Subgroups — C₂×#30656567 #45666112

Order 32896.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22551528	C₂	1	2	yes	maximal
2	#30656567	C₂	1	4	yes	maximal, direct product factor
3	C₂²×C₁₆×C₂₅₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#16861602	—	2	2	no	—
5	#22551528	—	2	2	no	—
6	#22551528	C₂²	1	4	yes	—
7	C₂²×C₈×C₂₅₇	C₂²	1	1	yes	characteristic
8	C₂×C₁₆×C₂₅₇	C₄	1	2	yes	—
9	C₂×C₁₆×C₂₅₇	C₄	1	2	yes	—
10	C₂×C₁₆×C₂₅₇	C₂²	1	2	yes	—
11	#16861602	—	2	4	no	—
12	#16861602	—	4	2	no	—
13	C₂×#14523339	—	4	2	no	—
14	C₂²×C₄×C₂₅₇	D₄	1	1	yes	characteristic
15	C₂×C₈×C₂₅₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₂₅₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×C₂₅₇	D₄	1	1	yes	characteristic
18	C₂×C₈×C₂₅₇	Q₈	1	1	yes	characteristic
19	C₂×C₈×C₂₅₇	Q₈	1	1	yes	characteristic
20	C₂×C₈×C₂₅₇	C₂³	1	1	yes	characteristic
21	C₁₆×C₂₅₇	C₂×C₄	1	4	yes	—
22	#14523339	—	4	4	no	—
23	C₂³×C₂₅₇	Q₁₆	1	1	yes	characteristic
24	C₂×C₄×C₂₅₇	C₄⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₂₅₇	C₄⋊C₄	1	1	yes	characteristic
26	C₂×C₄×C₂₅₇	D₈	1	1	yes	characteristic
27	C₂×C₄×C₂₅₇	D₈	1	1	yes	characteristic
28	C₂×C₄×C₂₅₇	Q₁₆	1	1	yes	characteristic
29	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
30	C₈×C₂₅₇	C₄⋊C₄	1	1	yes	characteristic
31	C₈×C₂₅₇	C₄⋊C₄	1	1	yes	characteristic
32	C₈×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
33	C₈×C₂₅₇	C₂×Q₈	1	1	yes	characteristic
34	C₂²×C₂₅₇	—	2	2	no	—
35	C₂²×C₂₅₇	#107	1	1	yes	characteristic
36	C₂²×C₂₅₇	#107	1	1	yes	characteristic
37	C₂²×C₂₅₇	C₂×Q₁₆	1	1	yes	characteristic
38	C₄×C₂₅₇	#107	1	1	yes	characteristic
39	C₄×C₂₅₇	#107	1	1	yes	characteristic
40	C₄×C₂₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
41	C₄×C₂₅₇	C₂×D₈	1	1	yes	characteristic
42	C₂×C₂₅₇	—	2	1	no	—
43	C₂×C₂₅₇	—	2	1	no	—
44	C₂×C₂₅₇	#368	1	2	yes	complemented
45	C₂×C₂₅₇	C₂×#107	1	1	yes	characteristic
46	C₂₅₇	C₂×#368	1	1	yes	characteristic, complemented, Sylow
47	C₂×#368	—	257	1	no	maximal, Sylow
48	C₂²×C₁₆	D₂₅₇	1	1	yes	characteristic
49	C₂×#108	—	257	2	no	—
50	#368	—	257	4	no	—
51	C₂×M₁₆	—	514	2	no	—
52	C₂²×C₈	C₂×D₂₅₇	1	1	yes	characteristic
53	C₂×C₁₆	Dic₂₅₇	1	2	yes	—
54	C₂×C₁₆	Dic₂₅₇	1	2	yes	—
55	C₂×C₁₆	C₂×D₂₅₇	1	2	yes	—
56	#108	—	257	4	no	—
57	#108	—	514	2	no	—
58	C₂²×C₄	D₁₀₂₈	1	1	yes	characteristic
59	M₁₆	—	514	4	no	—
60	M₁₆	—	1028	2	no	—
61	C₂×C₈	—	1028	2	no	—
62	C₂×C₈	C₂×Dic₂₅₇	1	1	yes	characteristic
63	C₂×C₈	C₂×Dic₂₅₇	1	1	yes	characteristic
64	C₂×C₈	D₁₀₂₈	1	1	yes	characteristic
65	C₂×C₈	Dic₅₁₄	1	1	yes	characteristic
66	C₂×C₈	Dic₅₁₄	1	1	yes	characteristic
67	C₂×C₈	C₂²×D₂₅₇	1	1	yes	characteristic
68	C₁₆	C₂×Dic₂₅₇	1	4	yes	—
69	C₂³	Dic₁₀₂₈	1	1	yes	characteristic
70	C₂×C₄	C₄⋊Dic₂₅₇	1	1	yes	characteristic
71	C₂×C₄	C₄⋊Dic₂₅₇	1	1	yes	characteristic
72	C₂×C₄	D₂₀₅₆	1	1	yes	characteristic
73	C₂×C₄	D₂₀₅₆	1	1	yes	characteristic
74	C₂×C₄	Dic₁₀₂₈	1	1	yes	characteristic
75	C₂×C₄	C₂×D₁₀₂₈	1	1	yes	characteristic
76	C₈	—	1028	4	no	—
77	C₈	C₄⋊Dic₂₅₇	1	1	yes	characteristic
78	C₈	C₄⋊Dic₂₅₇	1	1	yes	characteristic
79	C₈	C₂²×Dic₂₅₇	1	1	yes	characteristic
80	C₈	C₂×Dic₅₁₄	1	1	yes	characteristic
81	C₂²	—	2	2	no	—
82	C₂²	#22551526	1	1	yes	characteristic
83	C₂²	#22551526	1	1	yes	characteristic
84	C₂²	C₂×Dic₁₀₂₈	1	1	yes	characteristic
85	C₄	#22551526	1	1	yes	characteristic
86	C₄	#22551526	1	1	yes	characteristic
87	C₄	C₂×C₄⋊Dic₂₅₇	1	1	yes	characteristic
88	C₄	C₂×D₂₀₅₆	1	1	yes	characteristic
89	C₂	—	2	1	no	—
90	C₂	—	2	1	no	—
91	C₂	#30656567	1	2	yes	direct product factor
92	C₂	C₂×#22551526	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₂₀₅₆
commutator	C₈×C₂₅₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₂₅₇
Fitting	C₂²×C₁₆×C₂₅₇	C₂
socle	C₂²×C₂₅₇	#107
prime radical	C₂³×C₂₅₇	Q₁₆
cosocle	C₂×C₈×C₂₅₇	C₂³
hypercenter	C₂²×C₁₆	D₂₅₇
hypocenter	C₂₅₇	C₂×#368
LCS	G ▷ C₈×C₂₅₇ ▷ C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
2. #30656567	C₂	direct (N × H)
44. C₂×C₂₅₇	#368	semidirect
46. C₂₅₇	C₂×#368	semidirect
91. C₂	#30656567	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#22551528	2
#30656567	2
C₂²×C₁₆×C₂₅₇	2
C₂×#368	257

Subgroups — C2×#30656567 #45666112

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#30656567 #45666112