Subgroups — #46830523

Order 34624.

118 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	#31772548	C₂	1	4	yes	maximal, complemented
3	C₄×#16864513	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄⋊Q₈×C₅₄₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₂₁₆₄	—	2	1	no	—
6	C₂×D₂₁₆₄	—	2	1	no	—
7	C₂×D₂₁₆₄	C₂²	1	2	yes	—
8	C₂×#16864513	C₂²	1	2	yes	—
9	C₂×Q₈×C₅₄₁	—	2	1	no	—
10	C₄⋊C₄×C₅₄₁	—	2	1	no	—
11	C₄⋊C₄×C₅₄₁	C₂²	1	2	yes	—
12	C₄²×C₅₄₁	C₂²	1	1	yes	characteristic
13	C₂²×D₅₄₁	—	2	2	no	—
14	D₂₁₆₄	—	2	4	no	—
15	D₂₁₆₄	—	4	2	no	—
16	D₂₁₆₄	—	4	2	no	—
17	#16864513	—	2	2	no	—
18	Q₈×C₅₄₁	—	4	2	no	—
19	C₂×C₄×C₅₄₁	—	2	2	no	—
20	C₂×C₄×C₅₄₁	D₄	1	1	yes	characteristic
21	C₂×C₄×C₅₄₁	D₄	1	1	yes	characteristic
22	C₂×C₄×C₅₄₁	C₂³	1	1	yes	characteristic
23	C₂²×C₅₄₁	C₂×D₄	1	1	yes	characteristic
24	C₂×D₅₄₁	—	4	2	no	—
25	C₂×D₅₄₁	—	4	2	no	—
26	C₂×D₅₄₁	—	4	2	no	—
27	C₄×C₅₄₁	—	4	2	no	—
28	C₄×C₅₄₁	D₈	1	2	yes	—
29	C₄×C₅₄₁	SD₁₆	1	2	yes	—
30	C₄×C₅₄₁	#41	1	2	yes	—
31	C₂×C₅₄₁	#124	1	1	yes	characteristic
32	C₂×C₅₄₁	C₂×D₈	1	1	yes	characteristic
33	C₂×C₅₄₁	C₂×SD₁₆	1	1	yes	characteristic
34	D₅₄₁	—	8	2	no	—
35	C₅₄₁	#486	1	1	yes	characteristic, complemented, Sylow
36	#486	—	541	1	no	maximal, Sylow
37	C₄⋊Q₈	D₅₄₁	1	1	yes	characteristic, complemented
38	#127	—	541	1	no	—
39	#102	—	541	4	no	—
40	C₄×C₈	—	541	1	no	—
41	C₂×Q₈	—	2	1	no	—
42	C₂×D₄	—	541	2	no	—
43	C₂×D₄	—	1082	1	no	—
44	C₂×D₄	—	1082	1	no	—
45	C₂×C₈	—	541	2	no	—
46	C₄⋊C₄	—	2	1	no	—
47	C₄⋊C₄	C₂×D₅₄₁	1	2	yes	—
48	C₄²	C₂×D₅₄₁	1	1	yes	characteristic
49	C₂³	—	1082	2	no	—
50	Q₈	—	4	2	no	—
51	D₄	—	1082	4	no	—
52	D₄	—	2164	2	no	—
53	D₄	—	2164	2	no	—
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	C₂²⋊D₅₄₁	1	1	yes	characteristic
56	C₂×C₄	C₂²⋊D₅₄₁	1	1	yes	characteristic
57	C₂×C₄	C₂²×D₅₄₁	1	1	yes	characteristic
58	C₈	—	1082	2	no	—
59	C₂²	—	2164	2	no	—
60	C₂²	—	2164	2	no	—
61	C₂²	—	2164	2	no	—
62	C₂²	C₂×C₂²⋊D₅₄₁	1	1	yes	characteristic
63	C₄	—	4	2	no	—
64	C₄	#22575821	1	2	yes	—
65	C₄	#22575823	1	2	yes	—
66	C₄	#22575837	1	2	yes	—
67	C₂	—	4328	2	no	—
68	C₂	#31772617	1	1	yes	characteristic
69	C₂	C₂×#22575821	1	1	yes	characteristic
70	C₂	C₂×#22575823	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₅₄₁
commutator, cosocle	C₂×C₄×C₅₄₁	C₂³
Frattini	C₂×C₄	C₂²×D₅₄₁
Fitting	C₄⋊Q₈×C₅₄₁	C₂
socle	C₂²×C₅₄₁	C₂×D₄
prime radical	#31772625	C₂
hypercenter	C₄⋊Q₈	D₅₄₁
hypocenter	C₅₄₁	#486
LCS	G ▷ C₂×C₄×C₅₄₁ ▷ C₂×C₅₄₁ ▷ C₅₄₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #31772548	C₂	semidirect
3. C₄×#16864513	C₂	semidirect
4. C₄⋊Q₈×C₅₄₁	C₂	semidirect
35. C₅₄₁	#486	semidirect
37. C₄⋊Q₈	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
#31772548	2
C₄×#16864513	2
C₄⋊Q₈×C₅₄₁	2
#486	541