Subgroups — #49142448

Order 36544.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#32912742	C₂	1	1	yes	maximal, characteristic
2	#32912725	C₂	1	2	yes	maximal, complemented
3	#32912622	C₂	1	2	yes	maximal, complemented
4	#32912613	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄⋊Q₈×C₅₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#22603734	—	2	2	no	—
7	#22603733	—	2	2	no	—
8	#22603733	C₂²	1	2	yes	—
9	Q₈×D₅₇₁	—	2	2	no	—
10	#22603723	—	2	2	no	—
11	#22603720	—	2	2	no	—
12	#22603717	C₂²	1	2	yes	—
13	C₂×Q₈×C₅₇₁	C₂²	1	1	yes	characteristic
14	C₂×Q₈×C₅₇₁	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×C₅₇₁	—	2	2	no	—
16	C₄²×C₅₇₁	C₂²	1	1	yes	characteristic
17	Dic₁₁₄₂	—	2	2	no	—
18	Dic₁₁₄₂	D₄	1	2	yes	—
19	D₂₂₈₄	—	2	2	no	—
20	D₂₂₈₄	D₄	1	2	yes	—
21	C₂²⋊D₅₇₁	—	2	2	no	—
22	C₄×D₅₇₁	—	2	4	no	—
23	C₄×D₅₇₁	—	2	2	no	—
24	C₅₇₁⋊C₈	—	2	2	no	—
25	Q₈×C₅₇₁	—	2	2	no	—
26	Q₈×C₅₇₁	—	2	2	no	—
27	C₂×C₄×C₅₇₁	—	2	1	no	—
28	C₂×C₄×C₅₇₁	—	2	1	no	—
29	C₂×C₄×C₅₇₁	D₄	1	2	yes	—
30	C₂×C₄×C₅₇₁	C₂³	1	1	yes	characteristic
31	C₂²×C₅₇₁	C₂×D₄	1	1	yes	characteristic
32	C₂×D₅₇₁	—	2	2	no	—
33	C₄×C₅₇₁	—	2	2	no	—
34	C₄×C₅₇₁	—	2	2	no	—
35	C₄×C₅₇₁	—	4	1	no	—
36	C₄×C₅₇₁	C₂×D₄	1	2	yes	—
37	Dic₅₇₁	—	2	2	no	—
38	C₂×C₅₇₁	—	2	1	no	—
39	C₂×C₅₇₁	#120	1	1	yes	characteristic
40	D₅₇₁	—	4	2	no	—
41	C₅₇₁	#456	1	1	yes	characteristic, complemented, Sylow
42	#456	—	571	1	no	maximal, Sylow
43	2₋¹⁺⁴	—	571	1	no	—
44	#137	—	571	2	no	—
45	C₄⋊Q₈	D₅₇₁	1	1	yes	characteristic, complemented
46	#104	—	571	2	no	—
47	#101	—	571	1	no	—
48	#41	—	571	2	no	—
49	#41	—	1142	2	no	—
50	#41	—	1142	2	no	—
51	C₂×Q₈	—	1142	2	no	—
52	C₂×Q₈	C₂×D₅₇₁	1	1	yes	characteristic
53	C₂×Q₈	C₂×D₅₇₁	1	1	yes	characteristic
54	Q₁₆	—	1142	2	no	—
55	SD₁₆	—	1142	2	no	—
56	M₁₆	—	571	2	no	—
57	C₄⋊C₄	—	2	2	no	—
58	C₄²	C₂×D₅₇₁	1	1	yes	characteristic
59	Q₈	—	2	2	no	—
60	Q₈	—	2	2	no	—
61	Q₈	—	571	2	no	—
62	Q₈	—	1142	2	no	—
63	D₄	—	571	2	no	—
64	D₄	—	1142	2	no	—
65	D₄	—	1142	2	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	1142	4	no	—
69	C₂×C₄	—	1142	2	no	—
70	C₂×C₄	C₂²⋊D₅₇₁	1	2	yes	—
71	C₂×C₄	C₂²×D₅₇₁	1	1	yes	characteristic
72	C₈	—	1142	2	no	—
73	C₂²	—	1142	2	no	—
74	C₂²	C₂×C₂²⋊D₅₇₁	1	1	yes	characteristic
75	C₄	—	2	2	no	—
76	C₄	—	2	2	no	—
77	C₄	—	4	1	no	—
78	C₄	—	1142	2	no	—
79	C₄	D₄×D₅₇₁	1	2	yes	—
80	C₂	—	2	1	no	—
81	C₂	—	2284	2	no	—
82	C₂	#32912662	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#32912662
commutator, cosocle	C₂×C₄×C₅₇₁	C₂³
Frattini	C₂×C₄	C₂²×D₅₇₁
Fitting	C₄⋊Q₈×C₅₇₁	C₂
socle	C₂×C₅₇₁	#120
prime radical	#32912742	C₂
hypercenter	C₄⋊Q₈	D₅₇₁
hypocenter	C₅₇₁	#456
LCS	G ▷ C₂×C₄×C₅₇₁ ▷ C₂×C₅₇₁ ▷ C₅₇₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #32912725	C₂	semidirect
3. #32912622	C₂	semidirect
4. #32912613	C₂	semidirect
5. C₄⋊Q₈×C₅₇₁	C₂	semidirect
41. C₅₇₁	#456	semidirect
45. C₄⋊Q₈	D₅₇₁	semidirect

Maximal subgroups

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

subgroup	index
#32912742	2
#32912725	2
#32912622	2
#32912613	2
C₄⋊Q₈×C₅₇₁	2
#456	571