	maximal, complemented
4	C₂×C₄×#11761640	C₂	1	3	yes	maximal, complemented
5	#41×D₂₆₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11761640	C₂²	1	6	yes	—
7	C₄⋊#11761640	C₂²	1	6	yes	—
8	C₂²⋊#11761640	C₂²	1	6	yes	—
9	C₄×#11761640	C₂²	1	1	yes	characteristic
10	C₄×#11761640	C₂²	1	3	yes	—
11	C₄×#11761640	C₂²	1	6	yes	—
12	#16864403	C₄	1	1	yes	characteristic, complemented
13	#16864402	C₄	1	3	yes	complemented
14	#16864401	C₄	1	3	yes	complemented
15	Q₈×D₂₆₉	C₂²	1	1	yes	characteristic
16	D₄×D₂₆₉	C₂²	1	3	yes	—
17	C₂×C₄×D₂₆₉	C₂²	1	3	yes	—
18	#41×C₂₆₉	C₄	1	1	yes	characteristic, complemented
19	C₂×#11761640	—	2	12	no	—
20	C₂×#11761640	C₂³	1	6	yes	—
21	C₂×#11761640	C₂³	1	2	yes	—
22	C₂²×D₂₆₉	C₂³	1	3	yes	—
23	Dic₅₃₈	C₂×C₄	1	3	yes	complemented
24	D₁₀₇₆	C₂×C₄	1	3	yes	complemented
25	C₂²⋊D₂₆₉	C₂×C₄	1	6	yes	complemented
26	C₂×Dic₂₆₉	C₂×C₄	1	3	yes	complemented
27	C₄×D₂₆₉	C₂×C₄	1	3	yes	complemented
28	C₄×D₂₆₉	C₂×C₄	1	3	yes	complemented
29	C₄×D₂₆₉	C₂³	1	1	yes	characteristic
30	C₄×D₂₆₉	C₂³	1	3	yes	—
31	Q₈×C₂₆₉	C₂×C₄	1	1	yes	characteristic, complemented
32	D₄×C₂₆₉	C₂×C₄	1	3	yes	complemented
33	C₂×C₄×C₂₆₉	C₂×C₄	1	3	yes	complemented
34	C₂²×C₂₆₉	C₂²×C₄	1	3	yes	—
35	C₂×D₂₆₉	—	2	6	no	—
36	C₂×D₂₆₉	C₂²×C₄	1	3	yes	—
37	C₂×D₂₆₉	C₂⁴	1	1	yes	characteristic
38	#11761640	—	2	6	no	—
39	#11761640	#41	1	4	yes	direct product factor
40	C₄×C₂₆₉	C₂²×C₄	1	3	yes	—
41	C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
42	Dic₂₆₉	C₂²×C₄	1	1	yes	characteristic
43	Dic₂₆₉	C₂²×C₄	1	3	yes	—
44	C₂×C₂₆₉	—	2	3	no	—
45	C₂×C₂₆₉	C₂³×C₄	1	1	yes	characteristic
46	D₂₆₉	—	2	3	no	—
47	D₂₆₉	C₂×#41	1	2	yes	—
48	C₂₆₉	C₄×#41	1	1	yes	characteristic, complemented, Sylow
49	C₄×#41	—	269	1	no	maximal, Sylow
50	C₂×#41	—	269	1	no	—
51	C₄×Q₈	—	269	2	no	—
52	C₄×D₄	—	269	6	no	—
53	#117	—	269	3	no	—
54	C₂×C₄²	—	269	3	no	—
55	#41	—	269	1	no	—
56	#41	—	269	3	no	—
57	#41	—	269	3	no	—
58	#41	#11761640	1	1	yes	characteristic, direct product factor
59	C₂×Q₈	—	269	1	no	—
60	C₂×D₄	—	269	3	no	—
61	C₂²×C₄	—	269	3	no	—
62	C₂²×C₄	—	269	6	no	—
63	C₄⋊C₄	—	269	6	no	—
64	C₂²⋊C₄	—	269	6	no	—
65	C₄²	—	269	1	no	—
66	C₄²	—	269	3	no	—
67	C₄²	—	269	6	no	—
68	C₂³	—	269	3	no	—
69	Q₈	—	269	3	no	—
70	Q₈	C₂×#11761640	1	1	yes	characteristic, complemented
71	D₄	—	269	6	no	—
72	D₄	—	269	3	no	—
73	D₄	C₂×#11761640	1	3	yes	complemented
74	C₂×C₄	—	269	2	no	—
75	C₂×C₄	—	269	3	no	—
76	C₂×C₄	—	269	3	no	—
77	C₂×C₄	—	269	3	no	—
78	C₂×C₄	—	269	1	no	—
79	C₂×C₄	—	269	3	no	—
80	C₂×C₄	—	269	6	no	—
81	C₂×C₄	—	538	12	no	—
82	C₂×C₄	C₂×#11761640	1	3	yes	complemented
83	C₂²	—	269	1	no	—
84	C₂²	—	269	3	no	—
85	C₂²	—	538	6	no	—
86	C₂²	C₂²×#11761640	1	3	yes	—
87	C₄	—	269	4	no	—
88	C₄	—	269	1	no	—
89	C₄	—	269	3	no	—
90	C₄	—	538	6	no	—
91	C₄	C₂²×#11761640	1	3	yes	—
92	C₄	C₂²×#11761640	1	1	yes	characteristic
93	C₂	—	2	3	no	—
94	C₂	—	269	2	no	—
95	C₂	—	538	3	no	—
96	C₂	C₂³×#11761640	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×#11761640
commutator, socle	C₂×C₂₆₉	C₂³×C₄
Frattini	C₂	C₂³×#11761640
Fitting	#41×C₂₆₉	C₄
prime radical	#41×D₂₆₉	C₂
cosocle	C₂×D₂₆₉	C₂⁴
hypercenter	#41	#11761640
hypocenter	C₂₆₉	C₄×#41
LCS	G ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₄ ◁ #41

Complements

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

N	H ≅ G/N	type
1. Q₈×#11761640	C₂	semidirect
2. D₄×#11761640	C₂	semidirect
3. #22575532	C₂	semidirect
4. C₂×C₄×#11761640	C₂	semidirect
12. #16864403	C₄	semidirect
13. #16864402	C₄	semidirect
14. #16864401	C₄	semidirect
18. #41×C₂₆₉	C₄	semidirect
23. Dic₅₃₈	C₂×C₄	semidirect
24. D₁₀₇₆	C₂×C₄	semidirect
25. C₂²⋊D₂₆₉	C₂×C₄	semidirect
26. C₂×Dic₂₆₉	C₂×C₄	semidirect
27. C₄×D₂₆₉	C₂×C₄	semidirect
28. C₄×D₂₆₉	C₂×C₄	semidirect
31. Q₈×C₂₆₉	C₂×C₄	semidirect
32. D₄×C₂₆₉	C₂×C₄	semidirect
33. C₂×C₄×C₂₆₉	C₂×C₄	semidirect
39. #11761640	#41	direct (N × H)
48. C₂₆₉	C₄×#41	semidirect
58. #41	#11761640	direct (N × H)
70. Q₈	C₂×#11761640	semidirect
73. D₄	C₂×#11761640	semidirect
82. C₂×C₄	C₂×#11761640	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#11761640	2
D₄×#11761640	2
#22575532	2
C₂×C₄×#11761640	2
#41×D₂₆₉	2
C₄×#41	269