Subgroups — #45662391

Order 32896.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#16861593	C₂	1	1	yes	maximal, characteristic
2	#30656490	C₂	1	1	yes	maximal, characteristic
3	#335×C₂₅₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#16861593	C₄	1	1	yes	characteristic
5	C₂×#16861593	C₄	1	1	yes	characteristic
6	C₄⋊#14523339	—	2	1	no	—
7	C₄⋊C₈×C₂₅₇	—	2	1	no	—
8	C₄×C₈×C₂₅₇	C₂²	1	1	yes	characteristic
9	C₂×#14523339	—	4	1	no	—
10	#16861593	—	2	1	no	—
11	#16861593	C₈	1	2	yes	complemented
12	C₂×C₈×C₂₅₇	—	4	1	no	—
13	C₂×C₈×C₂₅₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₂₅₇	Q₈	1	1	yes	characteristic
15	C₄²×C₂₅₇	D₄	1	1	yes	characteristic
16	#14523339	—	8	1	no	—
17	C₂×C₄×C₂₅₇	C₄⋊C₄	1	1	yes	characteristic
18	C₂×C₄×C₂₅₇	D₈	1	1	yes	characteristic
19	C₂×C₄×C₂₅₇	Q₁₆	1	1	yes	characteristic
20	C₈×C₂₅₇	—	2	1	no	—
21	C₈×C₂₅₇	—	8	1	no	—
22	C₈×C₂₅₇	C₂×C₈	1	1	yes	characteristic
23	C₈×C₂₅₇	M₁₆	1	1	yes	characteristic
24	C₂²×C₂₅₇	#107	1	1	yes	characteristic
25	C₄×C₂₅₇	—	2	1	no	—
26	C₄×C₂₅₇	C₄⋊C₈	1	1	yes	characteristic
27	C₄×C₂₅₇	#108	1	1	yes	characteristic
28	C₄×C₂₅₇	D₁₆	1	1	yes	characteristic
29	C₄×C₂₅₇	Q₃₂	1	1	yes	characteristic
30	C₂×C₂₅₇	#335	1	1	yes	characteristic
31	C₂×C₂₅₇	#366	1	1	yes	characteristic
32	C₂×C₂₅₇	#368	1	1	yes	characteristic
33	C₂₅₇	#1371	1	1	yes	characteristic, complemented, Sylow
34	#1371	—	257	1	no	maximal, Sylow
35	C₄×C₁₆	—	257	1	no	—
36	#335	—	257	1	no	—
37	#335	D₂₅₇	1	1	yes	characteristic
38	C₂×C₁₆	—	257	1	no	—
39	C₂×C₁₆	—	257	1	no	—
40	C₄⋊C₈	—	2	1	no	—
41	C₄⋊C₈	—	514	1	no	—
42	C₄×C₈	C₂×D₂₅₇	1	1	yes	characteristic
43	C₂×C₈	—	4	1	no	—
44	C₂×C₈	—	1028	1	no	—
45	C₂×C₈	C₄×D₂₅₇	1	1	yes	characteristic
46	C₂×C₈	Dic₅₁₄	1	1	yes	characteristic
47	C₄²	C₂²⋊D₂₅₇	1	1	yes	characteristic
48	C₁₆	—	257	2	no	—
49	C₁₆	—	514	1	no	—
50	C₂×C₄	#16861597	1	1	yes	characteristic
51	C₂×C₄	#16861603	1	1	yes	characteristic
52	C₂×C₄	#16861608	1	1	yes	characteristic
53	C₈	—	2	1	no	—
54	C₈	—	8	1	no	—
55	C₈	—	2056	1	no	—
56	C₈	C₈×D₂₅₇	1	1	yes	characteristic
57	C₈	#16861601	1	1	yes	characteristic
58	C₂²	#22551525	1	1	yes	characteristic
59	C₄	—	2	1	no	—
60	C₄	#22551521	1	1	yes	characteristic
61	C₄	#22551527	1	1	yes	characteristic
62	C₄	#22551533	1	1	yes	characteristic
63	C₄	#22551538	1	1	yes	characteristic
64	C₂	#30656489	1	1	yes	characteristic
65	C₂	#30656562	1	1	yes	characteristic
66	C₂	#30656566	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#16861603
commutator	C₈×C₂₅₇	C₂×C₈
Frattini	C₄×C₈	C₂×D₂₅₇
Fitting	#335×C₂₅₇	C₂
socle, prime radical	C₂²×C₂₅₇	#107
cosocle	C₄×C₈×C₂₅₇	C₂²
hypercenter	#335	D₂₅₇
hypocenter	C₂₅₇	#1371
LCS	G ▷ C₈×C₂₅₇ ▷ C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #335

Complements

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

N	H ≅ G/N	type
11. #16861593	C₈	semidirect
33. C₂₅₇	#1371	semidirect

Maximal subgroups

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

subgroup	index
C₄×#16861593	2
#30656490	2
#335×C₂₅₇	2
#1371	257