Subgroups — #30657754

Order 16448.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22551659	C₂	1	2	yes	maximal, complemented
2	C₂²⋊#14523346	C₂	1	2	yes	maximal
3	C₄×#14523346	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×Dic₂₅₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523346	C₂²	1	4	yes	—
6	C₂²×Dic₂₅₇	C₄	1	1	yes	characteristic
7	C₂²×Dic₂₅₇	C₂²	1	1	yes	characteristic
8	C₄×Dic₂₅₇	C₄	1	1	yes	characteristic
9	C₄×Dic₂₅₇	C₄	1	1	yes	characteristic
10	C₄×Dic₂₅₇	C₂²	1	1	yes	characteristic
11	C₄×Dic₂₅₇	C₂²	1	1	yes	characteristic
12	C₂²×C₄×C₂₅₇	C₄	1	1	yes	characteristic
13	#14523346	—	2	4	no	—
14	C₂×Dic₂₅₇	—	2	2	no	—
15	C₂×Dic₂₅₇	C₈	1	4	yes	complemented
16	C₂×Dic₂₅₇	C₂×C₄	1	1	yes	characteristic
17	C₂×Dic₂₅₇	C₂×C₄	1	1	yes	characteristic
18	C₂×Dic₂₅₇	C₂×C₄	1	1	yes	characteristic
19	C₂×Dic₂₅₇	C₂³	1	1	yes	characteristic
20	C₂³×C₂₅₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₂₅₇	—	2	2	no	—
22	C₂×C₄×C₂₅₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₂₅₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₂₅₇	—	2	1	no	—
25	C₂²×C₂₅₇	—	2	1	no	—
26	C₂²×C₂₅₇	C₂×C₈	1	2	yes	—
27	C₂²×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
28	C₄×C₂₅₇	—	2	1	no	—
29	C₄×C₂₅₇	M₁₆	1	2	yes	—
30	Dic₂₅₇	—	2	1	no	—
31	Dic₂₅₇	C₂×C₈	1	2	yes	—
32	Dic₂₅₇	C₂×C₈	1	2	yes	—
33	Dic₂₅₇	#41	1	2	yes	—
34	C₂×C₂₅₇	—	2	2	no	—
35	C₂×C₂₅₇	#117	1	1	yes	characteristic
36	C₂×C₂₅₇	C₂²×C₈	1	1	yes	characteristic
37	C₂×C₂₅₇	C₂×M₁₆	1	1	yes	characteristic
38	C₂₅₇	#431	1	1	yes	characteristic, complemented, Sylow
39	#431	—	257	1	no	maximal, Sylow
40	C₂×C₄²	—	257	1	no	—
41	C₄⋊C₈	—	257	2	no	—
42	C₂²⋊C₈	—	257	2	no	—
43	C₄×C₈	—	257	2	no	—
44	C₂²×C₄	—	257	1	no	—
45	C₂²×C₄	—	257	1	no	—
46	C₂²×C₄	#11759967	1	1	yes	characteristic
47	C₂×C₈	—	257	4	no	—
48	C₄²	—	257	1	no	—
49	C₄²	—	257	1	no	—
50	C₄²	—	257	1	no	—
51	C₄²	—	257	1	no	—
52	C₂³	C₂×#11759967	1	1	yes	characteristic
53	C₂×C₄	—	257	4	no	—
54	C₂×C₄	—	257	1	no	—
55	C₂×C₄	—	257	1	no	—
56	C₂×C₄	—	257	1	no	—
57	C₂×C₄	—	257	1	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	514	2	no	—
60	C₂×C₄	C₂×#11759967	1	1	yes	characteristic
61	C₂×C₄	C₂×#11759967	1	1	yes	characteristic
62	C₈	—	514	4	no	—
63	C₂²	—	2	1	no	—
64	C₂²	—	2	1	no	—
65	C₂²	#16861626	1	2	yes	—
66	C₂²	C₂²×#11759967	1	1	yes	characteristic
67	C₄	—	257	2	no	—
68	C₄	—	257	2	no	—
69	C₄	—	257	2	no	—
70	C₄	—	2	1	no	—
71	C₄	—	514	1	no	—
72	C₄	#16861627	1	2	yes	—
73	C₂	—	2	2	no	—
74	C₂	#22551671	1	1	yes	characteristic
75	C₂	C₂×#16861626	1	1	yes	characteristic
76	C₂	C₂×#16861627	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#11759967
commutator	C₂×C₂₅₇	C₂²×C₈
Frattini	C₂²	C₂²×#11759967
Fitting	C₂²×C₄×C₂₅₇	C₄
socle	C₂²×C₂₅₇	C₂²×C₄
prime radical	C₂³×C₂₅₇	C₂×C₄
cosocle	C₂×Dic₂₅₇	C₂³
hypercenter	C₂²×C₄	#11759967
hypocenter	C₂₅₇	#431
LCS	G ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
1. #22551659	C₂	semidirect
3. C₄×#14523346	C₂	semidirect
15. C₂×Dic₂₅₇	C₈	semidirect
38. C₂₅₇	#431	semidirect

Maximal subgroups

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

subgroup	index
#22551659	2
C₂²⋊#14523346	2
C₄×#14523346	2
C₂×C₄×Dic₂₅₇	2
#431	257