Subgroups — #39910014

Order 26176.

258 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27189724	C₂	1	8	yes	maximal, complemented
2	C₂×#20266972	C₂	1	6	yes	maximal, complemented
3	C₂×#41×C₄₀₉	C₂	1	1	yes	maximal, characteristic
4	#20266972	C₂²	1	4	yes	—
5	#20266972	C₂²	1	12	yes	complemented
6	C₂×#15692543	C₂²	1	12	yes	complemented
7	#41×C₄₀₉	C₄	1	4	yes	—
8	#41×C₄₀₉	C₂²	1	4	yes	—
9	C₂×Q₈×C₄₀₉	C₄	1	1	yes	characteristic
10	C₂×D₄×C₄₀₉	C₄	1	3	yes	—
11	C₂²×C₄×C₄₀₉	C₂²	1	3	yes	—
12	#15692543	C₂³	1	8	yes	—
13	C₂³×C₄₀₉	C₂×C₄	1	3	yes	—
14	Q₈×C₄₀₉	C₂×C₄	1	4	yes	—
15	D₄×C₄₀₉	C₂×C₄	1	12	yes	—
16	C₂×C₄×C₄₀₉	C₂×C₄	1	3	yes	—
17	C₂×C₄×C₄₀₉	C₂×C₄	1	6	yes	—
18	C₂×C₄×C₄₀₉	C₂³	1	1	yes	characteristic
19	C₂×C₄×C₄₀₉	C₂³	1	6	yes	—
20	C₂²×C₄₀₉	—	4	3	no	—
21	C₂²×C₄₀₉	C₂²×C₄	1	1	yes	characteristic
22	C₂²×C₄₀₉	C₂²×C₄	1	6	yes	—
23	C₄×C₄₀₉	C₂²×C₄	1	1	yes	characteristic
24	C₄×C₄₀₉	C₂²×C₄	1	6	yes	—
25	C₄×C₄₀₉	C₂⁴	1	1	yes	characteristic
26	C₂×C₄₀₉	—	2	1	no	—
27	C₂×C₄₀₉	—	2	6	no	—
28	C₂×C₄₀₉	C₂³×C₄	1	1	yes	characteristic
29	C₄₀₉	#568	1	1	yes	characteristic, complemented, Sylow
30	#568	—	409	1	no	maximal, Sylow
31	C₂×#41	D₄₀₉	1	1	yes	characteristic
32	#131	—	409	8	no	—
33	C₂×M₁₆	—	409	6	no	—
34	#41	Dic₄₀₉	1	4	yes	—
35	#41	C₂×D₄₀₉	1	4	yes	—
36	C₂×Q₈	Dic₄₀₉	1	1	yes	characteristic
37	C₂×D₄	Dic₄₀₉	1	3	yes	—
38	C₂²×C₄	C₂×D₄₀₉	1	3	yes	—
39	M₁₆	—	409	4	no	—
40	M₁₆	—	409	12	no	—
41	C₂×C₈	—	409	12	no	—
42	C₂³	C₂×Dic₄₀₉	1	3	yes	—
43	Q₈	C₂×Dic₄₀₉	1	4	yes	—
44	D₄	C₂×Dic₄₀₉	1	12	yes	—
45	C₂×C₄	C₂×Dic₄₀₉	1	3	yes	—
46	C₂×C₄	C₂×Dic₄₀₉	1	6	yes	—
47	C₂×C₄	C₂²×D₄₀₉	1	6	yes	—
48	C₂×C₄	C₂²×D₄₀₉	1	1	yes	characteristic
49	C₈	—	409	8	no	—
50	C₂²	—	4	3	no	—
51	C₂²	C₂²×Dic₄₀₉	1	1	yes	characteristic
52	C₂²	C₂²×Dic₄₀₉	1	6	yes	—
53	C₄	C₂²×Dic₄₀₉	1	1	yes	characteristic
54	C₄	C₂²×Dic₄₀₉	1	6	yes	—
55	C₄	C₂³×D₄₀₉	1	1	yes	characteristic
56	C₂	—	2	1	no	—
57	C₂	—	2	6	no	—
58	C₂	C₂³×Dic₄₀₉	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂³×D₄₀₉
commutator, socle	C₂×C₄₀₉	C₂³×C₄
Fitting, prime radical	C₂×#41×C₄₀₉	C₂
cosocle	C₄×C₄₀₉	C₂⁴
hypercenter	C₂×#41	D₄₀₉
hypocenter	C₄₀₉	#568
LCS	G ▷ C₂×C₄₀₉ ▷ C₄₀₉
UCS	C₁ ◁ C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. #27189724	C₂	semidirect
2. C₂×#20266972	C₂	semidirect
5. #20266972	C₂²	semidirect
6. C₂×#15692543	C₂²	semidirect
29. C₄₀₉	#568	semidirect

Maximal subgroups

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

subgroup	index
#27189724	2
C₂×#20266972	2
C₂×#41×C₄₀₉	2
#568	409