Subgroups — #39910126

Order 26176.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄₀₉⋊M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#13109888	C₂	1	1	yes	maximal, characteristic
3	M₁₆×D₄₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×#13109888	C₄	1	2	yes	—
5	#20266998	—	2	1	no	—
6	#20266997	C₄	1	2	yes	complemented
7	#20266996	—	2	1	no	—
8	C₄⋊#13109888	—	2	1	no	—
9	C₄⋊#13109888	—	2	1	no	—
10	C₂×C₄×D₄₀₉	C₂²	1	1	yes	characteristic
11	#20266972	C₄	1	1	yes	characteristic, complemented
12	#20266971	—	2	1	no	—
13	C₈×D₄₀₉	—	2	1	no	—
14	M₁₆×C₄₀₉	C₄	1	1	yes	characteristic, complemented
15	C₂×#13109888	—	2	2	no	—
16	C₂×#13109888	—	4	2	no	—
17	#15692550	—	2	2	no	—
18	C₂²×D₄₀₉	C₂×C₄	1	1	yes	characteristic
19	C₂×Dic₄₀₉	C₂×C₄	1	1	yes	characteristic
20	C₄×D₄₀₉	D₄	1	1	yes	characteristic
21	C₄×D₄₀₉	D₄	1	1	yes	characteristic
22	C₄×D₄₀₉	D₄	1	1	yes	characteristic
23	C₄×D₄₀₉	Q₈	1	1	yes	characteristic
24	#15692543	—	2	1	no	—
25	C₂×C₄×C₄₀₉	C₂×C₄	1	1	yes	characteristic
26	C₈×C₄₀₉	—	2	1	no	—
27	C₂²×C₄₀₉	C₄²	1	1	yes	characteristic
28	C₂×D₄₀₉	—	2	1	no	—
29	C₂×D₄₀₉	—	2	1	no	—
30	C₂×D₄₀₉	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×D₄₀₉	C₄⋊C₄	1	1	yes	characteristic
32	#13109888	—	4	2	no	—
33	C₄×C₄₀₉	C₂²⋊C₄	1	1	yes	characteristic
34	C₄×C₄₀₉	C₄⋊C₄	1	1	yes	characteristic
35	Dic₄₀₉	C₂²⋊C₄	1	1	yes	characteristic
36	Dic₄₀₉	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄₀₉	—	2	1	no	—
38	C₂×C₄₀₉	#95	1	1	yes	characteristic
39	D₄₀₉	—	2	1	no	—
40	D₄₀₉	#100	1	1	yes	characteristic
41	D₄₀₉	#101	1	1	yes	characteristic
42	C₄₀₉	#343	1	1	yes	characteristic, complemented, Sylow
43	#343	—	409	1	no	maximal, Sylow
44	C₂×M₁₆	—	409	1	no	—
45	C₂×M₁₆	—	409	1	no	—
46	C₂×C₄⋊C₄	—	409	1	no	—
47	C₂²×C₄	—	409	2	no	—
48	C₂²×C₄	—	409	1	no	—
49	M₁₆	—	818	1	no	—
50	M₁₆	—	818	1	no	—
51	M₁₆	—	409	2	no	—
52	M₁₆	—	409	1	no	—
53	M₁₆	#13109888	1	1	yes	characteristic, complemented
54	C₂×C₈	—	818	1	no	—
55	C₂×C₈	—	818	1	no	—
56	C₄⋊C₄	—	818	1	no	—
57	C₄⋊C₄	—	818	1	no	—
58	C₂³	—	409	1	no	—
59	C₂×C₄	—	818	2	no	—
60	C₂×C₄	—	1636	2	no	—
61	C₂×C₄	—	409	1	no	—
62	C₂×C₄	—	409	1	no	—
63	C₂×C₄	—	409	1	no	—
64	C₂×C₄	—	409	1	no	—
65	C₂×C₄	—	409	1	no	—
66	C₂×C₄	C₂×#13109888	1	1	yes	characteristic
67	C₈	—	2	1	no	—
68	C₈	—	818	2	no	—
69	C₈	—	818	1	no	—
70	C₂²	—	818	1	no	—
71	C₂²	—	818	1	no	—
72	C₂²	—	409	1	no	—
73	C₂²	—	409	1	no	—
74	C₂²	C₄×#13109888	1	1	yes	characteristic
75	C₄	—	1636	2	no	—
76	C₄	—	409	1	no	—
77	C₄	—	409	1	no	—
78	C₄	C₂²⋊#13109888	1	1	yes	characteristic
79	C₄	C₄⋊#13109888	1	1	yes	characteristic
80	C₂	—	2	1	no	—
81	C₂	—	818	1	no	—
82	C₂	—	409	1	no	—
83	C₂	—	409	1	no	—
84	C₂	(C₂×#13109888)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	(C₂×#13109888)⋊C₄
commutator	C₂²×C₄₀₉	C₄²
Frattini	C₂×C₄	C₂×#13109888
Fitting	M₁₆×C₄₀₉	C₄
socle	C₂×C₄₀₉	#95
prime radical	C₂²×D₄₀₉	C₂×C₄
cosocle	C₂×C₄×D₄₀₉	C₂²
hypercenter	M₁₆	#13109888
hypocenter	C₄₀₉	#343
LCS	G ▷ C₂²×C₄₀₉ ▷ C₂×C₄₀₉ ▷ C₄₀₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
6. #20266997	C₄	semidirect
11. #20266972	C₄	semidirect
14. M₁₆×C₄₀₉	C₄	semidirect
42. C₄₀₉	#343	semidirect
53. M₁₆	#13109888	semidirect

Maximal subgroups

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

subgroup	index
D₄₀₉⋊M₁₆	2
C₂×C₄⋊#13109888	2
M₁₆×D₄₀₉	2
#343	409