Subgroups — #30657467

Order 16448.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22551591	C₂	1	4	yes	maximal
2	#22551590	C₂	1	4	yes	maximal
3	C₄×Dic₅₁₄	C₂	1	2	yes	maximal
4	#22551557	C₂	1	4	yes	maximal
5	C₄⋊Q₈×C₂₅₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₁₄	C₂²	1	4	yes	—
7	C₂×Dic₅₁₄	C₂²	1	2	yes	—
8	C₄⋊Dic₂₅₇	C₂²	1	2	yes	—
9	#16861597	C₂²	1	8	yes	—
10	#16861597	C₂²	1	4	yes	—
11	C₄×Dic₂₅₇	C₂²	1	4	yes	—
12	C₄×Dic₂₅₇	C₂²	1	4	yes	—
13	C₂×Q₈×C₂₅₇	C₂²	1	2	yes	—
14	C₄⋊C₄×C₂₅₇	C₂²	1	4	yes	—
15	C₄²×C₂₅₇	C₂²	1	1	yes	characteristic
16	Dic₅₁₄	—	4	4	no	—
17	Dic₅₁₄	Q₈	1	8	yes	complemented
18	C₂×Dic₂₅₇	C₂³	1	4	yes	—
19	C₂×Dic₂₅₇	C₂³	1	4	yes	—
20	Q₈×C₂₅₇	—	4	2	no	—
21	C₂×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
22	C₂×C₄×C₂₅₇	C₂³	1	4	yes	—
23	C₂×C₄×C₂₅₇	C₂³	1	2	yes	—
24	C₂²×C₂₅₇	C₂⁴	1	1	yes	characteristic
25	C₄×C₂₅₇	—	2	1	no	—
26	C₄×C₂₅₇	—	2	4	no	—
27	C₄×C₂₅₇	C₂×Q₈	1	4	yes	—
28	Dic₂₅₇	—	2	4	no	—
29	Dic₂₅₇	C₂×Q₈	1	8	yes	—
30	C₂×C₂₅₇	C₂²×Q₈	1	2	yes	—
31	C₂×C₂₅₇	H(5,𝔽₂)	1	1	yes	characteristic
32	C₂₅₇	Q₈²	1	1	yes	characteristic, complemented, Sylow
33	Q₈²	—	257	1	no	maximal, Sylow
34	C₄⋊Q₈	—	257	4	no	—
35	C₄⋊Q₈	—	257	4	no	—
36	C₄⋊Q₈	D₂₅₇	1	1	yes	characteristic
37	C₄×Q₈	—	257	4	no	—
38	C₄×Q₈	—	257	2	no	—
39	C₂×Q₈	—	257	2	no	—
40	C₂×Q₈	—	257	4	no	—
41	C₂×Q₈	C₂×D₂₅₇	1	2	yes	—
42	C₄⋊C₄	—	257	8	no	—
43	C₄⋊C₄	—	257	4	no	—
44	C₄⋊C₄	—	257	2	no	—
45	C₄⋊C₄	C₂×D₂₅₇	1	4	yes	—
46	C₄²	—	257	4	no	—
47	C₄²	—	257	4	no	—
48	C₄²	C₂×D₂₅₇	1	1	yes	characteristic
49	Q₈	—	257	8	no	—
50	Q₈	—	4	2	no	—
51	Q₈	—	1028	4	no	—
52	C₂×C₄	—	257	4	no	—
53	C₂×C₄	—	257	4	no	—
54	C₂×C₄	C₂²×D₂₅₇	1	4	yes	—
55	C₂×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
56	C₂×C₄	C₂²×D₂₅₇	1	2	yes	—
57	C₂²	C₂³×D₂₅₇	1	1	yes	characteristic
58	C₄	—	257	8	no	—
59	C₄	—	2	1	no	—
60	C₄	—	2	4	no	—
61	C₄	—	514	4	no	—
62	C₄	Q₈×D₂₅₇	1	4	yes	—
63	C₂	C₂×Q₈×D₂₅₇	1	2	yes	—
64	C₂	#22551637	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₅₇
commutator, socle, prime radical, cosocle	C₂²×C₂₅₇	C₂⁴
Fitting	C₄⋊Q₈×C₂₅₇	C₂
hypercenter	C₄⋊Q₈	D₂₅₇
hypocenter	C₂₅₇	Q₈²
LCS	G ▷ C₂²×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
17. Dic₅₁₄	Q₈	semidirect
32. C₂₅₇	Q₈²	semidirect

Maximal subgroups

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

subgroup	index
#22551591	2
#22551590	2
C₄×Dic₅₁₄	2
#22551557	2
C₄⋊Q₈×C₂₅₇	2
Q₈²	257