Subgroups — #30656799

Order 16448.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22551549	C₂	1	1	yes	maximal, characteristic
2	#22551521	C₂	1	2	yes	maximal, complemented
3	#22551504	C₂	1	2	yes	maximal
4	C₄×#14523339	C₂	1	1	yes	maximal, characteristic, complemented
5	#97×C₂₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₂₅₇	C₂²	1	1	yes	characteristic
7	C₂×#14523339	C₂²	1	2	yes	—
8	#16861597	C₄	1	2	yes	—
9	#16861595	C₄	1	2	yes	—
10	C₄×Dic₂₅₇	C₂²	1	1	yes	characteristic
11	C₂×C₈×C₂₅₇	C₂²	1	2	yes	—
12	C₄²×C₂₅₇	C₂²	1	1	yes	characteristic
13	C₂²×D₂₅₇	C₂×C₄	1	1	yes	characteristic
14	C₂×Dic₂₅₇	C₂×C₄	1	2	yes	—
15	C₂×Dic₂₅₇	C₂×C₄	1	1	yes	characteristic
16	C₄×D₂₅₇	—	2	2	no	—
17	#14523339	—	2	2	no	—
18	C₂×C₄×C₂₅₇	C₂×C₄	1	2	yes	—
19	C₂×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
20	C₈×C₂₅₇	—	2	2	no	—
21	C₂²×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
22	C₂×D₂₅₇	—	2	1	no	—
23	C₂×D₂₅₇	—	2	2	no	—
24	C₄×C₂₅₇	—	2	2	no	—
25	C₄×C₂₅₇	#41	1	2	yes	—
26	Dic₂₅₇	—	2	1	no	—
27	Dic₂₅₇	—	2	2	no	—
28	C₂×C₂₅₇	#117	1	1	yes	characteristic
29	C₂×C₂₅₇	#131	1	2	yes	—
30	D₂₅₇	—	4	1	no	—
31	C₂₅₇	#433	1	1	yes	characteristic, complemented, Sylow
32	#433	—	257	1	no	maximal, Sylow
33	#117	—	257	1	no	—
34	C₄⋊C₈	—	257	2	no	—
35	C₂²⋊C₈	—	257	2	no	—
36	#97	D₂₅₇	1	1	yes	characteristic, complemented
37	C₄×C₈	—	257	1	no	—
38	C₂²×C₄	—	257	1	no	—
39	C₂×C₈	—	257	2	no	—
40	C₂×C₈	C₂×D₂₅₇	1	2	yes	—
41	C₄⋊C₄	—	257	2	no	—
42	C₂²⋊C₄	—	257	2	no	—
43	C₄²	—	257	1	no	—
44	C₄²	C₂×D₂₅₇	1	1	yes	characteristic
45	C₂³	—	257	1	no	—
46	C₂×C₄	—	257	2	no	—
47	C₂×C₄	—	257	1	no	—
48	C₂×C₄	—	514	2	no	—
49	C₂×C₄	C₄×D₂₅₇	1	2	yes	—
50	C₂×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
51	C₈	—	2	2	no	—
52	C₈	—	514	2	no	—
53	C₂²	—	514	2	no	—
54	C₂²	—	514	1	no	—
55	C₂²	C₂×C₄×D₂₅₇	1	1	yes	characteristic
56	C₄	—	2	2	no	—
57	C₄	—	514	2	no	—
58	C₄	—	514	1	no	—
59	C₄	#16861618	1	2	yes	—
60	C₂	—	1028	1	no	—
61	C₂	#22551549	1	1	yes	characteristic
62	C₂	#22551600	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₂₅₇
commutator, socle	C₂²×C₂₅₇	C₂²×C₄
Fitting	#97×C₂₅₇	C₂
prime radical	C₂²×D₂₅₇	C₂×C₄
cosocle	C₂×C₄×C₂₅₇	C₂³
hypercenter	#97	D₂₅₇
hypocenter	C₂₅₇	#433
LCS	G ▷ C₂²×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂×C₄ ◁ #97

Complements

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

N	H ≅ G/N	type
2. #22551521	C₂	semidirect
4. C₄×#14523339	C₂	semidirect
5. #97×C₂₅₇	C₂	semidirect
31. C₂₅₇	#433	semidirect
36. #97	D₂₅₇	semidirect

Maximal subgroups

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

subgroup	index
#22551549	2
#22551521	2
#22551504	2
C₄×#14523339	2
#97×C₂₅₇	2
#433	257