Subgroups — #39910147

Order 26176.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27189777	C₂	1	1	yes	maximal, characteristic, complemented
2	#27189775	C₂	1	1	yes	maximal, characteristic, complemented
3	#27189686	C₂	1	1	yes	maximal, characteristic
4	#20266997	—	2	1	no	—
5	C₂²⋊#13109888	—	2	1	no	—
6	C₂×C₂²⋊D₄₀₉	—	2	1	no	—
7	C₂×C₂²⋊D₄₀₉	C₂²	1	1	yes	characteristic
8	C₂²⋊Dic₄₀₉	—	2	1	no	—
9	C₂²⋊Dic₄₀₉	C₄	1	1	yes	characteristic, complemented
10	C₂⁴×C₄₀₉	C₄	1	1	yes	characteristic, complemented
11	C₂×#13109888	—	4	1	no	—
12	#15692550	—	4	1	no	—
13	C₂²×D₄₀₉	D₄	1	1	yes	characteristic
14	C₂²⋊D₄₀₉	—	4	1	no	—
15	C₂²⋊D₄₀₉	—	4	2	no	—
16	C₂×Dic₄₀₉	—	2	1	no	—
17	C₂×Dic₄₀₉	D₄	1	1	yes	characteristic
18	C₂³×C₄₀₉	—	2	1	no	—
19	C₂³×C₄₀₉	—	4	2	no	—
20	C₂³×C₄₀₉	—	4	1	no	—
21	C₂³×C₄₀₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄₀₉	—	2	2	no	—
23	C₂²×C₄₀₉	—	2	1	no	—
24	C₂²×C₄₀₉	—	4	1	no	—
25	C₂²×C₄₀₉	—	4	1	no	—
26	C₂²×C₄₀₉	—	4	2	no	—
27	C₂²×C₄₀₉	—	4	2	no	—
28	C₂²×C₄₀₉	—	4	1	no	—
29	C₂²×C₄₀₉	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×D₄₀₉	—	2	1	no	—
31	C₂×D₄₀₉	—	4	1	no	—
32	#13109888	—	8	1	no	—
33	Dic₄₀₉	—	2	1	no	—
34	Dic₄₀₉	—	4	1	no	—
35	C₂×C₄₀₉	—	2	1	no	—
36	C₂×C₄₀₉	—	4	2	no	—
37	C₂×C₄₀₉	—	4	1	no	—
38	C₂×C₄₀₉	#99	1	1	yes	characteristic
39	D₄₀₉	—	4	1	no	—
40	C₄₀₉	AΣL(1,𝔽₁₆)	1	1	yes	characteristic, complemented, Sylow
41	AΣL(1,𝔽₁₆)	—	409	1	no	maximal, Sylow
42	#120	—	409	1	no	—
43	#100	—	409	1	no	—
44	#99	—	409	1	no	—
45	C₂⁴	#13109888	1	1	yes	characteristic, complemented
46	C₂×D₄	—	818	1	no	—
47	C₂×D₄	—	409	1	no	—
48	M₁₆	—	818	1	no	—
49	C₂²⋊C₄	—	818	1	no	—
50	C₂²⋊C₄	—	818	1	no	—
51	C₂²⋊C₄	—	409	1	no	—
52	C₂³	—	2	1	no	—
53	C₂³	—	4	2	no	—
54	C₂³	—	4	1	no	—
55	C₂³	—	409	1	no	—
56	C₂³	C₂×#13109888	1	1	yes	characteristic
57	D₄	—	1636	1	no	—
58	D₄	—	1636	2	no	—
59	C₂×C₄	—	818	1	no	—
60	C₂×C₄	—	1636	1	no	—
61	C₂×C₄	—	409	1	no	—
62	C₈	—	1636	1	no	—
63	C₂²	—	2	2	no	—
64	C₂²	—	2	1	no	—
65	C₂²	—	4	1	no	—
66	C₂²	—	4	1	no	—
67	C₂²	—	4	2	no	—
68	C₂²	—	4	2	no	—
69	C₂²	—	4	1	no	—
70	C₂²	—	818	1	no	—
71	C₂²	—	1636	1	no	—
72	C₂²	C₂²⋊#13109888	1	1	yes	characteristic
73	C₄	—	818	1	no	—
74	C₄	—	1636	1	no	—
75	C₄	—	3272	1	no	—
76	C₂	—	2	1	no	—
77	C₂	—	4	2	no	—
78	C₂	—	4	1	no	—
79	C₂	—	1636	1	no	—
80	C₂	#27189775	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#27189775
commutator	C₂³×C₄₀₉	C₂×C₄
Frattini	C₂³	C₂×#13109888
Fitting	C₂⁴×C₄₀₉	C₄
socle	C₂×C₄₀₉	#99
prime radical	#27189686	C₂
cosocle	C₂×C₂²⋊D₄₀₉	C₂²
hypercenter	C₂⁴	#13109888
hypocenter	C₄₀₉	AΣL(1,𝔽₁₆)
LCS	G ▷ C₂³×C₄₀₉ ▷ C₂²×C₄₀₉ ▷ C₂×C₄₀₉ ▷ C₄₀₉
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂³ ◁ C₂⁴

Complements

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

N	H ≅ G/N	type
1. #27189777	C₂	semidirect
2. #27189775	C₂	semidirect
9. C₂²⋊Dic₄₀₉	C₄	semidirect
10. C₂⁴×C₄₀₉	C₄	semidirect
40. C₄₀₉	AΣL(1,𝔽₁₆)	semidirect
45. C₂⁴	#13109888	semidirect

Maximal subgroups

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

subgroup	index
#27189777	2
#27189775	2
#27189686	2
AΣL(1,𝔽₁₆)	409