Subgroups — C₂×#22551521 #30656757

Order 16448.

138 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523339	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₂₅₇	C₂	1	1	yes	maximal, characteristic
3	#22551521	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₈×C₂₅₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₅₇	C₄	1	1	yes	characteristic
6	C₂²×Dic₂₅₇	C₄	1	1	yes	characteristic
7	C₂×#14523339	—	2	2	no	—
8	C₂×#14523339	C₂²	1	2	yes	—
9	C₄×Dic₂₅₇	C₄	1	2	yes	—
10	C₄×Dic₂₅₇	C₂²	1	2	yes	—
11	C₂²×C₄×C₂₅₇	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₂₅₇	—	2	2	no	—
13	C₂×C₈×C₂₅₇	C₂²	1	2	yes	—
14	C₂×Dic₂₅₇	—	2	2	no	—
15	C₂×Dic₂₅₇	C₈	1	4	yes	complemented
16	C₂×Dic₂₅₇	C₂×C₄	1	2	yes	—
17	C₂×Dic₂₅₇	C₂×C₄	1	2	yes	—
18	#14523339	—	2	2	no	—
19	C₂³×C₂₅₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₂₅₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₂₅₇	D₄	1	2	yes	—
22	C₂×C₄×C₂₅₇	Q₈	1	2	yes	—
23	C₂×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
24	C₈×C₂₅₇	—	2	2	no	—
25	C₂²×C₂₅₇	C₄⋊C₄	1	2	yes	—
26	C₂²×C₂₅₇	C₂×C₈	1	2	yes	—
27	C₂²×C₂₅₇	M₁₆	1	2	yes	—
28	C₂²×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
29	C₄×C₂₅₇	C₄⋊C₄	1	2	yes	—
30	C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
31	C₄×C₂₅₇	C₂×Q₈	1	1	yes	characteristic
32	Dic₂₅₇	—	2	2	no	—
33	Dic₂₅₇	C₂×C₈	1	4	yes	complemented
34	C₂×C₂₅₇	C₄⋊C₈	1	4	yes	complemented
35	C₂×C₂₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₂₅₇	C₂²×C₈	1	1	yes	characteristic
37	C₂×C₂₅₇	C₂×M₁₆	1	1	yes	characteristic
38	C₂₅₇	C₂×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
39	C₂×C₄⋊C₈	—	257	1	no	maximal, Sylow
40	C₂²×C₈	—	257	1	no	—
41	C₂²×C₈	D₂₅₇	1	1	yes	characteristic
42	C₂×C₄²	—	257	1	no	—
43	C₄⋊C₈	—	257	4	no	—
44	C₂²×C₄	—	257	1	no	—
45	C₂²×C₄	—	257	1	no	—
46	C₂²×C₄	C₂×D₂₅₇	1	1	yes	characteristic
47	C₂×C₈	—	257	2	no	—
48	C₂×C₈	—	2	2	no	—
49	C₂×C₈	—	514	2	no	—
50	C₂×C₈	C₂×D₂₅₇	1	2	yes	—
51	C₄²	—	257	2	no	—
52	C₄²	—	257	2	no	—
53	C₂³	C₄×D₂₅₇	1	1	yes	characteristic
54	C₂×C₄	—	257	4	no	—
55	C₂×C₄	—	257	2	no	—
56	C₂×C₄	—	257	2	no	—
57	C₂×C₄	—	514	2	no	—
58	C₂×C₄	C₄×D₂₅₇	1	1	yes	characteristic
59	C₂×C₄	C₂²⋊D₂₅₇	1	2	yes	—
60	C₂×C₄	Dic₅₁₄	1	2	yes	—
61	C₂×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
62	C₈	—	2	2	no	—
63	C₈	—	514	2	no	—
64	C₂²	#16861597	1	2	yes	—
65	C₂²	C₈×D₂₅₇	1	2	yes	—
66	C₂²	#16861601	1	2	yes	—
67	C₂²	C₂×C₄×D₂₅₇	1	1	yes	characteristic
68	C₄	—	257	4	no	—
69	C₄	—	514	2	no	—
70	C₄	#16861597	1	2	yes	—
71	C₄	C₂×C₂²⋊D₂₅₇	1	1	yes	characteristic
72	C₄	C₂×Dic₅₁₄	1	1	yes	characteristic
73	C₂	#22551521	1	4	yes	direct product factor
74	C₂	C₂×#16861597	1	1	yes	characteristic
75	C₂	C₂×C₈×D₂₅₇	1	1	yes	characteristic
76	C₂	C₂×#16861601	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₅₇
commutator	C₂×C₂₅₇	C₂²×C₈
Frattini	C₂×C₄	C₂²×D₂₅₇
Fitting	C₂²×C₈×C₂₅₇	C₂
socle, prime radical	C₂³×C₂₅₇	C₂×C₄
cosocle	C₂×C₄×C₂₅₇	C₂³
hypercenter	C₂²×C₈	D₂₅₇
hypocenter	C₂₅₇	C₂×C₄⋊C₈
LCS	G ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
3. #22551521	C₂	direct (N × H)
15. C₂×Dic₂₅₇	C₈	semidirect
33. Dic₂₅₇	C₂×C₈	semidirect
34. C₂×C₂₅₇	C₄⋊C₈	semidirect
38. C₂₅₇	C₂×C₄⋊C₈	semidirect
73. C₂	#22551521	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14523339	2
C₂×C₄×Dic₂₅₇	2
#22551521	2
C₂²×C₈×C₂₅₇	2
C₂×C₄⋊C₈	257

Subgroups — C2×#22551521 #30656757

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22551521 #30656757