Subgroups — #39909365

Order 26176.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27189739	C₂	1	1	yes	maximal, characteristic
2	#27189723	C₂	1	2	yes	maximal, complemented
3	#27189653	C₂	1	1	yes	maximal, characteristic, complemented
4	#27189645	C₂	1	2	yes	maximal, complemented
5	C₄×C₈×C₄₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#20266988	C₂²	1	2	yes	—
7	Dic₁₆₃₆	C₄	1	1	yes	characteristic, complemented
8	#20266977	C₄	1	2	yes	complemented
9	D₃₂₇₂	C₄	1	1	yes	characteristic, complemented
10	#20266972	C₂²	1	2	yes	—
11	#20266971	—	2	2	no	—
12	C₈×D₄₀₉	—	2	2	no	—
13	C₂×C₈×C₄₀₉	C₂²	1	1	yes	characteristic
14	C₂×C₈×C₄₀₉	C₂²	1	1	yes	characteristic
15	C₄²×C₄₀₉	C₂²	1	1	yes	characteristic
16	Dic₈₁₈	C₂×C₄	1	2	yes	—
17	D₁₆₃₆	C₂×C₄	1	2	yes	—
18	C₂²⋊D₄₀₉	—	2	2	no	—
19	C₄×D₄₀₉	—	2	2	no	—
20	#15692543	—	2	2	no	—
21	C₂×C₄×C₄₀₉	—	2	1	no	—
22	C₂×C₄×C₄₀₉	C₂³	1	1	yes	characteristic
23	C₈×C₄₀₉	C₂×C₄	1	1	yes	characteristic
24	C₈×C₄₀₉	C₂×C₄	1	1	yes	characteristic
25	C₈×C₄₀₉	D₄	1	1	yes	characteristic
26	C₈×C₄₀₉	D₄	1	1	yes	characteristic
27	C₂²×C₄₀₉	#41	1	1	yes	characteristic
28	C₂×D₄₀₉	—	2	2	no	—
29	C₄×C₄₀₉	—	2	2	no	—
30	C₄×C₄₀₉	C₂²×C₄	1	1	yes	characteristic
31	C₄×C₄₀₉	C₂×D₄	1	1	yes	characteristic
32	Dic₄₀₉	—	2	2	no	—
33	C₂×C₄₀₉	—	2	1	no	—
34	C₂×C₄₀₉	C₄×D₄	1	1	yes	characteristic
35	D₄₀₉	—	4	2	no	—
36	C₄₀₉	#443	1	1	yes	characteristic, complemented, Sylow
37	#443	—	409	1	no	maximal, Sylow
38	#135	—	409	1	no	—
39	#131	—	409	2	no	—
40	#108	—	409	1	no	—
41	#104	—	409	2	no	—
42	C₄×C₈	D₄₀₉	1	1	yes	characteristic, complemented
43	#41	—	409	2	no	—
44	Q₁₆	—	409	1	no	—
45	SD₁₆	—	409	2	no	—
46	D₈	—	409	1	no	—
47	M₁₆	—	818	2	no	—
48	M₁₆	—	409	2	no	—
49	C₂×C₈	—	818	2	no	—
50	C₂×C₈	C₂×D₄₀₉	1	1	yes	characteristic
51	C₂×C₈	C₂×D₄₀₉	1	1	yes	characteristic
52	C₄²	C₂×D₄₀₉	1	1	yes	characteristic
53	Q₈	—	409	2	no	—
54	D₄	—	818	2	no	—
55	D₄	—	409	2	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	818	2	no	—
58	C₂×C₄	C₂²×D₄₀₉	1	1	yes	characteristic
59	C₈	—	818	2	no	—
60	C₈	C₄×D₄₀₉	1	1	yes	characteristic
61	C₈	C₄×D₄₀₉	1	1	yes	characteristic
62	C₈	D₁₆₃₆	1	1	yes	characteristic
63	C₈	D₁₆₃₆	1	1	yes	characteristic
64	C₂²	—	818	2	no	—
65	C₂²	#20266988	1	1	yes	characteristic
66	C₄	—	2	2	no	—
67	C₄	—	818	2	no	—
68	C₄	C₂×C₄×D₄₀₉	1	1	yes	characteristic
69	C₄	C₂×D₁₆₃₆	1	1	yes	characteristic
70	C₂	—	2	1	no	—
71	C₂	—	1636	2	no	—
72	C₂	C₄×D₁₆₃₆	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₁₆₃₆
commutator	C₄×C₄₀₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₄₀₉
Fitting	C₄×C₈×C₄₀₉	C₂
socle	C₂×C₄₀₉	C₄×D₄
prime radical	#27189739	C₂
cosocle	C₂×C₄×C₄₀₉	C₂³
hypercenter	C₄×C₈	D₄₀₉
hypocenter	C₄₀₉	#443
LCS	G ▷ C₄×C₄₀₉ ▷ C₂×C₄₀₉ ▷ C₄₀₉
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #27189723	C₂	semidirect
3. #27189653	C₂	semidirect
4. #27189645	C₂	semidirect
5. C₄×C₈×C₄₀₉	C₂	semidirect
7. Dic₁₆₃₆	C₄	semidirect
8. #20266977	C₄	semidirect
9. D₃₂₇₂	C₄	semidirect
36. C₄₀₉	#443	semidirect
42. C₄×C₈	D₄₀₉	semidirect

Maximal subgroups

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

subgroup	index
#27189739	2
#27189723	2
#27189653	2
#27189645	2
C₄×C₈×C₄₀₉	2
#443	409