Subgroups — #30657013

Order 16448.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22551625	C₂	1	2	yes	maximal
2	C₂×Dic₁₀₂₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#22551598	C₂	1	1	yes	maximal, characteristic
4	#22551510	C₂	1	2	yes	maximal, complemented
5	#108×C₂₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#16861618	C₂²	1	1	yes	characteristic
7	C₂×Dic₅₁₄	C₂²	1	2	yes	—
8	Dic₁₀₂₈	—	2	2	no	—
9	Dic₁₀₂₈	—	2	1	no	—
10	Dic₁₀₂₈	—	2	1	no	—
11	#16861607	—	2	2	no	—
12	#16861602	C₂²	1	1	yes	characteristic
13	#16861601	—	2	1	no	—
14	C₈×D₂₅₇	—	2	1	no	—
15	M₁₆×C₂₅₇	C₂²	1	2	yes	—
16	C₂×C₈×C₂₅₇	C₂²	1	1	yes	characteristic
17	Dic₅₁₄	—	2	2	no	—
18	Dic₅₁₄	—	2	2	no	—
19	Dic₅₁₄	D₄	1	1	yes	characteristic
20	D₁₀₂₈	D₄	1	1	yes	characteristic
21	C₂²⋊D₂₅₇	—	2	1	no	—
22	C₂×Dic₂₅₇	—	2	2	no	—
23	C₄×D₂₅₇	—	2	1	no	—
24	#14523339	—	2	1	no	—
25	C₂×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
26	C₈×C₂₅₇	—	2	2	no	—
27	C₈×C₂₅₇	D₄	1	1	yes	characteristic
28	C₈×C₂₅₇	D₄	1	1	yes	characteristic
29	C₂²×C₂₅₇	#41	1	1	yes	characteristic
30	C₂×D₂₅₇	—	2	1	no	—
31	C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
32	C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
33	Dic₂₅₇	—	2	1	no	—
34	Dic₂₅₇	—	4	2	no	—
35	C₂×C₂₅₇	—	2	1	no	—
36	C₂×C₂₅₇	#121	1	1	yes	characteristic
37	D₂₅₇	—	4	1	no	—
38	C₂₅₇	#473	1	1	yes	characteristic, complemented, Sylow
39	#473	—	257	1	no	maximal, Sylow
40	#137	—	257	2	no	—
41	C₂×Q₁₆	—	257	1	no	—
42	#131	—	257	1	no	—
43	#108	D₂₅₇	1	1	yes	characteristic, complemented
44	#101	—	257	2	no	—
45	#41	—	257	1	no	—
46	C₂×Q₈	—	257	2	no	—
47	Q₁₆	—	514	2	no	—
48	Q₁₆	—	514	1	no	—
49	Q₁₆	—	514	1	no	—
50	SD₁₆	—	514	2	no	—
51	M₁₆	—	257	1	no	—
52	M₁₆	—	514	1	no	—
53	M₁₆	C₂×D₂₅₇	1	2	yes	—
54	C₂×C₈	—	514	1	no	—
55	C₂×C₈	C₂×D₂₅₇	1	1	yes	characteristic
56	Q₈	—	257	1	no	—
57	Q₈	—	514	2	no	—
58	Q₈	—	514	2	no	—
59	D₄	—	257	1	no	—
60	D₄	—	514	1	no	—
61	C₂×C₄	—	514	1	no	—
62	C₂×C₄	—	514	2	no	—
63	C₂×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
64	C₈	—	2	2	no	—
65	C₈	—	514	1	no	—
66	C₈	D₁₀₂₈	1	1	yes	characteristic
67	C₈	D₁₀₂₈	1	1	yes	characteristic
68	C₂²	—	514	1	no	—
69	C₂²	#16861619	1	1	yes	characteristic
70	C₄	—	514	1	no	—
71	C₄	—	1028	2	no	—
72	C₄	D₄×D₂₅₇	1	1	yes	characteristic
73	C₄	C₂×D₁₀₂₈	1	1	yes	characteristic
74	C₂	—	2	1	no	—
75	C₂	—	1028	1	no	—
76	C₂	#22551567	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22551567
commutator, cosocle	C₂×C₄×C₂₅₇	C₂³
Frattini	C₂×C₄	C₂²×D₂₅₇
Fitting	#108×C₂₅₇	C₂
socle	C₂×C₂₅₇	#121
prime radical	#16861618	C₂²
hypercenter	#108	D₂₅₇
hypocenter	C₂₅₇	#473
LCS	G ▷ C₂×C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #108

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₁₀₂₈	C₂	semidirect
4. #22551510	C₂	semidirect
5. #108×C₂₅₇	C₂	semidirect
38. C₂₅₇	#473	semidirect
43. #108	D₂₅₇	semidirect

Maximal subgroups

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

subgroup	index
#22551625	2
C₂×Dic₁₀₂₈	2
#22551598	2
#22551510	2
#108×C₂₅₇	2
#473	257