Subgroups — #30657116

Order 16448.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861608	C₂	1	4	yes	maximal
2	#22551592	C₂	1	1	yes	maximal, characteristic
3	C₄×#14523339	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Q₈×C₂₅₇	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₅₁₄	C₂²	1	2	yes	—
6	#16861608	—	2	8	no	—
7	C₂×#14523339	C₂²	1	2	yes	—
8	C₄⋊Dic₂₅₇	—	2	2	no	—
9	C₂×Q₈×C₂₅₇	C₂²	1	2	yes	—
10	C₄⋊C₄×C₂₅₇	—	2	2	no	—
11	C₄²×C₂₅₇	C₂²	1	1	yes	characteristic
12	Dic₅₁₄	—	2	4	no	—
13	C₂×Dic₂₅₇	—	2	2	no	—
14	#14523339	D₄	1	4	yes	—
15	Q₈×C₂₅₇	—	2	4	no	—
16	C₂×C₄×C₂₅₇	—	2	2	no	—
17	C₂×C₄×C₂₅₇	D₄	1	2	yes	—
18	C₂×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
19	C₂²×C₂₅₇	C₂×D₄	1	1	yes	characteristic
20	C₄×C₂₅₇	—	4	2	no	—
21	C₄×C₂₅₇	Q₁₆	1	4	yes	complemented
22	C₄×C₂₅₇	C₂×D₄	1	2	yes	—
23	Dic₂₅₇	—	4	2	no	—
24	C₂×C₂₅₇	#127	1	1	yes	characteristic
25	C₂×C₂₅₇	C₂×Q₁₆	1	2	yes	—
26	C₂₅₇	#494	1	1	yes	characteristic, complemented, Sylow
27	#494	—	257	1	no	maximal, Sylow
28	C₂×Q₁₆	—	257	4	no	—
29	C₄⋊Q₈	—	257	1	no	—
30	C₄⋊Q₈	D₂₅₇	1	1	yes	characteristic
31	C₄×C₈	—	257	1	no	—
32	C₂×Q₈	—	257	2	no	—
33	C₂×Q₈	C₂×D₂₅₇	1	2	yes	—
34	Q₁₆	—	514	8	no	—
35	C₂×C₈	—	257	2	no	—
36	C₄⋊C₄	—	2	2	no	—
37	C₄⋊C₄	—	514	2	no	—
38	C₄²	C₂×D₂₅₇	1	1	yes	characteristic
39	Q₈	—	2	4	no	—
40	Q₈	—	514	4	no	—
41	C₂×C₄	—	2	2	no	—
42	C₂×C₄	—	514	2	no	—
43	C₂×C₄	C₂²⋊D₂₅₇	1	2	yes	—
44	C₂×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
45	C₈	—	257	4	no	—
46	C₂²	C₂×C₂²⋊D₂₅₇	1	1	yes	characteristic
47	C₄	—	4	2	no	—
48	C₄	—	1028	2	no	—
49	C₄	#16861608	1	4	yes	complemented
50	C₄	D₄×D₂₅₇	1	2	yes	—
51	C₂	#22551588	1	1	yes	characteristic
52	C₂	C₂×#16861608	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₅₇
commutator, cosocle	C₂×C₄×C₂₅₇	C₂³
Frattini	C₂×C₄	C₂²×D₂₅₇
Fitting	C₄⋊Q₈×C₂₅₇	C₂
socle, prime radical	C₂²×C₂₅₇	C₂×D₄
hypercenter	C₄⋊Q₈	D₂₅₇
hypocenter	C₂₅₇	#494
LCS	G ▷ C₂×C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
21. C₄×C₂₅₇	Q₁₆	semidirect
26. C₂₅₇	#494	semidirect
49. C₄	#16861608	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16861608	2
#22551592	2
C₄×#14523339	2
C₄⋊Q₈×C₂₅₇	2
#494	257