Subgroups — #44417503

Order 29248.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28410551	C₂	1	3	yes	maximal
2	#28410544	C₂	1	3	yes	maximal
3	#28410543	C₂	1	3	yes	maximal
4	Q₈×Dic₄₅₇	C₂	1	1	yes	maximal, characteristic
5	Q₈×Dic₄₅₇	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₉₁₄	C₂	1	3	yes	maximal
7	C₄×Q₈×C₄₅₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₉₁₄	C₂²	1	3	yes	—
9	C₄⋊Dic₄₅₇	C₂²	1	3	yes	—
10	C₄⋊Dic₄₅₇	C₂²	1	3	yes	—
11	C₄⋊Dic₄₅₇	C₂²	1	3	yes	—
12	#20300720	C₂²	1	1	yes	characteristic
13	#20300720	C₂²	1	3	yes	—
14	#20300720	C₂²	1	6	yes	—
15	C₄×Dic₄₅₇	C₂²	1	3	yes	—
16	C₄×Dic₄₅₇	C₂²	1	3	yes	—
17	C₂×Q₈×C₄₅₇	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₄₅₇	C₂²	1	3	yes	—
19	C₄²×C₄₅₇	C₂²	1	3	yes	—
20	Dic₉₁₄	—	2	6	no	—
21	C₂×Dic₄₅₇	C₂³	1	1	yes	characteristic
22	C₂×Dic₄₅₇	C₂³	1	3	yes	—
23	C₂×Dic₄₅₇	C₂³	1	3	yes	—
24	C₂×Dic₄₅₇	C₂³	1	1	yes	characteristic
25	Q₈×C₄₅₇	Q₈	1	4	yes	complemented
26	C₂×C₄×C₄₅₇	C₂³	1	1	yes	characteristic
27	C₂×C₄×C₄₅₇	C₂³	1	3	yes	—
28	C₂×C₄×C₄₅₇	C₂³	1	3	yes	—
29	C₂²×C₄₅₇	C₂⁴	1	1	yes	characteristic
30	C₄×C₄₅₇	—	2	3	no	—
31	C₄×C₄₅₇	—	2	1	no	—
32	C₄×C₄₅₇	C₂×Q₈	1	6	yes	—
33	Dic₄₅₇	—	2	3	no	—
34	Dic₄₅₇	—	2	1	no	—
35	Dic₄₅₇	—	2	3	no	—
36	Dic₄₅₇	#41	1	2	yes	—
37	C₂×C₄₅₇	C₂²×Q₈	1	1	yes	characteristic
38	C₂×C₄₅₇	C₂×#41	1	1	yes	characteristic
39	C₂×C₄₅₇	2₋¹⁺⁴	1	1	yes	characteristic
40	C₄₅₇	#557	1	1	yes	characteristic, complemented, Sylow
41	#557	—	457	1	no	maximal, Sylow
42	C₄⋊Q₈	—	457	3	no	—
43	#125	—	457	3	no	—
44	#125	—	457	3	no	—
45	C₄×Q₈	—	457	1	no	—
46	C₄×Q₈	—	457	1	no	—
47	C₄×Q₈	—	457	3	no	—
48	C₄×Q₈	D₄₅₇	1	1	yes	characteristic
49	C₂×Q₈	—	457	3	no	—
50	C₂×Q₈	C₂×D₄₅₇	1	1	yes	characteristic
51	C₄⋊C₄	—	457	1	no	—
52	C₄⋊C₄	—	457	3	no	—
53	C₄⋊C₄	—	457	3	no	—
54	C₄⋊C₄	—	457	3	no	—
55	C₄⋊C₄	—	457	3	no	—
56	C₄⋊C₄	—	457	6	no	—
57	C₄⋊C₄	C₂×D₄₅₇	1	3	yes	—
58	C₄²	—	457	3	no	—
59	C₄²	—	457	3	no	—
60	C₄²	C₂×D₄₅₇	1	3	yes	—
61	Q₈	—	914	6	no	—
62	Q₈	Dic₉₁₄	1	4	yes	complemented
63	C₂×C₄	—	457	1	no	—
64	C₂×C₄	—	457	3	no	—
65	C₂×C₄	—	457	3	no	—
66	C₂×C₄	—	457	1	no	—
67	C₂×C₄	C₂²×D₄₅₇	1	1	yes	characteristic
68	C₂×C₄	C₂²×D₄₅₇	1	3	yes	—
69	C₂×C₄	C₂²×D₄₅₇	1	3	yes	—
70	C₂²	C₂³×D₄₅₇	1	1	yes	characteristic
71	C₄	—	2	3	no	—
72	C₄	—	2	1	no	—
73	C₄	—	914	3	no	—
74	C₄	—	914	1	no	—
75	C₄	—	914	3	no	—
76	C₄	—	457	2	no	—
77	C₄	C₂×Dic₉₁₄	1	6	yes	—
78	C₂	C₂²×Dic₉₁₄	1	1	yes	characteristic
79	C₂	#41×D₄₅₇	1	1	yes	characteristic
80	C₂	#28410600	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₅₇
commutator, socle, prime radical, cosocle	C₂²×C₄₅₇	C₂⁴
Fitting	C₄×Q₈×C₄₅₇	C₂
hypercenter	C₄×Q₈	D₄₅₇
hypocenter	C₄₅₇	#557
LCS	G ▷ C₂²×C₄₅₇ ▷ C₄₅₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
25. Q₈×C₄₅₇	Q₈	semidirect
40. C₄₅₇	#557	semidirect
62. Q₈	Dic₉₁₄	semidirect

Maximal subgroups

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

subgroup	index
#28410551	2
#28410544	2
#28410543	2
Q₈×Dic₄₅₇	2
C₄×Dic₉₁₄	2
C₄×Q₈×C₄₅₇	2
#557	457