Subgroups — #46829918

Order 34624.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31772625	C₂	1	1	yes	maximal, characteristic
2	C₄⋊#16864513	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊C₈×C₅₄₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₂₁₆₄	—	2	1	no	—
5	C₂×D₂₁₆₄	—	2	1	no	—
6	C₂×D₂₁₆₄	C₄	1	2	yes	—
7	C₂×#16864513	—	2	1	no	—
8	C₂×C₈×C₅₄₁	—	2	1	no	—
9	C₄²×C₅₄₁	C₂²	1	1	yes	characteristic
10	C₂²×D₅₄₁	—	2	2	no	—
11	D₂₁₆₄	—	4	2	no	—
12	D₂₁₆₄	—	4	1	no	—
13	D₂₁₆₄	—	4	1	no	—
14	D₂₁₆₄	—	4	1	no	—
15	D₂₁₆₄	—	4	1	no	—
16	#16864513	—	4	1	no	—
17	C₂×C₄×C₅₄₁	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₅₄₁	D₄	1	1	yes	characteristic
19	C₂×C₄×C₅₄₁	D₄	1	1	yes	characteristic
20	C₈×C₅₄₁	—	4	1	no	—
21	C₂²×C₅₄₁	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×D₅₄₁	—	4	2	no	—
23	C₂×D₅₄₁	—	4	2	no	—
24	C₂×D₅₄₁	—	4	2	no	—
25	C₄×C₅₄₁	—	2	1	no	—
26	C₄×C₅₄₁	D₈	1	1	yes	characteristic
27	C₄×C₅₄₁	D₈	1	1	yes	characteristic
28	C₄×C₅₄₁	SD₁₆	1	1	yes	characteristic
29	C₄×C₅₄₁	SD₁₆	1	1	yes	characteristic
30	C₂×C₅₄₁	#100	1	1	yes	characteristic
31	C₂×C₅₄₁	#102	1	1	yes	characteristic
32	C₂×C₅₄₁	#102	1	1	yes	characteristic
33	D₅₄₁	—	8	2	no	—
34	C₅₄₁	#331	1	1	yes	characteristic, complemented, Sylow
35	#331	—	541	1	no	maximal, Sylow
36	#127	—	541	1	no	—
37	C₄⋊C₈	—	541	1	no	—
38	C₄⋊C₈	D₅₄₁	1	1	yes	characteristic, complemented
39	C₂×D₄	—	541	2	no	—
40	C₂×D₄	—	1082	1	no	—
41	C₂×D₄	—	1082	1	no	—
42	C₂×C₈	—	2	1	no	—
43	C₂×C₈	—	1082	1	no	—
44	C₄²	C₂×D₅₄₁	1	1	yes	characteristic
45	C₂³	—	1082	2	no	—
46	D₄	—	2164	2	no	—
47	D₄	—	2164	1	no	—
48	D₄	—	2164	1	no	—
49	D₄	—	2164	1	no	—
50	D₄	—	2164	1	no	—
51	C₂×C₄	C₄×D₅₄₁	1	1	yes	characteristic
52	C₂×C₄	C₂²⋊D₅₄₁	1	1	yes	characteristic
53	C₂×C₄	D₂₁₆₄	1	1	yes	characteristic
54	C₈	—	4	1	no	—
55	C₈	—	2164	1	no	—
56	C₂²	—	2164	2	no	—
57	C₂²	—	2164	2	no	—
58	C₂²	—	2164	2	no	—
59	C₂²	#22575813	1	1	yes	characteristic
60	C₄	—	2	1	no	—
61	C₄	#22575821	1	1	yes	characteristic
62	C₄	D₄₃₂₈	1	1	yes	characteristic
63	C₄	#22575823	1	1	yes	characteristic
64	C₄	#22575825	1	1	yes	characteristic
65	C₂	—	4328	2	no	—
66	C₂	#31772544	1	1	yes	characteristic
67	C₂	#31772548	1	1	yes	characteristic
68	C₂	#31772550	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#22575813
commutator	C₂×C₄×C₅₄₁	C₂×C₄
Frattini	C₄²	C₂×D₅₄₁
Fitting	C₄⋊C₈×C₅₄₁	C₂
socle	C₂²×C₅₄₁	C₂²⋊C₄
prime radical	#31772625	C₂
cosocle	C₄²×C₅₄₁	C₂²
hypercenter	C₄⋊C₈	D₅₄₁
hypocenter	C₅₄₁	#331
LCS	G ▷ C₂×C₄×C₅₄₁ ▷ C₂²×C₅₄₁ ▷ C₅₄₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₄⋊#16864513	C₂	semidirect
3. C₄⋊C₈×C₅₄₁	C₂	semidirect
34. C₅₄₁	#331	semidirect
38. 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
#31772625	2
C₄⋊#16864513	2
C₄⋊C₈×C₅₄₁	2
#331	541