Subgroups — #30657363

Order 16448.

242 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22551571	C₂	1	4	yes	maximal, complemented
2	#22551569	C₂	1	8	yes	maximal, complemented
3	C₂×C₂²⋊Dic₂₅₇	C₂	1	2	yes	maximal
4	C₂×C₂²⋊C₄×C₂₅₇	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₅₁₄	C₂²	1	4	yes	complemented
6	C₂²×Dic₂₅₇	C₂²	1	4	yes	—
7	C₄⋊Dic₂₅₇	C₂²	1	4	yes	complemented
8	#16861597	C₂²	1	8	yes	complemented
9	C₂²⋊Dic₂₅₇	C₂²	1	4	yes	—
10	C₂²⋊Dic₂₅₇	C₂²	1	4	yes	—
11	C₂⁴×C₂₅₇	C₂²	1	1	yes	characteristic
12	C₂²×C₄×C₂₅₇	C₂²	1	2	yes	—
13	C₂²⋊C₄×C₂₅₇	C₂²	1	4	yes	—
14	Dic₅₁₄	—	4	4	no	—
15	C₂×Dic₂₅₇	—	4	4	no	—
16	C₂×Dic₂₅₇	C₂³	1	8	yes	—
17	C₂³×C₂₅₇	—	4	2	no	—
18	C₂³×C₂₅₇	Q₈	1	4	yes	complemented
19	C₂³×C₂₅₇	C₂³	1	2	yes	—
20	C₂³×C₂₅₇	C₂³	1	1	yes	characteristic
21	C₂×C₄×C₂₅₇	—	4	2	no	—
22	C₂×C₄×C₂₅₇	C₂³	1	4	yes	—
23	C₂²×C₂₅₇	—	2	2	no	—
24	C₂²×C₂₅₇	—	2	4	no	—
25	C₂²×C₂₅₇	—	4	4	no	—
26	C₂²×C₂₅₇	C₂×Q₈	1	4	yes	—
27	C₂²×C₂₅₇	C₂×Q₈	1	2	yes	—
28	C₂²×C₂₅₇	C₂⁴	1	1	yes	characteristic
29	C₄×C₂₅₇	—	2	4	no	—
30	Dic₂₅₇	—	2	8	no	—
31	C₂×C₂₅₇	—	2	2	no	—
32	C₂×C₂₅₇	—	2	4	no	—
33	C₂×C₂₅₇	C₂²×Q₈	1	1	yes	characteristic
34	C₂×C₂₅₇	H(5,𝔽₂)	1	2	yes	—
35	C₂₅₇	#543	1	1	yes	characteristic, complemented, Sylow
36	#543	—	257	1	no	maximal, Sylow
37	C₂²⋊Q₈	—	257	4	no	—
38	C₂²⋊Q₈	—	257	8	no	—
39	C₂×C₂²⋊C₄	—	257	2	no	—
40	C₂×C₂²⋊C₄	D₂₅₇	1	1	yes	characteristic
41	C₂⁴	C₂×D₂₅₇	1	1	yes	characteristic
42	C₂×Q₈	—	257	4	no	—
43	C₂²×C₄	—	257	4	no	—
44	C₂²×C₄	C₂×D₂₅₇	1	2	yes	—
45	C₄⋊C₄	—	257	8	no	—
46	C₄⋊C₄	—	257	4	no	—
47	C₂²⋊C₄	—	257	4	no	—
48	C₂²⋊C₄	—	257	4	no	—
49	C₂²⋊C₄	C₂×D₂₅₇	1	4	yes	—
50	C₂³	—	4	2	no	—
51	C₂³	Dic₅₁₄	1	4	yes	complemented
52	C₂³	C₂²×D₂₅₇	1	1	yes	characteristic
53	C₂³	C₂²×D₂₅₇	1	2	yes	—
54	Q₈	—	1028	4	no	—
55	C₂×C₄	—	257	8	no	—
56	C₂×C₄	—	4	2	no	—
57	C₂×C₄	—	1028	4	no	—
58	C₂×C₄	C₂²×D₂₅₇	1	4	yes	—
59	C₂²	—	2	2	no	—
60	C₂²	—	2	4	no	—
61	C₂²	—	4	4	no	—
62	C₂²	C₂×Dic₅₁₄	1	2	yes	—
63	C₂²	C₂×Dic₅₁₄	1	4	yes	—
64	C₂²	C₂³×D₂₅₇	1	1	yes	characteristic
65	C₄	—	2	4	no	—
66	C₄	—	514	8	no	—
67	C₂	—	2	2	no	—
68	C₂	—	2	4	no	—
69	C₂	C₂²×Dic₅₁₄	1	1	yes	characteristic
70	C₂	#22551637	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₅₇
commutator, socle, cosocle	C₂²×C₂₅₇	C₂⁴
Fitting	C₂×C₂²⋊C₄×C₂₅₇	C₂
prime radical	C₂⁴×C₂₅₇	C₂²
hypercenter	C₂×C₂²⋊C₄	D₂₅₇
hypocenter	C₂₅₇	#543
LCS	G ▷ C₂²×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. #22551571	C₂	semidirect
2. #22551569	C₂	semidirect
5. C₂×Dic₅₁₄	C₂²	semidirect
7. C₄⋊Dic₂₅₇	C₂²	semidirect
8. #16861597	C₂²	semidirect
18. C₂³×C₂₅₇	Q₈	semidirect
35. C₂₅₇	#543	semidirect
51. C₂³	Dic₅₁₄	semidirect

Maximal subgroups

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

subgroup	index
#22551571	2
#22551569	2
C₂×C₂²⋊Dic₂₅₇	2
C₂×C₂²⋊C₄×C₂₅₇	2
#543	257