Subgroups — #30657069

Order 16448.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22551572	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₄×D₂₅₇	C₂	1	1	yes	maximal, characteristic
3	#22551523	C₂	1	1	yes	maximal, characteristic, complemented
4	#22551517	C₂	1	1	yes	maximal, characteristic, complemented
5	#22551515	C₂	1	1	yes	maximal, characteristic, complemented
6	#22551504	C₂	1	1	yes	maximal, characteristic
7	#106×C₂₅₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₅₁₄	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₂₅₇	—	2	1	no	—
10	C₂×C₄×D₂₅₇	C₂²	1	1	yes	characteristic
11	C₂×#14523339	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₂₅₇	—	2	1	no	—
13	C₄⋊Dic₂₅₇	C₂²	1	1	yes	characteristic
14	#16861597	—	2	1	no	—
15	#16861595	—	2	1	no	—
16	C₂×C₈×C₂₅₇	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₂₅₇	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₂₅₇	C₂²	1	1	yes	characteristic
19	C₂²×D₂₅₇	D₄	1	1	yes	characteristic
20	Dic₅₁₄	—	2	1	no	—
21	Dic₅₁₄	—	2	1	no	—
22	C₂×Dic₂₅₇	—	2	1	no	—
23	C₂×Dic₂₅₇	—	2	1	no	—
24	C₂×Dic₂₅₇	D₄	1	1	yes	characteristic
25	C₄×D₂₅₇	—	2	1	no	—
26	C₄×D₂₅₇	—	2	1	no	—
27	C₄×D₂₅₇	—	4	2	no	—
28	#14523339	—	2	1	no	—
29	C₂×C₄×C₂₅₇	—	2	1	no	—
30	C₂×C₄×C₂₅₇	—	2	1	no	—
31	C₂×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
32	C₈×C₂₅₇	—	2	1	no	—
33	C₂²×C₂₅₇	C₂×D₄	1	1	yes	characteristic
34	C₂×D₂₅₇	—	2	1	no	—
35	C₂×D₂₅₇	—	2	1	no	—
36	C₂×D₂₅₇	SD₁₆	1	2	yes	—
37	C₄×C₂₅₇	—	4	1	no	—
38	C₄×C₂₅₇	—	4	1	no	—
39	C₄×C₂₅₇	#41	1	1	yes	characteristic
40	C₄×C₂₅₇	#41	1	1	yes	characteristic
41	Dic₂₅₇	—	2	1	no	—
42	Dic₂₅₇	—	4	1	no	—
43	Dic₂₅₇	—	4	1	no	—
44	C₂×C₂₅₇	#123	1	1	yes	characteristic
45	C₂×C₂₅₇	C₂×SD₁₆	1	1	yes	characteristic
46	C₂×C₂₅₇	#137	1	1	yes	characteristic
47	D₂₅₇	—	2	2	no	—
48	C₂₅₇	#483	1	1	yes	characteristic, complemented, Sylow
49	#483	—	257	1	no	maximal, Sylow
50	C₂²⋊Q₈	—	257	1	no	—
51	C₂×C₄⋊C₄	—	257	1	no	—
52	#106	—	257	1	no	—
53	#106	D₂₅₇	1	1	yes	characteristic, complemented
54	#103	—	257	1	no	—
55	#103	—	257	1	no	—
56	C₂²⋊C₈	—	257	1	no	—
57	C₂×Q₈	—	257	1	no	—
58	C₂²×C₄	—	257	1	no	—
59	C₂²×C₄	—	514	1	no	—
60	C₂×C₈	—	257	1	no	—
61	C₂×C₈	C₂×D₂₅₇	1	1	yes	characteristic
62	C₄⋊C₄	—	257	1	no	—
63	C₄⋊C₄	—	514	1	no	—
64	C₄⋊C₄	—	514	1	no	—
65	C₄⋊C₄	C₂×D₂₅₇	1	1	yes	characteristic
66	C₄⋊C₄	C₂×D₂₅₇	1	1	yes	characteristic
67	C₂²⋊C₄	—	514	1	no	—
68	C₂³	—	257	1	no	—
69	Q₈	—	514	1	no	—
70	Q₈	—	514	1	no	—
71	C₂×C₄	—	257	1	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	514	1	no	—
75	C₂×C₄	—	514	1	no	—
76	C₂×C₄	—	514	1	no	—
77	C₂×C₄	—	514	1	no	—
78	C₂×C₄	—	1028	2	no	—
79	C₂×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
80	C₈	—	2	1	no	—
81	C₈	—	514	1	no	—
82	C₂²	—	257	2	no	—
83	C₂²	—	514	1	no	—
84	C₂²	—	514	1	no	—
85	C₂²	D₄×D₂₅₇	1	1	yes	characteristic
86	C₄	—	514	1	no	—
87	C₄	—	4	1	no	—
88	C₄	—	4	1	no	—
89	C₄	—	1028	1	no	—
90	C₄	—	1028	1	no	—
91	C₄	#16861618	1	1	yes	characteristic
92	C₄	#16861619	1	1	yes	characteristic
93	C₂	—	514	2	no	—
94	C₂	#22551577	1	1	yes	characteristic
95	C₂	SD₁₆×D₂₅₇	1	1	yes	characteristic
96	C₂	#22551621	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₅₇
commutator, cosocle	C₂×C₄×C₂₅₇	C₂³
Frattini	C₂×C₄	C₂²×D₂₅₇
Fitting	#106×C₂₅₇	C₂
socle	C₂²×C₂₅₇	C₂×D₄
prime radical	C₂²×D₂₅₇	D₄
hypercenter	#106	D₂₅₇
hypocenter	C₂₅₇	#483
LCS	G ▷ C₂×C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

Complements

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

N	H ≅ G/N	type
3. #22551523	C₂	semidirect
4. #22551517	C₂	semidirect
5. #22551515	C₂	semidirect
7. #106×C₂₅₇	C₂	semidirect
48. C₂₅₇	#483	semidirect
53. #106	D₂₅₇	semidirect

Maximal subgroups

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

subgroup	index
#22551572	2
C₄⋊C₄×D₂₅₇	2
#22551523	2
#22551517	2
#22551515	2
#22551504	2
#106×C₂₅₇	2
#483	257