Subgroups — #46830293

Order 34624.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31772650	C₂	1	1	yes	maximal, characteristic
2	#31772634	C₂	1	2	yes	maximal, complemented
3	#31772564	C₂	1	1	yes	maximal, characteristic, complemented
4	#31772556	C₂	1	2	yes	maximal, complemented
5	C₄×C₈×C₅₄₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#22575836	C₂²	1	2	yes	—
7	Dic₂₁₆₄	C₄	1	1	yes	characteristic, complemented
8	#22575825	C₄	1	2	yes	complemented
9	D₄₃₂₈	C₄	1	1	yes	characteristic, complemented
10	#22575820	C₂²	1	2	yes	—
11	#22575819	—	2	2	no	—
12	C₈×D₅₄₁	—	2	2	no	—
13	C₂×C₈×C₅₄₁	C₂²	1	1	yes	characteristic
14	C₂×C₈×C₅₄₁	C₂²	1	1	yes	characteristic
15	C₄²×C₅₄₁	C₂²	1	1	yes	characteristic
16	Dic₁₀₈₂	C₂×C₄	1	2	yes	—
17	D₂₁₆₄	C₂×C₄	1	2	yes	—
18	C₂²⋊D₅₄₁	—	2	2	no	—
19	C₄×D₅₄₁	—	2	2	no	—
20	#16864513	—	2	2	no	—
21	C₂×C₄×C₅₄₁	—	2	1	no	—
22	C₂×C₄×C₅₄₁	C₂³	1	1	yes	characteristic
23	C₈×C₅₄₁	C₂×C₄	1	1	yes	characteristic
24	C₈×C₅₄₁	C₂×C₄	1	1	yes	characteristic
25	C₈×C₅₄₁	D₄	1	1	yes	characteristic
26	C₈×C₅₄₁	D₄	1	1	yes	characteristic
27	C₂²×C₅₄₁	#41	1	1	yes	characteristic
28	C₂×D₅₄₁	—	2	2	no	—
29	C₄×C₅₄₁	—	2	2	no	—
30	C₄×C₅₄₁	C₂²×C₄	1	1	yes	characteristic
31	C₄×C₅₄₁	C₂×D₄	1	1	yes	characteristic
32	Dic₅₄₁	—	2	2	no	—
33	C₂×C₅₄₁	—	2	1	no	—
34	C₂×C₅₄₁	C₄×D₄	1	1	yes	characteristic
35	D₅₄₁	—	4	2	no	—
36	C₅₄₁	#443	1	1	yes	characteristic, complemented, Sylow
37	#443	—	541	1	no	maximal, Sylow
38	#135	—	541	1	no	—
39	#131	—	541	2	no	—
40	#108	—	541	1	no	—
41	#104	—	541	2	no	—
42	C₄×C₈	D₅₄₁	1	1	yes	characteristic, complemented
43	#41	—	541	2	no	—
44	Q₁₆	—	541	1	no	—
45	SD₁₆	—	541	2	no	—
46	D₈	—	541	1	no	—
47	M₁₆	—	541	2	no	—
48	M₁₆	—	1082	2	no	—
49	C₂×C₈	—	1082	2	no	—
50	C₂×C₈	C₂×D₅₄₁	1	1	yes	characteristic
51	C₂×C₈	C₂×D₅₄₁	1	1	yes	characteristic
52	C₄²	C₂×D₅₄₁	1	1	yes	characteristic
53	Q₈	—	541	2	no	—
54	D₄	—	541	2	no	—
55	D₄	—	1082	2	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	1082	2	no	—
58	C₂×C₄	C₂²×D₅₄₁	1	1	yes	characteristic
59	C₈	—	1082	2	no	—
60	C₈	C₄×D₅₄₁	1	1	yes	characteristic
61	C₈	C₄×D₅₄₁	1	1	yes	characteristic
62	C₈	D₂₁₆₄	1	1	yes	characteristic
63	C₈	D₂₁₆₄	1	1	yes	characteristic
64	C₂²	—	1082	2	no	—
65	C₂²	#22575836	1	1	yes	characteristic
66	C₄	—	2	2	no	—
67	C₄	—	1082	2	no	—
68	C₄	C₂×C₄×D₅₄₁	1	1	yes	characteristic
69	C₄	C₂×D₂₁₆₄	1	1	yes	characteristic
70	C₂	—	2	1	no	—
71	C₂	—	2164	2	no	—
72	C₂	C₄×D₂₁₆₄	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₂₁₆₄
commutator	C₄×C₅₄₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₅₄₁
Fitting	C₄×C₈×C₅₄₁	C₂
socle	C₂×C₅₄₁	C₄×D₄
prime radical	#31772650	C₂
cosocle	C₂×C₄×C₅₄₁	C₂³
hypercenter	C₄×C₈	D₅₄₁
hypocenter	C₅₄₁	#443
LCS	G ▷ C₄×C₅₄₁ ▷ C₂×C₅₄₁ ▷ C₅₄₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #31772634	C₂	semidirect
3. #31772564	C₂	semidirect
4. #31772556	C₂	semidirect
5. C₄×C₈×C₅₄₁	C₂	semidirect
7. Dic₂₁₆₄	C₄	semidirect
8. #22575825	C₄	semidirect
9. D₄₃₂₈	C₄	semidirect
36. C₅₄₁	#443	semidirect
42. C₄×C₈	D₅₄₁	semidirect

Maximal subgroups

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

subgroup	index
#31772650	2
#31772634	2
#31772564	2
#31772556	2
C₄×C₈×C₅₄₁	2
#443	541