Subgroups — #30657044

Order 16448.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22551584	C₂	1	1	yes	maximal, characteristic
2	#22551525	C₂	1	1	yes	maximal, characteristic
3	#22551523	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#14523339	C₂	1	1	yes	maximal, characteristic
5	#22551516	C₂	1	1	yes	maximal, characteristic
6	#22551516	C₂	1	1	yes	maximal, characteristic
7	C₄×Q₈×C₂₅₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#14523339	C₂²	1	1	yes	characteristic
9	C₂×#14523339	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₂₅₇	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₂₅₇	C₂²	1	1	yes	characteristic
12	#16861597	—	2	1	no	—
13	#16861597	—	2	1	no	—
14	C₂×Q₈×C₂₅₇	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×C₂₅₇	—	2	1	no	—
16	C₄⋊C₄×C₂₅₇	C₂²	1	1	yes	characteristic
17	C₄²×C₂₅₇	—	2	1	no	—
18	C₄²×C₂₅₇	C₂²	1	1	yes	characteristic
19	C₂×Dic₂₅₇	—	2	1	no	—
20	C₂×Dic₂₅₇	—	2	1	no	—
21	#14523339	—	2	1	no	—
22	#14523339	—	2	1	no	—
23	Q₈×C₂₅₇	—	2	1	no	—
24	Q₈×C₂₅₇	Q₈	1	2	yes	—
25	C₂×C₄×C₂₅₇	—	2	1	no	—
26	C₂×C₄×C₂₅₇	—	2	1	no	—
27	C₂×C₄×C₂₅₇	D₄	1	1	yes	characteristic
28	C₂×C₄×C₂₅₇	D₄	1	1	yes	characteristic
29	C₂×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
30	C₂²×C₂₅₇	C₂×D₄	1	1	yes	characteristic
31	C₄×C₂₅₇	—	2	2	no	—
32	C₄×C₂₅₇	—	2	1	no	—
33	C₄×C₂₅₇	—	2	1	no	—
34	C₄×C₂₅₇	—	4	1	no	—
35	C₄×C₂₅₇	C₂×Q₈	1	1	yes	characteristic
36	C₄×C₂₅₇	#41	1	1	yes	characteristic
37	Dic₂₅₇	—	4	1	no	—
38	Dic₂₅₇	—	4	1	no	—
39	C₂×C₂₅₇	C₂²⋊Q₈	1	1	yes	characteristic
40	C₂×C₂₅₇	#135	1	1	yes	characteristic
41	C₂×C₂₅₇	#137	1	1	yes	characteristic
42	C₂₅₇	#479	1	1	yes	characteristic, complemented, Sylow
43	#479	—	257	1	no	maximal, Sylow
44	#125	—	257	1	no	—
45	C₄×Q₈	D₂₅₇	1	1	yes	characteristic
46	#107	—	257	1	no	—
47	#106	—	257	1	no	—
48	C₄⋊C₈	—	257	1	no	—
49	#103	—	257	1	no	—
50	#103	—	257	1	no	—
51	C₂×Q₈	C₂×D₂₅₇	1	1	yes	characteristic
52	C₂×C₈	—	257	1	no	—
53	C₂×C₈	—	257	1	no	—
54	C₄⋊C₄	—	257	1	no	—
55	C₄⋊C₄	—	257	1	no	—
56	C₄⋊C₄	—	2	1	no	—
57	C₄⋊C₄	—	514	1	no	—
58	C₄⋊C₄	—	514	1	no	—
59	C₄⋊C₄	C₂×D₂₅₇	1	1	yes	characteristic
60	C₄²	—	2	1	no	—
61	C₄²	C₂×D₂₅₇	1	1	yes	characteristic
62	Q₈	—	2	1	no	—
63	Q₈	Dic₅₁₄	1	2	yes	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	514	1	no	—
67	C₂×C₄	—	514	1	no	—
68	C₂×C₄	C₂²⋊D₂₅₇	1	1	yes	characteristic
69	C₂×C₄	C₂²⋊D₂₅₇	1	1	yes	characteristic
70	C₂×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
71	C₈	—	514	1	no	—
72	C₈	—	514	1	no	—
73	C₂²	C₂×C₂²⋊D₂₅₇	1	1	yes	characteristic
74	C₄	—	2	2	no	—
75	C₄	—	2	1	no	—
76	C₄	—	2	1	no	—
77	C₄	—	4	1	no	—
78	C₄	—	1028	1	no	—
79	C₄	—	1028	1	no	—
80	C₄	C₂×Dic₅₁₄	1	1	yes	characteristic
81	C₄	#16861618	1	1	yes	characteristic
82	C₂	#22551571	1	1	yes	characteristic
83	C₂	#22551612	1	1	yes	characteristic
84	C₂	#22551623	1	1	yes	characteristic

	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	#479
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
42. C₂₅₇	#479	semidirect

Maximal subgroups

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

subgroup	index
#22551584	2
#22551525	2
#22551523	2
C₄⋊#14523339	2
#22551516	2
C₄×Q₈×C₂₅₇	2
#479	257