Subgroups — #46830281

Order 34624.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₂₁₆₄	C₂	1	1	yes	maximal, characteristic
2	#31772593	C₂	1	2	yes	maximal
3	#31772551	C₂	1	2	yes	maximal
4	#31772538	C₂	1	1	yes	maximal, characteristic
5	#106×C₅₄₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₀₈₂	C₂²	1	2	yes	—
7	Dic₂₁₆₄	C₄	1	4	yes	complemented
8	C₂×#16864513	C₂²	1	1	yes	characteristic
9	#22575815	—	2	2	no	—
10	C₄×Dic₅₄₁	—	2	2	no	—
11	C₄×Dic₅₄₁	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₅₄₁	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₅₄₁	C₂²	1	2	yes	—
14	Dic₁₀₈₂	—	2	2	no	—
15	Dic₁₀₈₂	C₂×C₄	1	4	yes	—
16	C₂×Dic₅₄₁	—	2	2	no	—
17	C₂×Dic₅₄₁	D₄	1	2	yes	—
18	#16864513	—	2	1	no	—
19	C₂×C₄×C₅₄₁	—	2	2	no	—
20	C₂×C₄×C₅₄₁	C₂³	1	1	yes	characteristic
21	C₈×C₅₄₁	C₂×C₄	1	2	yes	—
22	C₂²×C₅₄₁	C₂×D₄	1	1	yes	characteristic
23	C₄×C₅₄₁	—	4	2	no	—
24	C₄×C₅₄₁	C₂²×C₄	1	1	yes	characteristic
25	C₄×C₅₄₁	#41	1	1	yes	characteristic
26	Dic₅₄₁	—	2	4	no	—
27	Dic₅₄₁	—	2	2	no	—
28	C₂×C₅₄₁	C₄×D₄	1	1	yes	characteristic
29	C₂×C₅₄₁	#137	1	2	yes	—
30	C₅₄₁	#441	1	1	yes	characteristic, complemented, Sylow
31	#441	—	541	1	no	maximal, Sylow
32	C₂×Q₁₆	—	541	1	no	—
33	C₄×Q₈	—	541	2	no	—
34	#106	D₅₄₁	1	1	yes	characteristic
35	#103	—	541	2	no	—
36	#97	—	541	1	no	—
37	C₂×Q₈	—	541	2	no	—
38	Q₁₆	—	541	4	no	—
39	C₂×C₈	—	541	1	no	—
40	C₂×C₈	C₂×D₅₄₁	1	1	yes	characteristic
41	C₄⋊C₄	—	1082	2	no	—
42	C₄⋊C₄	C₂×D₅₄₁	1	2	yes	—
43	C₄²	—	541	1	no	—
44	C₄²	—	1082	2	no	—
45	Q₈	—	541	4	no	—
46	Q₈	—	1082	2	no	—
47	C₂×C₄	—	2	2	no	—
48	C₂×C₄	—	541	2	no	—
49	C₂×C₄	—	1082	2	no	—
50	C₂×C₄	C₂²×D₅₄₁	1	1	yes	characteristic
51	C₈	—	1082	1	no	—
52	C₈	C₄×D₅₄₁	1	2	yes	—
53	C₂²	D₄×D₅₄₁	1	1	yes	characteristic
54	C₄	—	4	2	no	—
55	C₄	—	1082	4	no	—
56	C₄	—	1082	2	no	—
57	C₄	C₂×C₄×D₅₄₁	1	1	yes	characteristic
58	C₄	#22575837	1	1	yes	characteristic
59	C₂	#31772590	1	1	yes	characteristic
60	C₂	#31772657	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₅₄₁
commutator	C₄×C₅₄₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₅₄₁
Fitting	#106×C₅₄₁	C₂
socle, prime radical	C₂²×C₅₄₁	C₂×D₄
cosocle	C₂×C₄×C₅₄₁	C₂³
hypercenter	#106	D₅₄₁
hypocenter	C₅₄₁	#441
LCS	G ▷ C₄×C₅₄₁ ▷ C₂×C₅₄₁ ▷ C₅₄₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

Complements

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

N	H ≅ G/N	type
7. Dic₂₁₆₄	C₄	semidirect
30. C₅₄₁	#441	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₂₁₆₄	2
#31772593	2
#31772551	2
#31772538	2
#106×C₅₄₁	2
#441	541