Subgroups — #39909686

Order 26176.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27189736	C₂	1	1	yes	maximal, characteristic
2	Q₁₆×D₄₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	Dic₃₂₇₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#27189663	C₂	1	1	yes	maximal, characteristic, complemented
5	#27189661	C₂	1	1	yes	maximal, characteristic, complemented
6	#27189656	C₂	1	1	yes	maximal, characteristic, complemented
7	SD₃₂×C₄₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#20266987	—	2	1	no	—
9	Q₈×D₄₀₉	—	2	1	no	—
10	Dic₁₆₃₆	C₂²	1	1	yes	characteristic
11	Dic₁₆₃₆	C₂²	1	1	yes	characteristic
12	#20266978	—	2	1	no	—
13	#20266976	—	2	1	no	—
14	C₈×D₄₀₉	C₂²	1	1	yes	characteristic
15	#20266963	C₂²	1	1	yes	characteristic
16	Q₁₆×C₄₀₉	C₂²	1	1	yes	characteristic
17	D₈×C₄₀₉	C₂²	1	1	yes	characteristic
18	C₁₆×C₄₀₉	C₂²	1	1	yes	characteristic
19	Dic₈₁₈	—	2	1	no	—
20	Dic₈₁₈	—	2	1	no	—
21	Dic₈₁₈	—	4	1	no	—
22	C₂²⋊D₄₀₉	—	4	1	no	—
23	C₂×Dic₄₀₉	—	4	1	no	—
24	C₄×D₄₀₉	—	4	1	no	—
25	C₄×D₄₀₉	D₄	1	1	yes	characteristic
26	#15692543	D₄	1	1	yes	characteristic
27	Q₈×C₄₀₉	—	2	1	no	—
28	D₄×C₄₀₉	—	2	1	no	—
29	C₈×C₄₀₉	C₂³	1	1	yes	characteristic
30	C₂²×C₄₀₉	—	4	1	no	—
31	C₂×D₄₀₉	D₈	1	1	yes	characteristic
32	C₄×C₄₀₉	—	4	1	no	—
33	C₄×C₄₀₉	C₂×D₄	1	1	yes	characteristic
34	Dic₄₀₉	—	4	1	no	—
35	Dic₄₀₉	—	4	1	no	—
36	Dic₄₀₉	D₈	1	1	yes	characteristic
37	C₂×C₄₀₉	—	8	1	no	—
38	C₂×C₄₀₉	C₂×D₈	1	1	yes	characteristic
39	D₄₀₉	—	2	1	no	—
40	C₄₀₉	#510	1	1	yes	characteristic, complemented, Sylow
41	#510	—	409	1	no	maximal, Sylow
42	#135	—	409	1	no	—
43	C₂×Q₁₆	—	409	1	no	—
44	Q₃₂	—	409	1	no	—
45	Q₃₂	—	409	1	no	—
46	SD₃₂	—	409	1	no	—
47	SD₃₂	D₄₀₉	1	1	yes	characteristic, complemented
48	M₃₂	—	409	1	no	—
49	#41	—	818	1	no	—
50	C₂×Q₈	—	818	1	no	—
51	Q₁₆	—	818	1	no	—
52	Q₁₆	—	409	1	no	—
53	Q₁₆	—	409	1	no	—
54	Q₁₆	C₂×D₄₀₉	1	1	yes	characteristic
55	SD₁₆	—	818	1	no	—
56	D₈	C₂×D₄₀₉	1	1	yes	characteristic
57	C₂×C₈	—	409	1	no	—
58	C₁₆	—	409	1	no	—
59	C₁₆	C₂×D₄₀₉	1	1	yes	characteristic
60	Q₈	—	2	1	no	—
61	Q₈	—	818	1	no	—
62	Q₈	—	818	1	no	—
63	Q₈	—	1636	1	no	—
64	D₄	—	2	1	no	—
65	D₄	—	1636	1	no	—
66	C₂×C₄	—	1636	1	no	—
67	C₂×C₄	—	1636	1	no	—
68	C₂×C₄	—	409	1	no	—
69	C₈	—	409	1	no	—
70	C₈	C₂²×D₄₀₉	1	1	yes	characteristic
71	C₂²	—	4	1	no	—
72	C₂²	—	409	1	no	—
73	C₄	—	4	1	no	—
74	C₄	—	1636	1	no	—
75	C₄	—	1636	1	no	—
76	C₄	—	409	1	no	—
77	C₄	D₄×D₄₀₉	1	1	yes	characteristic
78	C₂	—	8	1	no	—
79	C₂	—	818	1	no	—
80	C₂	D₈×D₄₀₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×D₄₀₉
commutator, cosocle	C₈×C₄₀₉	C₂³
Frattini	C₈	C₂²×D₄₀₉
Fitting	SD₃₂×C₄₀₉	C₂
socle	C₂×C₄₀₉	C₂×D₈
prime radical	#27189736	C₂
hypercenter	SD₃₂	D₄₀₉
hypocenter	C₄₀₉	#510
LCS	G ▷ C₈×C₄₀₉ ▷ C₄×C₄₀₉ ▷ C₂×C₄₀₉ ▷ C₄₀₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ SD₃₂

Complements

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

N	H ≅ G/N	type
2. Q₁₆×D₄₀₉	C₂	semidirect
3. Dic₃₂₇₂	C₂	semidirect
4. #27189663	C₂	semidirect
5. #27189661	C₂	semidirect
6. #27189656	C₂	semidirect
7. SD₃₂×C₄₀₉	C₂	semidirect
40. C₄₀₉	#510	semidirect
47. SD₃₂	D₄₀₉	semidirect

Maximal subgroups

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

subgroup	index
#27189736	2
Q₁₆×D₄₀₉	2
Dic₃₂₇₂	2
#27189663	2
#27189661	2
#27189656	2
SD₃₂×C₄₀₉	2
#510	409