Subgroups — #30657004

Order 16448.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22551618	C₂	1	2	yes	maximal, complemented
2	#22551598	C₂	1	1	yes	maximal, characteristic, complemented
3	#22551528	C₂	1	1	yes	maximal, characteristic, complemented
4	#22551509	C₂	1	2	yes	maximal, complemented
5	C₂×D₈×C₂₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#16861618	C₂²	1	1	yes	characteristic
7	#16861606	—	2	2	no	—
8	#16861603	—	2	2	no	—
9	#16861602	C₂²	1	1	yes	characteristic
10	#16861602	C₂²	1	2	yes	—
11	#16861601	—	2	1	no	—
12	C₈×D₂₅₇	—	2	1	no	—
13	C₂×D₄×C₂₅₇	C₂²	1	2	yes	—
14	D₈×C₂₅₇	—	2	1	no	—
15	D₈×C₂₅₇	—	2	1	no	—
16	C₂×C₈×C₂₅₇	C₂²	1	1	yes	characteristic
17	Dic₅₁₄	D₄	1	1	yes	characteristic
18	D₁₀₂₈	D₄	1	1	yes	characteristic
19	C₂²⋊D₂₅₇	—	2	1	no	—
20	C₄×D₂₅₇	—	2	1	no	—
21	#14523339	—	2	1	no	—
22	#14523339	—	2	2	no	—
23	C₂³×C₂₅₇	—	2	2	no	—
24	D₄×C₂₅₇	—	2	2	no	—
25	D₄×C₂₅₇	—	2	2	no	—
26	C₂×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
27	C₈×C₂₅₇	D₄	1	1	yes	characteristic
28	C₈×C₂₅₇	D₄	1	1	yes	characteristic
29	C₂²×C₂₅₇	—	4	2	no	—
30	C₂²×C₂₅₇	—	8	2	no	—
31	C₂²×C₂₅₇	#41	1	1	yes	characteristic
32	C₂×D₂₅₇	—	2	1	no	—
33	C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
34	C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
35	Dic₂₅₇	—	2	1	no	—
36	C₂×C₂₅₇	—	2	1	no	—
37	C₂×C₂₅₇	—	8	2	no	—
38	C₂×C₂₅₇	#121	1	1	yes	characteristic
39	D₂₅₇	—	4	1	no	—
40	C₂₅₇	#472	1	1	yes	characteristic, complemented, Sylow
41	#472	—	257	1	no	maximal, Sylow
42	AGL(1,ℤ/8)	—	257	2	no	—
43	C₂×D₈	D₂₅₇	1	1	yes	characteristic, complemented
44	#131	—	257	1	no	—
45	#108	—	257	1	no	—
46	#100	—	257	2	no	—
47	#41	—	257	1	no	—
48	C₂×D₄	C₂×D₂₅₇	1	2	yes	—
49	SD₁₆	—	514	2	no	—
50	D₈	—	2	1	no	—
51	D₈	—	2	1	no	—
52	D₈	—	514	2	no	—
53	M₁₆	—	257	1	no	—
54	M₁₆	—	257	2	no	—
55	M₁₆	—	514	1	no	—
56	C₂×C₈	—	514	1	no	—
57	C₂×C₈	C₂×D₂₅₇	1	1	yes	characteristic
58	C₂³	—	2	2	no	—
59	Q₈	—	257	1	no	—
60	D₄	—	257	1	no	—
61	D₄	—	2	2	no	—
62	D₄	—	2	2	no	—
63	D₄	—	514	1	no	—
64	C₂×C₄	—	514	1	no	—
65	C₂×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
66	C₈	—	514	2	no	—
67	C₈	—	514	1	no	—
68	C₈	C₂²⋊D₂₅₇	1	1	yes	characteristic
69	C₈	C₂²⋊D₂₅₇	1	1	yes	characteristic
70	C₂²	—	514	1	no	—
71	C₂²	—	4	2	no	—
72	C₂²	—	8	2	no	—
73	C₂²	#16861617	1	1	yes	characteristic
74	C₄	—	514	1	no	—
75	C₄	D₄×D₂₅₇	1	1	yes	characteristic
76	C₄	C₂×C₂²⋊D₂₅₇	1	1	yes	characteristic
77	C₂	—	2	1	no	—
78	C₂	—	1028	1	no	—
79	C₂	—	8	2	no	—
80	C₂	#22551563	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22551563
commutator, cosocle	C₂×C₄×C₂₅₇	C₂³
Frattini	C₂×C₄	C₂²×D₂₅₇
Fitting	C₂×D₈×C₂₅₇	C₂
socle	C₂×C₂₅₇	#121
hypercenter	C₂×D₈	D₂₅₇
hypocenter	C₂₅₇	#472
LCS	G ▷ C₂×C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×D₈

Complements

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

N	H ≅ G/N	type
1. #22551618	C₂	semidirect
2. #22551598	C₂	semidirect
3. #22551528	C₂	semidirect
4. #22551509	C₂	semidirect
5. C₂×D₈×C₂₅₇	C₂	semidirect
40. C₂₅₇	#472	semidirect
43. C₂×D₈	D₂₅₇	semidirect

Maximal subgroups

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

subgroup	index
#22551618	2
#22551598	2
#22551528	2
#22551509	2
C₂×D₈×C₂₅₇	2
#472	257