Subgroups — #38786232

Order 25664.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27165759	C₂	1	1	yes	maximal, characteristic
2	#27165743	C₂	1	2	yes	maximal, complemented
3	#27165673	C₂	1	1	yes	maximal, characteristic, complemented
4	#27165665	C₂	1	2	yes	maximal, complemented
5	C₄×C₈×C₄₀₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#20264366	C₂²	1	2	yes	—
7	Dic₁₆₀₄	C₄	1	1	yes	characteristic, complemented
8	#20264355	C₄	1	2	yes	complemented
9	D₃₂₀₈	C₄	1	1	yes	characteristic, complemented
10	#20264350	C₂²	1	2	yes	—
11	#20264349	—	2	2	no	—
12	C₈×D₄₀₁	—	2	2	no	—
13	C₂×C₈×C₄₀₁	C₂²	1	1	yes	characteristic
14	C₂×C₈×C₄₀₁	C₂²	1	1	yes	characteristic
15	C₄²×C₄₀₁	C₂²	1	1	yes	characteristic
16	Dic₈₀₂	C₂×C₄	1	2	yes	—
17	D₁₆₀₄	C₂×C₄	1	2	yes	—
18	C₂²⋊D₄₀₁	—	2	2	no	—
19	C₄×D₄₀₁	—	2	2	no	—
20	#15691999	—	2	2	no	—
21	C₂×C₄×C₄₀₁	—	2	1	no	—
22	C₂×C₄×C₄₀₁	C₂³	1	1	yes	characteristic
23	C₈×C₄₀₁	C₂×C₄	1	1	yes	characteristic
24	C₈×C₄₀₁	C₂×C₄	1	1	yes	characteristic
25	C₈×C₄₀₁	D₄	1	1	yes	characteristic
26	C₈×C₄₀₁	D₄	1	1	yes	characteristic
27	C₂²×C₄₀₁	#41	1	1	yes	characteristic
28	C₂×D₄₀₁	—	2	2	no	—
29	C₄×C₄₀₁	—	2	2	no	—
30	C₄×C₄₀₁	C₂²×C₄	1	1	yes	characteristic
31	C₄×C₄₀₁	C₂×D₄	1	1	yes	characteristic
32	Dic₄₀₁	—	2	2	no	—
33	C₂×C₄₀₁	—	2	1	no	—
34	C₂×C₄₀₁	C₄×D₄	1	1	yes	characteristic
35	D₄₀₁	—	4	2	no	—
36	C₄₀₁	#443	1	1	yes	characteristic, complemented, Sylow
37	#443	—	401	1	no	maximal, Sylow
38	#135	—	401	1	no	—
39	#131	—	401	2	no	—
40	#108	—	401	1	no	—
41	#104	—	401	2	no	—
42	C₄×C₈	D₄₀₁	1	1	yes	characteristic, complemented
43	#41	—	401	2	no	—
44	Q₁₆	—	401	1	no	—
45	SD₁₆	—	401	2	no	—
46	D₈	—	401	1	no	—
47	M₁₆	—	802	2	no	—
48	M₁₆	—	401	2	no	—
49	C₂×C₈	—	802	2	no	—
50	C₂×C₈	C₂×D₄₀₁	1	1	yes	characteristic
51	C₂×C₈	C₂×D₄₀₁	1	1	yes	characteristic
52	C₄²	C₂×D₄₀₁	1	1	yes	characteristic
53	Q₈	—	401	2	no	—
54	D₄	—	802	2	no	—
55	D₄	—	401	2	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	802	2	no	—
58	C₂×C₄	C₂²×D₄₀₁	1	1	yes	characteristic
59	C₈	—	802	2	no	—
60	C₈	C₄×D₄₀₁	1	1	yes	characteristic
61	C₈	C₄×D₄₀₁	1	1	yes	characteristic
62	C₈	D₁₆₀₄	1	1	yes	characteristic
63	C₈	D₁₆₀₄	1	1	yes	characteristic
64	C₂²	—	802	2	no	—
65	C₂²	#20264366	1	1	yes	characteristic
66	C₄	—	2	2	no	—
67	C₄	—	802	2	no	—
68	C₄	C₂×C₄×D₄₀₁	1	1	yes	characteristic
69	C₄	C₂×D₁₆₀₄	1	1	yes	characteristic
70	C₂	—	2	1	no	—
71	C₂	—	1604	2	no	—
72	C₂	C₄×D₁₆₀₄	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₁₆₀₄
commutator	C₄×C₄₀₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₄₀₁
Fitting	C₄×C₈×C₄₀₁	C₂
socle	C₂×C₄₀₁	C₄×D₄
prime radical	#27165759	C₂
cosocle	C₂×C₄×C₄₀₁	C₂³
hypercenter	C₄×C₈	D₄₀₁
hypocenter	C₄₀₁	#443
LCS	G ▷ C₄×C₄₀₁ ▷ C₂×C₄₀₁ ▷ C₄₀₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #27165743	C₂	semidirect
3. #27165673	C₂	semidirect
4. #27165665	C₂	semidirect
5. C₄×C₈×C₄₀₁	C₂	semidirect
7. Dic₁₆₀₄	C₄	semidirect
8. #20264355	C₄	semidirect
9. D₃₂₀₈	C₄	semidirect
36. C₄₀₁	#443	semidirect
42. C₄×C₈	D₄₀₁	semidirect

Maximal subgroups

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

subgroup	index
#27165759	2
#27165743	2
#27165673	2
#27165665	2
C₄×C₈×C₄₀₁	2
#443	401